# The global stability of the Kaluza–Klein spacetime

Cécile Huneau  École Polytechnique & CNRS<EMAIL_ADDRESS>,
Annalaura Stingo  École Polytechnique<EMAIL_ADDRESS>and
Zoe Wyatt  King’s College London<EMAIL_ADDRESS>
###### Abstract.

In this paper we show the classical global stability of the flat Kaluza–Klein
spacetime, which corresponds to Minkowski spacetime in $\mathbb{R}^{1+4}$ with
one direction compactified on a circle. We consider small perturbations which
are allowed to vary in all directions including the compact direction. These
perturbations lead to the creation of massless modes and Klein–Gordon modes.
On the analytic side, this leads to a PDE system coupling wave equations to an
infinite sequence of Klein–Gordon equations with different masses. The
techniques we use are based purely in physical space using the vectorfield
method.

##  1\. Introduction

The goal of the present article is to prove the global stability of the
Kaluza–Klein spacetime for the Einstein vacuum equations

(1.1) $R_{\mu\nu}[g]=0$

where $R_{\mu\nu}$ denotes the Ricci tensor of an unknown Lorentzian metric
$g$. The Kaluza–Klein spacetime is a solution of (1.1) on
${\mathbb{R}}^{1+3}\times{\mathbb{S}}^{1}$ and consists of a Lorentzian metric
$\overline{g}$, given in the standard coordinates
$(t,x)\in{\mathbb{R}}^{1+3}$, $y\in{\mathbb{S}}^{1}$ by

$\overline{g}=-(dt)^{2}+\sum_{{\bm{i}}=1}^{3}(dx^{\bm{i}})^{2}+(dy)^{2}.$

The Einstein equations in this higher dimensional setting have, as in the
standard $3+1$ setting, a well-posed initial value formulation. The data
consist of a triplet $(\Sigma_{0},g_{0},K_{0})$ where $\Sigma_{0}$ is a
4-dimensional manifold diffeomorphic to
${\mathbb{R}}^{3}\times{\mathbb{S}}^{1}$ equipped with a Riemannian metric
$g_{0}$ and $K_{0}$ is a symmetric two-tensor. Solving (1.1) with initial data
$(\Sigma_{0},g_{0},K_{0})$ means that one looks for a 5-dimensional manifold
$\mathscr{M}$ with a Lorentzian metric $g$ satisfying (1.1) and an embedding
$\Sigma_{0}\hookrightarrow\mathscr{M}$ such that $g_{0}$ is the pullback of
$g$ to $\Sigma_{0}$ and $K_{0}$ is the second fundamental form of
$\Sigma_{0}$. The initial value problem is overdetermined and the data must
satisfy the _constraint equations_ 111We use the Einstein summation convention
over repeated indexes. Greek indexes run from 0 to 4 while Latin indexes run
from 1 to 4. Bold Greek and Latin indexes run up to 3. We use the notation
$x^{0}=t$ and $x^{4}=y$ so that $\partial_{\mu}=\partial/\partial x^{\mu}$ for
$\mu=0,\dots,4$ denotes any derivative along the coordinate axes.

$R[g_{0}]-K_{0}^{ij}{K_{0}}_{ij}+{K_{0}}_{i}^{i}{K_{0}}^{j}_{j}=0,\quad\nabla^{j}{K_{0}}_{ij}-\nabla_{i}{K_{0}}^{j}_{j}=0$

where $R[g_{0}]$ is the scalar curvature of $g_{0}$ and $\nabla$ is the Levi-
Civita connection of $g_{0}$. These equations simply come from the vanishing
of the time components of the Einstein tensor

$R_{0i}=0,\qquad R_{00}-\frac{1}{2}Rg_{00}=0.$

In PDE terminology, the _local well-posedness_ of the Einstein equations was
proved in the seminal works of Choquet-Bruhat [8] and Choquet-Bruhat and
Geroch [10], who show the existence and uniqueness (up to diffeomorphisms) of
a maximal globally hyperbolic spacetime arising from any set of smooth initial
data satisfying the constraint equations. This is a local result in the sense
that it does not guarantee that the spacetime solution $(\mathscr{M},g)$ is
causally geodesically complete. We observe that their proofs, which are
performed in a 4-dimensional setting, do not actually depend on the particular
manifold $\mathscr{M}$ considered (nor on its dimension, or whether or not it
is compact or a product with compact factors) and therefore apply to the
Kaluza–Klein setting. We also mention the recent work of the first author with
Vâlcu [22] in which initial data for the Einstein equations on manifolds of
the form ${\mathbb{R}}^{1+n}\times\mathbb{T}^{m}$ are constructed.

The articles mentioned above constitute the starting point to investigate and
prove the global stability of the flat metric $\overline{g}$. An informal
statement of our main result is the following

######  Theorem 1.1.

Let $(\Sigma_{0},g_{0},K_{0})$ be an arbitrary set of smooth asymptotically
flat initial data satisfying the constraint equations, with
$\Sigma_{0}\cong{\mathbb{R}}^{3}\times{\mathbb{S}}^{1}$,

$\begin{gathered}{g_{0}}=\begin{pmatrix}(1+\chi(r)M/r)I_{3}&0\\\
0&1\end{pmatrix}+{g_{0}^{1}},\quad(I_{3})_{{\bm{i}}{\bm{j}}}=\delta_{{\bm{i}}{\bm{j}}}\\\
\text{where }{g_{0}^{1}}_{ij}=O(r^{-1-\kappa}),\
{K_{0}}_{ij}=O(r^{-2-\kappa})\ \text{as }r=|x|\rightarrow\infty,\
\kappa>0\end{gathered}$

and such that $g_{0}-\delta$ and $K_{0}$ satisfy global smallness assumptions.
Then, there exists a causally geodesically complete spacetime asymptotically
converging to the Kaluza–Klein spacetime.

In the above theorem, $\chi$ is a cut-off function supported outside some ball
centered at 0 and $M$ is a positive constant corresponding to the ADM mass. We
refer to the work of Dai [12] on the positive mass theorem for manifolds
including those of Kaluza–Klein type.

The global stability problem for the flat metric $\overline{g}$ can be cast
into the form of a small data global existence problem for quasilinear wave
equations. The Einstein equations can be written as a system of quasilinear
wave equations for the unknown metric coefficients $g_{\alpha\beta}$ if one
works with a standard gauge, called the _harmonic_ or _wave coordinate_ or _De
Donder_ gauge, in which the (harmonic) coordinates
$\\{x^{\alpha}\\}_{\alpha=0,\dots,4}$ are defined to be solutions of the
geometric wave equation222$g^{\mu\nu}$ and $\overline{g}^{\mu\nu}$ denote
respectively the coefficients of the inverse metric of $g$ and $\overline{g}$.
Unless differently specified, we lower and raise indexes using the metric
$\overline{g}$, i.e. for any tensor $\pi_{\alpha\beta}$ we define
$\pi^{\alpha\beta}:=\overline{g}^{\alpha\mu}\overline{g}^{\beta\nu}\pi_{\mu\nu}$.
$\Box_{g}x^{\alpha}=g^{\mu\nu}\nabla_{\mu}\nabla_{\nu}x^{\alpha}=0$, where
$\nabla$ denotes the Levi-Civita connection of $g$. Relative to these
coordinates the metric $g$ satisfies the so-called _wave condition_

(1.2)
$g^{\alpha\beta}g_{\nu\mu}\Gamma^{\nu}_{\alpha\beta}=g^{\alpha\beta}\partial_{\beta}g_{\alpha\mu}-\frac{1}{2}g^{\alpha\beta}\partial_{\mu}g_{\alpha\beta}=0,\quad\mu=0,\dots,4$

under which the wave operator $\Box_{g}$ on functions coincides with the
reduced wave operator
$\tilde{\Box}_{g}=g^{\mu\nu}\partial_{\mu}\partial_{\nu}$. In this gauge the
equations (1.1) become

(1.3) $\tilde{\Box}_{g}g_{\alpha\beta}=\tilde{F}_{\alpha\beta}(g)(\partial
g,\partial g)\quad\text{on }{\mathbb{R}}^{1+3}\times{\mathbb{S}}^{1}$

where $\tilde{F}_{\alpha\beta}(u)(v,v)$ depends quadratically on $v$. A
straightforward computation shows that these source terms decompose into the
sum of the following

$\displaystyle\tilde{P}(\partial_{\alpha}g,\partial_{\beta}g)$
$\displaystyle:=\frac{1}{4}g^{\mu\nu}g^{\rho\sigma}\left(\partial_{\alpha}g_{\mu\nu}\partial_{\beta}g_{\rho\sigma}-2\partial_{\alpha}g_{\mu\rho}\partial_{\beta}g_{\nu\sigma}\right)$
$\displaystyle\tilde{Q}_{\alpha\beta}(\partial g,\partial g)$
$\displaystyle:=g^{\mu\nu}g^{\rho\sigma}\partial_{\mu}g_{\rho\alpha}\partial_{\nu}g_{\sigma\beta}-g^{\mu\nu}g^{\rho\sigma}Q_{\mu\sigma}(\partial
g_{\rho\alpha},\partial
g_{\nu\beta})+g^{\mu\nu}g^{\rho\sigma}Q_{\alpha\mu}(\partial
g_{\nu\sigma},\partial g_{\rho\beta})$
$\displaystyle\quad+g^{\mu\nu}g^{\rho\sigma}Q_{\beta\mu}(\partial
g_{\nu\sigma},\partial
g_{\rho\alpha})+\frac{1}{2}g^{\mu\nu}g^{\rho\sigma}Q_{\sigma\alpha}(\partial
g_{\mu\nu},\partial
g_{\rho\beta})+\frac{1}{2}g^{\mu\nu}g^{\rho\sigma}Q_{\sigma\beta}(\partial
g_{\mu\nu},\partial g_{\rho\alpha})$

where $Q_{\mu\nu}$ denotes the quadratic null form333The quadratic form
$Q_{0}(\partial\phi,\partial\psi)=\overline{g}^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\psi$
is also a null form.

$Q_{\mu\nu}(\partial\phi,\partial\psi)=\partial_{\mu}\phi\partial_{\nu}\psi-\partial_{\nu}\phi\partial_{\mu}\psi.$

The initial conditions
$(g_{\alpha\beta}|_{t=0},\partial_{t}g_{\alpha\beta}|_{t=0})$ for (1.3) are
defined from $(\Sigma_{0},g_{0},K_{0})$ as follows

(1.4) $\displaystyle g_{ij}|_{t=0}={g_{0}}_{ij},\qquad
g_{00}|_{t=0}=-a^{2},\qquad g_{0i}|_{t=0}=g_{i0}|_{t=0}=0,$
$\displaystyle(\partial_{t}g_{ij})|_{t=0}=-2a{K_{0}}_{ij},\qquad(\partial_{t}g_{00})|_{t=0}=2a^{3}g_{0}^{ij}{K_{0}}_{ij},$
$\displaystyle(\partial_{t}g_{0i})_{t=0}=a^{2}g_{0}^{kl}\partial_{l}{g_{0}}_{ki}-\frac{1}{2}a^{2}g_{0}^{kl}\partial_{i}{g_{0}}_{kl}-a\partial_{i}a$

where $a^{2}:=(1-M\chi(r)r^{-1})$ denotes the lapse function, so that they are
compatible with the constraint equations and satisfy the wave condition. In
particular the constraint equations yield a decay for $g_{ij}$ of the form

$g_{{\bm{i}}{\bm{j}}}=\big{(}1+M\chi(r)r^{-1}\big{)}\delta_{{\bm{i}}{\bm{j}}}+O(r^{-1-\kappa}),\qquad
g_{44}=1+O(r^{-1-\kappa})$

The initial data for $g_{00}$ and $g_{0i}$ are free and we set them as in
(1.4), following what was done by Lindblad and Rodnianski in their work [40],
for compatibility with the wave coordinates for Schwarzschild. The condition
$(\partial_{t}g_{ij})|_{t=0}=-2a{K_{0}}_{ij}$ is given so that $K_{0}$ is the
second fundamental form of $\Sigma_{0}$, i.e.
$K_{0}(X,Y)=-g|_{t=0}(\nabla_{X}\partial_{t},Y)$ for any vector fields $X,Y$.

Any solution to the Einstein equations (1.1) with smooth data
$(\Sigma_{0},g_{0},K_{0})$ satisfies (1.3)-(1.4) when written in harmonic
coordinates. Conversely, any solution $g_{\alpha\beta}$ of (1.3)-(1.4) with
initial data compatible with the constraint equations and satisfying the wave
condition (1.2) will satisfy (1.2) for all times and hence gives rise to a
solution of (1.1) with data $(\Sigma_{0},g_{0},K_{0})$ defined from (1.4). We
refer to Ringström [49] for more details on the subject. From now on, we will
then entirely focus on the formulation (1.3)-(1.4).

###  1.1. State of the art

There is a vast literature in general relativity concerning the stability of
physical solutions to the Einstein equations. In the 4-dimensional setting,
the global stability of the simplest solution, the Minkowski metric, was
proved in a monumental work by Chistodoulou and Klainerman [11] and later
revisited in the works of Lindblad and Rodnianski [39, 40] using the harmonic
gauge. See also the results by Klainerman and Nicolò [32], Bieri and Zipser
[5], Hintz and Vasy [20], Choquet-Bruhat, Chruściel and Loizelet [9] for
Einstein-Maxwell systems and by Speck [50] for Einstein equations coupled to a
family of nonlinear electromagnetic field equations.

Analogous global stability results have also been proved for other
4-dimensional coupled Einstein matter systems. Einsten-Klein–Gordon systems
were investigated by LeFloch and Ma [37] in the case of restricted data
coinciding with the Schwarzschild metric outside a compact set, and global
stability was later proved by Ionescu and Pausader [25] in the case of
unrestricted data. We also cite the works by Fajman, Joudioux and Smulevici
[17] and Lindblad and Taylor [41] proving a global stability result for
Einstein-Vlasov systems for a class of restricted data, and the result by
Bigorgne, Fajman, Joudioux, Smulevici and Thaller [6] about the asymptotic
stability of Minkowski spacetime with non-compactly supported massless Vlasov
matter. There is also a very rich literature concerning the stability of other
explicit 4-dimensional solutions to the Einstein equations, for instance the
Kerr solution or solutions to the Einstein equations with positive
cosmological constants, but it is not our purpose to list such references
here.

Higher dimensional solutions of the Einstein equations, in particular
spacetimes with additional compact directions
${\mathbb{R}}^{1+3}\times\mathscr{K}$, have attracted substantial attention
from the theoretical physics community throughout the past century. Theories
of higher dimensional gravity are in fact of great interest in supergravity
and string theory as possible models for quantum gravity and are possible
candidates for providing a unified description of all the fundamental forces
in nature (gravity, electromagnetism, weak force and strong force). A guiding
philosophy of supergravity theories is that one should be able to recover
4-dimensional physics from higher-dimensional models, hence to perform some
sort of dimensional reduction by assuming the extra directions to be compact.

The classical mathematical approach to the unification of general relativity
with electromagnetism goes back to the works of physicists Kaluza [26] and
Klein [34]. In their original works, one extra dimension is considered and the
five spacetime dimensional gravity is compactified on a circle
${\mathbb{S}}^{1}_{R}$ of radius $R$ to obtain at low energies a $1+3$
dimensional Einstein-Maxwell-Scalar field system. We will briefly discuss the
reduction from the 5-dimensional to the 4-dimensional model in the next
subsection.

In a seminal work by Witten [59] it was proved that the Kaluza–Klein spacetime
$\overline{g}$ is unstable at the semiclassical level. However, classical
global stability was conjectured to hold true and such a result was proved by
the third author [60] for small perturbations that do not depend on the
compact direction. The goal of this paper is to extend the result of [60] and
to prove the global stability of $\overline{g}$ for more general perturbations
that can a-priori depend also on the compact direction. We mention that a
result analogous to [60] for cosmological Kaluza–Klein spacetimes, where the
Minkowski spacetime is replaced by the 4-dimensional Milne spacetime, has also
recently been shown by Branding, Fajman and Kröncke [7]. Furthermore global
existence, without a restriction to ${\mathbb{S}}^{1}$-independent data, was
shown on a quasilinear system of wave equations by the first two authors in
[21] and on a semilinear wave equation on a cosmological Kaluza–Klein
spacetime in [56]. In the context of higher-dimensional gravity we also cite a
result by Ettinger [16] on the global well-posedness of a 11-dimensional,
semilinear, gauge-invariant wave equation, and a global stability result by
Andersson, Blue, Yau and the third author [2] for spacetimes with a
supersymmetric compactification: that is, spacetimes $(\mathscr{M},\hat{g})$
with $\mathscr{M}={\mathbb{R}}^{1+n}\times K$ and $\hat{g}=\eta_{1+n}+k$,
where $\eta_{1+n}$ is the $(1+n)$-dimensional Minkowski metric and $(K,k)$ is
a compact Riemannian manifold that admits a spin structure and a nonzero
parallel spinor. Their proof uses the assumption $n\geq 9$ but the result is
conjectured to hold true for $n\geq 3$.

###  1.2. The zero-mode truncation

The Einstein equations in harmonic coordinates reduce to (1.3), which is a
system of wave equations on the product space
${\mathbb{R}}^{1+3}\times{\mathbb{S}}^{1}$. Assuming for a moment that the
compactifying circle ${\mathbb{S}}^{1}$ is replaced by the circle
${\mathbb{S}}^{1}_{R}$ of radius $R>0$, and by Fourier expanding the solution
$g$ of (1.3) along the periodic coordinate

$g_{\alpha\beta}(t,x,y)=\sum_{k\in\mathbb{Z}}e^{iky}g^{k}_{\alpha\beta}(t,x),$

it turns out that

$(-\partial^{2}_{t}+\Delta_{x}+\partial^{2}_{y})g_{\alpha\beta}=\sum_{k\in\mathbb{Z}}e^{iky}(-\partial^{2}_{t}+\Delta_{x}-(|k|/R)^{2})g^{k}_{\alpha\beta}$

which shows that the zero-modes $g^{0}_{\alpha\beta}$ of the metric
coefficients are massless waves while the non-zero modes $g^{k}_{\alpha\beta}$
are massive (Klein–Gordon) waves with mass $|k|/R$ for $k\neq 0$. Equations
(1.3) are hence equivalent to a system on ${\mathbb{R}}^{1+3}$ which couples
wave equations to an infinite sequence of Klein–Gordon equations with mass
$|k|/R$, $k\in\mathbb{Z}\setminus\\{0\\}$.

The heuristic physics argument, as explained by Pope in [48], to deal with
this phenomenon is to assume the radius $R$ to be very small (a choice that
would justify why we “don’t see” the additional dimensions) so that the masses
$|k|/R$ are too large to be physically observable. The non-zero modes are then
neglected and the solution is truncated to the massless mode, in other words
one assumes that $g_{\alpha\beta}(t,x,y)=g_{\alpha\beta}(t,x)$ is independent
of the $y$ coordinate.

Under the zero-mode truncation assumption, one can reduce the Kaluza–Klein
model to a three-dimensional Einstein-Maxwell-scalar field system. As
explained in[48], this is done using the following standard ansatz, in which
the higher dimensional metric coefficients $g_{\alpha\beta}$ are expressed in
terms of three-dimensional fields
$\hat{g}_{{\bm{\alpha}}{\bm{\beta}}},\phi,\mathscr{A}_{{\bm{\alpha}}}$ by

$g_{{\bm{\alpha}}{\bm{\beta}}}=e^{2\kappa\phi}\hat{g}_{{\bm{\alpha}}{\bm{\beta}}}+e^{2\rho\phi}\mathscr{A}_{{\bm{\alpha}}}\mathscr{A}_{{\bm{\beta}}},\quad
g_{{\bm{\alpha}}4}=e^{2\rho\phi}\mathscr{A}_{{\bm{\alpha}}},\quad
g_{44}=e^{2\rho\phi}$

where $\kappa=\sqrt{12}/12$ and $\rho=-2/\sqrt{12}$. The Einstein vacuum
equations (1.1) reduce then to the following minimally coupled
$(1+3)$-dimensional Einstein-Maxwell-Scalar field system

$\displaystyle
R_{{\bm{\alpha}}{\bm{\beta}}}=\frac{1}{2}\partial_{{\bm{\alpha}}}\phi\,\partial_{{\bm{\beta}}}\phi+\frac{1}{2}e^{-6\kappa\phi}\big{(}\mathscr{F}_{{\bm{\alpha}}{\bm{\mu}}}{\mathscr{F}_{{\bm{\beta}}}}^{{\bm{\mu}}}-\frac{1}{4}\mathscr{F}_{{\bm{\mu}}{\bm{\nu}}}\mathscr{F}^{{\bm{\mu}}{\bm{\nu}}}\hat{g}_{{\bm{\alpha}}{\bm{\beta}}}\big{)}$
$\displaystyle\nabla^{{\bm{\alpha}}}\big{(}e^{-6\kappa\phi}\mathscr{F}_{{\bm{\alpha}}{\bm{\beta}}}\big{)}=0$
$\displaystyle\tilde{\Box}_{\hat{g}}\phi=-\frac{3}{2}\kappa
e^{-6\kappa\phi}\mathscr{F}_{{\bm{\mu}}{\bm{\nu}}}\mathscr{F}^{{\bm{\mu}}{\bm{\nu}}}$

where
$\mathscr{F}_{{\bm{\alpha}}{\bm{\beta}}}=\partial_{\bm{\alpha}}\mathscr{A}_{\bm{\beta}}-\partial_{\bm{\beta}}\mathscr{A}_{\bm{\alpha}}$.
The above reduction can be also performed in higher dimensional settings where
$\mathscr{M}={\mathbb{R}}^{1+3}\times\mathbb{T}^{d}$. In the Kaluza–Klein
setting, this truncation to the zero mode is consistent in the sense that a
solution to the above Einstein-Maxwell-Scalar field system will be a solution
to the original vacuum Einstein equations in 5 dimensions.

The full global stability of the Kaluza–Klein spacetime to general
perturbations, that may a-priori depend on the compact direction, involves
studying solutions to a significantly more complicated PDE system than the
simpler dynamics of the above Einstein-Maxwell-Scalar field system studied in
[60]. This is the goal of the present article. We point out that we do not
want to focus here on the dependence of the solution on the radius $R$ and,
since there is no canonical choice of the radius $R$, we set $R=1$.

###  1.3. 4D Wave-Klein–Gordon systems

The dependence of the metric coefficients $g_{\alpha\beta}$ on the periodic
coordinate $y$ and their Fourier decomposition along this direction reveal
that system (1.3) is equivalent to a system coupling wave equations to an
infinite sequence of Klein–Gordon equations with different masses. The new
system is also quasilinear and the coupling between the wave and Klein–Gordon
components of the solution is strong.

