footnote

# Einstein-Klein-Gordon spacetimes
in the harmonic near-Minkowski regime

Philippe G. LeFloch111Laboratoire Jacques-Louis Lions and Centre National de
la Recherche Scientifique, Sorbonne Université, 4 Place Jussieu, 75252 Paris,
France. Email<EMAIL_ADDRESS>
† School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an,
710049 Shaanxi, People’s Republic of China. Email<EMAIL_ADDRESS>Key words and phrases. Einstein equations; nonlinear stability of Minkowski
spacetime; self-gravitating massive field; harmonic decay; near-Minkowski
regime; Euclidean–hyperboloidal foliation method. Version: August 2022.  and
Yue Ma†

###### Abstract

We study the initial value problem for the Einstein-Klein-Gordon system and
establish the global nonlinear stability of massive matter in the near-
Minkowski regime when the initial geometry is a perturbation of an
asymptotically flat, spacelike hypersurface in Minkowski spacetime and the
metric enjoys the harmonic decay $1/r$ (in term of a suitable distance
function $r$ at spatial infinity). Our analysis encompasses matter fields that
have small energy norm and solely enjoys a slow decay at spacelike infinity.
Our proof is based on the Euclidean-hyperboloidal foliation method recently
introduced by the authors, and distinguishes between the decay along
asymptotically hyperbolic slices and the decay along asymptotically Euclidean
slices. We carefully analyze the decay of metric component at the harmonic
level $1/r$, especially the metric component in the direction of the light
cone. In presence of such a slow-decaying matter field, we establish a global
existence theory for the Einstein equations expressed as a coupled system of
nonlinear wave and Klein-Gordon equations.

###### Contents

  1. 1 Introduction
  2. 2 Euclidean-hyperboloidal foliation and functional inequalities
  3. 3 Nonlinear stability of Einstein-Klein-Gordon spacetimes
  4. 4 Consequences of the energy estimates
  5. 5 Estimates based on the structure of the Einstein equations
  6. 6 Closing the bootstrap estimates

##  1 Introduction

###  1.1 Global evolution problem for the Einstein-matter equations

#### Main purpose.

We consider solutions to the Einstein equations in the near-Minkowski regime
and study the global nonlinear evolution problem when a suitable initial data
set is prescribed, consisting of a Riemannian metric, a symmetric two-tensor,
and matter data. These data are assumed to be close to a sufficiently flat and
asymptotically Schwarzschild-like perturbation of a hypersurface in Minkowski
spacetime. For the vacuum Einstein equations, this problem was solved first by
Christodoulou and Klainerman [9], while an alternative proof in wave
coordinates was given later on by Lindblad and Rodnianski [35, 36]. Solutions
with lower decay at spacelike infinity were constructed by Bieri [1] (and
[2]), while most recent contributions are due to Hintz and Vasy [15, 16]. All
of these results easily extend to massless matter fields.

Our aim in the present paper is to solve the global evolution problem for the
Einstein-matter system when the matter field under consideration is massive.
We are going to describe the problem and first state a simplified version of
our main result (Theorem 1.1) when the data have Schwarzschild decay. Later in
this paper (cf. Theorem 3.1) we state a more general result, when the metric
at spatial infinity enjoys harmonic-type decay.

A recent literature on the global dynamics of self-gravitating massive fields
is now available. The present paper is a companion to our work [30] and, on
the one hand, provides the arguments that are necessary in order to establish
the Schwarzschild-type decay of solutions to the Einstein equations and, on
the other hand, can also be considered as an overview of the more general
proof in [30] (see also [31]). These results grew from earlier work by the
authors in [26, 29].

While our project came under completion we learned that Ionescu and Pausader
simultaneously solved the same problem by introducing a completely different
methodology, which is based on the notion of spacetime resonances; see [21]. A
somewhat different class of initial data sets is covered therein, as far as
the functional norms and the spatial decay of solutions are concerned
(specially since the regularity in [21] is stated in weighted Fourier spaces).
We will not attempt to review here the vast literature on the subject, and we
refer the reader to our detailed overview in [30]. Recent work includes, among
others, the major contributions by Bigorgne [3, 4], Fajman et al. [11, 13],
Lindblad et al. [37], Smulevici [42] and Wang [44].

