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\DOI{10.5802/crmeca.273}
\datereceived{2024-10-18}
\dateaccepted{2024-11-04}
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\dateposted{2025-01-08}
\begin{document}

\begin{noXML}

\CDRsetmeta{articletype}{review}

\title{Some aspects of spectral and microlocal methods in Mathematical
General Relativity}

\alttitle{Quelques aspects des m\'ethodes spectrales et microlocales en
relativit\'e g\'en\'erale math\'ematique}

\author{\firstname{Dietrich} \lastname{H\"afner}\CDRorcid{0000-0002-5152-2277}}
\address{Universit\'e Grenoble Alpes, Institut Fourier, 100 rue des
maths, 38610 Gi\`eres, France}
\email{Dietrich.Hafner@univ-grenoble-alpes.fr}

\begin{abstract}
We review some of the important ideas of spectral and microlocal analysis which have
been applied to problems in Mathematical General Relativity over the last decades. 
\end{abstract}

\begin{altabstract}
Nous passons en revue quelques-unes des id\'ees importantes de
l'analyse spectrale et microlocale qui ont \'et\'e appliqu\'ees 
\`a des probl\`emes de relativit\'e g\'en\'erale math\'ematique
au cours des derni\`eres d\'ecennies. \looseness=-1
\end{altabstract}

\keywords{\kwd{Spectral theory}\kwd{Microlocal analysis}\kwd{General Relativity}}

\altkeywords{\kwd{Th\'eorie spectrale}\kwd{Analyse microlocale}\kwd{Relativit\'e g\'en\'erale}}

\thanks{ANR (ANR-20-CE40-0018-01)}

\maketitle

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{\vspace*{-2pt}}

\section{Introduction}

The aim of this review is to give an overview of important spectral and
microlocal methods that have been applied to questions in Mathematical
General Relativity over the last decades. We will try to explain some
of them using the wave equation on the De Sitter Schwarzschild metric
as a leading example.   

It is hard to say exactly when spectral and microlocal methods appear
first in the context of General Relativity. Their application was
certainly pushed by the desire of understanding quantum fields on a
curved background starting in the 1970s. Right from the beginning the
interaction was very rich, not only existing methods were applied in a
new context, but also this new context contributed to developing new
methods. For example in their celebrated paper \textit{Fourier Integral
Operators II}~\cite{DuHo} Duistermaat and H\"ormander cite the
construction of Feynman propagators for the Klein Gordon equation on a
general Lorentzian manifold as a motivation. It later turned out that
an important condition on states on curved spacetime, the so called
Hadamard condition, is naturally formulated in terms of microlocal
analysis, see the work of Radzikowski~\cite{Ra96}. Today these states
have been constructed by pseudodifferential calculus, see G\'erard and
Wrochna~\cite{GeWr14}.

In the construction of preferred quantum states on a curved background
scattering theory plays an important role and a whole program
concerning scattering theory on black hole type backgrounds was
initiated by Dimock and Kay in the 1980s, see e.g.~\cite{DiKa87}. This
program was later pushed forward by Bachelot in the 1990s, see
e.g.~\cite{Ba94} and culminated in his rigorous proof of the Hawking
effect in the spherically symmetric setting in 1999~\cite{Ba99}. This
picture was later completed by Dappiaggi, Moretti and
Pinamonti~\cite{DMP11} who showed that the Unruh state which is linked
to the Hawking effect fulfills the Hadamard condition. Today our
understanding of quantum fields is also very complete for fermions on
the Kerr spacetime, see H\"afner~\cite{Ha09} for the Hawking effect and
G\'erard, H\"afner and Wrochna~\cite{GHW23} for the Hadamard property
of the corresponding Unruh state. These results rely on classical
asymptotic completeness results obtained by H\"afner and Nicolas
in~\cite{HaNi04}. For bosons the picture is less complete, but see
Klein~\cite{Kl23} for the Hadamard property of the Unruh state in the
De Sitter Kerr metric. Whereas classical asymptotic completeness
results are also known for Klein Gordon fields on the (De Sitter) Kerr
spacetime (see H\"afner~\cite{Ha03}, Dafermos, Rodnianski,
Shlapentokh-Rothman\cite{DaRoSR18} and G\'erard, Georgescu,
H\"afner\cite{GGH17}), the results of Klein rather use decay estimates
as obtained e.g.\ in Hintz and Vasy~\cite{HV17}. 

Scattering theory for hyperbolic equations is very much linked to
dispersive properties of these equations. The importance of these
estimates for non linear problems becomes manifest in the fundamental
work on the non linear stability of Minkowski spacetime by
Christodoulou and Klainerman~\cite{ChKl93}.  

It is very fortunate that the so complicated Einstein equations have
explicit solutions like the (De Sitter) Kerr family of solutions
describing rotating black holes. However to be sure of the physical
importance of these solutions, one has to show that these families are
stable as solutions of the Einstein vacuum equations. Again dispersive
estimates for the linearized problem play a key role in this context.  
 
Whereas the scattering theory developed in the 1980s and 1990s only
showed that the local energy of the underlying equations decays to
zero, quantitative decay estimates were now needed. Roughly speaking
decay of the solution translates to regularity properties of the
underlying resolvent and vice-versa. A particular favorable situation
in this context is the case when the resolvent in suitable function
spaces has a meromorphic extension in a strip beyond the real axis. The
poles of this extension are then called resonances. It turns out that
this is the case for the De Sitter Kerr metric.   

The mathematical study of resonances for black hole spacetimes was
initiated by Bachelot and Motet-Bachlot in 1993~\cite{BaMB93}. A bit
later S\'a Barreto and Zworski~\cite{SZ97} succeeded to localize the
high frequency resonances of a De Sitter Schwarzschild black hole.
Later on Bony and H\"afner~\cite{BoHa08} obtained an expansion of the
local propagator of the wave equation on the De Sitter Schwarzschild
metric. This expansion in particular gives very precise dispersive
estimates. The analysis of Bony and H\"afner was extended to include
also the horizon by Melrose, S\'a Barreto and Vasy~\cite{MSBV14}.
Dyatlov then obtained similar results for the De Sitter Kerr
metric~\cite{Dy11a,Dy11b}. The most general version of this
theorem including De Sitter Kerr spacetime for small angular momentum
of the black hole  is due to Vasy~\cite{Va13}. The work of Vasy is an
important breakthrough in this context as it doesn't use the symmetries
of Kerr spacetime a part from the time invariance. More generally it
develops a robust Fredholm theory for non elliptic problems, that
turned out to be crucial in the following. This came out as a kind of
surprise since Fredholm theory so far seemed to be reserved for
elliptic problems. Not surprisingly, these new methods rapidly also
gave interesting results in quantum field theory, see  e.g.\ Vasy and
Wrochna~\cite{VW18}. 

An important obstacle for showing dispersive estimates on black hole
spacetimes is trapping that appears in these spacetimes. More precisely
to show the resonance expansion theorems one has to control the high
energy regime of the resolvent which is directly linked to the
trapping. Fortunately the trapping on (De Sitter) Kerr spacetimes is
very mild, more precisely it is $r$-normally hyperbolic. Suitable
resolvent estimates in the presence of $r$-normally hyperbolic
trapping have been shown by Wunsch-Zworski in~\cite{WZ11} and
Dyatlov~\cite{Dy16}, see also Hintz~\cite{Hi21b}.      

The program on dispersive estimates on the De Sitter Kerr family
culminated in the proof of the non linear stability of this  family for
small angular momentum of the black holes by Hintz and Vasy in
2016~\cite{HiVa18}. Whereas the restriction to small angular momentum
has been removed in the linear framework, see Petersen and
Vasy~\cite{PV23}, it remains for the non linear framework, because
growing modes for the linearized problem can't be excluded so far,  but
see Hintz~\cite{Hi21} for important progress in this direction.  

In the context of the Kerr family new difficulties appear. On the
linear level, the underlying resolvent can't be extended any more
meromorphically beyond the real axis. This is a low frequency problem
and very close to what happens when considering evolution equations on
an asymptotically euclidean manifold. It turns out that the resolvent
has nevertheless certain $C^k$ properties down to the real axis
permitting to show polynomial decay for several evolution equations,
see e.g.\ the series of papers by Bony and
H\"afner~\cite{BoHa10,BoHa12}. However, for the wave equation the low
frequency analysis becomes significantly simpler if one uses a
hyperboloidal foliation, see Vasy~\cite{Va21a,Va21b}. We also refer to
Baskin, Vasy and Wunsch~\cite{BVW15,BVW18} for the asymptotically
Minkowskian context. Let us also mention the work of Gajic and
Warnick~\cite{GW24} and Stucker~\cite{Stu24} about the definition of
quasinormal modes in this context. The above mentioned low frequency
problems are very different from the above mentioned high frequency
problems and one of the advantages of spectral and microlocal analysis
is that these problems can be very clearly separated and then treated
by very different methods.       

All this has then been used to show linear stability of the Kerr family
for small angular momentum, see H\"afner, Hintz and Vasy~\cite{HHV21}.
Recall that most black holes are rapidly spinning, see e.g.\ 
Thorne~\cite{Th74}, such that it would be important to remove the
restriction to small angular momentum. The very robust Fredholm
framework makes that the restriction to small angular momentum only
comes from the mode analysis.  Indeed all essential features for the
analysis like the structure of the horizon or the nature of trapping
are the same in the case of large angular momentum (see e.g.\ 
Dyatlov~\cite{Dy15}). In contrast to the De Sitter Kerr case, mode
stability for the Teukolsky equation on the Kerr background is known in
the full subextreme range in the Kerr case, see Whiting~\cite{Wh89}.
This was then used by Andersson, H\"afner and Whiting~\cite{AHW22} to
give a full analysis of the mode solutions for the linearized Einstein
equations. As an illustration how the whole analysis can be carried out
in the full subextreme case we cite Millet~\cite{Mi23} who obtains
optimal decay results for the Teukolsky equation on the Kerr
background, see also the work of Hintz~\cite{Hi22} in this context.  

The relatively weak decay for linearized gravity\footnote{More
precisely the solution of the linearized problem will decay to some
linearized Kerr plus gauge terms.} makes the analysis of the non linear
problem for the Kerr family more difficult than for the the De Sitter
Kerr family. In a series of papers
Klainerman--Szeftel~\cite{KS23,KS22a,KS22b} and
Giorgi--Klainerman--Szeftel~\cite{GKS22} give a proof of non linear
stability of the Kerr family in the case of small angular momentum, see
also Shen~\cite{Sh23}. We also refer to the work of Dafermos, Holzegel,
Rodnianski and Taylor~\cite{DaHoRoTa22} for the non linear stability of
the Schwarzschild metric.  

Let us also briefly mention the case of negative cosmological constant.
If one imposes for example Dirichlet boundary conditions for the Klein
Gordon equation on the Anti De Sitter Kerr metric, the trapping becomes
more severe. As a consequence the decay for the equation becomes
weaker, see e.g.\ Gannot~\cite{Ga14} and Holzegel,
Smulevici~\cite{HoSm14}.  

Microlocal methods also play an important role in the context of strong
cosmic censorship in particular in the case of positive cosmological
constant. Considering a (De Sitter) Kerr or (De Sitter)
Reissner--Nordstr\"om black hole, one can see that there exists a
stabilizing effect at the black hole horizon, often called redshift
effect and an inverse effect at the Cauchy horizon, called blueshift.
It is interesting to see that these redshift and blueshift effects are
directly linked to radial point estimates which play an important role
in establishing the Fredholm setting for linear equations on the
background, see Hintz and Vasy~\cite{HV17}. In the case of zero
cosmological constant the asymptotic behavior of the metric at infinity
leads to only inverse polynomial decay at the black hole horizon and
the blueshift effect at the Cauchy horizon seems to be stronger leading
in the end to an instability  of the Cauchy horizon, see the work of
Dafermos~\cite{Da05} as well as Luk and
Oh~\cite{LuOh19a,LuOh19b}. In the case of positive cosmological
constant there is a subtle competition between both effects. For
example the question if the solution of the wave equation is $H^{1}$
near the cosmological horizon is directly linked to the distance to the
real axis of the resonance which is closest to the real
axis\footnote{There is in fact a zero resonance but which is not
included in this discussion.}, see Cardoso, Costa, Destounis and
Hintz~\cite{CCDH18}. Unfortunately the exact position of these low
frequency resonances is very hard to compute. Nevertheless it is
possible that strong cosmic censorship fails for certain black hole
parameters. Interestingly, it seems that it is always valid on a
quantum level, see the work of Hollands, Wald and Zahn~\cite{HoWaZa20}.
At first glance it might be surprising that only the resonance position
that comes somehow from the analysis outside the black hole determines
what is going on on the Cauchy horizon. There is indeed a scattering
process between the black hole horizons and the Cauchy horizons and
scattering theory for the wave equation has been developed in this
context by Kehle and Shlapentokh-Rothman~\cite{KSR19} in the
spherically symmetric case. Remarkably as already observed by
Chandresekhar and Hartle~\cite{ChHa82} the poles of the corresponding
scattering matrix can be computed and they lie too far from the real
axis to contribute to the instability although one can give data on the
black hole horizons which lead to certain instabilities. Note that
asymptotic completeness results inside the black hole can also be
obtained by time dependent methods, see the work of H\"afner, Nicolas
and Mokdad~\cite{HMN21} and Mokdad and Provci~\cite{MP23}. However this
doesn't give a scattering matrix which is needed to study the behavior
in the directions transverse to the Cauchy horizon.     

We concentrate in this review on spectral analysis, resolvent analysis
and Fourier transform. This however doesn't mean that microlocal
analysis cannot be applied in non stationary situations. Indeed in
general one will be faced with situations where the operators are not
stationary but settle down in some sense to a stationary situation. In
this situation the here described analysis is the first of two
ingredients for the global analysis of the operator. The second
ingredient is a control of regularity of solutions of the non
stationary equation and requires in addition other techniques which can
however be fully microlocal, see e.g.\ the recent work of
Hintz~\cite{Hi23}. We also refer to Hintz and Vasy~\cite{HV23} in this
context.   

In the above review we have concentrated on the mathematical literature
on the subject. There is also a very important physics literature that
unfortunately we can't review here. Also it goes without saying that
only some of the aspects of microlocal analysis in General Relativity
can be treated in this review and inevitable important aspects and
important contributions will be missing. We hope that the review gives
nevertheless an idea of some of the important ideas that have been
developed over the last years. 

The review is organized as follows. In Section~\ref{SecBH} we review
some of the important aspects of the (De Sitter) Kerr family of
spacetimes which are used in the microlocal approach. A few of the
above mentioned results are presented in some detail in Sections
\ref{SecRes},~\ref{SecMicro} gives some minimal notions of
mircolocal and semiclassical analysis. In Section~\ref{SecWavSdS} we
discuss the wave equation on the De Sitter Schwarzschild metric.
Through this relatively simple example we try to explain two of the
main ideas for showing decay estimates for hyperbolic equations via
microlocal methods. Firstly the Fredholm setting and the underlying
estimates (in particular radial point estimates) and secondly the high
frequency estimates in the presence of trapping.   