The study of systems coupling (a finite number of) wave and Klein–Gordon
equations has attracted considerable interest from the mathematical community,
especially in the past three decades. In terms of small data global well-
posedness results in $1+3$ spacetime dimensions we cite the initial results by
Georgiev [19] and Katayama [27], followed by LeFloch and Ma [36], Wang [57,
58] and Ionescu and Pausader [24] who study such systems as a model for the
full Einstein-Klein–Gordon equations, see [37] and [25]. In [36] and [57]
global well-posedness is proved for compactly supported initial data and
quadratic quasilinear nonlinearities that satisfy some suitable conditions,
including the null condition of Klainerman [31] for self-interactions between
the wave components of the solution. An idea used in these works is that of
employing hyperbolic coordinates in the forward light cone; this was first
introduced by Klainerman [29] for Klein–Gordon equations and Tataru in the
wave context [52], and later reintroduced by LeFloch and Ma in [36] under the
name of hyperboloidal foliation method. In [24] global regularity and
scattering is proved in the case of small smooth initial data that decay at a
suitable rate at infinity and nonlinearities that do not verify the null
condition but present a particular resonant structure. We also cite the work
by Dong and the third author [14], who prove global well-posedness for a
quadratic semilinear interaction in which there are no derivatives on the
massless wave component. Other related results are [4, 47, 53, 54, 55, 33,
13]. See also [42, 46, 44, 43, 51, 23, 15, 45] for results about wave-
Klein–Gordon systems in lower dimensions, in particular a work by the second
author [51] and a subsequent result in collaboration with Ifrim [23], which
are the only ones where 2-dimensional strongly coupled quadratic wave-
Klein–Gordon systems with small mildly decaying data are investigated.
Advanced techniques, among which semiclassical microlocal analysis,
para/pseudo-differential calculus, wave packets, modified quasilinear
energies, are employed there to tackle a problem that is critical, quasilinear
and very weakly dispersive.

A now-standard tool used in most of the aforementioned works is the vector
field method. Linear wave and Klein–Gordon equations on ${\mathbb{R}}^{1+n}$
are invariant under translations, Euclidean rotations and hyperbolic rotations
(linear wave equations are also scale-invariant). These symmetries provide a
family of admissible vector fields (in the common terminology they are also
referred to as _Killing_ vector fields of Minkowski spacetime),

$\partial_{\mu},\qquad\Omega_{{\bm{i}}{\bm{j}}}=x_{\bm{i}}\partial_{\bm{j}}-x_{\bm{j}}\partial_{\bm{i}},\qquad\Omega_{0{\bm{i}}}=t\partial_{\bm{i}}+x_{\bm{i}}\partial_{t}$

which commute with the linear wave and Klein–Gordon operators and are used to
define higher order energy functionals which control the Sobolev regularity of
the solution as well as its decay (and that of its derivatives) in space at
infinity. The rotations $\Omega_{{\bm{i}}{\bm{j}}}$ and $\Omega_{0{\bm{i}}}$
are also usually referred to as _Klainerman vector fields_. In the absence of
Klein–Gordon equations, that is in the case of wave equations only, one can
also consider the scaling vector field
$\mathscr{S}=t\partial_{t}+x^{\bm{i}}\partial_{\bm{i}}$ (a _conformal Killing_
vector field of Minkowski) and use the control on higher order energies to
derive fixed-time pointwise decay bounds for the solution via the so-called
Klainerman-Sobolev inequalities (see Klainerman [30])

(1.5) $(1+|t|+|x|)^{n-1}(1+||t|-|x||)|u(t,x)|^{2}\leq
C\sum_{|I|\leq(n+2)/2}\|Z^{I}u(t,\cdot)\|^{2}_{L^{2}({\mathbb{R}}^{n})}.$

In the above inequality $Z$ denotes any of the vector fields
$\partial_{\mu},\Omega_{{\bm{i}}{\bm{j}}},\Omega_{0{\bm{i}}},\mathscr{S}$ and
$Z^{I}$ is any product of $|I|$ such vector fields. Suitable energy estimates
and pointwise decay bounds are subsequently used to control the nonlinear
terms in the energy inequality and are essential to close the continuity
argument which is at the core of the proof of a long-time/global existence
result for small data.

The inequality (1.5) is, however, useless when dealing with Klein–Gordon
equations. The scaling vector field does not commute well with the linear
operator and one cannot generally expect to have a good control of the $L^{2}$
norm of $\mathscr{S}u$ when $u$ is a Klein–Gordon solution. Instead, if $u$ is
compactly supported inside the light cone444Any cone $t=|x|+c$ with $c>0$
would do. $t=|x|+1$ one can define higher order energy functionals on
hyperboloids $t^{2}-|x|^{2}=s^{2}$ and exploit Klainerman-Sobolev inequalities
on hyperboloids (see for instance [18])

(1.6) $\sup_{\mathscr{H}_{s}}t^{n/2}|u(t,x)|\leq
C\sum_{|I|\leq(n+2)/2}\|B^{I}u\|_{L^{2}(\mathscr{H}_{s})}$

where now $B^{I}$ are products involving hyperbolic rotations only, to get a
good pointwise control on the solution. This approach has been largely used in
the case of compactly supported initial data thanks to the finite speed of
propagation satisfied by both wave and Klein–Gordon equations, but it is not
adapted to treat the case of initial data that only enjoys some decay at
infinity. Other methods have been employed to handle such cases, based on
Fourier analysis, normal forms and/or microlocal analysis: see for instance
the work by Ionescu and Pausader [24] in the 1+3 dimensional setting, by the
second author [51] and in collaboration with Ifrim [23] for the 1+2
dimensional case, and references therein. See also a recent work by LeFloch
and Ma [35] using a foliation that merges hyperboloids with constant time
slices.

###  1.4. The 5D problem: main theorem and overview of the proof

According to the positive mass theorem, the solution $g_{\alpha\beta}$ of the
Cauchy problem (1.3)-(1.4) must have a non-trivial tail at spacelike
infinity555We choose to write this tail so that $g_{\alpha\beta}$ corresponds
to the Schwarzschild metric in wave coordinates at leading order. which
suggests to set
$g_{\alpha\beta}=\overline{g}_{\alpha\beta}+h^{0}_{\alpha\beta}+h^{1}_{\alpha\beta}$
where

(1.7) $\displaystyle
h^{0}_{{\bm{\alpha}}{\bm{\beta}}}=\chi\Big{(}\frac{r}{t}\Big{)}\chi(r)\frac{M}{r}\delta_{{\bm{\alpha}}{\bm{\beta}}},\qquad
h^{0}_{44}=0,$ $\displaystyle\chi\in\mathscr{C}^{\infty}({\mathbb{R}})\text{
with }\chi(s)=0\text{ for }s\leq 1/2,\ \chi(s)=1\text{ for }s\geq 3/4,\ r=|x|$

and look for $h^{1}_{\alpha\beta}$ the solution to the following system of
quasilinear wave equations

(1.8) $\tilde{\Box}_{g}h^{1}_{\alpha\beta}=F_{\alpha\beta}(h)(\partial
h,\partial h)-\tilde{\Box}_{g}h^{0}_{\alpha\beta},\qquad\text{on
}{\mathbb{R}}^{1+3}\times{\mathbb{S}}^{1}$

with data $(h^{1}|_{\alpha\beta},\partial_{t}h^{1}_{\alpha\beta})|_{t=2}$
being small and sufficiently decaying in space. The semilinear source term in
the above right hand side decompose into the following sum

$F_{\alpha\beta}(h)(\partial h,\partial h)=P_{\alpha\beta}(\partial h,\partial
h)+\mathbf{Q}_{\alpha\beta}(\partial h,\partial h)+G_{\alpha\beta}(h)(\partial
h,\partial h)$

where

  * -

$P_{\alpha\beta}(\partial h,\partial h)$ are quadratic _weak null_ terms

$P_{\alpha\beta}(\partial h,\partial
h)=\frac{1}{4}\bar{g}^{\mu\rho}\bar{g}^{\nu\sigma}\left(\partial_{\alpha}h_{\mu\rho}\partial_{\beta}h_{\nu\sigma}-2\partial_{\alpha}h_{\mu\nu}\partial_{\beta}h_{\rho\sigma}\right),$

  * -

$\mathbf{Q}_{\alpha\beta}(\partial h,\partial h)$ is a linear combination of
the classical quadratic null forms,

  * -

$G_{\alpha\beta}(h)(\partial h,\partial h)$ are cubic terms. More precisely,
they are quadratic in $\partial h$ with smooth coefficients depending on $h$
so that $G_{\alpha\beta}(0)(\partial h,\partial h)=0$.

The reduced wave operator can be written as
$\tilde{\Box}_{g}=\Box_{xy}+H^{\mu\nu}\partial_{\mu}\partial_{\nu}$, where
$\Box_{xy}=-\partial^{2}_{t}+\Delta_{x}+\partial^{2}_{y}$ is the flat wave
operator and $H^{\mu\nu}:=g^{\mu\nu}-\overline{g}^{\mu\nu}$ is the formal
inverse of $h_{\mu\nu}$ for small $h$, i.e.

(1.9)
$H^{\mu\nu}=-h^{\mu\nu}+\mathscr{O}^{\mu\nu}(h^{2})=-\overline{g}^{\mu\rho}\overline{g}^{\nu\sigma}h_{\rho\sigma}+\mathscr{O}^{\mu\nu}(h^{2}).$

We can now give a more precise statement of our main result.

######  Theorem 1.2.

Let $\kappa>0$. There exists $N\in{\mathbb{N}}$ sufficiently large and
$\epsilon_{0}>0$ small such that, for any $0<\epsilon<\epsilon_{0}$ and
initial data $g_{0},K_{0}$ solving the constraint equations and satisfying

(1.10) $\begin{gathered}\sum_{m\leq
N}\sum_{i+j=m}\|(1+r)^{\frac{1}{2}+i+\kappa}\partial_{y}^{j}\nabla_{x}^{i}(g_{0}-g^{0})\|_{\dot{H}^{1}_{x,y}}+\|(1+r)^{\frac{1}{2}+i+\kappa}\partial_{y}^{j}\nabla_{x}^{i}K_{0}\|_{L^{2}_{x,y}}\leq\epsilon,\\\
\sum_{m\leq
N-1}\sum_{i+j=m}\|(1+r)^{\frac{3}{2}+i+\kappa}\partial_{y}^{j}\nabla_{x}^{i}(g_{0}-g^{0})\|_{\dot{H}^{2}_{x,y}}+\|(1+r)^{\frac{3}{2}+i+\kappa}\partial_{y}^{j}\nabla_{x}^{i}K_{0}\|_{\dot{H}^{1}_{x,y}}\leq\epsilon\end{gathered}$

together with the $L^{2}$ estimate

$\|(1+r)^{-\frac{1}{2}+\kappa}(g_{0}-g^{0})\|_{L^{2}_{x,y}}\leq\epsilon$

with $r=|x|$ and $g^{0}$ defined by

$g^{0}_{{\bm{i}}{\bm{j}}}=(1+M\chi(r)r^{-1})\delta_{{\bm{i}}{\bm{j}}},\quad
g^{0}_{44}=1,\quad g^{0}_{4{\bm{i}}}=0,$

there exists a unique global solution $g_{\alpha\beta}$ to (1.3) with initial
data given by (1.4). This solution obeys the Einstein equations and decomposes
as
$g_{\alpha\beta}=\bar{g}_{\alpha\beta}+h^{0}_{\alpha\beta}+h^{1}_{\alpha\beta}$,
with $h^{0}_{\alpha\beta}$ defined by (1.7) and $h^{1}_{\alpha\beta}$
satisfying the pointwise estimate

$|h^{1}_{\alpha\beta}|\leq\frac{C_{0}\epsilon}{(1+t+|x|)^{1-\gamma}}$

with $C_{0}$ a numerical constant and $\gamma>0$ arbitrarily small but fixed.

The proof of the above result is based on a bootstrap argument, i.e. on the
propagation of some suitable a-priori energy estimates and pointwise decay
bounds on the solution, which is performed in two main steps:

_Step 1_ : deduction of higher order energy inequalities and of sharp
pointwise estimates from the a-priori energy assumptions;

_Step 2_ : estimates of the trilinear and quartic terms appearing in the right
hand side of the energy inequalities. In particular, deduction of suitable
higher order $L^{2}$ estimates of the source terms from the a-priori energy
assumptions and the pointwise decay bounds.

In order to run the above argument and in view of the issues discussed in the
previous subsection, one needs to find a strategy to obtain (at least in the
first instance) pointwise decay bounds on the solution from the a-priori
assumptions, knowing that inequality (1.5) cannot be used and (1.6) is valid
only in the interior of some light cone.

Similar to [21], the approach we take in the present paper is to decompose the
whole spacetime and study the problem separately in two regions, corresponding
to the interior and exterior of a hyperboloid666In this curved background, the
Minkowski cone $\\{t=|x|+1\\}$ is in fact only asymptotically spacelike.
asymptotically approaching the cone $\\{t=|x|+1\\}\times{\mathbb{S}}^{1}$.
This decomposition is quite natural, in that the analysis in the exterior is
totally independent of that in the interior and requires different tools. It
also allows us to explain our arguments with more clarity.

####  1.4.1. Exterior region: the bootstrap assumptions.

The bootstrap assumptions in the exterior region are higher order weighted
energy estimates on the solution

$\displaystyle E^{\text{e},\kappa}(t,Z^{\leq N}h^{1})^{1/2}\leq 2C_{0}\epsilon
t^{\sigma},$ $\displaystyle E^{e,1+\kappa}(t,\partial Z^{\leq
N-1}h^{1})^{1/2}\leq 2C_{0}\epsilon t^{\sigma}$

where the weighted energy functional is defined, for any $\lambda>0$, as

$E^{e,\lambda}(t,h^{1}_{\alpha\beta})=\iint_{\\{|x|\geq
t-1\\}\times{\mathbb{S}}^{1}}(2+|x|-t)^{1+2\lambda}|\nabla_{txy}h^{1}_{\alpha\beta}(t,x,y)|^{2}dxdy\
\\\
+\int_{2}^{t}\iint_{\\{|x|\geq\tau-1\\}\times{\mathbb{S}}^{1}}(2+|x|-t)^{2\lambda}|\overline{\nabla}h^{1}_{\alpha\beta}(\tau,x,y)|^{2}dxdyd\tau.$

In the above integrals, $\nabla_{txy}$ denotes the spacetime gradient while
$\overline{\nabla}=(\overline{\partial}_{0},\dots,\overline{\partial}_{4})=(\partial_{t}+\partial_{r},\not{\partial}_{{\bm{i}}},\partial_{y})$
denotes the tangent gradient to the cones $\\{t=r+1\\}\times{\mathbb{S}}^{1}$,
with $\not{\partial}_{\bm{i}}=\partial_{i}-\frac{x_{\bm{i}}}{r}\partial_{t}$
being the angular derivatives. The parameter $\kappa$ in the above a-priori
estimates is related to the asymptotic decay of the data, $N\in{\mathbb{N}}$
is assumed to be sufficiently large and $0<\sigma<\kappa$ sufficiently small.
Weighted Sobolev and Hardy inequalities allow us to obtain fixed-time
pointwise decay bounds on the solution from the assumptions on the weighted
energies, as for any given smooth function $U$

$\begin{gathered}|\nabla_{txy}U(t,x,y)|\leq
C(1+|x|)^{-1}(2+|x|-t)^{-\frac{1}{2}-\lambda}\sum_{|I|\leq
3}E^{e,\lambda}(t,Z^{I}U)^{1/2},\\\ |U(t,x,y)|\leq
C(1+|x|)^{-1}(2+|x|-t)^{-\lambda}\sum_{|I|\leq
2}E^{e,\lambda}(t,Z^{I}U)^{1/2}.\end{gathered}$

They also allow us to uncover faster spacetime decay for the tangential
derivatives, since their weighted $L^{2}$-spacetime norm is controlled by the
energy, and to recover the well-known property of waves that higher order
derivatives enjoy better decay in terms of the distance from the outgoing
Minkowski cones, which follows from the second energy assumption above. We
point out that, in the context of waves on ${\mathbb{R}}^{1+3}$ where the full
range of vector fields
$\Gamma\in\\{\Omega_{{\bm{i}}{\bm{j}}},\Omega_{0{\bm{i}}},\mathscr{S}\\}$ is
available, the latter two properties are easily derived from algebraic
relations. In particular, one can use that

$|\overline{\partial}\psi|\lesssim\sum_{|I|\leq
1}\frac{|\Gamma^{I}\psi|}{1+t+|t-|x||},\qquad|\partial^{2}\psi|\lesssim\sum_{|I|\leq
1}\frac{|\partial\Gamma^{I}\psi|}{1+|t-|x||}.$

####  1.4.2. Interior region: the bootstrap assumptions.

The bootstrap assumptions in the interior region are bounds on higher order
energies defined on truncated hyperboloids

$\mathscr{H}_{s}=\\{(t,x):t^{2}-|x|^{2}=s^{2}\text{ and }t\geq
1+\sqrt{1+|x|^{2}}\\}\times{\mathbb{S}}^{1},\qquad s\geq 2$

which are the branches of hyperboloids contained in the interior region, and
pointwise decay bounds on differentiated metric coefficients carrying only
Klainerman vector field derivatives. We denote the zero-mode (respectively
zero-average) component of coefficient $h^{1}_{\alpha\beta}$ by

$h^{1,\flat}_{\alpha\beta}=\fint_{{\mathbb{S}}^{1}}h^{1}_{\alpha\beta}dy,\quad\text{
and }\quad
h^{1,\natural}_{\alpha\beta}=h^{1}_{\alpha\beta}-h^{1,\flat}_{\alpha\beta}.$

We assume that, for some large integers $1\ll N_{1}\ll N$ and some small777In
practice, $\zeta,\gamma$ and $\delta$ are going to be replaced with a
hierarchy of increasing $\zeta_{k},\gamma_{k}$ and $\delta_{k}$, where $k$
accounts for the number of Klainerman vector fields in the product $Z^{I}$, so
that $\zeta_{i}\ll\gamma_{j}\ll\delta_{k}$ for any $i,j,k$ and the algebraic
relation $\gamma_{i}+\delta_{j}<\delta_{k}$ whenever $j<k$.
$0<\zeta<\gamma\ll\delta$, the following bounds are satisfied

$\displaystyle E^{i}(s,\partial^{\leq 1}Z^{\leq N}h^{1}_{\alpha\beta})\leq
C\epsilon^{2}s^{1+\zeta},$ $\displaystyle E^{i}(s,Z^{\leq
N}h^{1,\flat}_{\alpha\beta})\leq C\epsilon^{2}s^{\zeta},$ $\displaystyle
E^{i}(s,\partial^{\leq N-N_{1}}Z^{\leq N_{1}}h^{1}_{\alpha\beta})\leq
C\epsilon^{2}s^{\delta}$

where

$\displaystyle E^{i}(s,h^{1}_{\alpha\beta})$
$\displaystyle:=\iint_{\mathscr{H}_{s}}\big{|}(s/t)\partial_{t}h^{1}_{\alpha\beta}\big{|}^{2}+|\underline{\nabla}h^{1}_{\alpha\beta}|^{2}dxdy$
$\displaystyle\,=\iint_{\mathscr{H}_{s}}\big{|}(s/t)\nabla_{x}h^{1}_{\alpha\beta}\big{|}^{2}+\big{|}(1/t)\mathscr{S}h^{1}_{\alpha\beta}\big{|}^{2}+\sum_{1\leq
i<j\leq
3}\big{|}(1/t)\Omega_{ij}h^{1}_{\alpha\beta}\big{|}^{2}+|\partial_{y}h^{1}_{\alpha\beta}|^{2}\,dxdy$

and with $\Gamma\in\\{\Omega_{{\bm{i}}{\bm{j}}},\Omega_{0{\bm{i}}}\\}$

$|\Gamma^{\leq N_{1}}h^{1,\flat}_{\alpha\beta}(t,x)|\leq
C\epsilon(1+t)^{-1+\gamma}(1+|t-|x||)^{\gamma},\quad$
$\|t^{\frac{3}{2}}\partial_{y}^{\leq
1}(\partial^{I}\Gamma^{J}h^{1,\natural}_{\alpha\beta})\|_{L^{\infty}_{x}L^{2}_{y}(\mathscr{H}_{s})}+\|t^{\frac{1}{2}}s\partial_{tx}(\partial^{I}\Gamma^{J}h^{1,\natural}_{\alpha\beta})\|_{L^{\infty}_{x}L^{2}_{y}(\mathscr{H}_{s})}\leq
C\epsilon s^{\gamma},\quad|I|+|J|\leq N_{1}+1,\ |J|\leq N_{1}.$

In the above energy functional,
$\underline{\nabla}=(\underline{\partial_{1}},\dots,\underline{\partial_{4}})$
denotes the tangent gradient (to the hyperboloids) with
$\underline{\partial}_{{\bm{i}}}=\partial_{{\bm{i}}}+(x_{{\bm{i}}}/t)\partial_{t}$
for ${\bm{i}}=1,2,3$ and $\underline{\partial}_{4}=\partial_{4}$. Klainerman-
Sobolev inequalities on hyperboloids permit us to deduce pointwise decay
bounds for the solution, as for any given smooth function $U$ one has

$\begin{gathered}\sup_{{\mathbb{S}}^{1}}|\nabla_{tx}U(t,x,y)|\leq
C(1+t)^{-1}(1+|t-|x||)^{-1/2}\sum_{|I|\leq 3}E^{i}(s,Z^{I}U)^{1/2},\\\
\sup_{{\mathbb{S}}^{1}}|\underline{\nabla}U(t,x,y)|\leq
C(1+t)^{-3/2}\sum_{|I|\leq 3}E^{i}(s,Z^{I}U)^{1/2}.\end{gathered}$

Note that the latter inequality shows, again, that tangential derivatives
enjoy better decay estimates than usual derivatives. We postpone the
explanation of why we use the above hierarchy of energy assumptions to later
in this section.

####  1.4.3. Estimates on inhomogeneities: null and weak-null terms

Once energy bounds and pointwise decay bounds are available, one has to
estimate the trilinear and quartic terms appearing in the right hand side of
the energy inequalities. These involve the source terms of the equation
satisfied by the differentiated coefficients $Z^{K}h^{1}_{\alpha\beta}$

$\tilde{\Box}_{g}Z^{K}h^{1}_{\alpha\beta}=F^{K}_{\alpha\beta}+F^{0,K}_{\alpha\beta}$

where

$F^{K}_{\alpha\beta}=Z^{K}F_{\alpha\beta}(h)(\partial h,\partial
h)-[Z^{K},H^{\mu\nu}\partial_{\mu}\partial_{\nu}]h^{1}_{\alpha\beta},\qquad
F^{0,K}_{\alpha\beta}=Z^{K}\tilde{\Box}_{g}h^{0}_{\alpha\beta}$

are semilinear quadratic interactions. The explicit inhomogeneous terms
$F^{0,K}_{\alpha\beta}$ and the differentiated cubic terms
$Z^{K}G_{\alpha\beta}(h)(\partial h,\partial h)$ are short range perturbations
of the linear equations. We do not discuss them here as they cause no issue in
the analysis. The differentiated null terms
$Z^{K}\mathbf{Q}_{\alpha\beta}(\partial h,\partial h)$ are also easily
controlled, thanks to the following well-known property

$\displaystyle|Q_{0}(\partial\psi,\partial\varphi)|+|Q_{\alpha\beta}(\partial\psi,\partial\varphi)|$
$\displaystyle\lesssim|\overline{\partial}\psi||\partial\varphi|+|\partial\psi||\overline{\partial}\varphi|$
$\displaystyle\lesssim|\underline{\partial}\psi||\partial\varphi|+|\partial\psi||\underline{\partial}\varphi|+(s/t)^{2}|\partial\psi||\partial\varphi|$

and the better behavior of tangential derivatives.