#### Einstein equations and massive fields.

We consider four-dimensional manifolds $(\mathscr{M},g)$ in which
$\mathscr{M}\simeq[0,+\infty)\times\mathbb{R}^{3}$ and $g$ is a Lorentzian
metric with signature $(-,+,+,+)$ and Levi-Civita connection
$\nabla_{g}=\nabla$. Greek indices $\alpha,\beta,\ldots$ describe $0,1,2,3$,
and we denote by $R_{\alpha\beta}$ the Ricci curvature of $g$ and by
$R=g^{\alpha\beta}R_{\alpha\beta}$ its scalar curvature. Throughout, we use
implicit summation over repeated indices, as well as raising and lowering
indices with respect to the metric $g_{\alpha\beta}$ and its inverse denoted
by $g^{\alpha\beta}$.

We impose that Einstein-matter equations hold, namely

$G_{\alpha\beta}=8\pi\,T_{\alpha\beta},$ (1.1)

in which the left-hand side is Einstein’s curvature tensor defined (in
abstract indices) as

$G_{\alpha\beta}=R_{\alpha\beta}-{R\over 2}\,g_{\alpha\beta}.$ (1.2)

The right-hand side of (1.1) is the energy-momentum tensor, which depends upon
the matter content and, for instance, is taken to vanish identically for
vacuum Einstein spacetimes, so that (1.1) then reduces to the Ricci-flat
condition $R_{\alpha\beta}=0$.

While our method of analysis should apply to many massive matter models, we
are interested here in Klein-Gordon scalar fields
$\phi:\mathscr{M}\to\mathbb{R}$ with energy-momentum tensor

$T_{\alpha\beta}=\nabla_{\alpha}\phi\nabla_{\beta}\phi-\Big{(}{1\over
2}\nabla_{\gamma}\phi\nabla^{\gamma}\phi+U(\phi)\Big{)}g_{\alpha\beta}.$ (1.3)

Here, the potential function $U=U(\phi)$ depends on the nature of the matter
under consideration and, throughout, we assume that

$U(\phi)={1\over 2}c^{2}\phi^{2}+\mathcal{O}(\phi^{3})$ (1.4)

for some (mass-like) constant $c>0$. Recalling the Bianchi identity
$\nabla^{\alpha}R_{\alpha\beta}={1\over 2}\nabla_{\beta}R$, we have
$\nabla^{\alpha}G_{\alpha\beta}=0$ and, consequently,

$\nabla^{\alpha}T_{\alpha\beta}=0.$ (1.5)

In turn, we deduce that $\phi$ satisfies the nonlinear Klein-Gordon equation

$\Box_{g}\phi-U^{\prime}(\phi)=0.$ (1.6)

For instance, when $U(\phi)={c^{2}\over 2}\phi^{2}$, this is
$\Box_{g}\phi-c^{2}\phi=0$, which is linear in $\phi$ but also involves the
unknown metric.

#### Initial value problem.

For suitable initial data, the equation (1.6) is expected to uniquely
determine the evolution of the matter. Our challenge is precisely to study the
nonlinear coupling problem when the metric $g$ itself is one of the unknowns
and solves Einstein equations with suitably prescribed initial data. When a
foliation by spacelike hypersurfaces is chosen and a suitable gauge choice is
made, we use Latin indices varying between $1$ and $3$ and the Einstein
equations decompose into and evolution equations

$G_{ab}=8\pi\,T_{ab},\qquad\quad a,b=1,2,3.$ (1.7)

and constraint equations

$G_{00}=8\pi\,T_{00},\qquad\quad G_{0a}=8\pi\,T_{0a},\qquad\quad a,b=1,2,3.$
(1.8)