\section{Black hole spacetimes}\label{SecBH}
\subsection{The De Sitter Schwarzschild metric}

We first consider the De Sitter Schwarzschild metric in Boyer Lindquist
coordinates. For $\bhm>0$ we put $b_0=(\bhm,0)$. The manifold and
metric $(M,g)$ are then given by
{\begin{equation*}
M = \R_t\times \tilde{X},\quad\!
\tilde{X} = \cI_{b_0}\times S^2_{\omega},\quad
\cI_{b_0}=(r_-,r_+),\quad
g=F_{b_0}\mathrm{d}t^2-F_{b_0}^{-1}\mathrm{d}r^2-r^2
\mathrm{d}\omega^2,\quad
F_{b_0}(r)=1-\frac{2\bhm}{r}-\frac{\Lambda r^2}{3}.
\end{equation*}}\unskip
Here $\bhm>0$ is the mass of the black hole and $\Lambda>0$ is the
cosmological constant. We require $9\bhm^2\Lambda<1$ in the following.
$F_{b_0}$ then has two positive roots that we call $r_-,\, r_+$. The
potential $V_{b_0}(r) = {F_{b_0}(r)}/{r^2}$ has a unique maximum on
$(r_-,r_+)$ given by $r_0=3\bhm$. The Regge--Wheeler coordinate $r_*$
is defined by the requirement ${\mathrm{d}r_*}/{\mathrm{d}r} = 
F_{b_0}^{-1}$ and that $r_*(3\bhm)=0$. The coordinate $t_*(r_*)$ is
defined as a smooth function of $r_*$ such that   
{\begin{equation*}
t_* = \left\{\begin{array}{@{}cc@{}}
t+r_* &\quad r\le 3\bhm,\\ 
t-r_* &\quad r\ge 4\bhm. 
\end{array}\right. 
\end{equation*}}\unskip
We find in the coordinates $(t_*,r,\omega)$ and for $r\le 3\bhm$:
{\begin{equation*}
g=F_{b_0}\,\mathrm{d}t_*^2-2\,\mathrm{d}t_* \,\mathrm{d}r-r^2\,\mathrm{d}\omega^2
\end{equation*}}\unskip
as well as a similar formula for $r\ge 4\bhm$. This shows that the
metric extends for some $\epsilon>0$ to $\cM=\R_{t_*}\times X$,
$X=(r_--\epsilon,r_++\epsilon)\times S^2_{\omega}$. The null
hypersurfaces $\cH^+=\R_{t_*}\times\{r_-\}\times S^2_{\omega}$ and
$\overline{\cH}^+=\R_{t_*}\times\{r_+\}\times S^2_{\omega}$ are called
the future event horizon and the future cosmological horizon, see
Figure~\ref{fig:coordinates}. A similar construction can be performed
in the past, constructing then the past event horizon $\cH^-$ and the
past cosmological horizon~$\overline{\cH}-$.  

\begin{figure}
\includegraphics{fig01}
\caption{\label{fig:coordinates}Choice of coordinates, 
source:~\cite{HiVa18}.}
\end{figure}

\subsection{The De Sitter Kerr metric}

The De Sitter Kerr metric describes black holes with non zero angular
momentum. For $\bhm>0,\, \bha\in \R^3$ we put $b=(\bhm,\bha)$.  Let
also $a=\vert \bha \vert$. We again start with Boyer Lindquist
coordinates. For a suitable interval $\cI_b\subset \R$ defined below
the metric takes the form 
{\begin{eqnarray*}
g_b &=& \frac{F_b}{(1+\lambda_b)^2\rho_b^2}
(\mathrm{d}t-a\sin^2\theta \,\mathrm{d}\phi)^2 - 
\frac{\kappa_b\sin^2\theta}{(1+\lambda_b)^2\rho_b^2}
(a\,\mathrm{d}t-(r^2+a^2)\,\mathrm{d}\phi)^2 - 
\rho_b^2\left(\frac{\mathrm{d}r^2}{F_b} + 
\frac{\mathrm{d}\theta^2}{\kappa_b}\right), \\ 
F_b &=& (r^2+a^2)\left(1-\frac{\Lambda r^2}{3}\right)-2\bhm r, \quad 
\rho_b^2 = r^2+a^2\cos^2\theta,\quad
\lambda_b = \frac{\Lambda a^2}{3},\quad
\kappa_b = 1 + \lambda_b\cos^2\theta. 
\end{eqnarray*}}\unskip
Under suitable conditions $F_b$ has two positive roots $r_{b,-},\,
r_{b,+}$ and $\cI_{b}=(r_{b,-},r_{b,+})$ in this case. We can construct
a similar coordinate $t_*$ as in the De Sitter Schwarzschild case. If
one also changes the coordinate $\phi$ to some coordinate ${\phi}^{*}$,
then the metric $g$ can be extended in the new coordinates
$(t_*,r,\theta,\phi^*)$      to the manifold $\cM=\R_{t_*}\times X$,
$X=(r_{b,-}-\epsilon,r_{b,+}+\epsilon)\times S^2_{\theta,\phi^*}$. 
Event horizons and cosmological horizons are defined in a similar way
to the De Sitter Schwarzschild case. 

\subsection{Schwarzschild and Kerr metric}

In Boyer-Lindquist coordinates the Schwarzschild and the Kerr metric
are formally obtained by putting $\Lambda=0$. Note that the interval
$\cI_{b_0}$ then becomes $\cI_{b_0}=(2\bhm,\infty)$ in the
Schwarzschild case and the interval $\cI_b$ becomes
$\cI_b=(\bhm+\sqrt{\bhm^2-a^2},\infty)$ in the Kerr case. The
spacetimes are now asymptotically Minkowskian, no cosmological horizon
exists. We then have $\cM=\R_{t_*}\times X,\,
X=(\bhm+\sqrt{\bhm^2-a^2}-\epsilon,\infty)$. Passing to the conformally
rescaled metric $\hat{g}=r^{-2}g$, one can construct future and past
null infinity in a similar way to the construction of the cosmological
horizons. In the Kerr case we will also need the coordinate $\ft$ which
equals $t_*$ near the future event horizon and $t$ near infinity.   

\subsection{Some important properties of black hole spacetimes}
\subsubsection{Asymptotic behavior}

The fundamental difference between positive and zero cosmological
constant is that the spacetimes are asymptotically De Sitter in the
first case and asymptotically Minkowskian in the second case. In the
asymptotically Minkowski case we have more precisely:
{\begin{equation*}
g_{b_0} = g_{(0,0)} + \cO(r^{-1}), \quad 
g_{b}=g_{b_0}+\cO(r^{-2}), \quad r\rightarrow \infty. 
\end{equation*}}\unskip
Here $b_0=(\bhm,0),\, b=(\bhm,\bha)$ and $g_{(0,0)}$ is the Minkowski
metric. As we will explain in the next section, the different
asymptotic behavior has important consequences for the dispersive
properties of solutions of the wave equation.   

\subsubsection{Superradiance} 

It is easy to see that the vector field $\partial_t$ becomes spacelike
in a region outside the horizon if the rotation $\bha$ is non zero.
This holds for the De Sitter Kerr as well as for the Kerr case. More
generally there doesn't exist a global timelike Killing vector field.
As a consequence there doesn't exist a positive conserved quantity for
the wave equation and the evolution can't be described by a selfadjoint
hamiltonian.  

\subsubsection{Trapping in the De Sitter Schwarzschild case}
\label{TdSS}

We first consider the De Sitter Schwarzschild case. The associated wave
operator is 
{\begin{equation*}
\Box_{g_{b_0}} = F_{b_0}^{-1}\partial^2_t - \frac{1}{r^2}
\partial_r r^2 F_{b_0}\partial_r-r^{-2}\Delta_{\Sph^2}. 
\end{equation*}}\unskip
Let $P(\sigma)=F_{b_0}\tilde{\Box}_{g_{b_0}}(\sigma)$, where
$\tilde{\Box}_{g_{b_0}}(\sigma)$ is the Fourier transform of
$\Box_{g_{b_0}}$ with respect to $t$. 
Then $P(\sigma)$ has symbol
{\begin{equation*}
p = \xi^2+V_{b_0}\vert \eta\vert^2-\sigma^2,
\end{equation*}}\unskip
where $\sigma,\, \xi,\eta$ are dual to $t,r_*,\omega$ and
$V_{b_0} = {F_{b_0}}/{r^2}$. The projections of the
bicharacteristics of $p$ on $M$ are null geodesics, but it is actually
more interesting to consider the bicharacteristics themselves. The
quantities $\sigma$ and $\vert \eta\vert^2$ are clearly conserved along
the evolution and the Hamiltonian equations reduce to 
{\begin{equation*}
\dot{r}_* = 2\xi,\quad
\dot{\xi} = -V_{b_0}'\vert \eta\vert^2. 
\end{equation*}}\unskip
A solution is clearly given by $(r_*,\xi)=(0,0)$. Linearization around
this solution leads to the equation:
{\begin{equation*}
\dot{r}_* = 2\xi,\quad
\dot{\xi} = -V_{b_0}''(0)\vert \eta\vert^2r_*. 
\end{equation*}}\unskip
The matrix 
{\begin{equation*}
\left(\begin{array}{@{}cc@{}}
0 & 2 \\ 
-V_{b_0}''\vert \eta\vert^2 & 0 
\end{array}\right) 
\end{equation*}}\unskip
has eigenvalues $\lambda _{\pm} = \pm
\sqrt{-2V_{b_{0}}^{\prime\prime}(0)}\vert \eta\vert$. The point (0, 0)
is therefore a hyperbolic fixed point of this reduced system. 

\subsubsection{$r$-normally hyperbolic trapping} \label{TG}

In this subsection we describe the trapping arising in black hole
spacetimes in a more general framework. We follow closely
Dyatlov~\cite{Dy16}. Let $X$ be a compact manifold, $p:T^*X\rightarrow
\R$ a Hamiltonian with associated Hamiltonian vector field $H_p$. We
mimic infinity by an absorption operator $Q$ with semiclassical symbol
$q$, see Section~\ref{Secsemi} for the definition. We define the
backward/forward trapped sets $\Gamma_{\pm}$ and the trapped set $K$ in
the following way 
\begin{enumerate}[(1)]
\item $\Gamma_{\pm}$ are codimension $1$ orientable $C^{\infty}$
submanifolds of $T^*X$ such that $\Gamma_{\pm}\cap \{p=0\}\cap\{q=0\}$
are compact. 

\item if $(x,\xi)\in \{p=0\}\setminus \Gamma_{\pm}$, then $\mathrm{e}^{\mp
tH_p}(x,\xi)\in \{q>0\}$ for some $t\ge 0$;

\item $H_p$ is tangent to $\Gamma_{\pm}$. 

\item $\Gamma_{\pm}$ intersect transversely, $K=\Gamma_+\cap \Gamma_-$
is called the trapped set, and we assume that $WF'_h(Q)\cap
K\cap\{\vert p\vert\le \delta\}=\emptyset$ for $\delta>0$ small enough,
see Section~\ref{Secsemi} for a definition of $WF'_h(Q)$. 

\item $K$ is a symplectic codimension $2$ submanifold of $T^*X$. 

\item if $v\in T_K\Gamma_{\pm}$, then $\mathrm{d}\mathrm{e}^{\mp
tH_p}v$ exponentially approaches $TK\subset T_K\Gamma_{\pm}$ as
$t\rightarrow \infty$. 
\end{enumerate}
We furthermore define $0<\nu_{\min}\le \nu_{\max}$ as the maximal and
the minimal numbers such that for each $\epsilon>0$ there exists a
constant $C$ such that for each $v\in T_K\Gamma_{\pm}$
{\begin{equation*}
C^{-1}\mathrm{e}^{-(\nu_{\max}+\epsilon)t}\vert \uppi(v)\vert\le \vert
\uppi(\mathrm{d}\mathrm{e}^{\mp tH_p}v)\vert\le
C\mathrm{e}^{-(\nu_{\min}-\epsilon)t}\vert \uppi(v)\vert,\quad t\ge 0,
\end{equation*}}\unskip
where $\uppi :T_K\Gamma_{\pm}\rightarrow T_K\Gamma_{\pm}$ is any fixed
smooth linear projection whose kernel is equal to $TK$. In other words,
$\nu_{\min}$ and $\nu_{\max}$ are the minimal and maximal expansion
rates in directions transversal to the trapped set. In the above
situation we will speak about normally hyperbolic trapping, 
see Figure~\ref{fig: normally hyperbolic trapping}. The notion
of $r$-normally hyperbolic trapping requires in addition that the rate
of expansion transversal to $K$ is much larger than the rate of
expansion on $K$. More precisely the trapping is $r$-normally
hyperbolic, if $\nu_{\min}>r\mu_{\max},$ where $\mu_{\max}$ is the
maximal expansion rate along $K$, namely
{\begin{equation*}
\Vert \mathrm{d}\mathrm{e}^{tH_p}\vert_{TK}\Vert = 
\cO(\mathrm{e}^{(\mu_{\max}+\epsilon)t}),\quad \mbox{as }
\vert t\vert\rightarrow \infty
\end{equation*}}\unskip
for all $\epsilon>0$. It can be shown that the trapping on the (De
Sitter) Kerr metric is $r$-normally hyperbolic for all $r$, see
Dyatlov~\cite{Dy15}. This is a structurally stable property, see
Fenichel~\cite{Fe71} as well as Hirsch, Pugh and Shub~\cite{HPS77} for
details.

\begin{figure}
\includegraphics{fig02}
\caption{\label{fig: normally hyperbolic trapping}Normally hyperbolic
trapping, source:~\cite{Dy16}.}
\end{figure}

\section{Stability results obtained via mircolocal methods}
\label{SecRes}

In this section we recall some linear and non linear stability results
which have been obtained via microlocal methods. 

\subsection{The Cauchy problem in General Relativity}

We consider the Einstein vacuum equations 
{\begin{equation}\label{Einst}
\mathrm{Ric}(g) + \Lambda g = 0. 
\end{equation}}\unskip
Initial data for this equation are a triplet $(\Sigma,h,k)$ consisting
of a 3-manifold $\Sigma$, a riemannian metric $h$ on $\Sigma$ and a
symmetric 2-tensor $k$ on $\Sigma$ subject to the \textit{constraint
equations}, which are the Gauss--Codazzi equations on $\Sigma$ implied
by (\ref{Einst}). Let us fix $\Sigma$ as a submanifold of $\cM$.  For a
Lorentzian metric $g$ on $\cM$ we define
$\tau(g) = (-g\vert_{\Sigma}, \Pi^g_{\Sigma})$, where
$\Pi^g_{\Sigma}$ is the second fundamental for of $\Sigma$ within
$\cM$. A metric $g$ is said to solve the initial value problem with
data $(\Sigma,h,k)$ if 
\begin{itemize}
\item $\Sigma$ is spacelike with respect to $g$;
\item $\tau(g)=(h,k)$. 
\end{itemize}    
The pioneering work of Choquet-Bruhat~\cite{CB52} shows that the
Einstein equations have a local solution for smooth data fulfilling the
constraint equations. The non linear stability problem for black hole
solutions consists now in slightly perturbing initial data of these
solutions and to ask if the initial value problem has a global solution
which stays close to one of the members of the family. This is an
orbital stability. Note that because of energy radiation each member of
the family is not expected to be individually stable.   