The quadratic interactions that are more delicate to treat and require special
attention are the differentiated null terms $Z^{K}P_{\alpha\beta}(\partial
h,\partial h)$ and the commutator terms
$[Z^{K},H^{\mu\nu}\partial_{\mu}\partial_{\nu}]h^{1}_{\alpha\beta}$. The
particular structure of such terms was first highlighted by Lindblad and
Rodnianski [38, 39, 40] in the 4-dimensional setting and shows all its
potential in the null frame
$\mathscr{U}=\\{L,\underline{L},S^{1},S^{2}\\}\cup\\{\partial_{y}\\}$, where
$L=\partial_{t}+\partial_{r}$, $\underline{L}=\partial_{t}-\partial_{r}$ and
$S^{1},S^{2}$ are smooth vector fields tangent to the spheres
${\mathbb{S}}^{2}=\\{u\in{\mathbb{R}}^{3}:u\cdot x/|x|=0\\}$.

As concerns the weak null terms, one sees that if the metric tensor is
expressed with respect to $\mathscr{U}$ then

$P_{\alpha\beta}(\partial h,\partial h)\sim\partial h_{TU}^{2}+\partial
h_{LL}\partial h_{\underline{L}\underline{L}},\qquad
T\in\mathscr{T},U\in\mathscr{U}$

where $\mathscr{T}=\\{L,S^{1},S^{2}\\}\cup\\{\partial_{y}\\}$ denotes the
frame tangent to the flat outgoing cones. On the one hand, the choice of gauge
(in particular the _wave coordinate condition_) ensures that the derivatives
of $h_{LT}$ coefficients are well behaved, as they satisfy

(1.11) $|\partial
h_{LT}|\lesssim|\overline{\partial}h|+\mathscr{O}(h\cdot\partial h).$

On the other hand, the metric coefficients $h_{TU}$ solve quasilinear wave
equations whose source terms are null or cubic. In the exterior region, we
exploit this property to prove that the higher order weighted energies of such
coefficients grow at a slower rate $t^{C\epsilon}$, where $\epsilon\ll\sigma$
is the size of the data. From this we infer an improved pointwise decay for
$\partial Z^{K}h_{TU}$ with $|K|\ll N$ and the following weighted $L^{2}$
bound for the differentiated weak null terms

$\sum_{i=0}^{1}\big{\|}(2+r-t)^{\frac{1}{2}+i+\kappa}\partial^{i}Z^{\leq
N-i}P_{\alpha\beta}\big{\|}_{L^{2}}\lesssim\epsilon^{2}t^{-1+C\epsilon}+\mathscr{O}(\epsilon^{2}t^{-1-}).$

The above estimate shows that the weak null terms contribute to a slow growth
of the exterior energies. In the interior region, the enhanced pointwise
bounds satisfied by the derivatives of $Z^{K}h^{1}_{TU}$ for $|K|\ll N$ are
instead obtained directly from the equations they satisfy, using integration
along characteristics as done in [40]. This approach is possible provided that
we already have at our disposal suitable bounds on the solution in the
exterior region.

####  1.4.4. Commutator terms in the exterior region

The commutator terms also display an important structure when expressed with
respect to the null frame. The tensor $H^{\mu\nu}$ is decomposed as follows

(1.12) $H^{\mu\nu}:=H^{0,\mu\nu}+H^{1,\mu\nu},\quad
H^{0,{\bm{\mu}}{\bm{\nu}}}:=-\chi\Big{(}\frac{r}{t}\Big{)}\chi(r)\frac{M}{r}\delta^{{\bm{\mu}}{\bm{\nu}}},\quad
H^{0,44}=0,$

where $H^{0,\mu\nu}$ is the “Schwarzschild part” of $H$. The estimates of
$[Z^{K},H^{0,\mu\nu}\partial_{\mu}\partial_{\nu}]h^{1}_{\alpha\beta}$ are
straightforward and, similar to the weak null terms discussed above,
responsible for a slow growth of the exterior energy. The estimates of the
commutator involving coefficients $H^{1,\mu\nu}$ are instead obtained using
the fact that, for any tensor $\pi^{\mu\nu}$ and function $\psi$,

$|\pi^{\mu\nu}\partial_{\mu}\partial_{\nu}\psi|\lesssim|\pi_{LL}||\partial^{2}\psi|+|\pi||\partial\overline{\partial}\psi|$

so that either the tensor coefficient is a “good” coefficient $\pi_{LL}$ or
one of the two derivatives acting on $\psi$ is a tangential derivative. As
highlighted above, in the exterior region the enhanced behaviour of second
order derivatives $\partial\overline{\partial}$ as well as of $\partial^{2}$
is encoded in the energy assumptions. What is more, weighted Hardy type
inequalities and weighted Sobolev-Hardy inequalities allow us to get a good
control of the higher order weighted $L^{2}$ norms, as well as to recover good
pointwise decay bounds, of the solution with no derivatives. Suitable higher
order weighted $L^{2}$ estimates for these commutator terms in the exterior
region follow then rather easily.

####  1.4.5. Commutator terms in the interior region

A much more delicate analysis of the commutator terms is required in the
interior region. On the one hand, the interior energy assumptions do not
provide us with additional information on the second order derivatives and the
interior energy functionals only give a $\dot{H}^{1}$ type control on the
differentiated solution. The classical Hardy inequality written on
hyperboloids is

$\|r^{-1}U\|_{L^{2}(\mathscr{H}_{s})}\lesssim\|\underline{\partial}U\|_{L^{2}(\mathscr{H}_{s})}+\|\partial
U\|_{L^{2}(\Sigma^{\text{e}}_{t_{s}})}$

where $\Sigma^{\text{e}}_{t_{s}}$ is the exterior constant time slice that
intersects the interior hyperboloid $\mathscr{H}_{s}$ on the boundary between
the two regions. Such an inequality provides us with a control of the $L^{2}$
norm of the undifferentiated solution at the costly expense of a $r^{-1}$
factor. On the other hand, no extra decay (in terms of the distance from the
outgoing cones) is expected for the second order derivatives of the solution.
In fact, the zero-average component of the solution
$h^{1,\natural}_{\alpha\beta}$ is a Klein–Gordon type function, in that each
of its Fourier mode along the $y$-direction is solution to a Klein–Gordon
equation (see subsection 1.2). As a consequence of this latter fact one only
has $|\partial^{2}h^{1,\natural}_{\alpha\beta}|+|\partial
h^{1,\natural}_{\alpha\beta}|\lesssim(1+t+r)^{-3/2}$, which coupled with the
above Hardy inequality gives

$\big{\|}Z^{K}h^{1,\flat}\cdot\partial^{2}h^{1,\natural}_{\alpha\beta}\big{\|}_{L^{2}(\mathscr{H}_{s})}\lesssim
s^{-1/2}\big{(}\|\underline{\partial}h^{1,\flat}\|_{L^{2}(\mathscr{H}_{s})}+\|\partial
h^{1,\flat}\|_{L^{2}(\Sigma^{\text{e}}_{t_{s}})}\big{)}\lesssim
s^{-1/2+\delta}.$

The same inequality holds if $h^{1,\flat}$ is replaced by
$(H^{1,\mu\nu})^{\flat}$. These “wave-Klein–Gordon” contributions to the
commutator are the ones responsible for the $s^{1+}$ growth of the higher
order energies on $\mathscr{H}_{s}$. They are, however, absent in the
equations satisfied by the zero-modes $Z^{K}h^{1,\flat}_{\alpha\beta}$, as for
any two functions $f,g$ one has

$(f\cdot g)^{\flat}=f^{\flat}\cdot g^{\flat}+\big{(}f^{\natural}\cdot
g^{\natural}\big{)}^{\flat},$

therefore a much slower growth is expected for the higher order energies of
$h^{1,\flat}_{\alpha\beta}$.

The above observation motivates the use of a hierarchy in the interior energy
assumptions and to separately propagate the higher order energy estimates for
the zero-modes. To propagate the different interior energy assumptions, we
then need to estimate the commutators
$[Z^{K},\pi^{1,\mu\nu}\partial_{\mu}\partial_{\nu}]\phi$ separately for
$\pi=H^{1,\flat},H^{1,\natural}$ and
$\phi=h^{1,\flat}_{\alpha\beta},h^{1,\natural}_{\alpha\beta}$. The analysis is
reasonably straightforward when $\pi=H^{1,\natural}$ as we can rely on the
Poincaré inequality. When $\pi=H^{1,\flat}$ the analysis is finer, as we
express the metric coefficients $H^{1,\mu\nu}$ relative to the null framework
and all derivatives in terms of $\partial_{t},\partial_{y}$ and of the
tangential derivatives $\underline{\partial}_{\bm{a}}$ to hyperboloids. Doing
this, we see that

$|[Z^{K},(H^{1,\mu\nu})^{\flat}\partial_{\mu}\partial_{\nu}]\phi|\\\
\lesssim|Z^{K}H^{1,\flat}_{LL}||\partial^{2}_{t}\phi|+|Z^{K}H^{1,\flat}_{4L}||\partial_{t}\partial_{y}\phi|+\frac{|t^{2}-r^{2}|}{t^{2}}|Z^{K}H^{1,\flat}||\partial^{2}_{t}\phi|+\frac{|Z^{K}H^{1,\flat}||\partial
Z\phi|}{1+t+r}+\dots{}$

The remarkable property of the above right hand side is that each quadratic
term either contains coefficients $H^{1,\flat}_{LL}$ and $H^{1,\flat}_{4L}$ \-
which are “good” as a consequence of the wave condition - or have an extra
decaying factor $(|t^{2}-r^{2}|/t)^{2}$ and $(1+t+r)^{-1}$. Then suitable
estimates on the $L^{2}(\mathscr{H}_{s})$ norms of the above terms are
obtained by using a Hardy inequality _à la_ Lindblad and Rodnianski with
weights in $t-r$, which allow us to better exploit the pointwise decay of our
solution. We point the reader to subsection 4.6 for further details.

####  1.4.6. The null framework

We emphasise the importance of choosing a framework which correctly highlights
the structure of the weak null terms. In fact, the absence of “bad”
interactions, such as $(\partial h_{\underline{L}\underline{L}})^{2}$ and
$\partial h_{LL}\cdot\partial h_{TU}$ in the expression of the weak null terms
with respect to the null framework, is crucially related to the fact that the
transversal field $\underline{L}$ is orthogonal to
${\mathbb{S}}^{2}\times{\mathbb{S}}^{1}$, i.e. that
$\bar{g}(\underline{L},A)=0$ for $A\in\\{S^{1},S^{2},\partial_{y}\\}$. On the
contrary, the framework arising naturally from hyperboloids

$\mathscr{F}=\Big{\\{}\partial_{t},\quad\underline{\partial}_{\bm{a}}=\partial_{a}+\frac{x^{\bm{a}}}{t}\partial_{t}\Big{\\}}\cup\\{\partial_{y}\\},$

in which the “transversal field” ($\partial_{t}$ in the above example) is not
orthogonal to ${\mathbb{S}}^{2}\times{\mathbb{S}}^{1}$, causes the analogue of
the bad interaction $\partial h_{\underline{L}\underline{L}}\cdot\partial
h_{TU}$ to appear and critically fails to give a useful expression for the
weak null terms. This consideration leads us to adopt the null frame
decomposition both in the exterior and the interior region and to combine it
with the foliation by hyperboloids in the latter region. Indeed in this
region, and when required, the metric coefficients are expressed with respect
to the null frame $\mathscr{U}$ (in order to use the enhanced behavior of
$h_{LT}$ and $h_{TU}$ coefficients) while derivatives are written in terms of
those in $\mathscr{F}$ (in order to distinguish between the “good” tangential
derivatives $\underline{\partial}_{\bm{a}},\partial_{y}$ and the “bad”
direction $\partial_{t}$). Note our approach is different from what was done
in previous works on Einstein-Klein–Gordon systems. We finally mention that a
different framework than the null one is used by Ionescu and Pausader [25],
which is reminiscent of the div-curl decomposition of vector-fields in fluid
models and is more compatible with the Fourier transform approach employed
there.

####  1.4.7. The Einstein-Klein–Gordon equations

We conclude by pointing out that our proof can be used, mutatis mutandis, to
provide a new proof of the stability of the Minkowski solution to the
Einstein-Klein–Gordon equations. To briefly illustrate this point, we recall
that in a harmonic gauge the Einstein-Klein–Gordon equations read

(1.13)
$\tilde{\Box}_{g}h_{{\bm{\alpha}}{\bm{\beta}}}=\tilde{F}_{{\bm{\alpha}}{\bm{\beta}}}(h)(\partial
h,\partial
h)-2\Big{(}\partial_{\bm{\alpha}}\phi\partial_{\bm{\beta}}\phi+\frac{m^{2}}{2}g_{{\bm{\alpha}}{\bm{\beta}}}\Big{)},\qquad\tilde{\Box}_{g}\phi=m^{2}\phi.$

These equations are posed on ${\mathbb{R}}^{1+3}$, $m>0$ is a constant
parameter and $h$ is a perturbation away from the Minkowski spacetime
$\mathbf{m}$ defined via
$g_{{\bm{\alpha}}{\bm{\beta}}}=\mathbf{m}_{{\bm{\alpha}}{\bm{\beta}}}-h_{{\bm{\alpha}}{\bm{\beta}}}$.
The system (1.13) is much simpler to treat than (1.3). For example, without
the ${\mathbb{S}}^{1}$, the metric tensor $h$ remains entirely wave-like and
so all problematic wave–Klein–Gordon commutators no longer occur. The only
Klein–Gordon field is $\phi$ and it couples into the equation for the metric
only via semilinear nonlinearities. This coupling is weak in the sense that
the bootstrap assumptions for $h_{{\bm{\alpha}}{\bm{\beta}}}$ and $\phi$ can
be propagated separately. To conclude, due to our choice of null framework,
combined with the separate analysis used in the interior and exterior regions,
our proof provides an alternative perspective from what was done in previous
works on Einstein-Klein–Gordon systems in [37, 25].

###  1.5. Notation

Below is a list of notation, some of which have already been introduced in the
introduction, that we will use throughout the paper.

Coordinates:

  * •

$\\{x^{\alpha}\\}_{\alpha=0,\dots,4}$ with $x^{0}=t\in{\mathbb{R}}$,
$x=(x^{1},x^{2},x^{3})\in{\mathbb{R}}^{3}$, $x^{4}=y\in{\mathbb{S}}^{1}$ are
the harmonic coordinates. They satisfy the geometric wave equation
$g^{\mu\nu}\nabla_{\mu}\nabla_{\nu}x^{\alpha}=0$. We will always denote by
$r=|x|$ the radial component of $x$;

  * •

$u=t+r$ and ${\underline{u}}=t-r$ are the null coordinates. They are used in
the exterior region.

Derivatives:

  * •

$\nabla_{txy}=(\partial_{0},\dots,\partial_{4})$ denotes the spacetime
gradient, with $\partial_{\mu}=\partial/\partial x^{\mu}$. $\nabla_{xy}$
denotes the full spatial gradient in ${\mathbb{R}}^{3}\times{\mathbb{S}}^{1}$
while $\nabla_{x}$ is the spatial gradient in ${\mathbb{R}}^{3}$.
$\nabla_{tx}$ is the 4D spacetime gradient;

  * •

$\partial_{xy}$ denotes any of the derivatives $\partial_{i}$ with
$i=1,\dots,4$, while $\partial_{x}$ denotes any of the derivatives
$\partial_{{\bm{i}}}$ with ${\bm{i}}=1,2,3$. The definition of $\partial_{tx}$
and $\partial_{txy}$ are similar. We will use $\partial$ and $\partial_{txy}$
interchangeably;

  * •

$\Box_{xy}=-\partial^{2}_{t}+\Delta_{x}+\partial^{2}_{y}$ and
$\Box_{x}=-\partial^{2}_{t}+\Delta_{x}$;

  * •

$\partial_{r}=(x^{{\bm{i}}}/r)\partial_{{\bm{i}}}$ denotes the radial
derivative in ${\mathbb{R}}^{3}$;

  * •

$\not{\partial}$ denotes any of the angular components
$\not{\partial}_{{\bm{i}}}=\partial_{\bm{i}}-(x_{\bm{i}}/r)\partial_{r}$ of
$\partial_{{\bm{i}}}$ for ${\bm{i}}=1,2,3$;

  * •

$\partial_{u}=(1/2)(\partial_{t}+\partial_{r})$ and
$\partial_{\underline{u}}=(1/2)(\partial_{t}-\partial_{r})$ denote the null
derivatives;

  * •

$\overline{\nabla}=(\overline{\partial}_{0},\dots,\overline{\partial}_{4})=(\partial_{t}+\partial_{r},\not{\partial}_{{\bm{i}}},\partial_{y})$
denotes the tangent gradient to the cones $\\{t=r+1\\}\times{\mathbb{S}}^{1}$.
Moreover
$\overline{\nabla}_{x}=(\overline{\partial}_{0},\dots,\overline{\partial}_{3})=(\partial_{t}+\partial_{r},\not{\partial}_{{\bm{i}}})$;

  * •

$\overline{\partial}$ denotes any of the tangent derivatives
$\overline{\partial}_{\alpha}$ in ${\mathbb{R}}^{1+3}\times{\mathbb{S}}^{1}$,
$\overline{\partial}_{x}$ denotes any of the tangent derivatives
$\overline{\partial}_{{\bm{\alpha}}}$ in ${\mathbb{R}}^{1+3}$;

  * •

$\underline{\nabla}=(\underline{\partial}_{1},\dots,\underline{\partial}_{4})$
denotes the tangent gradient to the hyperboloids in
${\mathbb{R}}^{1+3}\times{\mathbb{S}}^{1}$, with
$\underline{\partial}_{{\bm{i}}}=\partial_{{\bm{i}}}+(x^{{\bm{i}}}/t)\partial_{t}$
and $\underline{\partial}_{4}=\partial_{y}$. Moreover
$\underline{\nabla}_{x}=(\underline{\partial}_{1},\underline{\partial}_{2},\underline{\partial}_{3})$;

  * •

$\underline{\partial}$ denotes any of the tangent derivatives
$\underline{\partial}_{\bm{a}},\underline{\partial}_{4}$ in
${\mathbb{R}}^{1+3}\times{\mathbb{S}}^{1}$, $\underline{\partial}_{x}$ denotes
any of tangent derivatives $\underline{\partial}_{{\bm{a}}}$ in
${\mathbb{R}}^{1+3}$. Sometimes we will use
$\underline{\partial}_{0}=\partial_{t}$.

Products:

  * •

Given a multi-index
$\alpha=(\alpha_{0},\alpha_{1},\dots,\alpha_{4})\in\mathbb{N}^{5}$, its length
is computed classically as $|\alpha|=\sum_{i=0}^{4}\alpha_{i}$. We set
$\partial^{\alpha}:=\partial_{0}^{\alpha_{0}}\partial_{1}^{\alpha_{1}}\partial_{2}^{\alpha_{2}}\partial_{3}^{\alpha_{3}}\partial_{4}^{\alpha_{4}}$
and
$\partial^{\alpha}_{x}:=\partial^{\alpha_{1}}_{1}\partial^{\alpha_{2}}_{2}\partial^{\alpha_{3}}_{3}$.
The definition of $\partial^{\alpha}_{xy}$ and $\partial^{\alpha}_{tx}$ are
analogous;

  * •

More generally, given a family of vector fields $\\{X_{1},\dots,X_{n}\\}$ and
a multi-index $\alpha=(\alpha_{1},\dots,\alpha_{n})\in\mathbb{N}^{n}$,
$X^{\alpha}=X^{\alpha_{1}}_{1}\dots X^{\alpha_{n}}_{n}$. With an abuse of
notation we will sometimes write $X^{k}$ (resp. $X^{\leq k}$) instead of
$\sum_{\alpha:|\alpha|=k}X^{\alpha}$ (resp. $\sum_{\alpha:|\alpha|\leq
k}X^{\alpha}$).

Metrics:

  * •

$\overline{g}=-(dt)^{2}+\sum_{\bm{i}}(dx^{\bm{i}})^{2}+(dy)^{2}$ denotes the
Kaluza–Klein metric on ${\mathbb{R}}^{1+3}\times{\mathbb{S}}^{1}$;

  * •

$g$ denotes a solution of the Einstein equations (1.1) on
${\mathbb{R}}^{1+3}\times{\mathbb{S}}^{1}$;

  * •

$\overline{g}^{\alpha\beta}$ and $g^{\alpha\beta}$ denote the inverse of the
metrics $\overline{g}_{\alpha\beta}$ and $g_{\alpha\beta}$ respectively. For
any other arbitrary $n$-tensor tensor $\pi_{\alpha_{1}\dots\alpha_{n}}$,
indices are raised and lowered using $\overline{g}$, e.g
${\pi^{\alpha_{1}}}_{\alpha_{2}\dots\alpha_{n}}=\overline{g}^{\alpha_{1}\mu}\pi_{\mu\alpha_{2}\dots\alpha_{n}}$;

  * •

$H^{\alpha\beta}=g^{\alpha\beta}-\overline{g}^{\alpha\beta}$ corresponds to
the formal inverse of
$h_{\alpha\beta}=g_{\alpha\beta}-\overline{g}_{\alpha\beta}$. When $h$ is
sufficiently small we have
$H^{\alpha\beta}=-h^{\alpha\beta}+\mathscr{O}^{\alpha\beta}(h^{2})$.