The initial value problem is then formulated by specifying a Riemannian metric
$g_{0}$ and a symmetric two-tensor field $k_{0}$, defined on a manifold with
topology $\mathbb{R}^{3}$, and satisfying the constraints (1.8), namely

$\displaystyle R_{g_{0}}+(\text{Tr}_{g_{0}}k_{0})^{2}-|k_{0}|_{g_{0}}^{2}$
$\displaystyle=16\pi
T_{00},\qquad\text{Div}_{g_{0}}\big{(}k_{0}-(\text{Tr}_{g_{0}}k_{0})g_{0}\big{)}=-8\pi
T_{0{\scriptscriptstyle\mspace{0.5mu}\bullet\mspace{0.5mu}}}.$ (1.9)

where $R_{g_{0}}$ denotes the scalar curvature, $\text{Tr}_{g_{0}}$ the trace,
and $\text{Div}_{g_{0}}$ the divergence of the metric $g_{0}$. We then seek a
globally hyperbolic Cauchy development [7, 8] consisting of a spacetime metric
$g$ together with a matter field $\phi$ satisfying Einstein’s evolution
equations and defining a manifold whose induced geometry (first- and second-
fundamental forms) on an initial hypersurface is given by the pair
$(g_{0},k_{0})$.

###  1.2 Self-gravitating fields in the asymptotically Schwarzschild regime

#### Wave-Klein-Gordon system of interest.

We work on the manifold
$\mathscr{M}\simeq\mathbb{R}^{1+3}_{+}:=\\{(t,x)\in\mathbb{R}^{1+3},t\geq
1\\}$ covered by a single coordinate chart $(t,x)=(t,x^{a})$ with $t\geq 1$
and $x\in\mathbb{R}^{3}$. We also write
$r^{2}:=|x|^{2}=\sum_{a=1}^{3}|x^{a}|^{2}$. We introduce the Minkowski metric
$g_{\textbf{Mink}}:=-dt^{2}+\sum_{a}(dx^{a})^{2}$, and observe that
$\mathscr{M}$ is the future of the initial hypersurface $\\{t=1\\}$. We also
introduce the outgoing light cone

$\qquad\mathscr{L}:=\big{\\{}r=t-1\big{\\}}\subset\mathbb{R}_{+}^{3+1}$ (1.10)

and its constant-$t$ slices are denoted by $\mathscr{L}_{t}$.

More precisely, we rely on global coordinate functions
$x^{\alpha}:\mathscr{M}\to\mathbb{R}$ (with $\alpha=0,1,2,3$) satisfying the
wave gauge conditions

$\Box_{g}x^{\alpha}=0,$ (1.11)

and we express the Einstein equations as a nonlinear system of second-order
partial differential equations, supplemented with second-order constraints.
The unknowns are the metric coefficients $g_{\alpha\beta}$ and the matter
field $\phi$. Specifically, we find

$\displaystyle\widetilde{\Box}_{g}g_{\alpha\beta}=\mathbb{F}_{\alpha\beta}(g,g;\partial
g,\partial
g)-16\pi\,\big{(}\partial_{\alpha}\phi\partial_{\beta}\phi+U(\phi)g_{\alpha\beta}\big{)},\qquad\widetilde{\Box}_{g}\phi-U^{\prime}(\phi)=\,0,$
(1.12)

where
$\widetilde{\Box}_{g}:=g^{\alpha^{\prime}\beta^{\prime}}\partial_{\alpha^{\prime}}\partial_{\beta^{\prime}}$
denotes a modified wave operator, and the wave gauge constraints take the form

$\displaystyle\Gamma^{\alpha}$
$\displaystyle=g^{\alpha\beta}\Gamma_{\alpha\beta}^{\lambda}=0,\qquad\Gamma_{\alpha\beta}^{\lambda}={1\over
2}\,g^{\lambda\lambda^{\prime}}\big{(}\partial_{\alpha}g_{\beta\lambda^{\prime}}+\partial_{\beta}g_{\alpha\lambda^{\prime}}-\partial_{\lambda^{\prime}}g_{\alpha\beta}\big{)}.$
(1.13)

The nonlinearities $\mathbb{F}=\mathbb{P}+\mathbb{Q}$ have an important
structure especially in connection with the Euclidian-hyperboloidal foliation,
which is described and play a central role in [30].