\subsection{The global nonlinear stability of the De Sitter Kerr
family}

The first non linear stability result for black holes was obtained by
Hintz and Vasy in 2016 for slowly rotating De Sitter Kerr family. We
state it here in its easiest form, see~\cite{HiVa18} for the full
precise formulation of the result.  Let us fix a De Sitter
Schwarzschild spacetime $(\cM, g_{b_0})$. Within this we consider a
compact spacelike hypersurface $\Sigma_0\subset\{t_*=0\}\subset \cM$
extending slightly beyond the event and cosmological horizons.  Denote
by $\Sigma_{t_*}$ the translates of $\Sigma_0$ by $\partial_{t_*}$ and
let $\Omega=\bigcup_{t_*\ge 0} \Sigma_{t_*}\subset \cM$ be the
spacetime region swept out by these, see Figure~\ref{fig:coordinates}. 

\begin{thm}[Hintz--Vasy 2016]\label{thmDSKerr}

Suppose that $(h,k)$ are smooth initial data on $\Sigma_0$, satisfying
the constraint equations, which are close to the data
$(h_{b_0},k_{b_0})$ of a De Sitter Schwarzschild spacetime in a high
regularity norm. Then there exists a solution $g$ of (\ref{Einst})
attaining these initial data at $\Sigma_0$, and black hole parameters
$b$ which are close to $b_0$, so that 
{\begin{equation*}
g-g_b=\cO(\mathrm{e}^{-\alpha t_*})
\end{equation*}}\unskip
for a constant $\alpha>0$ independent of the initial data; that is, $g$
decays exponentially fast to the De Sitter Kerr metric $g_b$. 
\end{thm}
Note that data for slowly rotating De Sitter Kerr black holes are
included so that the above result is a non linear stability result for
the De Sitter Kerr family in case of slow rotation of the black holes. 

\subsection{Linear stability of the Kerr family}

As seen above, the decay in the De Sitter Kerr case is very fast. As a
consequence passing from the linear to the non linear stability result
is not too difficult. This is different in the Kerr case and the full
non linear stability of the Kerr family has not been shown so far via
spectral and mircolocal methods, but see the series of papers by
Klainerman--Szeftel~\cite{KS23},
Giorgi--Klainerman--Szeftel~\cite{GKS22} for a proof via different
methods. However a linear stability result has been obtained by
H\"afner--Hintz--Vasy via spectral and microlocal methods which we state
now in its easiest version. Again we refer to the original
version~\cite{HHV21} for full details\footnote{Note that a small error
in~\cite{HHV21} has been corrected in~\cite{HHV24}, the main theorem
remains unchanged.}. Consider black hole parameters $b$ close to
$b_0=(\bhm_0,0)$.

\begin{thm}[Linear stability of the Kerr family, H-Hintz--Vasy 2019]
\label{thmKerr}

Let $\gamma',k'$ be symmetric 2-tensors on $\Sigma=\ft^{-1}(0)$
satisfying the linearized constraint equations and 
{\begin{equation*}
|\gamma'| \lesssim r^{-1-\alpha}, \quad
|k'| \lesssim r^{-2-\alpha},\; 0 <\alpha < 1,
\end{equation*}}\unskip
(and similar bounds for derivatives $\partial_{\omega},\, r\partial_r$
up to order $8$).  Then there exists a symmetric 2-tensor $h$ on $\cM$
such that
{\[
D_{g_b}\Ric(h) = 0, \quad
D_{g_b}\tau(h)=(\gamma',k'),
\]}\unskip
which decays to a linearized Kerr metric,
{\[
h = \dot{g}_b(\dot{b}) + \tilde h, \quad 
|\tilde h|\lesssim t_*^{-1-\alpha+},\quad
\left(\left.\dot{g}_b(\dot{b}) := 
\frac{\mathrm d}{{\mathrm d}s}g_{b+s \dot{b}}\right|_{s=0}\right).
\]}\unskip
\end{thm}
Note that Andersson et~al.~\cite{ABBM19} also
obtained a linear stability result for the Kerr family, but their decay
assumptions on the data are stronger, which eliminates the linearized
Kerr solution. Earlier results by Dafermos, Holzegel and
Rodnianski~\cite{DaHoRo19} as well as  Hung, Keller and
Wang~\cite{HKW20} establish linear stability of the Schwarzschild
solution. 

\subsection{Remarks on gauge issues}

To carry out the analysis in the linear or non linear case one has to
fix a gauge and one usually then considers a gauge fixed operator. On
the linear level one can then consider this gauge fixed operator with
general initial data. This is an advantage in an iteration scheme to
solve the non linear equation because one doesn't have to worry about
the linearized constraint equations at each step. On the other hand the
gauge fixed linear operator can now have growing modes which are not
physical in the sense that they are not solutions of the linearized
Einstein equations without gauge.    Unfortunately this kind of
scenario is produced when one uses the usual wave map gauge. In the De
Sitter Kerr case one even expects exponentially growing modes. In the
Kerr case the situation is more favorable and only generalized
quadratically growing modes can arise. The handling of these gauge
issues is beyond the scope of this review. Let us just say that in the
Kerr case one can get rid of these modes by the so called constraint
damping, we refer to~\cite{HiVa18} for the De Sitter Kerr case. In the
Kerr case once the constraint damping implemented we obtain a
linearized operator $L_b$ and the gauge fixed version of Theorem
\ref{thmKerr} is the following.    

\begin{thm}[H-Hintz--Vasy 2019]
Let $\alpha\in(0,1)$, and let $h_0,h_1\in\CI(\Sigma;S^2 T_\Sigma^*\cM)$,
{\[
|h_0| \lesssim r^{-1-\alpha}, \quad
|h_1| \lesssim r^{-2-\alpha}, 0 < \alpha < 1
\]}\unskip
(and similar bounds for derivatives). Let
{\[
\begin{cases}
L_b h = 0, \\
(h|_\Sigma, \mathcal{L}_{\partial_{t}}h|_\Sigma) = (h_0,h_1).
\end{cases}
\]}\unskip
Then there exist $\dot{b}\in\R\times\R^3$ and vector fields $W, V\in$
on $\cM$ such that
{\[
h = \dot{g}_b'(\dot{b}) + \cL_V g_b + \cL_W g_b + \tilde h, \quad
|\tilde h|\lesssim t_*^{-1-\alpha+}, \quad
\vert \cL_W g_b\vert\lesssim t_*^{-\alpha+}.
\]}\unskip
Here $\dot{g}'_b(\dot{b})$ is a gauge fixed version of
$\dot{g}_b(\dot{b})$ and $V$ lies in a fixed $7$-dimensional vector
space consisting of \textit{asymptotic translations}, \textit{asymptotic
boosts} and a Coulomb type vector field. 
\end{thm}    
A part from the Coulomb type solutions, the vector field $V$ lies
exactly in the space one would expect. In principle one can read off
the change of the black hole parameters and the movement of the black
hole in the given gauge. Let us also mention that one can get rid of
the gauge solution $\cL_Wg_b$ by admitting a gauge source term.  

\subsection{The large $a$ case}

As will be explained later on, the proof of these theorems relies on a
very robust Fredholm framework. The only ingredient which is not stable
with respect to perturbations is the mode analysis. This mode analysis
for the linearized Einstein equations has been carried out by
Andersson, H\"afner and Whiting~\cite{AHW22} also in the large $a$
case. Precise decay estimates for the Teukolsky equation (including the
computation of the leading order term) have been obtained  by these
techniques by Millet in~\cite{Mi23}. We give here a rough version of
the theorem without defining precisely the Teukolsky operator $T_s$. We
refer to~\cite{Mi24} for a review of the geometric background for the
Teukolsky operator.   

\begin{thm}[Millet 2023]\label{thmTeukolsky}
Let $0\le a<\bhm$. We fix $s\in \{0,\pm 1/2,\pm 1,\pm 3/2,\pm 2\}$. Let
$u_0,u_1$ \emph{smooth and compactly supported} on $\Sigma_0$. The
solution $u$ of the Cauchy problem
{\[
\begin{cases}
T_s u = 0\\
u(t=0) = u_0\\
\dfrac{\partial}{\partial t}u (t=0) = u_1
\end{cases}
\]}\unskip
satisfies:
{\begin{equation*}
|u(r,t_*,\theta, \phi) - \mathfrak{p}_{u_0,u_1}(r,t_*,\theta,\phi)|\leq 
Cr^{-1+}t_*^{-2-|s|+s-\epsilon}\left(\frac{t_*}{r}+1\right)^{-1-s-|s|}
\end{equation*}}\unskip
where $\epsilon>0$. We have
{\begin{equation*}
\mathfrak{p} = t_*^{-3-2|s|}
\dfrac{(2|s|+2)\left(\frac{t_*}{r}\right)^{2+|s|+s}+2(|s|-s+1)\left(\frac{t_*}{r}\right)^{1+|s|+s}}
{\left(\frac{t_*}{r}+2\right)^{2+|s|+s}}F_{u_0,u_1},
\end{equation*}}\unskip
\end{thm}
\noindent
where $F_{u_0,u_1}(r,\theta,\phi)$ can be expressed in terms of
hypergeometric functions and spin weighted spherical harmonics. The
hypotheses of the above theorem can be relaxed. In particular $H^N$
regularity for $N$ sufficiently large is sufficient. Also the compact
support of initial data is not necessary, inverse polynomial decay can
be permitted and the exact decay rate then depends on the exact decay
rate of the initial data. We refer to~\cite{Mi23} for details. Similar
results also in the large $a$ case have been obtained by Shlapentokh
Rothman and Teixeira Da Costa~\cite{SRTdC20,SRTdC23} but the precise
decay rates are weaker. For entire spin and small $a$ the above result
has been obtained before by Ma--Zhang in~\cite{MZ23}. The leading order
term has already been computed for the wave equation by
Hintz~\cite{Hi22} in the large $a$ case.  

\section{Microlocal and semiclassical analysis} \label{SecMicro}

In standard microlocal analysis the asymptotic parameter is given by
$\vert \xi\vert$, where $\xi$ is the fiber variable. We start our
representation with a review of that theory. In the semiclassical
theory a small parameter $h$ is added to  measure the wave length of
oscillations. We are then concerned with asymptotics as $h\rightarrow
0,\, \xi\rightarrow \infty$ and to make the link with standard
microlocal analysis $h$ can be thought as $h = {1}/{\vert \xi\vert}$.
A fundamental reference for the subject is H\"ormander~\cite{HoIII}. We
also refer to Zworski~\cite{Zwsemi} for the semiclassical part. Our
presentation owes a lot to the lecture notes of Wunsch ~\cite{Wumicro}
and Hintz~\cite{Himicro} as well as to the thesis of
Hintz~\cite{Hithesis}. 

\subsection{Microlocal analysis}\label{SecMA}

In this section we present some of the main objects of microlocal
analysis we will need. Let us start with a differential operator on
$\R^n$ of order $m$,
{\begin{equation*}
A = \sum_{\vert \alpha\vert\le m}a_{\alpha}(x)D_x^{\alpha}\in 
\mathrm{Diff}^m(\R^n). 
\end{equation*}}\unskip
We can rewrite this expression using the Fourier transform ${\mathcal
F}$:
{\begin{equation*}
(Au)(x) = \sum_{\vert \alpha\vert\le m}a_{\alpha}(x)
{\mathcal F}^{-1}(\xi^{\alpha}({\mathcal F}u)(\xi)) = 
\frac{1}{(2\uppi)^n}\int \mathrm{e}^{\mathrm{i}\la x,\xi\ra}
\sum_{\vert \alpha\vert\le m}a_{\alpha}(x)\xi^{\alpha}\hat{u}(
\xi)\,\mathrm{d}\xi=:(Op(a)u)(x), 
\end{equation*}}\unskip 
where $a(x,\xi)=\sum_{\vert \alpha\vert\le m}a_{\alpha}(x)\xi^{\alpha}$
is the symbol of the operator.  The above writing suggests that one can
permit more general symbols $a(x,\xi)$ and then define the
corresponding operator $Op(a)$ by 
{\begin{equation}\label{Op(a)}
(Op(a)u)(x) = \dfrac{1}{(2\uppi)^n}\int 
\mathrm{e}^{\mathrm{i}x\xi}a(x,\xi)\hat{u}(\xi)\,\mathrm{d}\xi 
= \frac{1}{(2\uppi)^n}\iint \mathrm{e}^{\mathrm{i}\la(x-y),\xi\ra}
a (x,\xi)u(y)\,\mathrm{d}y\,\mathrm{d}\xi.
\end{equation}}\unskip
A smooth function $a$ on $T^*\R^n$ is a symbol of order $m$ if the
following estimate holds:
{\begin{equation*}
\vert \partial_x^{\alpha}\partial_{\xi}^{\beta}a(x,\xi)\vert
\le C_{\alpha\beta}\la \xi\ra^{m-\vert\beta\vert},\quad
\forall \alpha, \beta\in \N^n. 
\end{equation*}}\unskip
Let $S^m(\R^n)$ be the set of symbols of order $m$. If $a\in S^m$ we
define $Op(a)$ by (\ref{Op(a)}). If $S^m$ is the set of symbols of
order $m$, we define the set of pseudodifferential operators of order
$m$ as
{\begin{equation*}
\Psi^m(\R^n) = \{Op(a),\, a\in S^m(\R^n)\}.
\end{equation*}}\unskip
For symbols in $S^m$ one can define other quantizations, for example
the Weyl quantization 
{\begin{equation*}
(a^w(x,D)u)(x) = \frac{1}{(2\uppi)^n}\iint
\mathrm{e}^{\mathrm{i}\la(x-y),\xi\ra}a\left(\frac{x+y}{2},\xi\right)
u(y)\,\mathrm{d}y\,\mathrm{d}\xi
\end{equation*}}\unskip
is often used. If $A\in \Psi^m(\R^n)$ we define the principal symbol
$\sigma_m(A)\in S^m/S^{m-1}$ by 
{\begin{equation*}
\sigma_m(A)=:a(x,\xi)\; \mbox{mod}\; S^{m-1}. 
\end{equation*}}\unskip
The principal symbol is independent of the choice of coordinates and it
doesn't depend on the particular quantization we use. We now collect
some important properties of pseudodifferential operators.

\begin{thm}[Fundamental properties of pseudodifferential operators]
\begin{enumerate}[(1)]
\item Algebra property. $\Psi^m$ is a vector space for each $m\in \R$.
If $A\in \Psi^m$ and $B\in \Psi^{m'}$, then $AB\in \Psi^{m+m'},\,
A^*\in \Psi^m$. The composition of operators is associative and
distributive. The identity operator is in $\Psi^0(X)$.