Null Frame and Decomposition:

  * •

$L=\partial_{t}+\partial_{r}$ denotes the vector field tangent to the outgoing
null cones in ${\mathbb{R}}^{1+3}\times{\mathbb{S}}^{1}$. In components,
$L^{0}=1$, $L^{{\bm{i}}}=x^{{\bm{i}}}/|x|$ and $L^{4}=0$;

  * •

$\underline{L}=\partial_{t}-\partial_{r}$ denotes the vector field tangent to
the incoming null cones in ${\mathbb{R}}^{1+3}\times{\mathbb{S}}^{1}$. In
components, $\underline{L}^{0}=1$,
$\underline{L}^{{\bm{i}}}=-x^{{\bm{i}}}/|x|$ and $\underline{L}^{4}=0$;

  * •

$S^{1}$ and $S^{2}$ denote orthogonal vector fields spanning the tangent space
of the spheres $t=const$, $r=const$, $y\in{\mathbb{S}}^{1}$;

  * •

$\mathscr{U}=\\{L,\underline{L},S^{1},S^{2},\partial_{y}\\}$ denotes the full
null frame in ${\mathbb{R}}^{1+3}\times{\mathbb{S}}^{1}$;

  * •

$\mathscr{T}=\\{L,S^{1},S^{2},\partial_{y}\\}$ denotes the tangent frame in
${\mathbb{R}}^{1+3}\times{\mathbb{S}}^{1}$;

  * •

$\mathscr{L}=\\{L\\}$;

  * •

For any vector field $X$ and frame vector $U$, $X_{U}=X_{\alpha}U^{\alpha}$
where $X_{\alpha}=\overline{g}_{\alpha\beta}X^{\beta}$;

  * •

For any arbitrary vector field
$X=X^{\alpha}\partial_{\alpha}=X^{L}L+X^{\underline{L}}\underline{L}+X^{S^{1}}S^{1}+X^{S^{2}}S^{2}+X^{\partial_{y}}\partial_{y}$
where $X^{L}=-(1/2)X_{\underline{L}}$, $X^{\underline{L}}=-(1/2)X_{L}$,
$X^{A}=X_{A}$ for $A=S^{1},S^{2},\partial_{y}$;

  * •

For any $(0,2)$ tensor $\pi$ and two vector fields $X,Y$

$\pi_{XY}=\pi_{\alpha\beta}X^{\alpha}Y^{\beta}.$

For any two families $\mathscr{V},\mathscr{W}$ of vector fields,
$|\pi|_{\mathscr{V}\mathscr{W}}:=\sum_{V\in\mathscr{V},W\in\mathscr{W}}|\pi_{VW}|$;

  * •

The metric $\overline{g}$ has the following form relative to the null frame,
note $A,B\in\\{S^{1},S^{2},\partial_{y}\\}$,

$\overline{g}_{LL}=\overline{g}_{\underline{L}\underline{L}}=\overline{g}_{LA}=\overline{g}_{\underline{L}A}=0,\quad\overline{g}_{L\underline{L}}=\overline{g}_{\underline{L}L}=-2,\quad\overline{g}_{AB}=\delta_{AB}.$

As concerns the inverse metric, we have

$\overline{g}^{LL}=\overline{g}^{\underline{L}\underline{L}}=\overline{g}^{LA}=\overline{g}^{\underline{L}A}=0,\quad\overline{g}^{L\underline{L}}=\overline{g}^{\underline{L}L}=-1/2,\quad\overline{g}^{AB}=\delta^{AB}.$

Admissible Vector Fields:

  * •

$\\{\Gamma\\}=\\{\Omega_{{\bm{i}}{\bm{j}}},\Omega_{0{\bm{i}}}\\}$ is the
family of Klainerman vector fields, where
$\Omega_{{\bm{i}}{\bm{j}}}=x_{\bm{i}}\partial_{\bm{j}}-x_{\bm{j}}\partial_{\bm{i}},$
and $\Omega_{0{\bm{i}}}=t\partial_{\bm{i}}+x_{\bm{i}}\partial_{t}$;

  * •

$\\{Z\\}=\\{\partial_{\mu},\Omega_{{\bm{i}}{\bm{j}}},\Omega_{0{\bm{j}}},\partial_{y}\\}$
is the family of admissible vector fields;

  * •

For any multi-index $K=(I,J)$, we set $Z^{K}=\partial^{I}\Gamma^{J}$. If
$|I|+|J|=n$ and $|J|=k$, we say that $K$ is a multi-index of type $(n,k)$.

Commutators with the null frame:

  * •

$[\Omega_{0{\bm{j}}},\partial_{t}+\partial_{r}]=-\not{\partial}_{\bm{j}}-\frac{x_{\bm{j}}}{r}(\partial_{t}+\partial_{r})$,
$[\Omega_{0{\bm{j}}},\not{\partial}_{\bm{k}}]=\big{(}-\delta_{{\bm{j}}{\bm{k}}}+\frac{x_{\bm{j}}x_{\bm{k}}}{r^{2}}\big{)}\big{[}(\partial_{t}+\partial_{r})+\frac{1}{r}\Omega_{0r}\big{]}$

  * •

$[\Omega_{{\bm{i}}{\bm{j}}},\partial_{t}+\partial_{r}]=0$ ,
$[\Omega_{{\bm{i}}{\bm{j}}},\not{\partial}_{\bm{k}}]=-\delta_{{\bm{i}}{\bm{k}}}\not{\partial}_{\bm{j}}+\delta_{{\bm{j}}{\bm{k}}}\not{\partial}_{\bm{i}}$

  * •

$[\partial_{k},\partial_{t}+\partial_{r}]=\frac{\delta_{jk}}{r}\partial_{j}-\frac{x_{j}x_{k}}{r^{3}}\partial_{j}$

  * •

$[\Omega_{0{\bm{j}}},\partial_{y}]=[\Omega_{{\bm{i}}{\bm{j}}},\partial_{y}]=[\partial_{\alpha},\partial_{y}]=0$

Commutators with the hyperbolic derivatives:

  * •

$[\Omega_{0{\bm{j}}},\partial_{t}]=-\partial_{\bm{j}}$,
$[\Omega_{0{\bm{j}}},\underline{\partial}_{{\bm{a}}}]=-\frac{x_{{\bm{a}}}}{t}\underline{\partial}_{\bm{j}}$,
$[\Omega_{0{\bm{j}}},\underline{\partial}_{4}]=0$

  * •

$[\Omega_{{\bm{i}}{\bm{j}}},\partial_{t}]=[\Omega_{{\bm{i}}{\bm{j}}},\underline{\partial}_{4}]=0$,
$[\Omega_{{\bm{i}}{\bm{j}}},\underline{\partial}_{{\bm{a}}}]=\delta_{{\bm{a}}{\bm{j}}}\underline{\partial}_{\bm{i}}-\delta_{{\bm{i}}{\bm{a}}}\underline{\partial}_{\bm{j}}$

Exterior Region:

  * •

$\widetilde{\mathscr{H}}=\\{(t,x):(t-1)^{2}-r^{2}=1\\}\times{\mathbb{S}}^{1}$
denotes the hyperboloid that separates the interior and exterior region. It
asymptotically approaches the cone $\\{t=r+1\\}\times{\mathbb{S}}^{1}$;

  * •

$\mathscr{D}^{\text{e}}:=\\{(t,x):2\leq t\leq
1+\sqrt{1+r^{2}}\\}\times{\mathbb{S}}^{1}$ denotes the exterior region;

  * •

$\mathscr{D}^{\text{e}}_{T}$ denotes the portion of exterior region in the
time slab $[2,T)$;

  * •

$\Sigma^{\text{e}}_{t}:=\\{x\in\mathbb{R}^{3}:|x|\geq\sqrt{(t-1)^{2}-1}\\}\times\mathbb{S}^{1}$
denotes a constant time slice in the exterior region;

Interior Region:

  * •

$\mathscr{D}^{\text{i}}:=\\{(t,x):t\geq
1+\sqrt{1+r^{2}}\\}\times{\mathbb{S}}^{1}$ denotes the interior region;

  * •

$\mathscr{H}_{s}:=\\{t^{2}-r^{2}=s^{2}\text{ and }t\geq
r+1\\}\times{\mathbb{S}}^{1}$ denotes a truncated hyperboloid in
${\mathbb{R}}^{1+3}\times{\mathbb{S}}^{1}$;

  * •

$S_{s,r}:=\mathscr{H}_{s}\cap\\{|x|=r\\}$ is the two-sphere of radius $r$ on
the hyperboloid $\mathscr{H}_{s}$;

  * •

$\mathscr{H}_{[s_{0},s]}:=\\{(t,x,y)\in\mathscr{D}^{\text{i}}:s_{0}^{2}\leq
t^{2}-|x|^{2}\leq s^{2}\\}$ denotes the hyperbolic slab in the interior region
between $\mathscr{H}_{s_{0}}$ and $\mathscr{H}_{s}$ when $s>2$;

  * •

$\mathscr{H}_{[s_{0},\infty)}:=\\{(t,x,y)\in\mathscr{D}^{\text{i}}:2\leq
t^{2}-|x|^{2}\\}$ is the unbounded portion of interior region above some
hyperboloid $\mathscr{H}_{s_{0}}$.

###  1.6. From the null frame to hyperbolic derivatives

Below are some useful formulas relating the null framework $\mathscr{U}$ to
the hyperbolic derivatives $\underline{\partial}_{\bm{a}}$. We recall that
$s=\sqrt{t^{2}-r^{2}}$. We have that

(1.14)
$L=\Big{(}1-\frac{r}{t}\Big{)}\partial_{t}+\frac{x^{\bm{j}}}{r}\underline{\partial}_{\bm{j}},\quad\underline{L}=\Big{(}1+\frac{r}{t}\Big{)}\partial_{t}-\frac{x^{\bm{j}}}{r}\underline{\partial}_{\bm{j}},\quad\not{\partial}_{\bm{j}}=\underline{\partial}_{\bm{j}}-\frac{x_{\bm{j}}x^{\bm{i}}}{r^{2}}\underline{\partial}_{\bm{i}}$

and

(1.15)
$UV=c^{00}_{UV}\partial^{2}_{t}+c^{a0}_{UV}\underline{\partial}_{a}\partial_{t}+c^{0b}_{UV}\partial_{t}\underline{\partial}_{b}+c^{ab}_{UV}\underline{\partial}_{a}\underline{\partial}_{b}+d^{0}_{UV}\partial_{t}+d^{c}_{UV}\underline{\partial}_{c},\qquad
U,V\in\mathscr{U}$

where

(1.16) $\begin{gathered}c^{00}_{LL}=(1-r/t)^{2},\hskip
5.0ptc^{00}_{L\underline{L}}=(1-r^{2}/t^{2}),\hskip
5.0ptc^{00}_{\underline{L}\underline{L}}=(1+r/t)^{2},\hskip
5.0ptc^{00}_{AU}=0\text{ for }A=\\{S^{1},S^{2},\partial_{y}\\},\\\
c^{04}_{L\partial_{y}}=1-r/t,\hskip
5.0ptc^{04}_{\underline{L}\partial_{y}}=(1+r/t),\hskip
5.0ptc^{04}_{UV}=0\text{ otherwise}\\\
c^{44}_{\partial_{y}\partial_{y}}=1,\hskip 5.0ptc^{44}_{UV}=0\text{
otherwise}\end{gathered}$

and

(1.17)
$\begin{gathered}|\partial^{I}\Gamma^{J}c^{\alpha\beta}_{\underline{L}\underline{L}}|\lesssim_{IJ}(1+t+r)^{-|I|},\quad|\partial^{I}\Gamma^{J}d^{\gamma}_{UV}|\lesssim_{IJ}(1+t+r)^{-1-|I|},\qquad|I|\geq
0\\\
|c^{\alpha\beta}_{LU}|\lesssim\frac{t^{2}-r^{2}}{t^{2}},\quad|\partial^{I}\Gamma^{J}c^{\alpha\beta}_{TU}|\lesssim_{IJ}(1+t+r)^{-|I|},\qquad|I|\geq
1.\end{gathered}$

For any tensor $\pi$ we have the following relations

(1.18)
$4(t/s)^{2}\pi^{UV}c^{00}_{UV}=\pi_{\underline{L}\underline{L}}\frac{s^{2}}{(t+r)^{2}}+\pi_{LL}\frac{(t+r)^{2}}{s^{2}}+\pi_{L\underline{L}}$

and

(1.19) $\displaystyle\pi^{\mu\nu}\partial_{\mu}\partial_{\nu}$
$\displaystyle=\pi^{UV}c^{\mu\nu}_{UV}\underline{\partial}_{\mu}\underline{\partial}_{\nu}+\pi^{UV}d_{UV}^{\mu}\underline{\partial}_{\mu}$
$\displaystyle=\pi^{UV}\big{[}c_{UV}^{00}\partial_{t}^{2}+c_{UV}^{{\bm{a}}\beta}\underline{\partial}_{\bm{a}}\underline{\partial}_{\beta}+c_{UV}^{\alpha{\bm{b}}}\underline{\partial}_{\alpha}\underline{\partial}_{\bm{b}}+c_{UV}^{4\beta}\partial_{y}\underline{\partial}_{\beta}+d_{UV}^{\mu}\underline{\partial}_{\mu}\big{]}$
$\displaystyle\pi^{{\bm{\mu}}{\bm{\nu}}}\partial_{\bm{\mu}}\partial_{\bm{\nu}}$
$\displaystyle=\pi^{UV}\big{[}c_{UV}^{00}\partial_{t}^{2}+c_{UV}^{{\bm{a}}{\bm{\beta}}}\underline{\partial}_{\bm{a}}\underline{\partial}_{\bm{\beta}}+c_{UV}^{{\bm{\alpha}}{\bm{b}}}\underline{\partial}_{\bm{\alpha}}\underline{\partial}_{\bm{b}}+d_{UV}^{{\bm{\mu}}}\underline{\partial}_{\bm{\mu}}\big{]}.$

For any smooth function $u=u(t,x)$, we have the following inequalities

(1.20)
$\begin{gathered}|\not{\partial}u|\lesssim|\underline{\partial}_{x}u|,\qquad|\overline{\partial}u|\lesssim\Big{(}\frac{s}{t}\Big{)}^{2}|\partial
u|+|\underline{\partial}_{x}u|\\\ |\overline{\partial}\partial
u|\lesssim\Big{(}\frac{s}{t}\Big{)}^{2}|\partial^{2}u|+|\underline{\partial}_{x}\partial
u|\lesssim\Big{(}\frac{s}{t}\Big{)}^{2}|\partial^{2}u|+\frac{1}{t}|\partial
Z^{\leq 1}u|\\\
|\not{\partial}\not{\partial}u|\lesssim\frac{1}{t}|\underline{\partial}_{x}Z^{\leq
1}u|,\qquad|\overline{\partial}\overline{\partial}u|\lesssim\Big{(}\frac{s}{t}\Big{)}^{4}|\partial^{2}u|+\Big{(}\frac{s}{t}\Big{)}^{2}\frac{1}{t}|\partial
Z^{\leq 1}u|+\frac{1}{t}|\underline{\partial}_{x}Z^{\leq 1}u|.\end{gathered}$

###  1.7. Outline of the paper

The rest of the paper is organized in three main sections. Section 2
introduces some properties the metric coefficients inherit from the wave
condition and which will be used throughout. Section 3 is devoted to perform
the bootstrap argument in the exterior region and hence to prove the global
existence of the solution to (1.8) there. In section 4 we perform the
bootstrap argument in the interior region and conclude the proof of the main
theorem. Two appendix sections follow: in section A we state and prove the
exterior and interior energy inequalities, while section B contains a list of
weighted Sobolev and Hardy inequalities.

##  2\. The wave condition

The metric solution $g$ to (1.1) satisfies, when written in harmonic
coordinates $\\{x^{\mu}\\}_{\mu}$, the _wave coordinate condition_

$g^{\mu\nu}{\Gamma^{\lambda}_{\mu\nu}}=0,\quad\lambda=\overline{0,4}$

where $\Gamma^{\lambda}_{\mu\nu}$ are the Christoffel symbols of $g$ in the
coordinates $\\{x^{\mu}\\}_{\mu}$. The above equations are equivalent to each
of the following

(2.1) $\partial_{\mu}\big{(}g^{\mu\nu}\sqrt{|\det g|}\big{)}=0,\quad
g^{\alpha\beta}\partial_{\alpha}g_{\beta\nu}=\frac{1}{2}g^{\alpha\beta}\partial_{\nu}g_{\alpha\beta},\quad\partial_{\alpha}g^{\alpha\nu}=\frac{1}{2}g_{\alpha\beta}g^{\nu\mu}\partial_{\mu}g^{\alpha\beta},\quad\nu=\overline{0,4}.$

These relations are particularly useful when written with respect to the null
framework, as they allow us to recover additional information on metric
coefficients $H_{LT}$ (and hence on $h_{LT}$) for any $T\in\mathscr{T}$, and
to show that their derivatives have a special behavior compared to those of
general coefficients $H^{\mu\nu}$. This is the content of the following
Lemmas, which are presented in a slightly different form than the ones in
[40].

######  Lemma 2.1.

Let $g$ be a Lorentzian metric satisfying the wave coordinate condition
relative to a coordinate system $\\{x^{\mu}\\}_{\mu=0}^{4}$. Let $K=(I,J)$ be
any multi-index with positive length and assume that the perturbation tensor
$H^{\mu\nu}=g^{\mu\nu}-\bar{g}^{\mu\nu}$ satisfies the following

$|Z^{K^{\prime}}H|\lesssim
C,\quad\forall\,|K^{\prime}|\leq\lfloor|K|/2\rfloor.$

Then

(2.2) $|\partial
H|_{\mathscr{L}\mathscr{T}}\lesssim|\overline{\partial}H|+|H||\partial
H|\lesssim\Big{(}\frac{s}{t}\Big{)}^{2}|\partial
H|+|\underline{\partial}_{xy}H|+|H||\partial H|$

and in any region where $r\gtrsim t\gtrsim 1$

(2.3) $\displaystyle|Z^{K}\partial H|_{\mathscr{L}\mathscr{T}}+|\partial
Z^{K}H|_{\mathscr{L}\mathscr{T}}\lesssim\sum_{|K^{\prime}|\leq|K|}(|\overline{\partial}Z^{K^{\prime}}H|+r^{-1}|Z^{K^{\prime}}H|)+\sum_{|K_{1}|+|K_{2}|\leq|K|}|Z^{K_{1}}H||\partial
Z^{K_{2}}H|$
$\displaystyle\lesssim\sum_{|K^{\prime}|\leq|K|}\Big{(}\frac{s}{t}\Big{)}^{2}|\partial
Z^{K^{\prime}}H|+|\underline{\partial}Z^{K^{\prime}}H|+r^{-1}|Z^{K^{\prime}}H|+\sum_{|K_{1}|+|K_{2}|\leq|K|}|Z^{K_{1}}H||\partial
Z^{K_{2}}H|$

Similar estimates hold for the metric tensor
$h_{\mu\nu}=g_{\mu\nu}-\bar{g}_{\mu\nu}$.

###### Proof.

We write $g^{\mu\nu}$ in terms of the perturbation metric $H^{\mu\nu}$. From
the following equality

$g^{\mu\nu}\sqrt{|\text{det}g|}=(\bar{g}^{\mu\nu}+H^{\mu\nu})\Big{(}1-\frac{1}{2}\text{tr}H+\mathscr{O}(H^{2})\Big{)}$

and the wave condition (2.1) we obtain that

(2.4)
$\partial_{\mu}\Big{(}H^{\mu\nu}-\frac{1}{2}\bar{g}^{\mu\nu}\text{tr}H+\mathscr{O}^{\mu\nu}(H^{2})\Big{)}=0,\quad\text{where
}\mathscr{O}^{\mu\nu}(H^{2})=\mathscr{O}(|H|^{2}).$

The divergence of a vector field can be expressed relative to the null frame
as follows

(2.5)
$\partial_{\mu}F^{\mu}=L_{\mu}\partial_{\underline{u}}F^{\mu}-\underline{L}_{\mu}\partial_{u}F^{\mu}+A_{\mu}\partial_{A}F^{\mu},\quad
A\in\\{S_{1},S_{2},\partial_{y}\\}$

so setting
$\tilde{H}^{\mu\nu}:=H^{\mu\nu}-\frac{1}{2}\bar{g}^{\mu\nu}\text{tr}H$ and
contracting (2.4) with any $T\in\mathscr{T}$ we deduce that

(2.6)
$\partial_{\underline{u}}H_{LT}=\partial_{\underline{u}}\tilde{H}_{LT}=\partial_{u}\tilde{H}_{\underline{L}T}-\partial_{A}\tilde{H}_{AT}+{\mathscr{O}}(H\cdot\partial
H).$

The first of the above equalities follows from the fact that
$\bar{g}^{LL}=g^{LA}=0$. Relation (2.6) and the first two inequalities in
(1.20) imply immediately (2.2).

We now recall the commutators between any admissible vector field $Z$ and the
null frame, which can be summarized in the following formula

$[Z,\overline{\partial}_{\alpha}]=\sum_{{\bm{\beta}}=0}^{3}c_{Z\alpha}^{{\bm{\beta}}}\overline{\partial}_{{\bm{\beta}}}+\sum_{{\bm{i}}=1}^{3}d_{Z\alpha}^{{\bm{i}}}\frac{\partial_{{\bm{i}}}}{r}+e_{Z\alpha}\frac{\Omega_{0r}}{r},\qquad
c_{Z\alpha}^{{\bm{\beta}}},d_{Z\alpha}^{{\bm{i}}},e_{Z\alpha}=\mathscr{O}\big{(}\frac{x}{r}\big{)}$

where $c_{Z\alpha}^{{\bm{\beta}}},d_{Z\alpha}^{{\bm{i}}},e_{Z\alpha}$ are
smooth homogeneous functions of $x$ such that
$c_{\partial\alpha}^{{\bm{\beta}}}=e_{\partial\alpha}=0$,
$d^{{\bm{i}}}_{\Gamma\alpha}=0$ and

$|\partial^{I}_{x}c_{Z\alpha}^{{\bm{\beta}}}|+|\partial^{I}d_{Z\alpha}^{{\bm{i}}}|+|\partial^{I}e_{Z\alpha}|\lesssim
r^{-|I|},\qquad|I|\geq 0.$

Using an induction argument on $|K|$, one can show that for any sufficiently
smooth function $w$ the following inequality holds true whenever $r\gtrsim t$

(2.7)
$|[Z^{K},\overline{\partial}]w|\lesssim\sum_{|K^{\prime}|<|K|}(|\overline{\partial}Z^{K^{\prime}}w|+r^{-1}|\partial
Z^{K^{\prime}}w|)+\sum_{|K^{\prime}|\leq|K|}r^{-1}|Z^{K^{\prime}}w|.$

As concerns the commutators with the transverse vector field, we simply have

(2.8)
$|[Z^{K},\partial_{t}-\partial_{r}]w|\lesssim\sum_{|K^{\prime}|<|K|}|\partial
Z^{K^{\prime}}w|\lesssim\sum_{|K^{\prime}|<|K|}|\partial_{\underline{u}}Z^{K^{\prime}}w|+|\overline{\partial}Z^{K^{\prime}}w|.$

In order to obtain (2.3), we apply $Z^{K}$ vector fields to both sides of
equality (2.6). Using (2.7) we find that

$\displaystyle|Z^{K}\partial_{\underline{u}}H|_{\mathscr{L}\mathscr{T}}$
$\displaystyle\lesssim\sum_{|K^{\prime}|\leq|K|}(|\overline{\partial}Z^{K^{\prime}}H|+r^{-1}|Z^{K^{\prime}}H|)+\sum_{|K_{1}|+|K_{2}|\leq|K|}|Z^{K_{1}}H||\partial
Z^{K_{2}}H|$

which, together with (2.8), yields

$\displaystyle|\partial_{\underline{u}}Z^{K}H|_{\mathscr{L}\mathscr{T}}$
$\displaystyle\lesssim\sum_{|K^{\prime}|\leq|K|}(|\overline{\partial}Z^{K^{\prime}}H|+r^{-1}|Z^{K^{\prime}}H|)+\sum_{|K_{1}|+|K_{2}|\leq|K|}|Z^{K_{1}}H||\partial
Z^{K_{2}}H|$
$\displaystyle+\sum_{|K^{\prime}|<|K|}|\partial_{\underline{u}}Z^{K^{\prime}}H|_{\mathscr{L}\mathscr{T}}.$

The conclusion of the proof of the first inequality in (2.3) then follows by
induction on $|K|$. The latter follows using also (1.20). Finally,
inequalities (2.2) and (2.3) for $h$ simply follow from the equality
$H^{\mu\nu}=-h^{\mu\nu}+\mathscr{O}(h^{2})$. ∎

Inequalities (2.2) and (2.3) hold true also for the tensor $H^{1,\mu\nu}$
introduced in (1.12).