#### Merging the Minkowski and the Schwarzschild solutions.

In wave coordinates the Schwarzschild metric $g_{\textbf{Sch}}$ takes the form
(with $\omega_{a}:=x_{a}/r$)

$\displaystyle g_{\textbf{Sch},00}$ $\displaystyle=-\frac{r-m}{r+m},\qquad
g_{\textbf{Sch},0a}=0,\qquad
g_{\textbf{Sch},ab}=\frac{r+m}{r-m}\omega_{a}\omega_{b}+\frac{(r+m)^{2}}{r^{2}}(\delta_{ab}-\omega_{a}\omega_{b}).$
(1.14)

Consider a (regular) cut-off function $\chi^{\star}(r)$ that vanishes for
$r\leq 1/2$ and is identically $1$ for all $r\geq 3/4$. Given a (small) mass
coefficient $m>0$ we introduce the reference metric

$g^{\star}=g_{\textbf{Mink}}+\chi^{\star}(r)\,\chi^{\star}(r/(t-1))(g_{\textbf{Sch}}-g_{\textbf{Mink}}),\qquad
t\geq 2.$ (1.15)

This metric coincides with $g_{\textbf{Mink}}$ in the cone
$\big{\\{}r/(t-1)<1/2\big{\\}}$ and coincides with $g_{\textbf{Sch}}$ in the
exterior $\big{\\{}r/(t-1)\geq 3/4\big{\\}}$ which contains the light cone
$\mathscr{L}$. This metric satisfies the so-called light-bending property,
saying by definition that the light cone coefficient

$r\,g^{\star}(\mathbf{l},\mathbf{l})=4m+\mathcal{O}(1/r)\qquad\text{ for the
metric }g^{\star},$ (1.16)

is positive, where the light cone direction is defined by

$\mathbf{l}:=\partial_{t}-(x^{a}/r)\partial_{a}.$ (1.17)

This is essentially the same construction as in [35].

#### Class of initial data sets.

We are interested in solutions to the Einstein-matter system in the harmonic
regime (corresponding to $\lambda=1$ in [30]) and we establish a sharp decay
property in the case where the reference metric is constructed from the
Schwarzschild metric. The initial metric $g_{0}$ is assumed to be sufficiently
close to the Euclidean metric while the initial second fundamental form
$k_{0}$ is sufficiently small. Let us introduce the following decomposition
(and $a,b=1,2,3$)

$g_{0ab}=g^{\star}_{0ab}+u_{0ab}=\delta_{ab}+h^{\star}_{0ab}+u_{0ab},\qquad\quad
k_{0ab}=k^{\star}_{0ab}+l_{0ab}.$ (1.18)

We aim at covering a variety of asymptotic behaviors and, at this juncture, it
is convenient to introduce the following terminology.

  * •

The part $h^{\star}_{0}$ is referred to as the initial reference and will be
assumed to be small in a (weighted, high-order) pointwise norm.

  * •

The part $u_{0}$ is referred to as the initial perturbation and will be
assumed to be small in (weighted, high-order) energy norm.

An example of a such decomposition is provided by the construction in Lindblad
and Rodnianski [35], where the initial data is decomposed as the sum of a
finite-energy perturbation plus an (asymptotically) Schwarzschild metric
outside of a compact set (with sufficiently small and positive mass). In our
theory, the two parts are treated differently. Indeed, $h_{0}^{\star}$ is the
initial trace of $h^{\star}$ while $u_{0}$ is propagated.