\item Boundedness. If $A\in \Psi^m$, then $A\in \cL(H^s,H^{s-m})$ for
all $s\in \R$. 

\item  Principal symbol homomorphism.
{\begin{equation*}
\sigma_{m+m'}(AB)=\sigma_m(A)\sigma_{m'}(B), \quad 
\sigma_m(A^*)=\overline{\sigma(A)}. 
\end{equation*}}\unskip
\item Symbol of commutator. If $A\in \Psi^m(X),\, B\in \Psi^{m'}(X)$,
then $[A,B]\in \Psi^{m+m'-1}(X)$ and we have 
{\begin{equation*}
\sigma_{m+m'-1}([A,B]) = \mathrm{i}\{\sigma_{m}(A),\sigma_{m'}(B)\}=H_ab,
\quad\mbox{if } a = \sigma_m(A), b = \sigma_m(B). 
\end{equation*}}\unskip
Here $\{.,.\}$ is the Poisson bracket and $H_a$ the hamiltonian vector
field associated to $a$. 
\end{enumerate}
\end{thm} 

\begin{definition}
\begin{enumerate}
\item The symbol $a\in S^m$ is elliptic at a point $(x_0,\xi_0)\in
\R^n\times(\R^n\setminus 0)$ if there exists $c,R>0$ and a conic (in
the $\xi$ variable) neighborhood $U$ of $(x_0,\xi_0)$ in
$\R^n\times(\R^n\setminus 0)$ such that $\vert a(x,\xi)\vert\ge c\la
\xi\ra^m$ for all $(x,\xi)\in U,\, \vert \xi\vert\ge R$. The set of all
points at which $a$ is elliptic is called the \textit{elliptic set} ${\rm
Ell}(a)$, and its complement the \textit{characteristic set} ${\rm
Char}(a)$. We say that $a$ is uniformly elliptic if it is elliptic at
every point with constants $R$ and $c$ uniform in $x$.   

\item The \textit{essential support} ess supp of $a\in S^m$ is the
complement of the set of all $(x_0,\xi_0)\in \R^n\times(\R^n\setminus
0)$ for which $a\in S^{-\infty}$ in a conic neighborhood of 
$(x_0,\xi_0)$. 

\item For $A=Op(a)$ we define its wavefront set as
$WF'(A):=\mbox{ess~supp } a$. 

\item Let $u\in \cD'(\R^n)$. We define the wave front set of $u$ as
follows. $(x,\xi)$ is not contained in $WF(u)$ if and only if there
exists an operator $A\in \Psi^0$ which is elliptic at $(x,\xi)$ and a
smooth cut-off $\psi\in C_c^{\infty}(\R^n),\, \psi(x)\neq 0$, such that
$A(\psi u)\in C^{\infty}(\R^n)$. 
\end{enumerate}
\end{definition}
We need two additional important results 

\begin{thm}
Let $P\in \Psi^m$ be a uniformly elliptic operator. Then there exists
$Q\in \Psi^{-m}$ such that
{\begin{equation*}
QP-1\in \Psi^{-\infty},\quad PQ-1\in \Psi^{-\infty}.
\end{equation*}}\unskip
\end{thm}
Another important result is the sharp G\aa rding inequality. 

\begin{thm}\label{thm4.2}
Let $a\in S^{2m+1}$ and ${\Re}a\ge 0$, then we have for $u\in \cS$:
{\begin{equation*}
Re\la Op(a)u,u\ra\ge -C\Vert u\Vert^2_{H^m}. 
\end{equation*}}\unskip
\end{thm}
We now radially compactify each fiber of $T^*\R^n$ to obtain
$\overline{T^*}\R^n$ and we will denote by
$S^*\R^n=\partial\overline{T^*}\R^n$ the cosphere bundle, which we thus
view as a boundary at fiber infinity of $\overline{T^*}\R^n$. We will
sometimes also need classical symbols. A classical symbol of order $m$
is then $\la\xi\ra^m$ times a smooth function on $\overline{T^*}\R^n$. 
For a classical symbol $a$ we can write down the Taylor series near
${1}/{\vert \xi\vert}=0$ to obtain 
{\begin{equation*}
a(x,\xi)\sim \sum_{j=0}^{\infty} a_{m-j}(x,\hat{\xi})\vert
\xi\vert^{m-j}, \quad  a_{m-j}\in C^{\infty}(\R^n\times \Sph^{n-1}),
\end{equation*}}\unskip
and where the tilde denotes an asymptotic expansion-truncating the
expansion at the $\vert \xi\vert^{m-N}$ term gives an error that is
$\cO(\vert \xi\vert^{m-N-1})$. 

Suppose that $A\in \Psi^m$ has compactly supported Schwartz kernel
supported in $U\times U$. Then under a coordinate change
$\kappa :U\rightarrow V$, the principal symbol of an operator
$a=\sigma_m(A)$ transforms as a function on the cotangent bundle and
the Hamiltonian vector field is invariantly defined under change of
coordinates. This allows to define pseudodifferential calculus on a
compact manifold $X$ and the above properties remain valid. In the non
compact case, some care has to be taken for example by imposing some
control at infinity or by requiring that the operators are properly
supported. 

\subsection{Propagation of singularities} 

Let $P\in \Psi^m(X)$ be a classical operator with real homogeneous
principal symbol $p_m=\sigma_m(P)\in S^m_{\mathrm{hom}}(T^*X)$. Fix a
boundary defining function $\rho$ of fiber infinity in
$\overline{T^*}X$. We rescale the Hamiltonian vector field ${\bf
H}_{p_m}=\rho^{m-1}H_{p_m}\in \cV(S^*X)$ and also rescale the principal
symbol $\mathbf{ p}=\rho^mp_m\in C^{\infty}(S^*X)$. Within the
characteristic set $\mathrm{Char}(P) = \mathbf{p}^{-1}(0)$, the rescaled
vector field induces a flow which is merely a rescaling of the
Hamiltonian flow of $p_m$ if we identify the subset ${\rm
Char}(P)\subset S^*X$ with the corresponding conic set of
$T^*X\setminus 0$. We remark that ${\rm H}_{p_m}$ vanishes at a point
$\zeta\in S^*X$ if and only if $H_{p_m}$ is radial at $\zeta$ (i.e.\ at
the ray in $T^*\setminus 0$ associated to $\zeta$), and in this case
the integral curve through $\zeta$ is constant. 

We suppose in the following  that $\zeta_0\in S^*X$ is such that ${\rm
H}_{p_m}\vert_{\zeta_0}\neq 0$, hence the hamiltonian flow in ${\rm
Char}(P)$ starting at $\zeta_0$ is non trivial; denote by
$\gamma :[0,T]\rightarrow S^*X$ a segment of a null bicharacteristic,
i.e.\ an integral curve of $\mathbf{H}_{p_m}$ in $\mathrm{Char}(P)$
starting at $\gamma(0)=\zeta_0$. The following theorem was first proven
by Duistermaat and H\"ormander in~\cite{DuHo} (Figure~\ref{fig3}
illustrates this theorem):       

\begin{figure}
\includegraphics{fig03}
\caption{\label{fig3}Propagation of singularites,
source:~\cite{Hithesis}.}
\end{figure}

\begin{thm}
Let $s,N\in \R$. Let $E,\, B,\, G\in \Psi^0$.  Assume that $E$ is
elliptic at $\zeta_0$, $B$ is elliptic at $\gamma(T)$, and $G$ is
elliptic at $\gamma([0,T])$, such that every backward null
bicharacteristic starting at a point in $WF'(B)$ reaches ${\rm Ell}(E)$
in finite time. Then 
{\begin{equation*}
\Vert Bu\Vert_{H^s}\le C (\Vert GPu\Vert_{H^{s-m+1}}+\Vert
Eu\Vert_{H^s}+\Vert u\Vert_{H^{N}}).
\end{equation*}}\unskip
\end{thm}

\begin{rmk}
\begin{enumerate}[(1)]
\item The estimates holds in the strong sense that if $u\in \cE'(X)$ is
such that the right hand side is finite, then so is the left hand side
and the estimate holds.  The theorem shows that $H^s$ regularity
propagates along null bicharacteristics in the following sense.
Assuming that we have microlocal $H^s$ control at $\gamma(0)$, we
conclude microlocal $H^s$ control at $\gamma(T)$ provided $Pu$ stays in
$H^{s-m+1}$ along the way.    

\item The hypothesis excludes radial points, we come back to radial
points in the special setting of the De Sitter Schwarzschild metric in
Section~\ref{Sec5.3.2}.  
\end{enumerate}
\end{rmk}

\begin{proof}[Sketch of the proof]
We sketch the proof in the case when $P$ is selfadjoint, $P=P^*$.  We
assume further $Pu=0$. We can suppose $T=1$. We begin by straightening
out the flow of $\mathbf{H}_{p_m}$. We introduce local coordinates
$(q_1,q')\in \R\times \R^{2n-2}$ on $S^*X$ near $\zeta_0$ such that
$\zeta_0=(0,0)$ and $\mathbf{H}_{p_m}=\partial_{q_1}$. Let for
$\delta>0$, $U_{\delta}=\{\vert q_1\vert,\vert q'\vert<\delta\}$. We
start with a formal computation and discuss regularity issues later.
For a commutant $A=Op(a)\in \Psi^{2s-m+1}$ such that $A=A^*$ to be
chosen later we compute:
{\begin{equation}\label{PS1}
0 = 2{\Im }\la Pu,Au\ra = 
\la \mathrm{i}(PA-AP)u,u\ra = 
\la \mathrm{i}[P,A]u,u\ra.
\end{equation}}\unskip
Let $\mathbf{ a}=\rho^{2s-m+1}a\in C^{\infty}(S^*X)$. We claim that we
can arrange 
{\begin{equation}\label{PS2}
\mathbf{ H}_{p_m}\mathbf{ a} = -\mathbf{ b}^2-M\mathbf{ a}+\mathbf{ e},
\end{equation}}\unskip
where $\mathbf{ b}\in C^{\infty}(S^*X)$ is non negative, and positive
in $\{q_1\in (\delta,1+\delta),\vert q'\vert<\delta\}$, while $\mathbf{
e}\in C^{\infty}(S^*X)$ is supported in the neighborhood $U_{\delta}$
and $M>0$ can be chosen large. We define $\mathbf{ a}$ as a product
$\mathbf{ a}=\mathbf{ a}_1(q_1)\mathbf{ a}'(q')$, where $\mathbf{ a}_1$
and $\mathbf{ a}'$ are constructed in the following way. Let
{\begin{eqnarray*}
\psi & \in & C_0^{\infty}(\R^{2n-2}), \mbox{ supp }\psi 
\subset\{\vert q'\vert\le 2\delta\},\, \psi(q') = 1 \quad
\mbox{ for } \vert q'\vert<\delta, \\
\chi_1 &\in & C^{\infty}(\R), \mbox{ supp }
\chi_1\subset(-\delta,\infty), \mbox{ supp} (1-\chi_1)
\subset(-\infty,\delta) \mbox{ (turn off function)},
\end{eqnarray*}}\unskip
{\vspace*{-5pt}}
{\begin{eqnarray*}
\chi_0(q_1) &=& \left\{\begin{array}{@{}cc@{}} 
\mathrm{e}^{-\frac{F}{1+\delta-q_1}} &\quad q_1<1+\delta,\\
0 &\quad q_1\ge 1+\delta, 
\end{array}\right.\\
\mathbf{ a}_1(q_1) &=& 
\chi_0(q_1)\chi^2_1(q_1), \quad\mathbf{ a}'(q') = \psi^2(q').  
\end{eqnarray*}}\unskip
It is not difficult to check that (\ref{PS2}) can be achieved for $M>0$
given if we choose $F$ sufficiently large.  Going back to $a$ we
compute 
{\begin{equation}\label{PS3}
H_{p_m}a = \rho^{-m+1}\mathbf{ H}_{p_m}(\rho^{-2s+m-1}\mathbf{ a})
= \rho^{-2s}(\mathbf{ H}_{p_m}\mathbf{ a}-(2s-m+1)\mathbf{ a}
\rho^{-1}\mathbf{ H}_{p_m}\rho).
\end{equation}}\unskip
Noting that $\rho^{-1}\mathbf{ H}_{p_m}\rho$ is bounded and choosing
$M$ in (\ref{PS2}) sufficiently large we see that 
{\begin{equation}\label{PS4}
H_{p_m}a = -b^2+e-M\tilde{a}+r\tilde{a},
\end{equation}}\unskip
where $b=\rho^{-s}\mathbf{ b}, e=\rho^{-2s}\mathbf{ e},
\tilde{a}=\rho^{-m+1}a, r\in S^0$. Let $\tilde{A}$ be a quantization
of $\tilde{a}$ and suppose that we arrange $\tilde{A}=C^*C$. Then
$-M\la \tilde{A}u,u\ra+\la Op(r\tilde{a})u,u\ra\le C \Vert
u\Vert^2_{H^{s-({1}/{2})}}$. Let $B\in \Psi^s(X)$ and $E\in
\psi^{2s}(X)$ be quantizations of $b$ and $e$, respectively. We then
obtain:
{\begin{equation}\label{PS5}
\la B^*Bu,u\ra \le \la Eu,u\ra+C\Vert u\Vert^2_{H^{s-\frac{1}{2}}}. 
\end{equation}}\unskip

It has to be pointed out that the integration by parts above are not
justified at the level of regularity supposed in the hypothesis. The
above argument therefore needs a regularization procedure and this can
be done for $u\in H^{s-1/2}$ microlocally near $\gamma([0,T])$,
see~\cite{Himicro} for details.  Supposing this done, then the estimate
(\ref{PS5}) implies $Bu\in L^2(X)$, therefore $u\in H^s$ microlocally
near $\gamma([0,T])$. Thus starting with the a priori knowledge $u\in
H^N(X)$, one can iteratively improve the regularity by ${1}/{2}$ in
each step, until one obtains $H^s$-regularity as desired. 
\end{proof}

\subsection{Semiclassical analysis}\label{Secsemi}

We briefly introduce some elements of semiclassical analysis that we
will need in the following. We refer to Zworski~\cite{Zwsemi} for
details. We start by introducing the algebra $\Psi^m_h(X)$ of 
\textit{semiclassical} pseudodifferential operators, depending on a
parameter $h$. The corresponding symbols $a(x,\xi;h)$ (denoted $a\in
S^m_h(X)$) satisfy $a(\cdot,\cdot;h)\in S^m(X)$ uniformly in $h$ as
$h\rightarrow 0$, with $S^m(X)$ defined in Section~\ref{SecMA}. The
semiclassical symbol of $A\in \Psi^m_h$, denoted $\sigma_h(A)$, lies in
the space $S^m_h(X)/hS_h^{m-1}(X)$. 