######  Lemma 2.2.

Under the same assumptions of the previous lemma, we have that

(2.9) $\displaystyle|Z^{K}\partial H^{1}|_{\mathscr{L}\mathscr{T}}+|\partial
Z^{K}H^{1}|_{\mathscr{L}\mathscr{T}}\lesssim\sum_{|K^{\prime}|\leq|K|}(|\overline{\partial}Z^{K^{\prime}}H^{1}|+r^{-1}|Z^{K^{\prime}}H^{1}|)$
$\displaystyle+\sum_{|K_{1}|+|K_{2}|\leq|K|}|Z^{K_{1}}H^{1}||\partial
Z^{K_{2}}H^{1}|+\frac{M\chi_{0}\big{(}t/2\leq r\leq 3t/4\big{)}}{(1+t+r)^{2}}$

and

(2.10) $\displaystyle|Z^{K}\partial H^{1}|_{\mathscr{L}\mathscr{T}}+|\partial
Z^{K}H^{1}|_{\mathscr{L}\mathscr{T}}\lesssim\sum_{|K^{\prime}|\leq|K|}\Big{(}\frac{s}{t}\Big{)}^{2}|\partial
Z^{K^{\prime}}H^{1}|+|\underline{\partial}Z^{K^{\prime}}H^{1}|+r^{-1}|Z^{K^{\prime}}H^{1}|$
$\displaystyle+\sum_{|K_{1}|+|K_{2}|\leq|K|}|Z^{K_{1}}H^{1}||\partial
Z^{K_{2}}H^{1}|+\frac{M\chi_{0}\big{(}t/2\leq r\leq 3t/4\big{)}}{(1+t+r)^{2}}$

where $\chi_{0}\big{(}t/2\leq r\leq 3t/4\big{)}$ is a cut-off function
supported for $t/2\leq r\leq 3t/4$. Similar estimates hold true for $h^{1}$.

###### Proof.

We set
$\tilde{H}^{0,\mu\nu}:=H^{0,\mu\nu}-\frac{1}{2}\bar{g}^{\mu\nu}\text{tr}(H^{0})$
and derive from the definition of $H^{0}$ that

(2.11)
$\partial_{\mu}\tilde{H}^{0,\mu\nu}=2\chi^{\prime}\Big{(}\frac{r}{t}\Big{)}\chi(r)\frac{M}{t^{2}}\delta^{\nu
0}.$

We inject the above formula into (2.4) and obtain that

$Z^{K}\partial_{\mu}(H^{1,\mu\nu})=-Z^{K}\partial_{\mu}\mathscr{O}^{\mu\nu}(H^{2})-Z^{K}\Big{(}2\chi^{\prime}\Big{(}\frac{r}{t}\Big{)}\chi(r)\frac{M}{t^{2}}\delta^{\nu
0}\Big{)}.$

Then the result of the statement follows using the same argument as in
previous lemma’s proof. Furthermore, from (1.12) a similar inequality can be
proved for $h^{1}_{\mu\nu}$. ∎

##  3\. The Exterior Region

The goal of this section is to prove the existence in the exterior region
$\mathscr{D}^{\text{e}}$ of the solution $h^{1}_{\alpha\beta}$ of (1.8) with
data satisfying the hypothesis of theorem 1.2. The proof is based on a
bootstrap argument in which the a-priori assumptions are bounds on the higher
order weighted energies of $h^{1}_{\alpha\beta}$, introduced below.

For any fixed $\kappa>0$, we define the exterior weighted energy functional of
$h^{1}_{\alpha\beta}$ as

$E^{\text{e},\kappa}(t,h^{1}_{\alpha\beta})=\iint_{\\{|x|\geq
t-1\\}\times{\mathbb{S}}^{1}}(2+|x|-t)^{1+2\kappa}|\nabla_{txy}h^{1}_{\alpha\beta}(t,x,y)|^{2}dxdy\
\\\
+\int_{2}^{t}\iint_{\\{|x|\geq\tau-1\\}\times{\mathbb{S}}^{1}}(2+|x|-t)^{2\kappa}|\overline{\nabla}h^{1}_{\alpha\beta}(\tau,x,y)|^{2}dxdyd\tau$

and denote
$E^{\text{e},\kappa}(t,h^{1})=\sum_{\alpha,\beta}E^{\text{e},\kappa}(t,h^{1}_{\alpha\beta})$.
We fix $N\in{\mathbb{N}}$ with $N\geq 7$ and assume the existence of a
positive constant $C_{0}$ and of some small parameters $0<\sigma<\kappa/3\ll
1$ such that the solution $h^{1}$ of (1.8) exists in
$\mathscr{D}^{\text{e}}_{T_{0}}$ and for all $t\in[2,T_{0})$ it satisfies

(3.1) $\displaystyle E^{\text{e},\kappa}(t,Z^{\leq N}h^{1})^{1/2}\leq
2C_{0}\epsilon t^{\sigma}$ (3.2) $\displaystyle E^{e,1+\kappa}(t,\partial
Z^{\leq N}h^{1})^{1/2}\leq 2C_{0}\epsilon t^{\sigma}.$

The result we want to prove here affirms the following

######  Proposition 3.1.

Let $N\in{\mathbb{N}}$ with $N\geq 6$ be fixed. There exists a constant
$C_{0}$ sufficiently large, $0<\epsilon_{0}\ll 1$ sufficiently small and a
universal positive constant $C$ such that, for every $0<\epsilon<\epsilon_{0}$
if $h^{1}$ is a solution of (1.8) in the time interval $[2,T_{0})$ and
satisfies the bounds (3.1)-(3.2) for all $t\in[2,T_{0})$, then in the same
interval it actually satisfies

(3.3) $\displaystyle E^{\text{e},\kappa}(t,Z^{\leq N}h^{1})^{1/2}\leq
C_{0}\epsilon t^{\frac{\sigma}{2}+CC_{0}\epsilon}$ (3.4) $\displaystyle
E^{e,1+\kappa}(t,\partial Z^{\leq N}h^{1})^{1/2}\leq C_{0}\epsilon
t^{\frac{\sigma}{2}+CC_{0}\epsilon}.$

The time $T_{0}$ in the statement of the above proposition is arbitrary and
one can hence infer that the solution exists globally in
$\mathscr{D}^{\text{e}}$. We also observe that, as a consequence of the energy
assumptions (3.1)-(3.2), there exists an integrable function $l\in
L^{1}([2,T_{0}))$ such that

(3.5) $\displaystyle\big{\|}(2+r-t)^{\frac{1}{2}+\kappa}\ \partial Z^{\leq
N}h^{1}\big{\|}_{L^{2}(\Sigma^{\text{e}}_{t})}\leq 2C_{0}\epsilon t^{\sigma}$
(3.6) $\displaystyle\big{\|}(2+r-t)^{\kappa}\ \overline{\partial}Z^{\leq
N}h^{1}\big{\|}_{L^{2}(\Sigma^{\text{e}}_{t})}\leq
2C_{0}\epsilon\sqrt{l(t)}t^{\sigma}$ (3.7)
$\displaystyle\big{\|}(2+r-t)^{\frac{3}{2}+\kappa}\ \partial^{2}Z^{\leq
N}h^{1}\big{\|}_{L^{2}(\Sigma^{\text{e}}_{t})}\leq 2C_{0}\epsilon t^{\sigma}$
(3.8) $\displaystyle\big{\|}(2+r-t)^{1+\kappa}\ \overline{\partial}\partial
Z^{\leq N}h^{1}\big{\|}_{L^{2}(\Sigma^{\text{e}}_{t})}\leq
2C_{0}\epsilon\sqrt{l(t)}t^{\sigma}.$

The first step to recover the enhanced bounds (3.3)-(3.4) is to compare the
equation satisfied by the differentiated unknown $Z^{K}h^{1}_{\alpha\beta}$
for any $K=(I,J)$ with $|K|\leq N+1$, with the linear inhomogeneous equation
(A.1). The commutation of $Z^{K}$ with equation (1.8) shows that
$Z^{K}h^{1}_{\alpha\beta}$ solves

(3.9)
$\tilde{\Box}_{g}Z^{K}h^{1}_{\alpha\beta}=F^{K}_{\alpha\beta}+F^{0,K}_{\alpha\beta},\qquad
F^{0,K}_{\alpha\beta}=Z^{K}\tilde{\Box}_{g}h^{0}_{\alpha\beta}$

with source term $F^{K}_{\alpha\beta}$ given by

(3.10) $F^{K}_{\alpha\beta}=Z^{K}F_{\alpha\beta}(h)(\partial h,\partial
h)-[Z^{K},H^{\mu\nu}\partial_{\mu}\partial_{\nu}]h^{1}_{\alpha\beta}.$

The second step consists in recovering suitable pointwise decay estimates and
$L^{2}$ estimates for tensors $h_{\alpha\beta}$ and $H^{\alpha\beta}$ and
their derivatives. Our aim is in fact to apply energy inequality (A.2) with
$\mathbf{W}=Z^{K}h^{1}_{\alpha\beta}$,
$\mathbf{F}=F^{K}_{\alpha\beta}+F^{0,K}_{\alpha\beta}$ and
$w(q)=(2+r-t)^{\frac{1}{2}+i+\kappa}$ with $i=0,1$ depending on $K$. Such
estimates allow us, on the one hand, to justify the use of (A.2) and, on the
other, to suitably estimate the different contributions to the right hand side
of such energies, and hence to propagate (3.1)-(3.2). The derivation of these
bounds is the content of the following subsections.

###  3.1. Pointwise bounds

A first set of pointwise decay bounds for the metric perturbation
$h^{1}_{\alpha\beta}$, as well as for tensor $H^{1,\alpha\beta}$, are obtained
from the a-priori energy assumptions (3.1)-(3.2) via the weighted Sobolev and
Hardy embeddings stated in appendix B. As concerns the mass term, a
straightforward computation using directly the expression of
$h^{0}_{\alpha\beta}$ in (1.7) shows that for all
$(t,x,y)\in{\mathbb{R}}^{1+3}\times{\mathbb{S}}^{1}$

(3.11)
$\sup_{{\mathbb{S}}^{1}}|\partial^{I}Z^{J}h^{0}|\lesssim\epsilon(1+r)^{-1-|I|}.$

######  Proposition 3.2.

Let us define the weighted pointwise norm

$|u(t)|_{\lambda}:=\sup_{(x,y)\in\Sigma^{\text{e}}_{t}}(1+t+r)(2+r-t)^{1+\lambda}|u(t,x,y)|.$

Assume that the solution $h^{1}_{\alpha\beta}$ of (1.8) exists in the time
interval $[2,T_{0})$ and satisfies (3.1)-(3.2) for all $t\in[2,T_{0})$. Then
the following estimates hold true in $\mathscr{D}^{\text{e}}_{T_{0}}$

(3.12) $|\partial Z^{\leq N-3}h^{1}|_{\kappa}+|\partial^{2}Z^{\leq
N-3}h^{1}|_{1/2+\kappa}\lesssim C_{0}\epsilon t^{\sigma}$ (3.13)
$|\overline{\partial}Z^{\leq
N-3}h^{1}|_{\kappa-1/2}+|\overline{\partial}\partial Z^{\leq
N-3}h^{1}|_{\kappa}\lesssim C_{0}\epsilon t^{\sigma}\sqrt{l(t)}$ (3.14)
$|Z^{\leq N-2}h^{1}|_{\kappa-1}\lesssim C_{0}\epsilon t^{\sigma}$ (3.15)
$|Z^{\leq N-3}h^{1,\natural}|_{\kappa-1/2}\lesssim C_{0}\epsilon t^{\sigma}.$

Estimates (3.12)-(3.14) hold true also for tensor $H^{1,\alpha\beta}$.

###### Proof.

Estimate (3.12) (resp. (3.13)) for the second order derivatives $\partial^{2}$
(resp. $\overline{\partial}\partial$) of $Z^{\leq N-3}h^{1}$ follows from the
energy bound (3.7) (resp. (3.8)) and from inequality (B.2) applied with
$\beta=1+\lambda$ and $\lambda=1/2+\kappa$ (resp. $\lambda=\kappa$).

Estimate (3.12) (resp. (3.13)) for the first order derivatives $\partial$
(resp. $\overline{\partial}$) of $Z^{\leq N-3}h^{1}$ follows from the energy
bound (3.7) (resp. (3.8)) and inequality (B.5) applied with $\beta=1+\kappa$
(resp. $\beta=1/2+\kappa$), and estimate (3.15) follows using in addition the
Poincaré inequality.

Estimate (3.14) follows from the energy bound (3.5) and inequality (B.5) with
$\beta=\kappa$.

Finally, one can show that estimates (3.12)-(3.14) hold true for
$H^{1,\alpha\beta}$ using (1.9). ∎

As a result of inequality (2.9) and the pointwise bounds we just obtained, we
can show that the metric coefficients $h^{1}_{LT}$ satisfy enhanced pointwise
decay estimates compared to those in Proposition 3.2.

######  Proposition 3.3.

Under the assumptions of Proposition 3.2, we have

(3.16) $\displaystyle|\partial Z^{\leq N-3}h^{1}_{LT}(t,x,y)|\lesssim
C_{0}\epsilon\big{[}(1+t+r)^{-1+\sigma}\sqrt{l(t)}(2+r-t)^{-\frac{1}{2}-\kappa}+(1+t+r)^{-2+2\sigma}(2+r-t)^{-\kappa}\big{]}$
(3.17) $\displaystyle|Z^{\leq N-4}h^{1}_{LT}(t,x,y)|\lesssim
C_{0}\epsilon(1+t+r)^{-\frac{3}{2}+2\sigma}(2+r-t)^{\frac{1}{2}-\kappa}$

and

(3.18) $\|Z^{\leq
N}h^{1}_{LT}(t,r)\|_{L^{2}(\mathbb{S}^{2}\times\mathbb{S}^{1})}\lesssim\frac{C_{0}\epsilon
t^{\sigma}}{t^{\kappa-\mu}(2+r-t)^{\mu}}\Big{(}\frac{\sqrt{l(t)}}{r^{1/2}}+\frac{1}{r}\Big{)}.$

The same bounds are satisfied by $H^{1}_{LT}$.

###### Proof.

From relation (1.9) and pointwise bounds (3.11), (3.12), (3.14), it is clear
that the estimates (3.16) and (3.17) for $h^{1}_{LT}$ are also satisfied by
$H^{1}_{LT}$.

Bound (3.16) follows immediately from inequality (2.9) coupled with
(3.12)-(3.14).

The proof of (3.17) requires more work because a naive integration of (3.16)
along the integral curves of $\partial_{t}-\partial_{r}$ does not produce the
required result due to the factor $\sqrt{l(t)}$ (it would if this was replaced
by the explicit decay $t^{-1/2}$). Of course, the estimate is satisfied in the
region where $r\geq 2t$ simply after (3.14). We then restrict our attention to
the portion of exterior region for which $r<2t$ and proceed as follows:

  * -

first, we recover a better bound for $\partial Z^{\leq N-4}h^{1}_{LT}$ than
the one in (3.16), in which $\sqrt{l(t)}$ is replaced by a decay
$t^{-\frac{1}{2}+}$. This is obtained by the integration of
$\underline{\partial}_{r}\partial Z^{\leq N-4}h^{1}_{LT}$ along hyperboloids
in some dyadic time slab, where
$\underline{\partial}_{r}=\frac{x^{\bm{j}}}{tr}\Omega_{0{\bm{j}}}$;

  * -

then, we deduce the desired estimate on $Z^{\leq N-4}h^{1}_{LT}$ by
integration of the bounds obtained in step 1 along the integral curves of
$\partial_{{\underline{u}}}$.

_Step 1._ From the relation
$\underline{\partial}_{r}=\frac{x^{\bm{j}}}{tr}\Omega_{0{\bm{j}}}$ and
inequality (3.16), we see that in fact

$|\underline{\partial}_{r}\partial Z^{\leq
N-4}h^{1}_{LT}|\lesssim(1+t+r)^{-1}|\partial Z^{\leq N-3}h^{1}_{LT}|\\\
\lesssim
C_{0}\epsilon(1+t+r)^{-2+\sigma}\sqrt{l(t)}(2+r-t)^{-\frac{1}{2}-\kappa}+C_{0}\epsilon(1+t+r)^{-3+2\sigma}(2+r-t)^{-\kappa}.$

Moreover, thanks to the pointwise bound (3.12) we have that on the cone $r=2t$

$|\partial Z^{\leq N-4}h^{1}_{LT}(t,x,y)|\lesssim
C_{0}\epsilon(1+t+r)^{-2-\kappa+\sigma}.$

We dyadically decompose the time interval
$[2,T_{0})=\cup_{k=1}^{k_{0}}[2^{k},2^{k+1})\cap[2,T_{0})$ where
$k_{0}\sim\ln_{2}T_{0}$ and denote $\mathscr{C}^{e}_{k}$ the portion of the
exterior region in the time slab $[2^{k},2^{k+1})$, so that
$\mathscr{C}^{e}=\cup_{k=1}^{k_{0}}\mathscr{C}^{e}_{k}$. Inequality (3.16) and
the fact that $l\in L^{1}([2,T_{0}))$ imply the existence, for every fixed
$k$, of a time $\tau_{k}\in[2^{k},2^{k+1})\cap[2,T_{0})$ such that

$|\partial Z^{\leq N-4}h^{1}_{LT}(\tau_{k},x,y)|\lesssim
C_{0}\epsilon\tau_{k}^{-\frac{3}{2}+\sigma}(2+r-\tau_{k})^{-\frac{1}{2}-\kappa}+C_{0}\epsilon\tau_{k}^{-2+2\sigma}(2+r-\tau_{k})^{-\kappa}.$

For every fixed $(t,x,y)\in\mathscr{C}^{e}_{k}$, we then integrate
$\underline{\partial}_{r}\partial Z^{\leq N-4}h^{1}_{LT}$ along the integral
curve $\tau\mapsto\gamma(\tau)$ of $\underline{\partial}_{r}$ passing through
$(t,x,y)$999These are the hyperboloids $\\{\tau^{2}-|w|^{2}=t^{2}-r^{2}\\}$.
If $t=r$ they degenerate into the cone $\\{\tau-|w|=t-r\\}$. until its first
intersection with $\\{\tau=\tau_{k}\\}\cup\\{|w|=2\tau\\}$. We denote
$(\tau^{*}_{k},x^{*}_{k},y)$ the point at which such an intersection occurs
first and observe that $\tau^{*}_{k}\sim 2^{k}\sim t$. We deduce that

(3.19) $\displaystyle|\partial Z^{\leq N-4}h^{1}_{LT}(t,x,y)|\leq|\partial
Z^{N-4}h^{1}_{LT}(\tau^{*}_{k},x^{*}_{k},y)|+\int_{\tau^{*}_{k}}^{t}|\underline{\partial}_{r}\partial
Z^{\leq N-4}h^{1}_{LT}(\gamma(\tau))|d\tau$ $\displaystyle\lesssim
C_{0}\epsilon(1+t+r)^{-\frac{3}{2}+\sigma}(2+r-t)^{-\frac{1}{2}-\kappa}+C_{0}\epsilon(1+t+r)^{-2+2\sigma}(2+r-t)^{-\kappa}.$

_Step 2._ We now integrate (3.19) along the integral lines of
$\partial_{{\underline{u}}}$, up to $\mathscr{B}=\\{r=2t\\}\cup\\{t=2\\}$.
After (3.14), we have that

$|Z^{\leq N-2}h^{1}|_{\mathscr{B}}\lesssim C_{0}\epsilon(1+t+r)^{-2}(2+r-t).$

Therefore, from (3.19) we get that

$\displaystyle|Z^{\leq N-4}h^{1}_{LT}(t,x,y)|\leq|Z^{\leq
N-4}h^{1}_{LT}(\lambda^{*}_{k},x^{*}_{k},y)|+\int_{\lambda^{*}_{k}}^{t}|\partial_{{\underline{u}}}Z^{\leq
N-4}h^{1}_{LT}(\gamma(\tau))|d\tau$ $\displaystyle\lesssim
C_{0}\epsilon(1+t+r)^{-2}(2+r-t)+(1+t+r)^{-\frac{3}{2}+\sigma}\int_{\lambda^{*}_{k}}^{t}C_{0}\epsilon(2+t+r-2\tau)^{-\frac{1}{2}-\kappa}d\tau$
$\displaystyle+(1+t+r)^{-2+2\sigma}\int_{\lambda^{*}_{k}}^{t}C_{0}\epsilon(2+t+r-2\tau)^{-\kappa}d\tau$

and

$|Z^{\leq N-4}h^{1}_{LT}(t,x,y)|\lesssim
C_{0}\epsilon(1+t+r)^{-\frac{3}{2}+2\sigma}(2+r-t)^{\frac{1}{2}-\kappa}.$

As concerns the proof of (3.18), we begin by applying inequality (B.6) with
$\beta=\mu$ to $Z^{\leq N}h^{1}_{LT}$. We get that

$(2+r-t)^{2\mu}r^{2}\|Z^{\leq
N}h^{1}_{LT}\|_{L^{2}(\mathbb{S}^{2}\times\mathbb{S}^{1})}^{2}\lesssim\iint_{\Sigma^{\text{e}}_{t}}(2+r-t)^{1+2\mu}(\partial
Z^{\leq N}h^{1}_{LT})^{2}dxdy.$

We decompose the above right hand side into

$\iint_{\Sigma^{\text{e}}_{t}\setminus\Sigma^{\text{e}}_{2t}}(2+r-t)^{1+2\mu}(\partial
Z^{\leq
N}h^{1}_{LT})^{2}dxdy+\iint_{\Sigma^{\text{e}}_{2t}}(2+r-t)^{1+2\mu}(\partial
Z^{\leq N}h^{1}_{LT})^{2}dxdy.$

The integral over $\Sigma^{\text{e}}_{2t}$ is simply estimated using the
energy bound (3.5) as follows

$\iint_{\Sigma^{\text{e}}_{2t}}(2+r-t)^{1+2\mu}(\partial Z^{\leq
N}h^{1}_{LT})^{2}dxdy\lesssim
t^{2(\mu-\kappa)}\iint_{\Sigma^{\text{e}}_{t}}(2+r-t)^{1+2\kappa}|\partial
Z^{\leq N}h^{1}|^{2}dxdy\lesssim\epsilon^{2}t^{2(\mu-\kappa+\sigma)}.$

The integral over $\Sigma^{\text{e}}_{t}\setminus\Sigma^{\text{e}}_{2t}$ is
estimated using (2.9) for $h^{1}$