For the metric perturbation, we introduce the energy norms

$\displaystyle\mathbf{E}^{\text{metric}}_{\kappa,N}(g_{0},k_{0})=$
$\displaystyle\sum_{|I|\leq N}\big{\|}\langle
r\rangle^{\kappa+|I|}\big{(}|\partial_{x}^{I}\partial_{x}u_{0}|+|\partial_{x}^{I}u_{1}|\big{)}\big{\|}_{L^{2}(\mathbb{R}^{3})},$
(1.19) $\displaystyle\mathbf{E}^{\text{matter}}_{\mu,N}(\phi_{0},\phi_{1}):=$
$\displaystyle\sum_{|I|\leq N}\big{\|}\langle
r\rangle^{\mu+|I|}(|\partial_{x}^{I}\partial_{x}\phi_{0}|+|\partial_{x}^{I}\phi_{0}|+|\partial_{x}^{I}\phi_{1}|)\big{\|}_{L^{2}(\mathbb{R}^{3})}.$

While our proof below will provide a more general result as stated below, we
find it convenient to state first our result for the perturbation of the
Schwarzschild solution.

Given an initial data set we decompose it according to (1.18) and we introduce
the linear development denoted by $u_{\textbf{init}}$ of the initial data set
$(u_{0\alpha\beta},u_{1\alpha\beta})$, that is, we introduce the solution to
the (free, linear) wave equation with this initial data. It was established in
[30, Proposition LABEL:prop1-12-01-2022] that, under the assumption that the
norm above is finite,

$|u_{\textbf{init}}|\lesssim C_{0}\epsilon(t+r+1)^{-1}.$ (1.20)

Our main assumption beyond the smallness on the norms (1.19) is the following
sign condition which we referred to as the light-bending condition

$\inf_{\mathscr{M}^{\textbf{near}}_{\ell}}\big{(}4m+r\,u_{\textbf{init}}(\mathbf{l},\mathbf{l})\big{)}\geq\epsilon_{\star}.$
(1.21)

Here, a parameter $\ell\in(0,1/2]$ being fixed once for all, we have defined
the near-light cone domain to be

$\mathscr{M}^{\textbf{near}}_{\ell}:=\Big{\\{}t\geq 2,\quad t-1\leq
r\leq\frac{t}{1-\ell}\Big{\\}},$ (1.22)

where in agreement with [30] it is convenient to restrict attention to $t\geq
2$. Interestingly, this condition can be easily satisfied when

  * •

either $\epsilon$ is small with respect to $\epsilon_{\star}$, so that the
contribution $m$ from the Schwarzschild metric dominates,

  * •

both initial data $u_{0}\geq 0$ and $u_{1}\geq 0$ are non-negative (as is
clear from the fact that fundamental solution to the wave equation is a non-
negative measure),

  * •

or yet a combination of the above two extreme examples, namely, the negative
contribution of the perturbation is small with respect to the Schwarzschild
mass.

#### Main statement.

We are in a position to state our main result. In fact, a more general
statement concerning perturbations of reference metrics with harmonic decay is
actually established in this paper; see Theorem 3.1.

######  Theorem 1.1 (Nonlinear stability of self-gravitating Klein-Gordon
fields. Perturbations of the Schwarzschild solution).

A constant $C_{\star}>0$ being fixed, the following result holds for all
sufficiently small $\epsilon,\epsilon_{\star}$ satisfying $\epsilon\leq
C_{\star}\epsilon_{\star}$. Consider the reference metric $g^{\star}$ defined
in (1.15) by merging together the Minkowski metric $g_{\textbf{Mink}}$ and the
Schwarzschild metric with mass $\epsilon_{\star}$ along a light cone. Consider
a set of contraint-satisfying initial data $(g_{0},k_{0},\phi_{0},\phi_{1})$,
a large integer $N$, and some exponents $(\kappa,\mu,\epsilon)$ satisfying

$\kappa\in(1/2,1),\qquad\mu\in(3/4,1),\qquad\kappa\leq\mu.$ (1.23)

Then provided the initial data satisfies the light bending condition (1.21)
together with the smallness condition