We will be concerned in particular with the class $\Psi^0_h(X)$. We
moreover require that symbols of the elements of $\Psi^0_h(X)$ are
classical in the sense that they have an asymptotic expansion in non
negative powers of $h$. More precisely we require that $a\in \Psi^0_h$
writes 
{\begin{equation*}
a\sim \sum_{j=0}^{\infty} h^ja_j, \mbox{ meaning } 
\left\vert \partial_x^{\alpha}\partial^{\beta}_{\xi}
\left(a-\sum_{j=0}^{N-1}h^ja_j\right)\right\vert\le 
C_{\alpha,\beta}h^{N}\la \xi\ra^{-N-\vert\beta\vert}
\end{equation*}}\unskip
for all multiindices $\alpha$, $\beta$ and $N\in \N$. For $A,A'\in
\Psi^0_h(X)$ with principal symbols $a,a'$ we have 
{\begin{equation*}
\sigma_h(A\circ A')=aa', \quad 
\sigma_h(A^*)=\bar{a}, \quad 
\sigma_{h}(\mathrm{i}h^{-1}[A,A']) = \{a,a'\}=H_aa'. 
\end{equation*}}\unskip
In the context of semiclassical analysis we will mostly use the Weyl
quantization:
{\begin{equation*}
a^w(x,hD)u(x) = \dfrac{1}{(2\uppi h)^n}
\iint a\left(\frac{x+y}{2},\xi\right)\mathrm{e}^{\frac{\mathrm{i}}{h}
\la x-y,\xi\ra}u(y)\,\mathrm{d}y\,\mathrm{d}\xi. 
\end{equation*}}\unskip
Moreover each $A=a^w(x,hD)\in \Psi^0_h(X)$ acts $L^2(X)\rightarrow
L^2(X)$ with norm uniformly bounded in $h$. In fact we have the bound
\cite[Theorem 13.13]{Zwsemi}
{\begin{equation}
\lim\sup _{h\rightarrow 0}\Vert A\Vert _{L^2(X)\rightarrow L^2(X)}\le 
\sup_{T^*X}\vert \sigma_h(A)\vert.  
\end{equation}}\unskip
Each $A\in \Psi^k_h(X)$ has a semiclassical wave front set $WF'_h(A)$,
a closed (and not necessarily conic) subset of the fiber-radially
compactified cotangent bundle $\overline{T^*}X$; a point $(x,\xi)\in
\overline{T^*}X$ does not lie in $WF'_h(A)$ if and only if the full
symbol of $A$ satisfies $a(x',\xi')=\cO(h^{\infty}\la
\xi'\ra^{-\infty})$ for $h$ small enough and $(x',\xi')\in T^*X$ in a
neighborhood of $(x,\xi)$ in $\overline{T^*}X$. We will mostly use the
subalgebra of \textit{compactly microlocalized} pseudodifferential
operators $\Psi_h^{\mathrm{comp}}(X)\subset \Psi^0_h(X)$. For $A\in
\Psi_h^{\mathrm{comp}}(X)$ the wave front set is a compact subset of
$T^*X$. We say that $A=B+\cO(h^{\infty})$ microlocally in some set
$U\subset T^*X$ if $WF'_h(A-B)\cap U=\emptyset.$ We will often consider
sequences $u(h_j)\in L^2(X)$, where $h_j\rightarrow 0$ is a sequence of
positive numbers; we typically suppress the dependence on $j$ and write
$u=u(h)$. For such a sequence, we say that $u=\cO(h^{\delta})$
microlocally in some set $U\subset T^*X$, if $\Vert
Au\Vert_{L^2}=\cO(h^{\delta})$ for each $A\in
\Psi^{\mathrm{comp}}_h(X)$ such that $WF'_h(A)\subset U$. The notion
$u=o(h^{\delta})$ is defined similarly.     

\subsection{Semiclassical defect measures}
\begin{definition}
Assume that $h_j\rightarrow 0$ and $u=u(h_j)\in L^2(X)$ is uniformly
bounded as $j\rightarrow \infty$. We say that $u$ converges to a
non-negative Radon measure $\mu$ on $T^*X$ if for each $A=A(h)\in
\Psi^{\mathrm{comp}}_h(X)$, we have 
{\begin{equation}
\la A(h_j)u(h_j),u(h_j)\ra\rightarrow 
\int_{T^*X}\sigma_h(A)\,\mathrm{d}\mu \quad \mbox{as }
j\rightarrow\infty.
\end{equation}}\unskip
See \cite[Chapter 5]{Zwsemi} for an introduction to defect measures, in
particular,
\begin{enumerate}[(1)]
\item for each sequence $u(h_j)$ uniformly bounded in $L^2$, there
exists a subsequence $h_{j_k}$ such that $u({h_{j_k}})$ converges to
some $\mu$ \cite[Theorem 5.2]{Zwsemi};
\item we have $\mu(U)=0$ for some open set $U\subset T^*X$ if and only
if $u=o(1)$ microlocally on $U$.    
\end{enumerate} 
\end{definition}
The following basic properties of microlocal measures are easy to
prove, see e.g.~\cite{Dy16}. 

\begin{lemmma}[Ellipticity]\label{lemmaSME}
Take $P\in \Psi^0_h(X)$ and denote $p=\sigma_h(P)$. Assume that
$u=u(h_j)$ converges to some measure $\mu$ and $Pu=o(1)$ microlocally
in some open set $U\subset T^*X$. Then $\mu(U\cap \{p\neq 0\})=0$. 
\end{lemmma}

\begin{lemmma}[Propagation]\label{lemmaSMP}
Take $P,W\in \Psi^0_h$, denote $p=\sigma_h(P),\, w=\sigma_h(W)$, and
assume $P^*=P$. Assume that $u=u(h_j)$ converges to some measure $\mu$
and denote $f:=(P-\mathrm{i}hW)u$. Then for each $a\in
C_0^{\infty}(T^*X)$ and for each $Y\in \Psi_h^{\mathrm{comp}}(X)$ such
that $Y=1+\cO(h^{\infty})$ microlocally in a neighborhood of $\supp a$
we have,
{\begin{equation}
\left\vert \int_{T^*X}(H_p-2{\Re w})a\,\mathrm{d}\mu\right\vert \le 
2\Vert a\Vert _{\infty}\lim\sup_{h\rightarrow 0}(h^{-1}
\Vert Yf\Vert_{L^2}\Vert Yu\Vert_{L^2}). 
\end{equation}}\unskip
In particular, if $f=o(h)$ microlocally in a neighborhood of $\supp a$,
then
{\begin{equation*}
\int_{T^*X}(H_p-2{\Re w}a)\,\mathrm{d}\mu=0.
\end{equation*}}\unskip
\end{lemmma}

\section{The wave equation on the De Sitter Schwarzschild metric}
\label{SecWavSdS}

In this section we explain the general strategy for showing decay
estimates via microlocal methods for a simple example: the wave
equation on the De Sitter Schwarzschild metric. Let $\cM=\R_{t_*}\times
X,\, X=(r_--\epsilon,r_++\epsilon)\times \Sph^2_{\omega}$. Near $r=r_-$
resp.\ $r=r_+$ the metric is given by 
{\begin{equation*}
g = F\,\mathrm{d}t_*^2 - 2\,\mathrm{d}t_* \,\mathrm{d}r - r^2\,\mathrm{d}\omega^2 
\quad\mbox{resp.}\quad 
g = F\,\mathrm{d}t_*^2 + 2\,\mathrm{d}t_* \,\mathrm{d}r - r^2\,\mathrm{d}\omega^2,
\end{equation*}}\unskip
where $F(r) = 1 - ({2\bhm}/{r}) - ({1}/{3}) \Lambda r^2.$ Recall that
the surface gravities are given by $\kappa_{\pm} = ({1}/{2})
F'(r_{\pm})$. The wave operator is given near $r=r_-$ resp.\ $r=r_+$ by
{\begin{equation*}
\Box_g=2D_{t_*}D_r+\frac{1}{r^2}D_rr^2FD_r-r^{-2}\Delta_{\Sph^2}
\quad\mbox{resp.}\quad
\Box_g=-2D_{t_*}D_r+\frac{1}{r^2}D_rr^2FD_r-r^{-2}\Delta_{\Sph^2}. 
\end{equation*}}\unskip
Fourier transforming this with respect to $t_*$, i.e. putting 
{\begin{equation*}
\hat{\Box}_g(\sigma) = \mathrm{e}^{\mathrm{i}t_*\sigma}
\Box_g\mathrm{e}^{-\mathrm{i}t_*\sigma}. 
\end{equation*}}\unskip
then gives 
{\begin{equation*}
\hat{\Box}_g(\sigma)=-2\sigma D_r+\frac{1}{r^2}D_rr^2FD_r-r^{-2}\Delta_{\Sph^2}
\quad \mbox{resp.}\quad 
\hat{\Box}_g(\sigma)=2\sigma D_r+\frac{1}{r^2}D_rr^2FD_r-r^{-2}\Delta_{\Sph^2}. 
\end{equation*}}\unskip
We will consider in the following the forward problem
{\begin{equation}\label{MS1}
\Box_gu = f,
\end{equation}}\unskip
where $f\in C_0^{\infty}(\cM)$ and we require that $\supp
u\subset\{t_*\ge -T\}$ for some $T>0$.  Let $\Hext^s(X)$ resp.\ 
$\Hsupp^s(X)$ be the Sobolev spaces based on extendible resp.\ supported
distributions. Initial value problems can easily be transformed into
forward problems and vice versa, see e.g.~\cite{Mi23}. The existence of
a forward solution is assured by the global hyperbolicity of the
spacetime.  

\subsection{A key theorem}
A first important result is that the Fourier transformed wave operators
form an analytic family of Fredholm operators. Let $\gamma = \max
\{({1}/{\kappa_-}),({1}/{\kappa_+})\}$. We have 
\begin{thm}\label{thm5.1}
Let $s\ge 1/2$, ${\cX}^s=\{u\in \Hext^s(X);\,\hat{\Box}_g(0)u\in
\Hext^{s-1}(X)\},\, \cY^{s-1}=\Hext^{s-1}(X).$  The operators 
{\begin{equation*}
\hat{\Box}_g(\sigma):\cX^s\rightarrow \cY^{s-1} 
\end{equation*}}\unskip
form an analytic family of Fredholm operators of index $0$ for all
$\sigma\in \mathcal{C}$ such that 
{\begin{equation}\label{CondImsigma}
{\Im}\sigma>\frac{1-2s}{2\gamma}. 
\end{equation}}\unskip
Moreover $\hat{\Box}_g(\sigma)$ is invertible if ${\Im}\sigma> 0$. 
\end{thm} 

\begin{thm}\label{thm5.2}
Let $t_{0}\in \R$. There are $C,\, \nu>0$ such that for $0<\delta<C$
and $s>({1}/{2})+\gamma \delta$ any solution to (\ref{MS1}) with $f\in
\mathrm{e}^{-\delta t_*}\bar{H}^{s-1+\nu}(M)$ and with
$\supp(u)\cup\supp(f)\subset \{t_*>t_{0}\}$ has an asymptotic expansion
{\begin{equation}\label{Resexp}
u(t_*,r,\omega) - \sum_{j=1}^N \sum_{k=0}^{k_j}t_*^k
\mathrm{e}^{-\mathrm{i}\sigma_jt_*}a_{j,k}(r,\omega)\in 
\mathrm{e}^{-\delta t_*}\Hext^s(M),
\end{equation}}\unskip
where $\sigma_1,\ldots,\sigma_N$ are the (finitely many) quasinormal
mode frequencies with 
{\begin{equation*}
{\Im}\sigma_j>-\delta
\end{equation*}}\unskip
and $k_j$ their multiplicities, and where
$\mathrm{e}^{-\mathrm{i}\sigma_jt_*}a_{k,j}$ are the $C^{\infty}$
generalized quasinormal modes with frequency $\sigma_j$. 
\end{thm}

\begin{rmk}
\upshape
\begin{enumerate}[(1)]
\item {Quasinormal mode frequencies are also called resonances, they
correspond to those $\sigma$ such that $\hat{\Box}_g(\sigma)$ has a non
trivial kernel. By analytic Fredholm theory one obtains a meromorphic
family of operators $\hat{\Box}^{-1}_g(\sigma):\cY^{s-1}\rightarrow
\cX^s$. Resonances or quasinormal mode frequencies are then the poles
of this family of operators. These poles may be of higher order which
gives rise to the $t_*^k$ coefficients. 

\item Note that close to the black hole horizon,
$\hat{\Box}_g(\sigma)=\mathrm{e}^{\mathrm{i}\sigma
r_*}\tilde{\Box}_g(\sigma)\mathrm{e}^{-\mathrm{i}\sigma r_*}$, where
$\tilde{\Box}_g(\sigma)$ is the Fourier transformed operator with
respect to $t$. Therefore $v$ is mode solution for
$\hat{\Box}_g(\sigma)$ ($\hat{\Box}_g(\sigma)v=0$) if and only if
$u=\mathrm{e}^{-\mathrm{i}\sigma r_*}v$ is mode solution for
$\tilde{\Box}_g(\sigma)$ and the requirement that $v$ is smooth at the
horizon translates to the usual outgoing condition $u\sim
\mathrm{e}^{-\mathrm{i}\sigma r_*}$, $r_*\rightarrow -\infty$ for $u$. 

\item The analogue of Theorem \ref{thm5.1} is false in the
Schwarzschild case. Nevertheless the resolvent has good regularity
properties down to the real axis, see Vasy~\cite{Va21a,Va21b}.
This can then be used to show polynomial decay estimates, see H\"afner,
Hintz, Vasy~\cite{HHV21}, Hintz~\cite{Hi22} and Millet~\cite{Mi23}.

\item There exists a generalized version for the De Sitter Kerr metric
(see Vasy~\cite{Va13}, Petersen-Vasy~\cite{PV23}), but a priori nothing
is said about the existence of resonances with strictly positive
imaginary part. The non existence of mode solutions with positive
imaginary part of $\sigma$ has been shown by Whiting in the whole
subextreme case for the Kerr solution. The analogue question for the De
Sitter Kerr case remains open, but see Hintz~\cite{Hi21} for results in
this direction. 

\item High frequency resonances (meaning for large real part of
$\sigma$) are well approximated by so called pseudopoles which can be
computed and which are given by 
{\begin{equation*}
\frac{\sqrt{1-9\Lambda\bhm^2}}{3^{\frac{3}{2}}\bhm}
(\pm(\ell+1/2)-\mathrm{i}(1/2+n)),\quad \ell\in \N, n\in \N_0.  
\end{equation*}}\unskip
These resonances are very closely linked to the trapping of the
hamiltonian flow. 