$\displaystyle\iint_{\Sigma^{\text{e}}_{t}\setminus\Sigma^{\text{e}}_{2t}}(2+r-t)^{1+2\mu}(\partial
Z^{\leq N}h^{1}_{LT})^{2}dxdy$
$\displaystyle\lesssim\iint_{\Sigma^{\text{e}}_{t}\setminus\Sigma^{\text{e}}_{2t}}(2+r-t)^{1+2\mu}M^{2}\chi_{0}^{2}(t/2\leq
r\leq 3t/4)r^{-4}dxdy$
$\displaystyle+\iint_{\Sigma^{\text{e}}_{t}\setminus\Sigma^{\text{e}}_{2t}}(2+r-t)^{1+2\mu}\Big{(}|\overline{\partial}Z^{\leq
N}h^{1}|^{2}+r^{-2}|Z^{\leq N}h^{1}|^{2}+\sum_{|K_{1}|+|K_{2}|\leq
N}|Z^{K_{1}}h^{1}|^{2}|\partial Z^{K_{2}}h^{1}|^{2}\Big{)}dxdy,$

where $\chi_{0}(t/2\leq r\leq 3t/4)$ is a smooth cut-off function supported in
$t/2\leq r\leq 3t/4$. We observe that the portion of such support contained in
the exterior region is bounded. Therefore we get the following:

\- from the smallness assumption on $M$ and the above observation

$\iint_{\Sigma^{\text{e}}_{t}\setminus\Sigma^{\text{e}}_{2t}}(2+r-t)^{1+2\mu}M^{2}\chi_{0}^{2}(t/2\leq
r\leq 3t/4)r^{-4}dxdy\lesssim\epsilon^{2}t^{-4};$

\- from the energy bound (3.6)

$\iint_{\Sigma^{\text{e}}_{t}\setminus\Sigma^{\text{e}}_{2t}}(2+r-t)^{1+2\mu}|\overline{\partial}Z^{\leq
N}h^{1}|^{2}dxdy\\\ \lesssim
t^{1+2(\mu-\kappa)}\iint_{\Sigma^{\text{e}}_{t}}(2+r-t)^{2\kappa}|\overline{\partial}Z^{\leq
N}h^{1}|^{2}dxdy\lesssim\epsilon^{2}t^{1+2(\mu-\kappa+\sigma)}l(t);$

\- from inequality (B.4) with $\beta=2\kappa-1$ and the energy bound (3.5)

$\displaystyle\iint_{\Sigma^{\text{e}}_{t}\setminus\Sigma^{\text{e}}_{2t}}(2+r-t)^{1+2\mu}r^{-2}|Z^{\leq
N}h^{1}|^{2}dxdy\lesssim
t^{2(\mu-\kappa)}\iint_{\Sigma^{\text{e}}_{t}\setminus\Sigma^{\text{e}}_{2t}}(2+r-t)^{2\kappa-1}|Z^{\leq
N}h^{1}|^{2}dxdy$ $\displaystyle\lesssim
t^{2(\mu-\kappa)}\iint_{\Sigma^{\text{e}}_{t}}(2+r-t)^{1+2\kappa}|\partial
Z^{\leq N}h^{1}|^{2}dxdy\lesssim\epsilon^{2}t^{2(\mu-\kappa+\sigma)};$

\- from the energy bound (3.5) and the pointwise bound (3.14) that

$\displaystyle\sum_{\begin{subarray}{c}|K_{1}|+|K_{2}|\leq N\\\
|K_{1}|\leq\lfloor N/2\rfloor\end{subarray}}$
$\displaystyle\iint_{\Sigma^{\text{e}}_{t}\setminus\Sigma^{\text{e}}_{2t}}(2+r-t)^{1+2\mu}|Z^{K_{1}}h^{1}|^{2}|\partial
Z^{K_{2}}h^{1}|^{2}dxdy$
$\displaystyle\lesssim\epsilon^{2}t^{2\sigma}\iint_{\Sigma^{\text{e}}_{t}\setminus\Sigma^{\text{e}}_{2t}}(2+r-t)^{1+2(\mu-\kappa)}r^{-2}|\partial
Z^{\leq N}h^{1}|^{2}dxdy\lesssim\epsilon^{4}t^{-2+4\sigma};$

\- finally, from the decay bound (3.12), inequality (B.4) and the energy bound
(3.5)

$\displaystyle\sum_{\begin{subarray}{c}|K_{1}|+|K_{2}|\leq N\\\
|K_{2}|\leq\lfloor N/2\rfloor\end{subarray}}$
$\displaystyle\iint_{\Sigma^{\text{e}}_{t}\setminus\Sigma^{\text{e}}_{2t}}(2+r-t)^{1+2\mu}|Z^{K_{1}}h^{1}|^{2}|\partial
Z^{K_{2}}h^{1}|^{2}dxdy$
$\displaystyle\lesssim\epsilon^{2}t^{2\sigma}\iint_{\Sigma^{\text{e}}_{t}\setminus\Sigma^{\text{e}}_{2t}}(2+r-t)^{-1+2(\mu-\kappa)}r^{-2}|Z^{\leq
N}h^{1}|^{2}dxdy$
$\displaystyle\lesssim\epsilon^{2}t^{-2+2\sigma}\iint_{\Sigma^{\text{e}}_{t}}(2+r-t)^{1+2\kappa}|\partial
Z^{\leq N}h^{1}|^{2}dxdy\lesssim\epsilon^{4}t^{-2+4\sigma}.$

Summing up,

$\iint_{\Sigma^{\text{e}}_{t}}(2+r-t)^{1+2\mu}(\partial Z^{\leq
N}h^{1}_{LT})^{2}dxdy\lesssim
C_{0}^{2}\epsilon^{2}t^{2(\mu-\kappa+\sigma)}(1+t\,l(t))$

which concludes the proof of (3.18). ∎

###  3.2. The null and cubic terms

The combination of the energy assumptions and the decay bounds obtained in
proposition 3.2 yield easily the following weighted
$L^{2}(\Sigma^{\text{e}}_{t})$ estimates of the differentiated null and cubic
terms.

######  Proposition 3.4.

Fix $i=0,1$. Under the a-priori energy assumptions (3.1)-(3.2) we have

(3.20) $\|(2+r-t)^{\frac{1}{2}+i+\kappa}\partial^{i}Z^{\leq N}{\bf
Q}_{\alpha\beta}(\partial h,\partial
h)\|_{L^{2}(\Sigma^{\text{e}}_{t})}\lesssim
C_{0}^{2}\epsilon^{2}t^{-1+2\sigma}\sqrt{l(t)}+C_{0}^{2}\epsilon^{2}t^{-2+2\sigma}$

and

(3.21) $\|(2+r-t)^{\frac{1}{2}+i+\kappa}\partial^{i}Z^{\leq
N}G_{\alpha\beta}(h)(\partial h,\partial
h)\|_{L^{2}(\Sigma^{\text{e}}_{t})}\lesssim
C_{0}^{3}\epsilon^{3}t^{-2+3\sigma}.$

###### Proof.

We write $h=h^{1}+h^{0}$ and inject this decomposition into ${\bf
Q}_{\alpha\beta}$ and $G_{\alpha\beta}$. We prove estimates (3.20) and (3.21)
for null and cubic interactions involving only $h^{1}$-factors. The remaining
interactions, i.e. those involving at least one $h^{0}$-factor, can be easily
treated thanks to (3.11) so we leave the details to the reader.

It is well-known that the admissible vector fields $Z$ preserve the null
structure, in the sense that for any null form $Q$

$ZQ(\partial\phi,\partial\psi)=Q(\partial
Z\phi,\partial\psi)+Q(\partial\phi,\partial
Z\psi)+\tilde{Q}(\partial\phi,\partial\psi)$

where $\tilde{Q}$ is also a null form. Together with the fundamental property

$|Q(\partial\phi,\partial\psi)|\lesssim|\overline{\partial}\phi||\partial\psi|+|\partial\phi||\overline{\partial}\psi|,$

it implies that for $i=0,1$

$|\partial^{i}Z^{\leq N}{\bf Q}_{\alpha\beta}(\partial h^{1},\partial
h^{1})|\lesssim\sum_{\begin{subarray}{c}|K_{1}|+|K_{2}|\leq N\\\
|I_{1}|+|I_{2}|=i\end{subarray}}|\overline{\partial}\partial^{I_{1}}Z^{K_{1}}h^{1}||\partial\partial^{I_{2}}Z^{K_{2}}h^{1}|.$

We observe that at least one of the two indexes in the above summation has
length smaller than $\lfloor N/2\rfloor$. Therefore, if $N$ is sufficiently
large (e.g. $N\geq 6$) so that $\lfloor N/2\rfloor\leq N-3$ we deduce the
following:

\- from (3.13) and (3.5)

$\sum_{\begin{subarray}{c}|K_{1}|+|K_{2}|\leq N\\\ |K_{1}|\leq\lfloor
N/2\rfloor\\\
|I_{1}|+|I_{2}|=i\end{subarray}}\|(2+r-t)^{\frac{1}{2}+i+\kappa}\overline{\partial}\partial^{I_{1}}Z^{K_{1}}h^{1}\
\partial\partial^{I_{2}}Z^{K_{2}}h^{1}\|_{L^{2}(\Sigma^{\text{e}}_{t})}\\\
\lesssim C_{0}\epsilon t^{-1+\sigma}\sqrt{l(t)}\sum_{j\leq
i}\|(2+r-t)^{\frac{(i+j)}{2}}\partial\partial^{j}Z^{\leq
N}h^{1}\|_{L^{2}(\Sigma^{\text{e}}_{t})}\lesssim
C_{0}^{2}\epsilon^{2}t^{-1+2\sigma}\sqrt{l(t)};$

\- from (3.12) and (3.6)

$\sum_{\begin{subarray}{c}|K_{1}|+|K_{2}|\leq N\\\ |K_{2}|\leq\lfloor
N/2\rfloor\end{subarray}}\|(2+r-t)^{\frac{1}{2}+i+\kappa}\overline{\partial}Z^{K_{1}}h^{1}\
\partial\partial^{i}Z^{K_{2}}h^{1}\|_{L^{2}(\Sigma^{\text{e}}_{t})}\\\
\lesssim C_{0}\epsilon t^{-1+\sigma}\|\overline{\partial}Z^{\leq
N}h^{1}\|_{L^{2}(\Sigma^{\text{e}}_{t})}\lesssim
C_{0}^{2}\epsilon^{2}t^{-1+2\sigma}\sqrt{l(t)};$

\- from (3.12) and (3.8)

$\sum_{\begin{subarray}{c}|K_{1}|+|K_{2}|\leq N-1\\\
|K_{2}|\leq\lfloor(N-1)/2\rfloor\end{subarray}}\|(2+r-t)^{\frac{3}{2}+\kappa}\overline{\partial}\partial
Z^{K_{1}}h^{1}\ \partial Z^{K_{2}}h^{1}\|_{L^{2}(\Sigma^{\text{e}}_{t})}\\\
\lesssim C_{0}\epsilon
t^{-1+\sigma}\|(2+r-t)^{\frac{1}{2}}\overline{\partial}\partial Z^{\leq
N-1}h^{1}\|_{L^{2}(\Sigma^{\text{e}}_{t})}\lesssim
C_{0}^{2}\epsilon^{2}t^{-1+2\sigma}\sqrt{l(t)}.$

As concerns the cubic terms, we have that

$|\partial^{i}Z^{\leq N}G_{\alpha\beta}(h^{1})(\partial h^{1},\partial
h^{1})|\lesssim\sum_{\begin{subarray}{c}|K_{1}|+|K_{2}|+|K_{3}|\leq N\\\
|I_{1}|+|I_{2}|+|I_{3}|=i\end{subarray}}|\partial^{I_{1}}Z^{K_{1}}h^{1}||\partial\partial^{I_{2}}Z^{K_{2}}h^{1}||\partial\partial^{I_{3}}Z^{K_{3}}h^{1}|.$

From (3.12), inequality (B.4) with $\beta=2\kappa-1$ and (3.5), we deduce

$\sum_{\begin{subarray}{c}|K_{1}|+|K_{2}|+|K_{3}|\leq N\\\
|K_{2}|+|K_{3}|\leq\lfloor N/2\rfloor\\\
|I_{2}|+|I_{3}|=i\end{subarray}}\|(2+r-t)^{\frac{1}{2}+i+\kappa}Z^{K_{1}}h^{1}\,\partial\partial^{I_{2}}Z^{K_{2}}h^{1}\,\partial\partial^{I_{3}}Z^{K_{3}}h^{1}\|_{L^{2}(\Sigma^{\text{e}}_{t})}\\\
\lesssim
C_{0}^{2}\epsilon^{2}t^{-2+2\sigma}\|(2+r-t)^{-\frac{3-i}{2}-\kappa}Z^{\leq
N}h^{1}\|_{L^{2}(\Sigma^{\text{e}}_{t})}\\\ \lesssim
C_{0}^{2}\epsilon^{2}t^{-2+2\sigma}\|(2+r-t)^{\frac{1}{2}+\kappa}\partial
Z^{\leq N}h^{1}\|_{L^{2}(\Sigma^{\text{e}}_{t})}\lesssim
C_{0}^{3}\epsilon^{3}t^{-2+3\sigma}.$

The other interactions are easier to treat and their estimates are obtained
similarly to what has been done above for the quadratic terms. We leave the
details to the reader. ∎

An immediate consequence of the pointwise bounds (3.12)-(3.14) is the
following:

######  Proposition 3.5.

Under the assumptions (3.1)-(3.2) we have that

(3.22) $\displaystyle|Z^{\leq N-3}\mathbf{Q}_{\alpha\beta}(\partial
h^{1},\partial h^{1})(t)|_{\frac{1}{2}+2\kappa}\lesssim
C_{0}^{2}\epsilon^{2}t^{-1+2\sigma}\sqrt{l(t)},$ (3.23) $\displaystyle|Z^{\leq
N-3}G_{\alpha\beta}(h^{1})(\partial h^{1},\partial
h^{1})(t)|_{1+3\kappa}\lesssim C_{0}^{3}\epsilon^{3}t^{-2+3\sigma}.$

###  3.3. The commutator terms

The goal of this section is to get suitable weighted
$L^{2}(\Sigma^{\text{e}}_{t})$ estimates of the commutator terms
$[Z^{K},H^{\mu\nu}\partial_{\mu}\partial_{\nu}]h^{1}_{\alpha\beta}$ for
$|K|\leq N$. Such terms have a remarkable property when written in the null
frame, which was first highlighted in [40]. We present below a slightly
different version of this, which involves expanding first in the null frame
before evaluating the commutators.

######  Lemma 3.6.

Let $K$ be any fixed multi-index and assume that $\pi^{\mu\nu}$ is a tensor
satisfying

$|Z^{K^{\prime}}\pi|\leq C,\quad\forall\ |K^{\prime}|\leq\lfloor|K|/2\rfloor.$

Then for any smooth function $\phi$

(3.24)
$\displaystyle|[Z^{K},\pi^{\mu\nu}\partial_{\mu}\partial_{\nu}]\phi|\lesssim\sum_{\begin{subarray}{c}|K_{1}|+|K_{2}|\leq|K|\\\
|K_{2}|<|K|\end{subarray}}|Z^{K_{1}}\pi|_{\mathscr{L}\mathscr{L}}|\partial^{2}Z^{K_{2}}\phi|+|Z^{K_{1}}\pi||\overline{\partial}\partial
Z^{K_{2}}\phi|$
$\displaystyle+\sum_{|K_{1}|+|K_{2}|\leq|K|}r^{-1}|Z^{K_{1}}\pi||\partial
Z^{K_{2}}\phi|$

and

(3.25)
$\displaystyle\Big{|}[Z^{K},\pi^{\mu\nu}\partial_{\mu}\partial_{\nu}]\phi-\sum_{\begin{subarray}{c}|K_{1}|+|K_{2}|\leq|K|\\\
|K_{2}|<|K|\end{subarray}}\big{(}Z^{K_{1}}\pi_{LL}\cdot\partial^{2}_{t}Z^{K_{2}}\phi+Z^{K_{1}}\pi_{4L}\cdot\partial_{y}\partial_{t}Z^{K_{2}}\phi+Z^{K_{1}}\pi_{44}\cdot\partial^{2}_{y}Z^{K_{2}}\phi\big{)}\Big{|}$
$\displaystyle\lesssim\sum_{\begin{subarray}{c}|K_{1}|+|K_{2}|\leq|K|\\\
|K_{2}|<|K|\end{subarray}}\frac{|t^{2}-r^{2}|}{t^{2}}|Z^{K_{1}}\pi||\partial^{2}Z^{K_{2}}\phi|+|Z^{K_{1}}\pi||\partial\underline{\partial}_{x}Z^{K_{2}}\phi|+\sum_{|K_{1}|+|K_{2}|\leq|K|}\frac{|Z^{K_{1}}\pi||\partial
Z^{K_{2}}\phi|}{1+t+r}.$

###### Proof.

Let $U,V$ denote any vector field in $\mathscr{U}$. Inequality (3.24) follows
from the following decomposition

(3.26)
$\displaystyle[Z^{K},\pi^{\mu\nu}\partial_{\mu}\partial_{\nu}]\phi=[Z^{K},\pi^{UV}UV]\phi=\sum_{\begin{subarray}{c}|K_{1}|+|K_{2}|\leq|K|\\\
|K_{2}|<|K|\end{subarray}}(Z^{K_{1}}\pi^{UV})UVZ^{K_{2}}\phi+Z^{K_{1}}\pi^{UV}\
[Z^{K_{2}},UV]\phi$

and the fact that

(3.27)
$|[Z^{J},TU]\phi|\lesssim\sum_{|J^{\prime}|<|J|}|\overline{\partial}\partial
Z^{J^{\prime}}\phi|+\sum_{|J^{\prime\prime}|\leq|J|}r^{-1}|\partial
Z^{J^{\prime\prime}}\phi|.$

Using instead (1.19) we find that101010 We recall that
$\underline{\partial}_{0}=\partial_{t}$

$\displaystyle[Z^{K},\pi^{\mu\nu}\partial_{\mu}\partial_{\nu}]\phi=[Z^{K},\pi^{{\bm{\mu}}{\bm{\nu}}}\partial_{\bm{\mu}}\partial_{\bm{\nu}}]\phi+[Z^{K},\pi^{4\nu}\partial_{4}\partial_{\nu}]\phi+[Z^{K},\pi^{\nu
4}\partial_{\nu}\partial_{4}]\phi$
$\displaystyle=[Z^{K},\pi^{UV}c^{00}_{UV}\partial^{2}_{t}]\phi+[Z^{K},\pi^{UV}(c_{UV}^{{\bm{a}}{\bm{\beta}}}\underline{\partial}_{\bm{a}}\underline{\partial}_{\bm{\beta}}+c_{UV}^{{\bm{\alpha}}{\bm{b}}}\underline{\partial}_{\bm{\alpha}}\underline{\partial}_{\bm{b}}+d_{UV}^{\mu}\underline{\partial}_{\mu})]\phi$
$\displaystyle\quad+[Z^{K},\pi^{4U}\partial_{y}U]\phi+[Z^{K},\pi^{U4}U\partial_{4}]\phi$

where, thanks to the fact that
$|\partial^{I}\Gamma^{J}c^{00}_{LL}|\lesssim_{I,J}(t^{2}-r^{2})/t^{2}$ and
$\partial_{\bm{i}}=\underline{\partial}_{\bm{i}}-\frac{x_{\bm{i}}}{t}\partial_{t}$,
and up to homogeneous zero-order coefficients, we schematically have the
following equalities

$[Z^{K},\pi^{UV}c^{00}_{UV}\partial^{2}_{t}]\phi=\sum_{\begin{subarray}{c}|K_{1}|+|K_{2}|\leq|K|\\\
|K_{2}|<|K|\end{subarray}}Z^{K_{1}}\pi_{LL}\cdot\partial^{2}_{t}Z^{K_{2}}\phi+\frac{|t^{2}-r^{2}|}{t^{2}}Z^{K_{1}}\pi\cdot\partial^{2}Z^{K_{2}}\phi+Z^{K_{1}}\pi\cdot\partial\underline{\partial}_{x}Z^{K_{2}}\phi,$
$[Z^{K},\pi^{UV}(c_{UV}^{{\bm{a}}{\bm{\beta}}}\underline{\partial}_{\bm{a}}\underline{\partial}_{\bm{\beta}}+c_{UV}^{{\bm{\alpha}}{\bm{b}}}\underline{\partial}_{\bm{\alpha}}\underline{\partial}_{\bm{b}}+d_{UV}^{\mu}\underline{\partial}_{\mu})]\phi=\sum_{\begin{subarray}{c}|K_{1}|+|K_{2}|\leq|K|\\\
|K_{2}|<|K|\end{subarray}}Z^{K_{1}}\pi\cdot\partial\underline{\partial}_{x}Z^{K_{2}}\phi+\frac{Z^{K_{1}}\pi\cdot\partial
Z^{K_{2}}\phi}{1+t+r},$
$[Z^{K},\pi^{4U}\partial_{y}U]\phi=\sum_{\begin{subarray}{c}|K_{1}|+|K_{2}|\leq|K|\\\
|K_{2}|<|K|\end{subarray}}Z^{K_{1}}\pi_{4L}\cdot\partial_{y}\partial_{t}Z^{K_{2}}\phi+Z^{K_{1}}\pi_{44}\cdot\partial^{2}_{y}Z^{K_{2}}\phi+Z^{K_{1}}\pi\cdot\partial_{y}\underline{\partial}_{x}Z^{K_{2}}\phi.$

∎

######  Proposition 3.7.

Under the energy assumptions (3.1)-(3.2) we have for $i=0,1$

(3.28) $\displaystyle\left\|(2+r-t)^{\frac{1}{2}+i+\kappa}[\partial^{i}Z^{\leq
N},H^{\mu\nu}\partial_{\mu}\partial_{\nu}]h^{1}_{\alpha\beta}\right\|_{L^{2}(\Sigma^{\text{e}}_{t})}\lesssim\epsilon
t^{-1}E^{\text{e},i+\kappa}(t,\partial^{i}Z^{\leq
N-i}h^{1}_{\alpha\beta})^{1/2}$
$\displaystyle+\epsilon^{2}t^{-(\kappa-\rho)+2\sigma}\big{(}t^{-1/2}\sqrt{l(t)}+t^{-1}\big{)}.$

###### Proof.

We set $\phi=h^{1}_{\alpha\beta}$ in (3.24) and begin by observing that, for
every $K$ with $|K|\leq N$, the terms in the last line of the right hand side
have already been estimated. In fact, the cubic terms satisfy (3.21) and the
following bound was obtained in the proof of proposition 3.14

$\sum_{\begin{subarray}{c}i=0,1\\\ |K_{1}|+|K_{2}|\leq N\\\
|I_{1}|+|I_{2}|=i\end{subarray}}\left\|(2+r-t)^{\frac{1}{2}+i+\kappa}\,r^{-1}\partial^{I_{1}}Z^{K_{1}}h\cdot\partial\partial^{I_{2}}Z^{K_{2}}h\right\|_{L^{2}(\Sigma^{\text{e}}_{t})}\lesssim\epsilon^{2}t^{-\frac{3}{2}+2\sigma}.$