$\displaystyle\mathbf{E}^{\text{metric}}_{\kappa,N}(g_{0},k_{0})+\mathbf{E}^{\text{matter}}_{\mu,N}(\phi_{0},\phi_{1})\leq\epsilon,$
(1.24)

the maximal globally hyperbolic Cauchy development of
$(g_{0},k_{0},\phi_{0},\phi_{1})$ associated with the Einstein-massive field
system (1.1) and (1.3) is future causally geodesically complete, and
asymptotically approaches Minkowski spacetime in all (timelike, null,
spacetime) directions. Moreover, the component $g(\mathbf{l},\mathbf{l})$ has
a harmonic decay and enjoys the light bending condition, namely

$|g(\mathbf{l},\mathbf{l})|\lesssim{\epsilon_{\star}+\epsilon\over
t+r+1},\qquad\quad\inf_{\mathscr{M}^{\textbf{near}}_{\ell}}r\,g(\mathbf{l},\mathbf{l})\geq\epsilon_{\star}/2.$
(1.25)

#### Harmonic decay.

Our proof of Theorems 1.1 and 3.1 follows from the general method in [30] in
which we now take the parameter $\lambda$ therein to be the critical value
$\lambda=1$. We would like to point out here several significant differences
and several new ingredients that are required for the proof in the present
paper. First of all, the use of a hierarchy property for commutators was not
necessary for the range $\lambda<1$, but turns out to be essential now. On the
other hand, we revisit [30, Section LABEL:section-8-added] concerning the
pointwise decay of metric components (and their derivatives) and we use a
decomposition of the wave operators in which the component
$H^{\mathcal{N}}{}^{00}$ is directly controlled. In turn, in the analysis in
[30, Section LABEL:section---12] we now decouple the contributions of the
Hessian and of the commutators. Interestingly, this leads to significant
simplifications in comparison with the arguments in [30], while simultaneously
new estimates are required. For instance, in the analysis of the (now
harmonic) decay of the null metric component in [30, Section LABEL:section---
13], we no longer introduce on the “loss” exponent $\theta$. In order to
recover the desired $1/r$ behavior for the metric, the application of the
Kirchhoff formula must now be done at the $1/r$ level of decay, and this
requires a sharp control of the source-terms in the wave equation. Such terms
are due to the Ricci curvature of the reference metric (assumed to have
sufficient decay, and even to vanish for the Schwarzschild metric), the
commutators (discussed earlier) and the quasilinear terms (also discussed
earlier). Observe that it is also important to distinguish between zero-order
estimates (i.e. without differentiation) and high-order estimates. Finally, we
point out that, interestingly, our results also apply in the vacuum spacetimes
by taking $\phi$ to vanish identically: our proof then is somewhat simpler
than the one in [36], thanks to the fact that it takes advantage of the light-
bending condition —a consequence of the assumed positivity of the mass of the
Schwarzschild metric.

##  2 Euclidean-hyperboloidal foliation and functional inequalities

###  2.1 Spacetime foliation and vector fields

#### Preliminary.

We provide here an overview of the technical tools introduced by the authors
in [30] when developing the Euclidean-Hyperboloidal Foliation Method. This
method generally applies to establish the global existence of solutions with
small amplitude for coupled systems of wave and Klein-Gordon equations. The
method relies on the following key ingredients.

  * •

Construction of a spacetime foliation consisting of asymptotically
hyperboloidal slices and asymptotically flat slices.

  * •

Functional analysis tools on such a foliation: admissible vector fields,
Sobolev inequalities, and energy estimates.

  * •

Sup-norm estimates for wave equations and Klein-Gordon equations.

For the full set of notions and results, we refer to [30].