\item As $\Box_g(0)1=0$, $0$ is a resonance, in particular the solution
\textit{does not} decay. However it can be shown (see~\cite{BoHa08}) that
this is the only real resonance. Together with the high frequency
analysis of Section~\ref{SecHFA} this gives the existence of a strip
beyond the real axis in which the only resonance is zero. As a
corollary one obtains the existence of a constant $c$ with}
{\begin{equation*}
u(t_*,r,\omega)-c\in \mathrm{e}^{-\delta t_*}\bar{H}^s(M). 
\end{equation*}}\unskip
\end{enumerate}
\end{rmk}
The first version of Theorem \ref{thm5.2} was shown by Bony and
H\"afner in~\cite{BoHa08}. This was formulated in the original Boyer
Lindquist coordinates and excluded the horizons. A generalization of
this including the horizons was subsequently obtained by Melrose,
S\'a-Barreto and Vasy in~\cite{MSBV14}. Later on the theorem was
generalized by Dyatlov to the De Sitter Kerr case
in~\cite{Dy11a,Dy11b}. The most general version of this was obtained by
Vasy in~\cite{Va13} first in the small $a$ case and later generalized
to the large $a$ case in collaboration with Petersen~\cite{PV23}. 

In the remaining of this section we will explain some of the principal
ideas in the proof of Theorems \ref{thm5.1} and \ref{thm5.2}.  Since
$\cM$ is globally hyperbolic there exists a unique forward solution of
(\ref{MS1}). Standard energy estimates give a first control of the
solution of the form
{\begin{equation*}
\vert u(t_*,r,\omega)\vert\lesssim \mathrm{e}^{Ct_*}. 
\end{equation*}}\unskip
It is then easy to see that the Fourier transform of $u$ 
{\begin{equation*}
\hat{u}(\sigma) = \int_{\R} \mathrm{e}^{\mathrm{i}\sigma t_*}u(t_*)
\,\mathrm{d}t_* 
\end{equation*}}\unskip
is holomorphic in ${\Im }\sigma>C$. We now take Fourier transform of
Equation (\ref{MS1}) to obtain:
{\begin{equation*}
\hat{\Box}_g(\sigma)\hat{u}(\sigma)=\hat{f}(\sigma). 
\end{equation*}}\unskip
We then can write 
{\begin{equation*}
u(t_*,x) = \dfrac{1}{2\uppi} \int_{{\Im \sigma}=C+1}
\mathrm{e}^{-\mathrm{i}\sigma t_*}\hat{\Box}^{-1}_g
(\sigma)\hat{f}(\sigma)\,\mathrm{d}\sigma.
\end{equation*}}\unskip
This doesn't give so far any better behavior of $u$, but
$\hat{\Box}^{-1}_g(\sigma)$ is a meromorphic family of bounded
operators and we can shift the contour to ${\Im}\sigma=-\delta$. The
residue theorem then gives the expansion in (\ref{Resexp}). The most
delicate part is the estimate of the error, which requires in
particular high frequency estimates of the resolvent
$\hat{\Box}^{-1}_g(\sigma)$. 

\begin{rmk}
\upshape
{By choosing $s$ large enough the strip where the operators form an
analytic Fredholm family can be made arbitrary large. For
${\Im}\sigma=-C$ and $C$ large enough this line lies below an infinite
number of resonances. The sum in (\ref{Resexp}) is then infinite, but
still converges in some appropriate sense, see~\cite{BoHa08}.}
\end{rmk} 

\subsection{Fredholm operators}

Let us recall that a Fredholm operator $T:X\rightarrow Y$ is a bounded
operator with closed rank such that kernel and cokernel $\mathrm{coker}
= Y / \mathrm{range} (T)$ are finite dimensional. The index is then
defined as  $\mathrm{ind} T = \mathrm{dimker} T - \mathrm{dimcoker} T$.
The situation is particularly interesting when we have an analytic
family $D\ni z\mapsto T(z)$ of Fredholm operators. The index is then a
continuous function of $z$ and the Fredholm alternative says that
either $T(z)$ is invertible for no $z\in D$ or $F^{-1}(z)$ form a
meromorphic family of operators. Typically elliptic operators are
Fredholm between suitable Sobolev spaces and vice versa the Fredholm
property between these Sobolev spaces requires ellipticity. We refer to
\cite[Section 19.2]{HoIII} for details. 

The operators $\hat{P}(\sigma)$ we are interested in are not elliptic.
An analytic Fredholm theory can nevertheless be developed as observed
by Vasy~\cite{Va13}. A central observation is that the Fredholm theory
follows from suitable estimates. A general setting is the following,
see e.g.\ \cite[Lemma 6.23]{Mi23}:

\begin{lemmma}\label{lemma5.4}
Let $X_0\subset X_1\subset X_2$ and $Y_0\subset Y_1\subset Y_2$ be
Banach spaces (with continuous dense inclusions). Let $P:X_1\rightarrow
Y_2$ be a bounded operator such that $P\vert_{X_0}$ is bounded from
$X_0$ to $Y_1$. We assume that both inclusions $X_1\subset X_2$ and
$Y_0\subset Y_1$ are compact and that there exists $C>0$ such that for
all $u\in X_1$ and all $v\in Y_1^*$:
{\begin{eqnarray*}
\Vert u\Vert_{X_1}   &\le& C(\Vert Pu\Vert_{Y_1}+\Vert u\Vert_{X_2}), \\
\Vert v\Vert_{Y_1^*} &\le& C(\Vert P^*v\Vert_{X_1^*}+\Vert v\Vert_{Y_0^*}).
\end{eqnarray*}}\unskip
Under these assumptions, $P$ is Fredholm as an operator between the
Banach space $\cX :=\{u\in X_1;\, Pu\in Y_1\}$ (endowed with the norm
$\Vert u\Vert_{\cX}^2=\Vert u\Vert_{X_1}^2+\Vert Pu\Vert_{Y_1}^2$) and
$Y_1$. 
\end{lemmma}
Let $\beta = \min \{({1}/{\kappa_-}), ({1}/{\kappa_+})\}$ if
${\Im}\sigma\ge 0$, $\beta = \max \{({1}/{\kappa_-}),
({1}/{\kappa_+})\}$ if ${\Im}\sigma\le 0$. We will obtain the following
global estimates.
\begin{enumerate}
\item Let $s>s_0>({1}/{2})-\beta {\Im}\sigma$. Then we have 
{\begin{equation}\label{GE1}
\Vert u\Vert_{\Hext^s(X)}\le C (\Vert
\hat{\Box}_g(\sigma)u\Vert_{\Hext^{s-1}(X)}+\Vert
u\Vert_{\Hext^{s_0}(X)}).
\end{equation}}\unskip
\item Let $s_0<s'<({1}/{2})+\beta{\Im }\sigma$. Then we have:
{\begin{equation}\label{GE2}
\Vert u\Vert_{\Hsupp^{s'}(X)}\le C (\Vert
\hat{\Box}^*_g(\sigma)u\Vert_{\Hsupp^{s-1}(X)}+\Vert
u\Vert_{\Hsupp^{s_0}(X)}).
\end{equation}}\unskip
\end{enumerate}
The compact Sobolev embedding $\Hext^s(X)\hookrightarrow
\Hext^{s_0}(X)$ and Lemma \ref{lemma5.4} then imply the Fredholm
property. In the following we will only show (\ref{GE1}), the arguments
for (\ref{GE2}) are quite similar.  

\subsection{Bounded frequency analysis}

Let us recall that we have near $r=r_{\pm}$:
{\begin{equation*}
\hat{\Box}_g(\sigma) = r^{-2}D_rr^2FD_r\pm 2\sigma
r^{-2}D_rr^2-r^{-2}\Delta_{\Sph^2}.
\end{equation*}}\unskip
We will apply the unitary transform $U:L^2((r_-,r_+)\times \Sph^2,
r^2\,\mathrm{d}r\,\mathrm{d}\omega)\rightarrow L^2((r_-,r_+)\times
\Sph^2, \mathrm{d}r\,\mathrm{d}\omega), u\mapsto ru$ and then work with
the operator  
{\begin{equation*}
r\hat{\Box}_g(\sigma)r^{-1} = r^{-1}D_rr^2FD_rr^{-1}\pm 2\sigma
r^{-1}D_rr-r^{-2}\Delta_{\Sph^2},
\end{equation*}}\unskip
which we call again $\hat{\Box}_g(\sigma)$ in the following. The above
operator has principal classical symbol ($\xi$ is now dual to $r$):
{\begin{equation*}
\sigma_2(\hat{\Box}_g(\sigma))=F\xi^2+r^{-2}\vert\eta\vert^2=:p. 
\end{equation*}}\unskip
The operator is therefore elliptic over all points $(r,\omega)$ of the
base manifold with $r_-<r<r_+$. We will distinguish the elliptic and
non elliptic regions. 

\subsubsection{Elliptic type estimates}\label{Sec5.2.1}

\begin{propos}\label{prop5.6}
Let $\chi\in C_0^{\infty}(r_-+\delta,r_+-\delta)$ for some $\delta>0$.
Then we have  
{\begin{equation*}
\Vert \chi u\Vert_{\bar{H}^s}\lesssim \Vert
\wh{\Box}_g(\sigma)u\Vert_{\bar{H}^{s-2}}+\Vert u\Vert_{\bar{H}^{s-1}}.
\end{equation*}}\unskip
\end{propos}

\begin{proof}
Let $\phi\in C_0^{\infty}((-1,1))$ with $\phi=1$ in a neighborhood of
$0$. We write 
{\begin{equation*}
\chi(x)=\chi(x)\phi(p)+\chi(x)(1-\phi(p)).
\end{equation*}}\unskip
Note that $\chi(r)\phi(p)\in S^{-\infty},\, \chi(x)(1-\phi(p))\in S^0$.
By pseudodifferential calculus we only have to consider the second
term. Now,
{\begin{equation*}
Op(\chi(x)(1-\phi(p))) = Op \left(\chi(x)\frac{1-\phi(p)}{p}\right)
\hat{\Box}_g(\sigma)+R,\, R\in \Psi^{-1}. 
\end{equation*}}\unskip
The estimate for $s=0$ then follows by pseudodifferential calculus.
Higher order estimates are obtained by applying
$Op(\xi^2+\vert\eta\vert^2)$.
\end{proof}

\subsubsection{Radial point estimates}\label{Sec5.3.2}

The characteristic manifold is 
{\begin{equation*}
\Sigma = \{(r,\omega,\xi,\eta)\in T^*X\setminus 0;
p(r,\omega,\xi,\eta)=0\}
\end{equation*}}\unskip
and it splits into positive and negative energy shells:
{\begin{equation*}
\Sigma=\Sigma_+\cup \Sigma_-,
\Sigma_{\pm}=\Sigma\cap \{\pm \xi>0\}.
\end{equation*}}\unskip
As $F(r_-)=0$ there is a lack of ellipticity at $r=r_-$. Note also that
we have 
{\begin{equation*}
H_p = (-F'\xi^2+2r^{-3}\vert \eta\vert^2)
\partial_{\xi}+2\xi F\partial_r+r^{-2}H_{\vert \eta\vert^2}. 
\end{equation*}}\unskip
The 
{\begin{equation*}
\Lambda^{\pm}_{\pm}=\left\{(r_{\pm},\omega,\xi,0);\xi\in \R^*,\,
\omega\in \Sph^2\right\}\cap \Sigma_{\pm} 
\end{equation*}}\unskip
are the radial points of $H_p$. The lower index indicates the point on
the base manifold, the upper index the energy level. It is not
difficult to see that $\Lambda^+_-$ and $\Lambda_+^-$ are sources of
the hamiltonian vector field whereas $\Lambda^-_-$ and $\Lambda^+_+$
are sinks.  

\begin{propos}\label{prop5.7}
Suppose that $u\in \Hext^{s}$ with
$s>(1/2)-({1}/{\kappa_{\pm}}){\Im}\sigma$. Let $\chi_{\pm}\in
C^{\infty}_0((r_{\pm}-\delta,r_{\pm}+\delta))$. Then for $\delta>0$
sufficiently small we have: 
{\begin{equation*}
\Vert \chi_{\pm} u\Vert_{\Hext^s}\lesssim \Vert
\hat{\Box}_g(\sigma)u\Vert_{\Hext^{s-1}}+\Vert
u\Vert_{\Hext^{s-\frac{1}{2}}}. 
\end{equation*}}\unskip
\end{propos}

\begin{rmk}
\begin{enumerate}
\item {\rm The condition $s > (1 / 2) - ({1}/{\kappa_{\pm}}){\Im}\sigma$ gives
the condition (\ref{CondImsigma}) in Theorem \ref{thm5.1}. 
\item Let us try to understand where the condition
$s>({1}/{2})-\beta{\Im}\sigma$ comes from. As a toy model we consider
{\begin{equation*}
P_{\sigma}=D_xxD_x-\sigma D_x,\quad x\in (-1,1).
\end{equation*}}\unskip
In this case $\beta=1$. We have 
{\begin{equation*}
P_{\sigma}(x\pm \mathrm{i}0)^{\mathrm{i}\sigma}=0.
\end{equation*}}\unskip
We have $(x\pm \mathrm{i}0)^{\mathrm{i}\sigma}\in
H^{(1/2-{\Im}\sigma)-}$  but not in $H^s$ for $s>(1/2)-{\Im}\sigma$.
Imposing Sobolev regularity $s>(1/2)-{\Im}\sigma$ thus excludes these
solutions.  

\item The estimate is a very simplified version of the radial point
estimate obtained by Vasy, see \cite[Proposition 2.3]{Va13}. This kind
of estimates goes back to Melrose, see~\cite{Me94}. They are central in
Vasy's method concerning the Fredholm theory for non elliptic problems.
We also refer to Zworski~\cite{Zw16} for a review of this method.}
\end{enumerate}
\end{rmk}

\begin{proof}[Main ideas of the proof]
We will prove the proposition only for a model problem. We will
consider the operator which is given near $r=r_{\pm}$ by 
{\begin{equation*}
\hat{P}(\sigma)=r^{-1}D_rr^2FD_rr^{-1}\pm 2\sigma r^{-1}D_rr
\end{equation*}}\unskip
and $\hat{P}(\sigma)$ acts on $L^2((r_--\epsilon,r_++\epsilon);dr)$.  

Also to avoid technical issues we will suppose 
{$u\in C_0^{\infty}((r_--\epsilon,r_++\epsilon))$.}
We prove the proposition
at $r=r_-,$ the proof at $r=r_+$ being essentially the same. We also
put $x=r-r_-$. Let $\chi_-$ be as in the hypotheses with
$x\chi_-'(x)\le 0$.  We drop the index---in the following. Let
$\psi\in C^{\infty}(\R),\, \psi(x)=0$ for $x\le 0$ and $\psi(x)=1$ for
$x\ge 1$.  
{\begin{equation*}
A=C^*C,\quad C=Op(c),\quad c:=\xi^{s-\frac{1}{2}}\psi(\xi)\chi(x). 
\end{equation*}}\unskip
Let $a:=\sigma_{2s-1}(A)$.  Now,
{\begin{eqnarray}
2{\Im}\la \hat{P}(\sigma)u,A u\ra 
&=&   \la \mathrm{i}(\hat{P}^*(\sigma)A-A\hat{P}(\sigma))u,u\ra \nonumber \\
&=&   \la \mathrm{i}[\hat{P}(\sigma),A]u,u\ra+\la \mathrm{i}(\hat{P}^*(\sigma)-\hat{P}(\sigma))A u,u\ra \nonumber \\
&=&   \la \mathrm{i}[\hat{P}(\sigma),A]u,u\ra-4{\Im}\sigma \la r^{-1}D_xr A u,u\ra. \label{MS4}
\end{eqnarray}}\unskip
The symbolic calculation gives 
{\begin{equation}\label{Hpa}
H_pa = -(2s-1)\xi^{2s}\psi^2(\xi)F'\chi^2(x) - 
2F'\psi(\xi)\psi'(\xi)\xi^{2s+1}\chi^2(x) + 
4\xi^{2s}\psi^2(\xi)F\chi(x)\chi'(x).
\end{equation}}\unskip
Note that the second term is in $S^{-\infty}$ and that the third term
is negative. 