Using (3.11) and (3.7), (3.8), it is straightforward to prove that for $i=0,1$

$\displaystyle\sum_{\begin{subarray}{c}|K_{1}|+|K_{2}|\leq N\\\
|I_{1}|+|I_{2}|=i\end{subarray}}\big{\|}(2+r-t)^{\frac{1}{2}+i+\kappa}\partial^{I_{1}}Z^{K_{1}}h^{0}\cdot\partial^{2}\partial^{I_{2}}Z^{K_{2}}h^{1}\big{\|}_{L^{2}(\Sigma^{\text{e}}_{t})}\lesssim\epsilon
t^{-1}E^{\text{e},i+\kappa}(\partial^{i}Z^{\leq N}h^{1})^{1/2},$

hence we only focus on estimating the terms of the first line in the right
hand side of (3.24) with $h$ replaced by $h^{1}$. We choose exponents
$(p_{1},p_{2})$ such that

$(p_{1},p_{2})=\begin{cases}(2,\infty),\quad&\text{if }|K_{1}|=N\\\
(\infty,2),\quad&\text{if }|K_{2}|=N-1\\\
(4,4),\quad&\text{otherwise}.\end{cases}$

From the Sobolev’s injections
$H^{2}({\mathbb{S}}^{2}\times{\mathbb{S}}^{1})\subset
L^{\infty}({\mathbb{S}}^{2}\times{\mathbb{S}}^{1})$ and
$H^{1}({\mathbb{S}}^{2}\times{\mathbb{S}}^{1})\subset
L^{4}({\mathbb{S}}^{2}\times{\mathbb{S}}^{1})$

$\sum_{\begin{subarray}{c}|K_{1}|+|K_{2}|\leq N\\\
|I_{2}|=i\end{subarray}}\big{\|}(2+r-t)^{\frac{1}{2}+i+\kappa}Z^{K_{1}}h^{1}\cdot\overline{\partial}\partial\partial^{I_{2}}Z^{K_{2}}h^{1}\big{\|}_{L^{2}(\Sigma^{\text{e}}_{t})}^{2}\\\\[-5.0pt]
\leq\sum_{\begin{subarray}{c}|K_{1}|+|K_{2}|\leq N\\\
|I_{2}|=i\end{subarray}}\int_{r\geq
t-1}(2+r-t)^{1+2i+2\kappa}\|Z^{K_{1}}h^{1}\|^{2}_{L^{p_{1}}({\mathbb{S}}^{2}\times{\mathbb{S}}^{1})}\|\overline{\partial}\partial\partial^{I_{2}}Z^{K_{2}}h^{1}\|^{2}_{L^{p_{2}}({\mathbb{S}}^{2}\times{\mathbb{S}}^{1})}r^{2}dr\\\
\lesssim\int_{r\geq t-1}(2+r-t)^{1+2i+2\kappa}\|Z^{\leq
N}h^{1}\|^{2}_{L^{2}({\mathbb{S}}^{2}\times{\mathbb{S}}^{1})}\|\overline{\partial}\partial
Z^{\leq
N-1}h^{1}\|^{2}_{L^{2}({\mathbb{S}}^{2}\times{\mathbb{S}}^{1})}r^{2}dr$

so using the inequality (B.6) with $\beta=\kappa$ and the energy assumptions
(3.5), (3.6) we get

$\displaystyle\lesssim\left\|(2+r-t)^{\frac{1}{2}+\kappa}\partial Z^{\leq
N}h^{1}\right\|^{2}_{L^{2}(\Sigma^{\text{e}}_{t})}\int_{r\geq
t-1}(2+r-t)^{1+2i}r^{-2}\|\overline{\partial}\partial Z^{\leq
N-1}h^{1}\|^{2}_{L^{2}({\mathbb{S}}^{2}\times{\mathbb{S}}^{1})}r^{2}dr$
$\displaystyle\lesssim
t^{-1-2\kappa}\left\|(2+r-t)^{\frac{1}{2}+\kappa}\partial Z^{\leq
N}h^{1}\right\|^{2}_{L^{2}(\Sigma^{\text{e}}_{t})}\left\|(2+r-t)^{1+\kappa}\overline{\partial}\partial
Z^{\leq
N-1}h^{1}\right\|^{2}_{L^{2}(\Sigma^{\text{e}}_{t})}\lesssim\epsilon^{4}t^{-1-2\kappa+4\sigma}l(t).$

The same Sobolev’s embeddings, coupled with the decay bound (3.18) and the
energy assumption (3.5), also yield for $i=0,1$

$\displaystyle\sum_{\begin{subarray}{c}|K_{1}|+|K_{2}|\leq N\\\
|I_{2}|=i\end{subarray}}\big{\|}(2+r-t)^{\frac{1}{2}+i+\kappa}Z^{K_{1}}h^{1}_{LL}\cdot\partial^{2}\partial^{I_{2}}Z^{K_{2}}h^{1}\big{\|}_{L^{2}(\Sigma^{\text{e}}_{t})}^{2}$
$\displaystyle\lesssim\int_{r\geq t-1}(2+r-t)^{1+2i+2\kappa}\|Z^{\leq
N}h^{1}_{LL}\|^{2}_{L^{2}({\mathbb{S}}^{2}\times{\mathbb{S}}^{1})}\|\partial\partial
Z^{\leq
N-1}h^{1}\|^{2}_{L^{2}({\mathbb{S}}^{2}\times{\mathbb{S}}^{1})}r^{2}dr$
$\displaystyle\lesssim\epsilon^{2}t^{-2(\kappa-\rho)+2\sigma}(t^{-1}l(t)+t^{-2})\left\|(2+r-t)^{\frac{1}{2}+i+\kappa}\partial\partial
Z^{\leq N-1}h^{1}\right\|_{L^{2}(\Sigma^{\text{e}}_{t})}^{2}$
$\displaystyle\lesssim\epsilon^{4}t^{-2(\kappa-\rho)+4\sigma}(t^{-1}l(t)+t^{-2}).$

Finally, when $i=1$ the remaining terms to discuss are of the form $\partial
Z^{K_{1}}h^{1}_{LL}\cdot\partial^{2}Z^{K_{2}}h^{1}$ and $\partial
Z^{K_{1}}h^{1}\cdot\overline{\partial}\partial Z^{K_{2}}h^{1}$ but those
behave like null terms (the former thanks to (2.9) for $h^{1}$) and hence
satisfy (3.20). The details are left to the reader.

∎

We also have the following pointwise estimate of the commutator terms
involving a smaller number of vector fields. It will be useful in the proof of
Lemma 4.12.

######  Lemma 3.8.

Under the energy assumptions (3.1)-(3.2), there exists $\delta^{\prime}>0$
such that

(3.29) $\big{|}[Z^{\leq
N-4},H^{1,\mu\nu}\partial_{\mu}\partial_{\nu}]h^{1}_{\alpha\beta}\big{|}_{\kappa}\lesssim
C_{0}^{2}\epsilon^{2}t^{-2+2\sigma}\sqrt{l(t)}+\epsilon^{2}t^{-2-\delta^{\prime}}(2+r-t)^{-\frac{1}{2}}.$

###### Proof.

We use (3.24) with $\pi=H^{1}$ and $\phi=h^{1}_{\alpha\beta}$. Pointwise
bounds (3.12) and (3.17) yield

$\sum_{|K_{1}|+|K_{2}|\leq
N-4}\big{|}Z^{K_{1}}H^{1}_{LL}\cdot\partial^{2}Z^{K_{2}}h^{1}_{\alpha\beta}\big{|}\lesssim
C_{0}^{2}\epsilon^{2}t^{-2-\delta+\sigma}(2+r-t)^{-\frac{3}{2}-\kappa}$

for some $\delta>\sigma$, bounds (3.13) and (3.14) give

$\sum_{|K_{1}|+|K_{2}|\leq
N-4}\big{|}Z^{K_{1}}H^{1}\cdot\partial\overline{\partial}Z^{K_{2}}h^{1}_{\alpha\beta}\big{|}\lesssim
C_{0}^{2}\epsilon^{2}t^{-2+2\sigma}\sqrt{l(t)}(2+r-t)^{-1-2\kappa}$

and finally (3.12) and (3.14) imply

$\sum_{|K_{1}|+|K_{2}|\leq N-4}r^{-1}\big{|}Z^{K_{1}}H^{1}\cdot\partial
Z^{K_{2}}h^{1}_{\alpha\beta}\big{|}\lesssim
C_{0}^{2}\epsilon^{2}t^{-3+2\sigma}(2+r-t)^{-1-2\kappa}.$

The result of the statement follows by setting
$\delta^{\prime}=\delta-\sigma$. ∎

###  3.4. The $h^{1}_{TU}$ coefficients

In this subsection we show that, for any $T\in\mathscr{T}$ and
$U\in\mathscr{U}$, the coefficients $h^{1}_{TU}$ satisfy better energy bounds
than (3.1), more precisely that for any fixed $0<\rho<\kappa$ there exists
some positive constant $C$ such that

(3.30) $E^{\text{e},\kappa-\rho}(t,Z^{\leq N}h^{1}_{TU})^{1/2}\lesssim
C_{0}\epsilon t^{C\epsilon},\qquad t\in[2,T_{0}).$

This estimate essentially follows from the fact that no weak null terms appear
among the source terms in the equation satisfied by $h^{1}_{TU}$. This can be
simply seen by applying $T^{\alpha}U^{\beta}$ to (1.8) and then commuting with
$Z^{K}$, which shows that $Z^{K}h^{1}_{TU}$ is solution to

(3.31) $\tilde{\Box}_{g}Z^{K}h^{1}_{TU}=F^{K}_{TU}+F^{0,K}_{TU},\qquad
F^{0,K}_{TU}=F^{0,K}_{\alpha\beta}T^{\alpha}U^{\beta}$

with source term $F^{K}_{TU}$ given by

(3.32) $\displaystyle F^{K}_{TU}$
$\displaystyle=-[Z^{K},H^{\mu\nu}\partial_{\mu}\partial_{\nu}]h^{1}_{TU}+Z^{K}F_{TU}(h)(\partial
h,\partial h)$
$\displaystyle+\sum_{|K^{\prime}|\leq|K|}C^{{\bm{i}}\alpha\beta}_{TU,K^{\prime}}\not{\partial}_{\bm{i}}Z^{K^{\prime}}h^{1}_{\alpha\beta}+D^{\alpha\beta}_{TU,K^{\prime}}Z^{K^{\prime}}h^{1}_{\alpha\beta}$
$\displaystyle+\sum_{|K_{1}|+|K_{2}|\leq|K|}E^{{\bm{i}}\alpha\beta}_{TU\mu\nu,K_{1}K_{2}}Z^{K_{1}}H^{\mu\nu}\cdot\not{\partial}_{\bm{i}}Z^{K_{2}}h^{1}_{\alpha\beta}+F^{\alpha\beta}_{TU\mu\nu,K_{1}K_{2}}Z^{K_{1}}H^{\mu\nu}\cdot
Z^{K_{2}}h^{1}_{\alpha\beta}$

and smooth coefficients
$C^{i\alpha\beta}_{TU,K^{\prime}},E^{i\alpha\beta}_{TU\mu\nu,K_{1}K_{2}}=O(r^{-1})$,
$D^{\alpha\beta}_{TU,K^{\prime}},F^{\alpha\beta}_{TU\mu\nu,K_{1}K_{2}}=O(r^{-2})$.
Besides the additional terms arising from the commutation of vector fields $T$
and $U$ with the reduced wave operator, the main difference between the source
terms $F^{K}_{\alpha\beta}$ and $F^{K}_{TU}$ lies in the fact that the latter
is a linear combination of quadratic null terms and cubic terms only, as
$P_{TU}=P_{\alpha\beta}T^{\alpha}U^{\beta}$ and

(3.33)
$|P_{TU}(\phi,\psi)|\lesssim|\overline{\partial}\phi||\partial\psi|+|\partial\psi||\overline{\partial}\psi|.$

We compare (3.31) with equation (A.1). Thanks to the smallness of $H$ provided
by (3.11) and (3.17), we can apply the result of proposition A.1 with
$\mathbf{W}=Z^{K}h^{1}_{TU}$, $\mathbf{F}=F^{K}_{TU}+F^{0,K}_{TU}$,
$w(q)=(2+r-t)^{1+2(\kappa-\rho)}$ and $t_{1}=2,t_{2}=t$. We obtain that

(3.34) $\displaystyle E^{\text{e},\kappa-\rho}(t,Z^{K}h^{1}_{TU})$
$\displaystyle+\int_{\tilde{\mathscr{H}}_{2,t}}(2+r-t)^{1+2(\kappa-\rho)}\Big{[}\Big{(}\frac{1}{2(1+r^{2})}+\chi\left(\frac{r}{t}\right)\chi(r)\frac{M}{2r}\Big{)}|\partial_{t}Z^{K}h^{1}_{TU}|^{2}+|\underline{\nabla}Z^{K}h^{1}_{TU}|^{2}\Big{]}dxdy$
$\displaystyle+\iint_{\mathscr{D}^{\text{e}}_{[2,t]}}(2+r-\tau)^{2(\kappa-\rho)}\big{(}|LZ^{K}h^{1}_{TU}|^{2}+|\not{\nabla}Z^{K}h^{1}_{TU}|^{2}\big{)}d\tau
dxdy\lesssim E^{\text{e},\kappa-\rho}(2,Z^{K}h^{1}_{TU})$
$\displaystyle+\iint_{\mathscr{D}^{\text{e}}_{[2,t]}}(2+r-\tau)^{1+2(\kappa-\rho)}|(F^{K}_{TU}+F^{0,K}_{TU}+{\partial^{\alpha}H_{\alpha}}^{\sigma}\,\partial_{\sigma}Z^{K}h^{1}_{TU})\partial_{t}Z^{K}h^{1}_{TU}|d\tau
dxdy$
$\displaystyle+\iint_{\mathscr{D}^{\text{e}}_{[2,t]}}(2+r-\tau)^{1+2(\kappa-\rho)}|{\partial_{t}H_{\alpha}}^{\sigma}\,\partial_{\sigma}Z^{K}h^{1}_{TU}\,\partial^{\alpha}Z^{K}h^{1}_{TU}|\,d\tau
dxdy$
$\displaystyle+\iint_{\mathscr{D}^{\text{e}}_{[2,t]}}(2+r-\tau)^{2(\kappa-\rho)}|H^{\rho\sigma}\partial_{\rho}Z^{K}h^{1}_{TU}\partial_{\sigma}Z^{K}h^{1}_{TU}|\
d\tau dxdy$
$\displaystyle+\iint_{\mathscr{D}^{\text{e}}_{[2,t]}}(2+r-\tau)^{2(\kappa-\rho)}|(-H^{0\sigma}+\omega_{\bm{j}}H^{{\bm{j}}\sigma})\partial_{\sigma}Z^{K}h^{1}_{TU}\,\partial_{t}Z^{K}h^{1}_{TU}|\
d\tau dxdy.$

We start by estimating the contributions coming from the source term
$F^{0,K}_{TU}$.

######  Lemma 3.9.

There exists $\delta>0$ such that, under the energy assumptions (3.1)-(3.2)
and for $i=0,1$, we have

(3.35) $\|(2+r-t)^{\frac{1}{2}+i+\kappa}\partial^{i}Z^{\leq
N}\tilde{\Box}_{g}h^{0}_{\alpha\beta}\|_{L^{2}(\Sigma^{\text{e}}_{t})}\lesssim\epsilon
t^{-3-i}+\epsilon^{2}t^{-2+\delta}.$

Consequently

$\|(2+r-t)^{\frac{1}{2}+\kappa-\rho}Z^{\leq
N}F^{0}_{TU}\|_{L^{2}(\Sigma^{\text{e}}_{t})}\lesssim\epsilon
t^{-3}+\epsilon^{2}t^{-2+\delta}.$

###### Proof.

We recall that definition (1.7) of $h^{0}$ and that
$\tilde{\Box}_{g}h^{0}_{\alpha\beta}=F^{00}+F^{01}+F^{02}$ with

(3.36) $F^{00}=\Box h^{0}_{\alpha\beta},\qquad
F^{01}=-h^{0,\mu\nu}\partial_{\mu}\partial_{\nu}h^{0}_{\alpha\beta},\qquad
F^{02}=(-h^{1,\mu\nu}+\mathscr{O}^{\mu\nu}(h^{2}))\partial_{\mu}\partial_{\nu}h^{0}_{\alpha\beta}.$

The fundamental remark is that $\Box(M/r)=0$ away from $r=0$, which implies
that $F^{00}$ is supported for $t/2\leq r\leq 3t/4$. Consequently,
$\text{supp}F^{00}\cap\mathscr{D}^{\text{e}}_{T_{0}}$ is bounded and from
(3.11)

$\begin{gathered}|\partial^{i}Z^{\leq
N}F^{00}|\lesssim\frac{\epsilon}{(1+r)^{3+i}},\ \qquad\
\|(2+r-t)^{\frac{1}{2}+i+\kappa}\partial^{i}Z^{\leq
N}F^{00}\|_{L^{2}(\Sigma^{\text{e}}_{t})}\lesssim\epsilon
t^{-3-i}.\end{gathered}$

From (3.11) we also see that

$\begin{gathered}|\partial^{i}Z^{\leq
N}F^{01}|\lesssim\frac{\epsilon^{2}}{(1+r)^{4+i}}\\\
\|(2+r-t)^{\frac{1}{2}+i+\kappa}\partial^{i}Z^{\leq
N}F^{01}\|_{L^{2}(\Sigma^{\text{e}}_{t})}\lesssim\epsilon^{2}\|r^{-\frac{7}{2}+\kappa}\|_{L^{2}(\Sigma^{\text{e}}_{t})}\lesssim\epsilon^{2}t^{-2+\kappa}.\end{gathered}$

From (3.11), the Hardy inequality (B.4) with $\beta=2\kappa-1$ and estimate
(3.5), we have that

$\displaystyle\sum_{\begin{subarray}{c}|K_{1}|+|K_{2}|\leq N\\\
|I_{2}|=i\end{subarray}}\|(2+r-t)^{\frac{1}{2}+i+\kappa}Z^{K_{1}}h^{1,\mu\nu}\partial_{\mu}\partial_{\nu}\partial^{I_{2}}Z^{K_{2}}h^{0}_{\alpha\beta}\|_{L^{2}(\Sigma^{\text{e}}_{t})}$
$\displaystyle\lesssim\epsilon\|(2+r-t)^{\frac{1}{2}+i+\kappa}r^{-3-i}Z^{K_{1}}h^{1,\mu\nu}\|_{L^{2}(\Sigma^{\text{e}}_{t})}\lesssim\epsilon
t^{-2}\|(2+r-t)^{\frac{1}{2}+\kappa}\partial
Z^{K_{1}}h^{1,\mu\nu}\|_{L^{2}(\Sigma^{\text{e}}_{t})}\lesssim\epsilon^{2}t^{-2+\sigma}.$

The cubic terms
$\mathscr{O}^{\mu\nu}(h^{2})\partial_{\mu}\partial_{\nu}h^{0}_{\alpha\beta}$
verify similar estimates, the details are left to the reader. ∎

In order to estimate the contributions due to the curved background, we first
highlight the following relations.

######  Lemma 3.10.

For any sufficiently smooth function $\phi$ we have

(3.37)
$\displaystyle{\partial^{\alpha}H_{\alpha}}^{\sigma}\,\partial_{\sigma}\phi=-\frac{1}{2}(\partial_{{\underline{u}}}H_{LL}-\partial_{u}H_{\underline{L}L}+\partial_{A}H_{AL})\underline{L}\phi+({\partial^{\alpha}H_{\alpha}}^{T})T\phi$
(3.38)
$\displaystyle{\partial_{t}H_{\alpha}}^{\sigma}\,\partial_{\sigma}\phi\,\partial^{\alpha}\phi=\frac{1}{4}\partial_{t}H_{LL}(\underline{L}\phi)^{2}+\partial_{t}H^{T\alpha}\,(\partial_{\alpha}\phi)(T\phi)$
(3.39) $\displaystyle
H^{\rho\sigma}\partial_{\rho}\phi\,\partial_{\sigma}\phi=\frac{1}{4}H_{LL}(\underline{L}\phi)^{2}+H^{T\alpha}(T\phi)(\partial_{\alpha}\phi)$
(3.40)
$\displaystyle(-H^{0\sigma}+\omega_{\bm{j}}H^{{\bm{j}}\sigma})\partial_{\sigma}\phi=-\frac{1}{2}H_{LL}\,\underline{L}\phi+{H_{L}}^{T}(T\phi)$

###### Proof.

The proof of the above equalities follows after expressing all vector fields
relative to the null frame
$\mathscr{U}=\\{L,\underline{L},S^{1},S^{2},\partial_{y}\\}$ and observing
that

$-H^{0\sigma}+\omega_{\bm{j}}H^{{\bm{j}}\sigma}=\bar{g}_{\mu\nu}L^{\mu}H^{\nu\sigma}={H_{L}}^{\sigma}.$

∎

######  Lemma 3.11.

Under the a-priori assumptions (3.1)-(3.2) we have that for $i=0,1$ and any
multi-index $K$ with $|K|\leq N$

(3.41)
$\displaystyle\iint_{\mathscr{D}^{\text{e}}_{[2,t]}}(2+r-\tau)^{1+2(i+\kappa)}|\partial^{\mu}{H_{\mu}}^{\sigma}\cdot\partial_{\sigma}\partial^{i}Z^{K}h^{1}_{\alpha\beta}\cdot\partial_{t}\partial^{i}Z^{K}h^{1}_{\alpha\beta}|\,d\tau
dxdy$
$\displaystyle\iint_{\mathscr{D}^{\text{e}}_{[2,t]}}(2+r-\tau)^{1+2(i+\kappa)}|\partial_{t}{H_{\mu}}^{\sigma}\cdot\partial_{\sigma}\partial^{i}Z^{K}h^{1}_{\alpha\beta}\cdot\partial^{\mu}\partial^{i}Z^{K}h^{1}_{\alpha\beta}|\,d\tau
dxdy\lesssim C_{0}^{3}\epsilon^{3}$

and

(3.42)
$\displaystyle\iint_{\mathscr{D}^{\text{e}}_{[2,t]}}(2+r-\tau)^{2(i+\kappa)}|H^{\rho\sigma}\partial_{\rho}\cdot\partial^{i}Z^{K}h^{1}_{\alpha\beta}\cdot\partial_{\sigma}\partial^{i}Z^{K}h^{1}_{\alpha\beta}|\,d\tau
dxdy$
$\displaystyle+\iint_{\mathscr{D}^{\text{e}}_{[2,t]}}(2+r-\tau)^{2(i+\kappa)}|(-H^{0\sigma}+\omega_{\bm{j}}H^{{\bm{j}}\sigma})\cdot\partial_{\sigma}\partial^{i}Z^{K}h^{1}_{\alpha\beta}\cdot\partial_{t}\partial^{i}Z^{K}h^{1}_{\alpha\beta}|d\tau
dxdy$ $\displaystyle\lesssim\int_{2}^{t}\frac{\epsilon
E^{\text{e},i+\kappa}(\tau,\partial^{i}Z^{K}h^{1}_{\alpha\beta})}{\tau}\,d\tau+C_{0}^{3}\epsilon^{3}.$

For any $0\leq\rho<\kappa$, the same inequalities hold with
$(\kappa,h^{1}_{\alpha\beta})$ replaced by $(\kappa-\rho,h^{1}_{TU})$.