#### Foliation of interest.

In order to foliate $\mathscr{M}$ by spacelike hypersurfaces, we introduce a
cut-off function $\chi:\mathbb{R}\to[0,1]$ satisfying

$\chi(x)=\begin{cases}0,&x\leq 0,\\\ 1,&x>1.\end{cases}$ (2.1a) We also
introduce the hyperboloidal and Euclidean radii at a time $s$
${\mathbf{r}}^{\mathcal{H}}(s):={1\over
2}(s^{2}-1),\qquad{\mathbf{r}}^{\mathcal{E}}(s):={1\over 2}(s^{2}+1),$ (2.1b)
Then, a function $\xi$ referred to as the foliation coefficient is defined by
$\xi(s,r):=1-\chi(r-{\mathbf{r}}^{\mathcal{H}}(s))=\begin{cases}1,\quad&r<{\mathbf{r}}^{\mathcal{H}}(s),\\\
0,\quad&r>{\mathbf{r}}^{\mathcal{E}}(s).\end{cases}$ (2.1c) We next define the
Euclidean–hyperboloidal time function $T=\mathbf{T}(s,r)$ by solving the
ordinary differential equation
$\partial_{r}\mathbf{T}(s,r)=\frac{r\,\xi(s,r)}{(s^{2}+r^{2})^{1/2}},\qquad\mathbf{T}(s,0)=s.$
(2.1d)

In turn, our foliation
$\mathscr{M}_{s}:=\\{(t,x^{a})\in\mathscr{M}\,/\,t=\mathbf{T}(s,r)\\}$ is a
one-parameter family of asymptotically Euclidean, spacelike hypersurfaces. The
future of the initial surface $\\{t=1\\}$ is decomposed as

$\\{t\geq 1\\}=\mathscr{M}^{\textbf{init}}\cup\bigcup_{s\geq
2}\mathscr{M}_{s},\qquad\mathscr{M}_{s}=\mathscr{M}^{\mathcal{H}}_{s}\cup\mathscr{M}^{\mathcal{M}}_{s}\cup\mathscr{M}^{\mathcal{E}}_{s},$
(2.2a) with $\mathscr{M}^{\textbf{init}}:=\\{(t,x)\,/\,1\leq t\leq T(2,r)\\}$
and $\displaystyle\mathscr{M}^{\mathcal{H}}_{s}:$
$\displaystyle=\big{\\{}t=\mathbf{T}(s,|x|),\quad\quad|x|\leq{\mathbf{r}}^{\mathcal{H}}(s)\big{\\}}\qquad$
asymptotically hyperboloidal, (2.2b)
$\displaystyle\mathscr{M}^{\mathcal{M}}_{s}:$
$\displaystyle=\big{\\{}t=\mathbf{T}(s,|x|),\quad\,{\mathbf{r}}^{\mathcal{H}}(s)\leq|x|\leq{\mathbf{r}}^{\mathcal{E}}(s)\big{\\}}\qquad$
merging (or transition), $\displaystyle\mathscr{M}^{\mathcal{E}}_{s}:$
$\displaystyle=\big{\\{}t=\mathbf{T}(s),\quad\qquad\quad{\mathbf{r}}^{\mathcal{E}}(s)\leq|x|\big{\\}}$
$\displaystyle\text{asymptotically Euclidean}.$ We also write
$\mathscr{M}^{\mathcal{EM}}_{s}:=\mathscr{M}^{\mathcal{E}}\cup\mathscr{M}^{\mathcal{M}}$.

#### Frames of interest.

The following terminology will be used.

  * •

The semi-hyperboloidal frame (SHF)

$\partial^{\mathcal{H}}_{0}:=\partial_{t},\qquad\qquad\partial^{\mathcal{H}}_{a}=\not{\partial}^{\mathcal{H}}_{a}:=\frac{x^{a}}{t}\partial_{t}+\partial_{a}$
(2.3a)

is defined in $\mathscr{M}_{s}$ and is the appropriate frame in the
hyperboloidal domain in order to establish the relevant decay properties in
timelike and null directions. Some of our arguments will involve radial
integration based on
$\not{\partial}^{\mathcal{H}}_{r}:=(x^{a}/r)\not{\partial}^{\mathcal{H}}_{a}$.

  * •

The semi-null frame (SNF)

$\partial^{\mathcal{N}}_{0}:=\partial_{t},\qquad\qquad\partial^{\mathcal{N}}_{a}=\not{\partial}^{\mathcal{N}}_{a}:={x^{a}\over
r}\partial_{t}+\partial_{a}$ (2.3b)

is defined in $\mathscr{M}_{s}$ except on the center line $r=0$, and is the
appropriate frame within the Euclidean-merging domain in order to exhibit the
structure of the nonlinearities of the field equations and, in turn, to
establish the relevant decay properties in spatial and null directions.