We have 
{\begin{equation*}
\la \mathrm{i}[\hat{P}(\sigma),A]u,u\ra = \la Op(H_pa)u,u\ra+\la Ru,u\ra, 
\quad R\in \Psi^{2s-1}. 
\end{equation*}}\unskip
Applying the sharp G\aa rding inequality, Theorem \ref{thm4.2}, we find
{\begin{equation*}
\la \mathrm{i}[\hat{P}(\sigma),A]u,u\ra-4{\Im}\sigma 
\la r^{-1}D_x rAu,u\ra \le 
(-(2s-1)2\kappa_-+\tilde{\epsilon}(\delta) - 
4{\Im}\sigma)\Vert Op(\psi(\xi)\chi(x))u
\Vert_{\bar{H}^s}^2+\Vert u\Vert^2_{\bar{H}^{s-\frac{1}{2}}},
\end{equation*}}\unskip
where $\tilde{\epsilon}(\delta)\rightarrow 0, \, \delta\rightarrow 0$.
Here and in the following
$\Hext^s=\Hext^s((r_--\epsilon,r_-+\epsilon))$.

We also have 
{\begin{equation*}
\vert 2{\Im}\sigma\la \hat{P}(\sigma)u,Au\ra\vert \le 
C_{\hat{\epsilon}}\Vert \hat{P}(\sigma)u\Vert^2_{\bar{H}^{s-1}} + 
\hat{\epsilon}\Vert Op(\psi(\xi)\chi(x))u\Vert^2_{\bar{H}^s}+C
\Vert u\Vert_{\bar{H}^{s-1}}^2.
\end{equation*}}\unskip
Putting everything together and choosing $\delta,\, \hat{\epsilon}$
small enough gives:
{\begin{equation}\label{Drcontrol}
\Vert Op(\psi(\xi)\chi(x)) u\Vert^2_{\bar{H}^s}\le 
C\left(\Vert \hat{P}(\sigma)u\Vert^2_{\bar{H}^s} + 
\Vert u\Vert_{\bar{H}^{s-\frac{1}{2}}}^2\right).
\end{equation}}\unskip
We now want to obtain the same estimate with $Op(\psi(\xi)\chi(x))$
replaced by $\chi(x)$. To show the estimate for $Op(\psi(-\xi)\chi(x))$
we can replace $a(x,\xi)$ by $a(x,-\xi)$, then the above proof works
for $-\hat{P}(\sigma)$ instead  of $\hat{P}(\sigma)$. Writing
$1=\psi(\xi)+\psi(-\xi)+\psi_0(\xi)$ and using that $\psi_0(\xi)\in
S^{-\infty}$ we see that we can indeed replace $Op(\psi(\xi)\chi(x))$
by $\chi(x)$.
\end{proof}

\begin{rmk}
Using a regularization procedure one can show that it is sufficient to
suppose $Cu\in L^2$. What is usually called a radial point estimate is
rather (\ref{Drcontrol}), the important property of
$B=Op(\psi(\xi)\chi(x))$ is that backward bicharacteristics starting in
$WF'(B)$ tend to $\Lambda_-^+$, where $C$ has to be elliptic. In this
very simplified model this is used in the proof via the sign of
$4\xi^{2s}\psi^2(\xi)F\chi(x)\chi'(x)$ in (\ref{Hpa}). Also
$\hat{P}(\sigma)$ can be replaced by $G\hat{P}(\sigma)$ with $G\in
\Psi^0$ elliptic at $\Lambda_-^+$ as long as the closure of all
backward bicharacteristics starting in $WF'(B)$ is contained in the
elliptic set of $G$. We refer to~\cite{Va13} for details.   
\end{rmk}

\subsubsection{Proof of Theorem \myRef{thm5.1}}

Putting the estimates in Sections~\ref{Sec5.2.1} and \ref{Sec5.3.2}
together we obtain the following global estimates:

\begin{propos}
Let $u\in \Hext^{s}$ and $s> 1/2-\beta{\Im}\sigma$. Then we have 
{\begin{equation*}
\Vert u\Vert_{\bar{H}^s(X)}\le C\left(\Vert \hat{\Box}_g
(\sigma)u\Vert_{\bar{H}^{s-1}(X)}+
\Vert u\Vert_{\bar{H}^{s-\frac{1}{2}}(X)}\right). 
\end{equation*}}\unskip
\end{propos}

\begin{proof}
Let us first choose $\chi_{\pm}$ as in Proposition \ref{prop5.7} and
$\chi_{\pm}=1$ on $(r_{\pm} - ({\delta}/{2}), r_{\pm}+
({\delta}/{2}))$. We then choose $\hat{\chi}_{\pm}$ with $\supp
\hat{\chi}_{-}\subset [r_--\epsilon,r_-- ({\delta}/{2})),\,
\supp\hat{\chi}_{+}\subset (r_++ ({\delta}/{2}),r_++\epsilon]$ and
$\chi_0\in C_0^{\infty}(r_-+ ({\delta}/{2}),r_+-({\delta}/{2}))$ with
$\hat{\chi}_-+\chi_-+\chi_0+\chi_++\hat{\chi}_+=1$. We then use
Proposition \ref{prop5.6} on $\supp \chi_0$ and Proposition
\ref{prop5.7} on $\sup \chi_{\pm}$. Concerning $\supp \hat{\chi}_{\pm}$
let us observe that the symbol $p=F\xi^2+r^{-2}\vert \eta\vert^2$ is
hyperbolic there, because $F$ is negative on this interval. The
estimate follows then from classical hyperbolic estimates, we refer
to~\cite{Mi23} for details. 
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm5.1}]
We obtain in a similar way a suitable estimate also for the adjoint
operator. The two estimates then give the Fredholm property. To show
that the operators are invertible for ${\Im}\sigma>0$ we have to show
that kernel and cokernel are empty. We only consider the kernel, the
proof for the cokernel is similar. Let ${\Im}\sigma>0,\, u\in \Hext^s$
and $\hat{\Box}_g(\sigma)u=0$. Then 
{\begin{equation*}
F\tilde{\Box}_g(\sigma)\mathrm{e}^{-\mathrm{i}\sigma
h(r)r_*(r)}u(r,\omega)=0,
\end{equation*}}\unskip
where $h$ is some smooth function with $h(r)=1$ for $r\le 3\bhm$ and
$h(r)=-1$ for $r\ge 4\bhm$ and $\tilde{\Box}_g(\sigma)$ denotes the
Fourier transform with respect to $t$:
{\begin{equation*}
F\tilde{\Box}_g(\sigma)=r^{-2}FD_r(r^2F)D_r-r^{-2}F\Delta_{\Sph^2}-\sigma^2.
\end{equation*}}\unskip
$F\tilde{\Box}_g(0)$ is a selfadjoint positive operator on
$L^2((r_-,r_+)\times
\Sph^2;F^{-1}r^2\,\mathrm{d}r\,\mathrm{d}\omega^2)$. But for
${\Im}\sigma>0$ we have $\mathrm{e}^{-\mathrm{i}\sigma
h(r)r_*(r)}u(r,\omega)\in L^2((r_-,r_+)\times
\Sph^2;F^{-1}r^2\,\mathrm{d}r\,\mathrm{d}\omega^2),$ which is a
contradiction. Thus $u\vert_{(r_-,r_+)\times \Sph^2}=0$. To see that
$u$ really is zero on the whole manifold
$(r_--\epsilon,r_++\epsilon)\times \Sph^2$ we can follow \cite[Lemma
1]{Zw16}, see \cite[Proof of Theorem 4.5]{AHW22} for details.  The
index $0$ property then follows from the continuous dependence of the
index on $\sigma$.
\end{proof}

\subsection{High frequency analysis: trapping} \label{SecHFA}

Whereas the bounded frequency analysis is sufficient to prove Theorem
\ref{thm5.1}, the proof of Theorem \ref{thm5.2} requires a precise
control of the estimates when ${\Re}\sigma\rightarrow \infty$. This
requires a precise analysis of the semiclassical flow whose projection
on the base manifold gives the geodesic flow. The analysis then becomes
more complicated, but the general structure of elliptic, propagation
and radial point estimates remains similar. However there is a new
aspect which we want to discuss in this subsection which is trapping.  
To analyze the trapping it is easier to go back to the initial Boyer
Lindquist coordinates. We will present a toy model, the proof for the
existence of a resonance free strip follows very closely the proof of
Dyatlov~\cite{Dy16} for the general situation. 

\subsubsection{Resonance free strip}

Fourier transforming with respect to $t$ and fixing the angular
momentum $\ell$ leads to considering the operator:
{\begin{equation}\label{T1.1}
\tilde{P}_{\ell}-\sigma^2=D_{r_*}^2+V\ell(\ell+1)-\sigma^2,\quad
V=\frac{F}{r^2}. 
\end{equation}}\unskip
We introduce the semiclassical parameter $h^2 ={1}/{\ell(\ell+1)}$.
As explained in Section~\ref{TdSS} trapping is concentrated at $r_*=0$.
We introduce a Taylor development of $V$ at $r_*=0$. We will put
$x=r_*$ in the following. This leads to considering the local model 
{\begin{equation}\label{T1.2}
\tilde{P}-\lambda=h^2D_x^2-x^2-\lambda,\quad \lambda=\cO(h), 
\end{equation}}\unskip
which we will use as a toy model in the following. Note that in the
model (\ref{T1.1}) the critical energies are $\sigma^2\sim
V(3\bhm)\ell(\ell+1)$, after rescaling this becomes
$\lambda\sim\sqrt{V(3\bhm)}$. In the toy model however the critical
energy is zero, we therefore consider $\lambda=\cO(h)$. We obtain
semiclassical estimates that can be translated into high frequency
estimates in the original scaling. 

The symbol writes $\xi^2-x^2=(\xi-x)(\xi+x)$ which suggests that the
operator is unitary equivalent to $P=h(xD_x+D_xx)$. We can indeed
introduce the transform 
{\begin{equation*}
Uu(x) = \dfrac{1}{\sqrt{\uppi}h} 
\mathrm{e}^{-\mathrm{i}\frac{x^2}{2h}} \int 
\mathrm{e}^{\mathrm{i}\frac{\sqrt{2}xy}{h}}
\mathrm{e}^{-\mathrm{i}\frac{y^2}{2h}}u(y)\,\mathrm{d}y. 
\end{equation*}}\unskip  

It can be checked that $U:L^2(\R)\rightarrow L^2(\R)$ is an isometry
and that $\tilde{P}U=UP.$ We will in the following consider the
operator $P = ({h}/{2})(xD_x+D_xx)$. Note that 
{\begin{equation*}
\sigma_h(P)=x\xi=:p,\quad
H_p=-\xi\partial_{\xi}+x\partial_x.
\end{equation*}}\unskip 
Thus the Hamiltonian equations can be explicitly solved,
$\xi(s)=\mathrm{e}^{-t}\xi_0$, $x(s)=\mathrm{e}^tx_0$, where
$(x_0,\xi_0)$ are the initial data. In the abstract setting of
Section~\ref{TG} this means that we have:
{\begin{equation*}
\Gamma_+=\{\xi=0\},\quad \Gamma_-=\{x=0\},\quad K=\{x=\xi=0\}. 
\end{equation*}}\unskip
To analyze the situation near the trapped set, we will consider the
situation on a compact manifold and add an absorption operator.
Concretely, we put $X=\Sph^1=\R/(6\Z)$; we view $X$ as the interval
$[-3,3]$ with the endpoints glued together. We add an absorption
operator $Q=Op(q),\, q\in C^{\infty}(T^*X)$ such that $q(x,\xi)=0$ for
all $\vert (x,\xi)\vert\le 1$ and $q(x,\xi)=1$ for all $\vert
(x,\xi)\vert\ge 2$. We will now consider the operator 
$P-\mathrm{i}Q-\lambda$ on $X$. Let
$R(\lambda)=(P-\mathrm{i}Q-\lambda)^{-1}$. 

\begin{lemmma}\label{lemma5.7}
For each $\epsilon>0$ and $h$ small enough depending on $\epsilon$ we
have 
{\begin{equation}\label{RE1}
\Vert R(\lambda)\Vert_{L^2\rightarrow L^2}=o(h^{-2})
\quad \mbox{if }\  \vert \lambda\vert=\cO(h),\quad
{\Im}\lambda >-(\tfrac{1}{2}-\epsilon)h. 
\end{equation}}\unskip
\end{lemmma}

\begin{rmk}
Let us note that the same bound on the resolvent holds in general
situations of normally hyperbolic trapping as described in
Section~\ref{TG}, see~\cite{Dy16}. We will give here the same proof for
the model operator, which has the advantage of avoiding the technical
issues while keeping the main ideas of the proof. Further gaps can be
obtained for $r$-normally hyperbolic trapping, see also~\cite{Dy16}.  
\end{rmk}
To prove the lemma we will proceed by contradiction. Suppose that
(\ref{RE1}) does not hold.  Then for some $\epsilon>0$ small enough,
there exist sequences $h_j,h_j\rightarrow 0,\, \lambda(h_j)$,
$u=u(h_j)\in L^2$ with the following properties 
{\begin{equation}\label{CA4}
\Vert u\Vert_{L^2}=1,\quad
\vert \lambda\vert =\cO(h),\quad
{\Im}\lambda >-(\tfrac{1}{2}-\epsilon)h
\end{equation}}\unskip
and 
{\begin{equation}\label{CA5}
(P-\mathrm{i}Q-\lambda)u=\cO(h^2).
\end{equation}}\unskip

The interesting properties are obtained when commuting $(P-\lambda)u$
with the defining functions of $\Gamma_{\pm}$. Indeed we have 
{\begin{equation*}
hD_x(P-\lambda)u=(P-\mathrm{i}h-\lambda)hD_xu.
\end{equation*}}\unskip
Now suppose that ${\Im \lambda}\ge -(({1}/{2})-\epsilon)h$ for some
$\epsilon>0$. Then $h+{\Im \lambda}\ge (({1}/{2})+\epsilon) h$ and  we
can write 
{\begin{equation}\label{EsthDu1}
hD_xu=(P-\mathrm{i}h-\lambda)^{-1}hD_x(P-\lambda)u.
\end{equation}}\unskip
A simple application of the spectral theorem gives 
{\begin{equation}\label{EsthDu}
\Vert hD_xu\Vert\lesssim h^{-1}\Vert hD_x(P-\lambda)u\Vert. 
\end{equation}}\unskip
Note that the corresponding commutation relation with $x$
{\begin{equation*}
x(P-\lambda)=(P+\mathrm{i}h-\lambda)xu. 
\end{equation*}}\unskip
doesn't give an analogous estimate because of the different sign. 