###### Proof.

We start by remarking that from inequality (2.2) and bounds (3.11) to (3.14)

$|\partial
H_{LL}|+|\overline{\partial}H|\lesssim|\overline{\partial}h^{1}|+|\partial
h^{0}|+|h||\partial h|\lesssim
C_{0}\epsilon\Big{(}\frac{t^{\sigma}\sqrt{l(t)}}{r(2+r-t)^{\kappa}}+\frac{t^{2\sigma}}{r^{2}}\Big{)},$

which together with the energy bounds (3.5) and (3.7) implies

$\|(2+r-t)^{\frac{1}{2}+i+\kappa}(|\partial
H_{LL}|+|\overline{\partial}H|)\partial\partial^{i}Z^{\leq
N}h^{1}\|_{L^{2}(\Sigma^{\text{e}}_{t})}\lesssim
C_{0}^{2}\epsilon^{2}t^{-1+2\sigma}\sqrt{l(t)}+C_{0}^{2}\epsilon^{2}t^{-2+3\sigma}.$

From the pointwise estimates (3.11), (3.12) and energy bounds (3.6), (3.8) we
also have

$\|(2+r-t)^{\frac{1}{2}+i+\kappa}\partial
H\cdot\overline{\partial}\partial^{i}Z^{\leq
N}h^{1}\|_{L^{2}(\Sigma^{\text{e}}_{t})}\lesssim
C_{0}^{2}\epsilon^{2}t^{-1+2\sigma}\sqrt{l(t)}.$

The Cauchy-Schwartz inequality, relation (3.37) and the above estimates yield
that for $t\in[2,T_{0})$

$\displaystyle\iint_{\mathscr{D}^{\text{e}}_{[2,t]}}(2+r-\tau)^{1+2(i+\kappa)}\partial^{\mu}{H_{\mu}}^{\sigma}\cdot\partial_{\sigma}\partial^{i}Z^{K}h^{1}_{\alpha\beta}\cdot\partial_{t}\partial^{i}Z^{K}h^{1}_{\alpha\beta}|\,d\tau
dxdy$
$\displaystyle\lesssim\int_{2}^{t}\|(2+r-\tau)^{\frac{1}{2}+i+\kappa}\partial^{\mu}{H_{\mu}}^{\sigma}\cdot\partial_{\sigma}\partial^{i}Z^{K}h^{1}_{\alpha\beta}\|_{L^{2}(\Sigma^{\text{e}}_{t})}E^{\text{e},i+\kappa}(\tau,\partial^{i}Z^{K}h^{1}_{\alpha\beta})^{1/2}d\tau\lesssim
C_{0}^{3}\epsilon^{3}.$

Similarly, using relation (3.38) we get

$\displaystyle\iint_{\mathscr{D}^{\text{e}}_{[2,t]}}(2+r-\tau)^{1+2(i+\kappa)}|\partial_{t}{H_{\mu}}^{\sigma}\cdot\partial_{\sigma}\partial^{i}Z^{K}h^{1}_{\alpha\beta}\cdot\partial^{\mu}\partial^{i}Z^{K}h^{1}_{\alpha\beta}|\,d\tau
dxdy$
$\displaystyle\lesssim\iint_{\mathscr{D}^{\text{e}}_{[2,t]}}\|(2+r-\tau)^{\frac{1}{2}+i+\kappa}\partial
H_{LL}\cdot\partial\partial^{i}Z^{K}h^{1}_{\alpha\beta}\|_{L^{2}(\Sigma^{\text{e}}_{t})}E^{\text{e},i+\kappa}(\tau,\partial^{i}Z^{K}h^{1}_{\alpha\beta})^{1/2}d\tau
dxdy$
$\displaystyle+\iint_{\mathscr{D}^{\text{e}}_{[2,t]}}\|(2+r-\tau)^{\frac{1}{2}+i+\kappa}\partial
H\cdot\overline{\partial}\partial^{i}Z^{K}h^{1}_{\alpha\beta}\|_{L^{2}(\Sigma^{\text{e}}_{t})}E^{\text{e},i+\kappa}(\tau,\partial^{i}Z^{K}h^{1}_{\alpha\beta})^{1/2}d\tau
dxdy\lesssim C_{0}^{3}\epsilon^{3}.$

Finally, from formulas (3.39) and (3.40) and pointwise bounds (3.11), (3.14)
and (3.18)

$\displaystyle\iint_{\mathscr{D}^{\text{e}}_{[2,t]}}(2+r-\tau)^{2(i+\kappa)}|H^{\rho\sigma}\cdot\partial_{\rho}\partial^{i}Z^{K}h^{1}_{\alpha\beta}\cdot\partial_{\sigma}\partial^{i}Z^{K}h^{1}_{\alpha\beta}|\,d\tau
dxdy$
$\displaystyle+\iint_{\mathscr{D}^{\text{e}}_{[2,t]}}(2+r-\tau)^{2(i+\kappa)}|(-H^{0\sigma}+\omega_{\bm{j}}H^{{\bm{j}}\sigma})\cdot\partial_{\sigma}\partial^{i}Z^{K}h^{1}_{\alpha\beta}\cdot\partial_{t}\partial^{i}Z^{K}h^{1}_{\alpha\beta}|d\tau
dxdy$
$\displaystyle\lesssim\iint_{\mathscr{D}^{\text{e}}_{[2,t]}}(2+r-t)^{2(i+\kappa)}\big{[}|H_{LL}||\partial\partial^{i}Z^{K}h^{1}_{\alpha\beta}|^{2}+|H||\overline{\partial}\partial^{i}Z^{K}h^{1}_{\alpha\beta}||\partial\partial^{i}Z^{K}h^{1}_{\alpha\beta}|\big{]}\,d\tau
dxdy$ $\displaystyle\lesssim\int_{2}^{t}\frac{\epsilon
E^{\text{e},i+\kappa}(\tau,\partial^{i}Z^{K}h^{1}_{\alpha\beta})}{\tau}\,d\tau+C_{0}^{3}\epsilon^{3}.$

∎

######  Proposition 3.12.

Let $0<\rho<\kappa$ be fixed. Under the energy assumptions (3.1)-(3.2) there
exists a constant $C>0$ such that (3.30) holds for all $t\in[2,T_{0})$.

###### Proof.

The result follows from inequality (3.34) and the estimates we have obtained
so far. The quadratic semilinear terms in $P_{TU}$ are now null, as observed
in (3.33), therefore from (3.20) and (3.21) and the smallness of $\epsilon$ it
follows that

$\|(2+r-t)^{\frac{1}{2}+\kappa-\rho}Z^{\leq
N}F_{TU}\|_{L^{2}(\Sigma^{\text{e}}_{t})}\lesssim
C_{0}^{2}\epsilon^{2}t^{-1+2\sigma}\sqrt{l(t)}+C_{0}^{2}\epsilon^{2}t^{-2+3\sigma}.$

The only terms that still need to be addressed are the contributions to (3.32)
arising from the commutation of the null frame with the reduced wave operator.
Using the energy bounds (3.6) we see that

$\displaystyle\|(2+r-t)^{\frac{1}{2}+\kappa-\rho}C^{{\bm{i}}\alpha\beta}_{TU,K^{\prime}}\cdot\not{\partial}_{\bm{i}}Z^{\leq
N}h^{1}_{\alpha\beta}\|_{L^{2}(\Sigma^{\text{e}}_{t})}\lesssim\|(2+r-t)^{\frac{1}{2}+\kappa-\rho}r^{-1}\overline{\nabla}Z^{\leq
N}h^{1}_{\alpha\beta}\|_{L^{2}(\Sigma^{\text{e}}_{t})}$ $\displaystyle\lesssim
t^{-\frac{1}{2}-\rho}\|(2+r-t)^{\kappa}\overline{\nabla}Z^{\leq
N}h^{1}_{\alpha\beta}\|_{L^{2}(\Sigma^{\text{e}}_{t})}\lesssim C_{0}\epsilon
t^{-\frac{1}{2}-\rho+\sigma}\sqrt{l(t)}$

while from (3.5) and the weighted Hardy inequality (B.4) with
$\beta=2\kappa-1$ we get

$\displaystyle\|(2+r-t)^{\frac{1}{2}+\kappa-\rho}D^{\alpha\beta}_{TU,K^{\prime}}\cdot
Z^{\leq
N}h^{1}_{\alpha\beta}\|_{L^{2}(\Sigma^{\text{e}}_{t})}\lesssim\|(2+r-t)^{\frac{1}{2}+\kappa-\rho}r^{-2}Z^{\leq
N}h^{1}_{\alpha\beta}\|_{L^{2}(\Sigma^{\text{e}}_{t})}$ $\displaystyle\lesssim
t^{-1-\rho}\|(2+r-t)^{\kappa-\frac{1}{2}}Z^{\leq
N}h^{1}_{\alpha\beta}\|_{L^{2}(\Sigma^{\text{e}}_{t})}\lesssim
t^{-1-\rho}\|(2+r-t)^{\frac{1}{2}+\kappa}\partial Z^{\leq
N}h^{1}_{\alpha\beta}\|_{L^{2}(\Sigma^{\text{e}}_{t})}$ $\displaystyle\lesssim
C_{0}\epsilon t^{-1-\rho+\sigma}.$

We then recall the decomposition (1.12) of the tensor $H$, with $H^{0,\mu\nu}$
satisfying (3.11) and $H^{1,\mu\nu}$ verifying the bounds (3.12)-(3.14). We
similarly get

$\displaystyle\sum_{\begin{subarray}{c}|K_{1}|+|K_{2}|\leq N\\\
|K_{1}|\leq\lfloor
N/2\rfloor\end{subarray}}\|(2+r-t)^{\frac{1}{2}+\kappa-\rho}E^{{\bm{i}}\alpha\beta}_{TU\mu\nu,K_{1}K_{2}}\cdot
Z^{K_{1}}H^{1,\mu\nu}\cdot\not{\partial}_{\bm{i}}Z^{K_{2}}h^{1}_{\alpha\beta}\|_{L^{2}(\Sigma^{\text{e}}_{t})}$
$\displaystyle+\sum_{|K_{1}|+|K_{2}|\leq|K|}\|(2+r-t)^{\frac{1}{2}+\kappa-\rho}E^{{\bm{i}}\alpha\beta}_{TU\mu\nu,K_{1}K_{2}}\cdot
Z^{K_{1}}H^{0,\mu\nu}\cdot\not{\partial}_{\bm{i}}Z^{K_{2}}h^{1}_{\alpha\beta}\|_{L^{2}(\Sigma^{\text{e}}_{t})}$
$\displaystyle\lesssim C_{0}\epsilon
t^{\sigma}\|(2+r-t)^{\frac{1}{2}+\kappa-\rho}r^{-2}\overline{\nabla}Z^{\leq
N}h^{1}_{\alpha\beta}\|_{L^{2}(\Sigma^{\text{e}}_{t})}\lesssim
C_{0}^{2}\epsilon^{2}t^{-\frac{3}{2}-\rho+2\sigma}\sqrt{l(t)}$

and

$\displaystyle\sum_{\begin{subarray}{c}|K_{1}|+|K_{2}|\leq N\\\
|K_{2}|\leq\lfloor
N/2\rfloor\end{subarray}}\|(2+r-t)^{\frac{1}{2}+\kappa-\rho}E^{{\bm{i}}\alpha\beta}_{TU\mu\nu,K_{1}K_{2}}\cdot
Z^{K_{1}}H^{1,\mu\nu}\cdot\not{\partial}_{\bm{i}}Z^{K_{2}}h^{1}_{\alpha\beta}\|_{L^{2}(\Sigma^{\text{e}}_{t})}$
$\displaystyle\lesssim C_{0}\epsilon
t^{\sigma}\sqrt{l(t)}\|(2+r-t)^{\frac{1}{2}-\rho}r^{-2}Z^{\leq
N}H^{1,\mu\nu}\|_{L^{2}(\Sigma^{\text{e}}_{t})}$ $\displaystyle\lesssim
C_{0}\epsilon
t^{-1-\rho-\kappa+\sigma}\sqrt{l(t)}\|(2+r-t)^{\frac{1}{2}+\kappa}\partial
Z^{\leq N}H^{1,\mu\nu}\|_{L^{2}(\Sigma^{\text{e}}_{t})}\lesssim
C_{0}^{2}\epsilon^{2}t^{-1-\rho-\kappa+2\sigma}\sqrt{l(t)}.$

Finally,

$\displaystyle\sum_{|K_{1}|+|K_{2}|\leq
N}\|(2+r-t)^{\frac{1}{2}+\kappa-\rho}F^{\alpha\beta}_{TU\mu\nu,K_{1}K_{2}}\cdot
Z^{K_{1}}H^{\mu\nu}\cdot
Z^{K_{2}}h^{1}_{\alpha\beta}\|_{L^{2}(\Sigma^{\text{e}}_{t})}$
$\displaystyle\lesssim C_{0}\epsilon
t^{\sigma}\|(2+r-t)^{\frac{1}{2}+\kappa-\rho}r^{-3}Z^{\leq
N}h^{1}\|_{L^{2}(\Sigma^{\text{e}}_{t})}\lesssim
C_{0}^{2}\epsilon^{2}t^{-2-\rho+2\sigma}.$

By substituting the above estimates together with (3.28), (3.35), (3.41) and
(3.42) into (3.34) and choosing $\epsilon_{0}\ll 1$ sufficiently small so that
$C_{0}\epsilon<1$ we finally find the existence of a universal constant $C$
such that

$\displaystyle E^{\text{e},\kappa-\rho}(t,Z^{K}h^{1}_{TU})$ $\displaystyle\leq
CE^{\text{e},\kappa-\rho}(2,Z^{K}h^{1}_{TU})+CC_{0}^{2}\epsilon^{2}+\int_{2}^{t}\frac{C\epsilon
E^{\text{e},\kappa-\rho}(\tau,Z^{K}h^{1}_{TU})}{\tau}\,d\tau.$

Observe that $E^{\text{e},\kappa-\rho}(2,Z^{K}h^{1}_{TU})\lesssim
E^{\text{e},\kappa}(t,h^{1})$. Grönwall’s inequality and the energy assumption
(3.1) allow us to obtain

$E^{\text{e},\kappa-\rho}(t,Z^{K}h^{1}_{TU})\leq
C(E^{\text{e},\kappa}(2,Z^{K}h^{1})+C_{0}^{2}\epsilon^{2})t^{C\epsilon}\leq
2CC_{0}^{2}\epsilon^{2}t^{C\epsilon}$

and hence conclude the proof. ∎

An immediate consequence of (3.30) are the following weighted $L^{2}$ bounds

(3.43) $\displaystyle\big{\|}(2+r-t)^{\frac{1}{2}+\kappa-\rho}\ \partial
Z^{\leq N}h^{1}_{TU}\big{\|}_{L^{2}(\Sigma^{\text{e}}_{t})}\lesssim
C_{0}\epsilon t^{C\epsilon}$ (3.44)
$\displaystyle\big{\|}(2+r-t)^{\kappa-\rho}\ \overline{\partial}Z^{\leq
N}h^{1}_{TU}\big{\|}_{L^{2}(\Sigma^{\text{e}}_{t})}\lesssim C_{0}\epsilon
t^{C\epsilon}\sqrt{l(t)}$

for all $t\in[2,T_{0})$, where $l\in L^{1}([2,T_{0}))$. The weighted Sobolev
injection (B.2) with $\beta=1+2(\kappa-\rho)$ also yields the following
pointwise bound

(3.45) $|\partial Z^{\leq N-3}h^{1}_{TU}|_{\kappa-\rho-\frac{1}{2}}\lesssim
C_{0}\epsilon t^{C\epsilon}.$

###  3.5. The weak null terms

The goal of this subsection is to recover suitable higher order weighted
$L^{2}(\Sigma^{\text{e}}_{t})$ estimates for the quadratic weak null terms
$P_{\alpha\beta}(\partial h,\partial h)$ defined as

$P_{\alpha\beta}(\partial h,\partial
h)=\frac{1}{4}\bar{g}^{\mu\rho}\bar{g}^{\nu\sigma}\left(\partial_{\alpha}h_{\mu\rho}\partial_{\beta}h_{\nu\sigma}-2\partial_{\alpha}h_{\mu\nu}\partial_{\beta}h_{\rho\sigma}\right).$

These estimates are based on the following remarkable property, highlighted in
the works of Lindblad and Rodnianski [38, 39, 40], on Lemma 2.2 and on the
bounds (3.43), (3.45) satisfied by $h^{1}_{TU}$.

######  Lemma 3.13.

Let $\pi,\theta$ be arbitrary 2-tensors and $P$ be the quadratic form defined
by

$P(\pi,\theta)=\frac{1}{4}\bar{g}^{\mu\rho}\bar{g}^{\nu\sigma}\left(\pi_{\mu\rho}\theta_{\nu\sigma}-2\pi_{\mu\nu}\theta_{\rho\sigma}\right).$

Then

$|P(\pi,\theta)|\lesssim|\pi|_{\mathscr{T}\mathscr{U}}|\theta|_{\mathscr{T}\mathscr{U}}+|\pi|_{\mathscr{L}\mathscr{L}}|\theta|+|\pi||\theta|_{\mathscr{L}\mathscr{L}}.$

######  Proposition 3.14.

Fix $i=0,1$. There exists some constant $C>0$ such that, under the a-priori
energy assumptions (3.1)-(3.2), we have

(3.46) $\left\|(2+r-t)^{\frac{1}{2}+i+\kappa}\partial^{i}Z^{\leq
N}P_{\alpha\beta}\right\|_{L^{2}(\Sigma^{\text{e}}_{t})}\lesssim
C_{0}^{2}\epsilon^{2}\big{[}t^{-1+C\epsilon}+t^{-1+2\sigma}\sqrt{l(t)}+\epsilon
t^{-\frac{3}{2}+2\sigma}\big{]}.$

###### Proof.

We write $h=h^{1}+h^{0}$ and plug this decomposition into
$P_{\alpha\beta}(\partial h,\partial h)$. Using (3.11) and the energy bounds
(3.5), (3.7) it is straightforward to prove that there exists some small
$\delta>0$ such that

$\sum_{\begin{subarray}{c}i,j=0,1\\\ |K_{1}|+|K_{2}|\leq N\\\
|I_{1}|+|I_{2}|=i\end{subarray}}\|(2+r-t)^{\frac{1}{2}+i+\kappa}\,\partial^{I_{1}}Z^{K_{1}}h^{0}\cdot\partial\partial^{I_{2}}Z^{K_{2}}h^{j}\|_{L^{2}(\Sigma^{\text{e}}_{t})}\lesssim
C_{0}^{2}\epsilon^{2}t^{-2+\delta}.$

Hence we focus on proving that estimate (3.46) holds true for
$P_{\alpha\beta}(\partial h^{1},\partial h^{1})$.

We start by noticing that for any multi-index $K$,
$Z^{K}P_{\alpha\beta}(\partial h^{1},\partial h^{1})$ is a linear combination
of terms of the form $P_{\mu\nu}(\partial Z^{K_{1}}h^{1},\partial
Z^{K_{2}}h^{1})$ for some multi-indexes $K_{1},K_{2}$ such that
$|K_{1}|+|K_{2}|\leq|K|$ and $\mu,\nu=0,\dots,4$. Applying Lemma 3.13 and
Lemma 2.2 we see that for $i=0,1$

(3.47) $\displaystyle|\partial^{i}Z^{\leq N}P_{\alpha\beta}(\partial
h^{1},\partial h^{1})|\lesssim\sum_{\begin{subarray}{c}|K_{1}|+|K_{2}|\leq
N\\\
|I_{1}|+|I_{2}|=i\end{subarray}}|\partial\partial^{I_{1}}Z^{K_{1}}h^{1}|_{\mathscr{T}\mathscr{U}}|\partial\partial^{I_{2}}Z^{K_{2}}h^{1}|_{\mathscr{T}\mathscr{U}}$
$\displaystyle+\sum_{\begin{subarray}{c}|K_{1}|+|K_{2}|\leq N\\\
|I_{1}|+|I_{2}|=i\end{subarray}}|\overline{\partial}\partial^{I_{1}}Z^{K_{1}}h^{1}||\partial\partial^{I_{2}}Z^{K_{2}}h^{1}|+r^{-1}|\partial^{I_{1}}Z^{K_{1}}h^{1}||\partial\partial^{I_{2}}Z^{K_{2}}h^{1}|$
$\displaystyle+\sum_{\begin{subarray}{c}|K_{1}|+|K_{2}|+|K_{3}|\leq N\\\
|I_{1}|+|I_{2}|+|I_{3}|=i\end{subarray}}|\partial^{I_{1}}Z^{K_{1}}h^{1}||\partial\partial^{I_{2}}Z^{K_{2}}h^{1}||\partial\partial^{I_{3}}Z^{K_{3}}h^{1}|$
$\displaystyle+\frac{M\chi_{0}(t/2\leq r\leq 3t/4)}{(1+t+r)^{2}}\sum_{j\leq
i}|\partial\partial^{j}Z^{\leq N}h^{1}|,$

where $\chi_{0}(t/2\leq r\leq 3t/4)$ is supported for $t/2\leq r\leq 3t/4$.
Since the intersection of this support with the exterior region in bounded, it
is immediate to see that the weighted $L^{2}(\Sigma^{\text{e}}_{t})$ norm of
the last term in the above right hand side is bounded by
$C_{0}\epsilon^{2}t^{-2}$.

The cubic terms and the quadratic terms involving a tangential derivative have
been estimated in proposition 3.4 and satisfy (3.21) and (3.20) respectively.
The weighted $L^{2}$ norm of the quadratic term with the extra $r^{-1}$ factor
is bounded by $C_{0}^{2}\epsilon^{2}t^{-3/2+2\sigma}$, we leave the details to
the reader. Finally, from (3.43) and (3.45) with $\rho>0$ such that $k>2\rho$

$\sum_{\begin{subarray}{c}i=0,1\\\ |K_{1}|+|K_{2}|\leq N\\\
|I_{1}|+|I_{2}|=i\end{subarray}}\left\|(2+r-t)^{\frac{1}{2}+i+\kappa}|\partial\partial^{I_{1}}Z^{K_{1}}h^{1}|_{\mathscr{T}\mathscr{U}}|\partial\partial^{I_{2}}Z^{K_{2}}h^{1}|_{\mathscr{T}\mathscr{U}}\right\|_{L^{2}(\Sigma^{\text{e}}_{t})}\\\
\lesssim C_{0}\epsilon
t^{-1+C\epsilon}\sum_{i=0,1}\left\|(2+r-t)^{i-\frac{1}{2}+\rho}|\partial
Z^{\leq
N}h^{1}|_{\mathscr{T}\mathscr{U}}\right\|_{L^{2}(\Sigma^{\text{e}}_{t})}\lesssim
C_{0}^{2}\epsilon^{2}t^{-1+2C\epsilon}.$

∎