The estimate (\ref{EsthDu}) can be microlocalized. Let for $\delta \ll 1$
{\begin{equation*}
U_{\delta}=\{\vert \xi\vert<\delta, \vert x\vert<\delta\}.
\end{equation*}}\unskip
Note that 
{\begin{equation}\label{5.11}
(P-\lambda)u = \cO(h^2)\mbox{ microlocally on }U_{2\delta}.
\end{equation}}\unskip 
Let $\phi\in C_0^{\infty}((-4\delta,4\delta)),\phi=1$ on
$[-3\delta,3\delta]$, $\theta_+=\xi\phi(x)\phi(\xi)$,
$\Theta_+=Op_h(\theta_+).$ Applying $\Theta_+$ to (\ref{5.11}) and
using that
$[\Theta_+,P]=\mathrm{i}hOp_h(H_p\theta_+)+\cO(h^2)
=-\mathrm{i}h\Theta_++\cO(h^2)$
microlocally on $U_{2\delta}$ together with (\ref{5.11}) gives an
appropriate microlocalized version of (\ref{EsthDu}): 
{\begin{equation}\label{Theta+umicro}
\Theta_+u=\cO(h),\quad\mbox{microlocally on } U_{2\delta}.  
\end{equation}}\unskip
A central lemma is then the following:

\begin{lemmma}\label{lemma5.11}
There exists a constant $C$ such that for each $\delta_0>0$,
{\begin{equation}\label{5.4}
\mu(U_{\delta}\cap\{\vert x\vert<\delta_0\})\le C\delta_0. 
\end{equation}}\unskip
\end{lemmma}

\begin{proof}
We first note that for $a\in C_0^{\infty}(\Gamma_+\cap U_{\delta})$ we
have:
{\begin{equation}\label{5.5}
\left\vert \int_{\Gamma_+\cap
U_{\delta}}H_{\theta_+}a\,\mathrm{d}\mu\right\vert \le C\Vert
a\Vert_{\infty}. 
\end{equation}}\unskip
This follows indeed from Lemma \ref{lemmaSMP} and (\ref{Theta+umicro}).
Now choose $a=\tilde{a}(x)\chi(\xi)$, with $\chi\in
C_0^{\infty}((-\delta,\delta)),\, \chi(0)=1$ and
{\begin{equation*}
\supp \tilde{a}\subset(-2\delta_0,\delta_1),\quad
\Vert \tilde{a}\Vert_{\infty}\le 1,\quad
\partial_x\tilde{a}\ge-\frac{2}{\delta_1},\quad
\partial_x\tilde{a}\ge \frac{1}{3\delta_0}\quad 
\mbox{for }\vert x\vert\le \delta_0  
\end{equation*}}\unskip  
for some fixed $\delta_1\in (0,\delta)$ independent of $\delta_0$.
Noting that $\theta_+=\xi$ on $U_{\delta}$ we find
{\begin{equation*}
\int_{{\Gamma_+}\cap U_{\delta}}H_{\theta_+}a\,\mathrm{d}\mu\ge
\frac{1}{3\delta_0}\mu(\{\vert x\vert\le
\delta_0\})-\frac{2}{\delta_1}\mu(\{\delta_0\le \vert x\vert\le
\delta_1\}). 
\end{equation*}}\unskip
The LHS is bounded by a $\delta_0$-independent constant by (\ref{5.5});
so is the second term on the RHS (since $\mu$ is a finite measure).
Multiplying both sides by $3\delta_0$ we obtain (\ref{5.4}).
\end{proof}

\begin{lemmma}\label{lemma5.12}
{\begin{equation*}
\mu(U_{\delta}\setminus \Gamma_+)=0,\quad \mu(U_{\delta})>0.
\end{equation*}}\unskip
\end{lemmma}

\begin{proof}
The lemma follows from two type of estimates 
\begin{enumerate}
\item Take $A\in \Psi^0_h(\R)$ such that $WF'_h(A)\cap\{p=0\}\cap
\Gamma_+=\emptyset$. Then 
{\begin{equation*}
\Vert A u\Vert_{L^2}\le
Ch^{-1}\Vert(P-\mathrm{i}Q-\lambda)u\Vert_{L^2}+\cO(h^{\infty})\Vert
u\Vert_{L^2}.
\end{equation*}}\unskip
This follows from the elliptic estimate \cite[Proposition 2.4]{DyZw13}
and propagation of singularities with a complex absorbing operator
\cite[Proposition 2.4]{DyZw13}, since for each $(x,\xi)\in WF'_h(A)$,
there exists $ t\ge 0$ such that
$\mathrm{e}^{-tH_p}(x,\xi)\in\{p-\mathrm{i}q\neq 0\}$. 

\item Let $B\in \Psi^{\mathrm{comp}}_h(X)$ such that $\sigma_h(B)\neq
0$ on $\{p=0\}\cap K$:
{\begin{equation*}
\Vert u\Vert_{L^2}\le
Ch^{-1}\Vert(P-\mathrm{i}Q-\lambda)u\Vert_{L^2}+\Vert Bu\Vert_{L^2}. 
\end{equation*}}\unskip
To prove this second estimate we use the elliptic estimate and
propagation of singularities, noting that for each $(x,\xi)\in T^*\R$,
there exists $t\ge 0$ such that $\mathrm{e}^{-tH_p}(x,\xi)\in
\{p-\mathrm{i}q\neq 0\}\cup\{\sigma_h(B)\neq 0\}$, see \cite[Lemma
4.1]{Dy15}.
\end{enumerate}
\end{proof}

\begin{proof}[Proof of Lemma \ref{lemma5.7}]
Starting with (\ref{CA5}) and passing to a subsequence, we may assume
that 
{\begin{equation*}
u\rightarrow \mu,\quad h^{-1}{\Im \lambda}\rightarrow \omega\in \mathcal{C}. 
\end{equation*}}\unskip
By Lemma \ref{lemma5.12} we know that  
{\begin{equation*}
\mu(U_{\delta})>0,\quad \mu(U_{\delta}\setminus \Gamma_+)=0.
\end{equation*}}\unskip
We can now apply Lemma \ref{lemmaSMP} to see that for each $a\in
C_0^{\infty}(U_{\delta})$ we have 
{\begin{equation*}
\int_{\Gamma_+\cap U_{\delta}}H_pa\,\mathrm{d}\mu =
2{\Im} \omega\int_{\Gamma_+\cap U_{\delta}} a\,\mathrm{d}\mu.
\end{equation*}}\unskip
For $t\ge 0$, since $\mathrm{e}^{-tH_p}(\Gamma_+\cap U_{\delta})\subset
\Gamma_+\cap U_{\delta}$, we have 
{\begin{equation*}
\mu(\mathrm{e}^{-tH_p}(U_{\delta})) = 
\mathrm{e}^{2t{\Im}\omega}\mu(U_{\delta}). 
\end{equation*}}\unskip
By Lemma \ref{lemma5.11}, as $t\rightarrow\infty$
{\begin{equation*}
\mu(\mathrm{e}^{-tH_p}(U_{\delta}))\le 
\mu(U_{\delta}\cap \{\vert x\vert<\delta \mathrm{e}^{-t}\})\le 
C\mathrm{e}^{-t}. 
\end{equation*}}\unskip
However, since ${\Im}\omega\ge - ({1}/{2})+\epsilon$, we see that
$\mathrm{e}^{2t{\Im}\omega}$ decays exponentially slower than
$\mathrm{e}^{-t}$, as $t\rightarrow\infty$. Since $\mu(U_{\delta})>0$
we arrive at a contradiction, finishing the proof of the Proposition.
\end{proof}

\subsubsection{Approximation of resonances generated by the trapping}

As we have just seen the relatively mild trapping leads to a resonance
free strip. Nevertheless the trapping creates resonances that we now
want to compute in a high frequency limit.  The resonances can be
computed as the eigenvalues of a scaled operator which is no longer
selfadjoint. The operator $P$ coming from the De Sitter Schwarzschild
metric is an operator with holomorphic coefficients 
{\begin{equation}\label{PSdS}
P(w,hD_w;h) = (hD_w)^2+V(w,h);\quad
V(w,h) = Fr^{-2}(1+h^2r\partial_rF). 
\end{equation}}\unskip
Here $w$ replaces the Regge--Wheeler coordinate $r_*$. We put
$\Gamma_{\theta}=\mathrm{e}^{\mathrm{i}\theta}\R$ and we parametrize
$\Gamma_{\theta}$ by $x$, $w=\mathrm{e}^{\mathrm{i}\theta}x$.  The
scaled operator is then defined by
$P_{\theta}(x,hD_x;h)=P(w,hD_w;h)\vert_{\Gamma_{\theta}}$.  Note that
the RHS of this equation is well defined because all the coefficients
are holomorphic. The important proposition is the following
(see~\cite{SZ97}):

\begin{propos}
For $-2\theta<arg z<2\uppi-2\theta$ the operator   
{\begin{equation*}
P_{\theta}(x,hD_x;h)-z:H^2(\Gamma_{\theta})\rightarrow
L^2(\Gamma_{\theta}),
\end{equation*}}\unskip
is a Fredholm operator of index $0$. The eigenvalues of
$P_{\theta}(x,hD_x;h)$ in $-2\theta<\arg z<2\uppi-2\theta$ coincide with
the poles of the meromorphic continuation of
$(P_{\theta}(x,hD_x;h)-z)^{-1}$ there.
\end{propos}
To figure out what is a good choice for $\theta$ we come back to our
toy model $\tilde{P}=h^2D_x^2-x^2$ in the previous section. We then see
that a good choice is $\theta = {\uppi}/{4}$. Indeed we then find
$P_{\theta}=-\mathrm{i}(h^2D_x^2+x^2)$, which is just $-\mathrm{i}$
times the (semiclassical) harmonic oscillator and its eigenvalues are
of course known. Putting all this together leads to the following
characterization of the high frequency resonances in the De Sitter
Schwarzschild case (see~\cite{SZ97}):

\begin{propos}
There exists $K>0$ and $\theta>0$ such that for any $C>0$ there exists
an injective map, $\tilde{b}$, from the sets of pseudo-poles
{\begin{equation*}
\frac{(1-9\Lambda\bhm^2)^{1/2}}{3^{\frac{3}{2}}\bhm}
\left(\pm \N\pm\frac{1}{2}-\mathrm{i}\frac{1}{2}\left(\N_0+
\frac{1}{2}\right)\right)
\end{equation*}}\unskip
into the set of poles of the meromorphic continuation of
$(P-\sigma^2)^{-1}$ such that the poles in 
{\begin{equation*}
\Omega_C=\{\sigma :{\Im} \sigma>-C,\, \vert \sigma\vert>K,\,
{\Im}\sigma>-\theta\vert {\Re}\sigma\vert\}
\end{equation*}}\unskip
are in the image of $\tilde{b}$ and for $\tilde{b}(\nu)\in \Omega_C$,
{\begin{equation*}
\tilde{b}(\nu)-\nu\rightarrow 0,\quad \mbox{when }
\vert \nu\vert\rightarrow\infty.
\end{equation*}}\unskip
If ${\Re}\nu = ({(1-9\Lambda\bhm^2)^{1/2}}/{3^{{3}/{2}}\bhm})
(\pm\ell\pm ({1}/{2}))$, $\ell=1,2,\ldots$, then the corresponding
pole, $\tilde{b}(\nu)$ has multiplicity $2\ell+1$. 
\end{propos}

\subsection{More general situations}
\subsubsection{De Sitter Kerr metric}

The proof for the equivalent of Theorems~\ref{thm5.1} and \ref{thm5.2}
for the De Sitter Kerr metric follows the same lines as in the De
Sitter Schwarzschild case. The radial point estimates are now
formulated in terms of propagation estimates which permit to propagate
regularity out of these points. As already mentioned the trapping
keeps the same structure and the proof for the high energy resolvent
estimates follows the same lines. At first glance the new aspect seems
to be superradiance. Nevertheless thanks to the very robust Fredholm
setting this point is now reduced to the analysis of mode solutions.  

\subsubsection{Kerr metric}

The Kerr case is asymptotically Minkowskian and one has to deal with
this aspect. As an immediate consequence, the underlying resolvent
doesn't possess a meromorphic extension in a strip beyond the real
axis. It has nevertheless good $C^k$ properties down to the real axis,
see~\cite{Va21a,Va21b} and~\cite{HHV21}. Another important aspect is
that we find a radial point structure of the Hamiltonian flow also at
infinity.    

\subsubsection{Other fields}

As long as the fields are massless and not charged, most of the here
presented aspects remain valid also for other hyperbolic field
equations than the wave equation. Note that the presence of mass or
charge of the field can also create growing modes, we refer to
Shlapentokh-Rothmann for the Klein Gordon equation on the Kerr
metric~\cite{SR14} and to Besset--H\"afner~\cite{BeHa21} for the charged
Klein Gordon equation on the De Sitter Reissner--Nordstr\"om or De
Sitter Kerr Newman black hole. In the case of linearized gravity, one
has of course to deal with the gauge. The presented methods are well
adapted to generalized wave map gauges. In these generalized wave map
gauges the Fourier linearized Einstein operator $\wh{L}_g(\sigma)$ will
have a kernel for $\sigma=0$ consisting of linearized (De Sitter) Kerr
solutions and some special gauge solutions. The existence of this
kernel makes the analysis of $\wh{L}^{-1}_g(\sigma)$ more delicate
close to~$\sigma=0$.      

\subsubsection{Perturbations of the (De Sitter) Kerr metric}

As long as the metric remains stationary and the important structure of
radial points, trapping etc. is not changed the analysis remains more
or less the same on the linear level. Note that in contrast to the
trapping the radial point structure is a priori not stable with respect
to perturbations; however the estimates are, see \cite[Remark
2.5]{Va13} for a discussion of this point. A delicate part is the
analysis of the mode solutions which most often is analyzed on the
exact spacetimes by separation of variables and ode techniques. This
can however often be analyzed in a perturbative regime.    

If the spacetime itself is not stationary but say decays to a (De
Sitter) Kerr solution, the here presented analysis has to be completed
by a second step. Note however that this second step can be purely
microlocal, see~\cite{Hi23}. These aspects go beyond the scope of this
review.  

\section*{Declaration of interests}

The authors do not work for, advise, own shares in, or receive funds
from any organization that could benefit from this article, and have
declared no affiliations other than their research organizations.

\section*{Acknowledgements}

The author thanks Peter Hintz and Pascal Millet for helpful discussions
on this review. He also acknowledges support from the ANR funding
ANR-20-CE40-0018-01.

\CDRGrant[ANR]{ANR-20-CE40-0018-01}

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