  * •

The Euclidean–hyperboloidal frame (EHF) is defined as

$\displaystyle\partial^{\mathcal{E\hskip-1.70709ptH}}_{0}:=\partial_{t},\qquad\qquad\partial^{\mathcal{E\hskip-1.70709ptH}}_{a}$
$\displaystyle=\not{\partial}^{\mathcal{EH}}_{a}:=\partial_{a}+(x^{a}/r)\partial_{r}\mathbf{T}\,\partial_{t}=\partial_{a}+x^{a}\xi(s,r)(s^{2}+r^{2})^{-1/2}\partial_{t}$
(2.3c)

consists of tangent vectors $\not{\partial}^{\mathcal{EH}}_{a}$ to the slices
$\mathscr{M}_{s}$. Observe that
$\partial^{\mathcal{E\hskip-1.70709ptH}}_{a}=\partial^{\mathcal{H}}_{a}$ in
$\mathscr{M}^{\mathcal{H}}_{s}$, while
$\partial^{\mathcal{E\hskip-1.70709ptH}}_{a}=\partial_{a}$ in
$\mathscr{M}^{\mathcal{E}}_{s}$. The expressions of the vectors
$\partial^{\mathcal{E\hskip-1.70709ptH}}_{a}$ are more involved in the merging
$\mathscr{M}^{\mathcal{M}}_{s}$, where the vectors
$\partial^{\mathcal{E\hskip-1.70709ptH}}_{a}$ interpolate between
$\partial^{\mathcal{H}}_{a}$ and $\partial_{a}$. Some of our arguments will
involve radial integration based on
$\not{\partial}^{\mathcal{EH}}_{r}:=(x^{a}/r)\not{\partial}^{\mathcal{EH}}_{a}$.

#### Admissible vector fields.

Minkowski spacetime admits three sets of Killing fields.

  * •

The spacetime translations generated by the vector fields $\partial_{\alpha}$
($\alpha=0,1,2,3$).

  * •

The Lorentz boosts generated by the vector fields
$L_{a}:=x_{a}\partial_{t}+t\,\partial_{a}$ ($a=1,2,3$).

  * •

The spatial rotations generated by the vector fields
$\Omega_{ab}:=x_{a}\partial_{b}-x_{b}\partial_{a}$ ($a,b=1,2,3$).

We refer to $\partial_{\alpha},L_{a},\Omega_{ab}$ as the admissible fields
which commute with the wave and Klein-Gordon operators in Minkowski spacetime.
In defining high-order norms, we combine admissible vector fields together. An
operator $Z=\partial^{I}L^{J}\Omega^{K}$ is called an ordered admissible
operator. To such an operator $Z=\partial^{I}L^{J}\Omega^{K}$, we associate
its order, degree, and rank by

$\textbf{ord}(Z)=|I|+|J|+|K|,\qquad\textbf{deg}(Z)=|I|,\qquad\textbf{rank}(Z)=|J|+|K|.$
(2.4)

###  2.2 Energy and Sobolev inequalities

#### Weighted energy inequality.

For the foliation under consideration, the fundamental energy functional
associated with the wave equation and, more generally, the Klein-Gordon
equations involve the weight $\zeta=\zeta(t,x)$ defined by

$\zeta(s,r)^{2}:=1-\frac{r^{2}\xi^{2}(s,r)}{s^{2}+r^{2}}=\begin{cases}s^{2}/t^{2},\quad&r<{\mathbf{r}}^{\mathcal{H}}(s),\\\
1,\quad&r>{\mathbf{r}}^{\mathcal{E}}(s).\end{cases}$ (2.5)

In addition, we introduce a weight which reduces to a constant in the interior
of the light cone and is essentially the distance to the light cone in the
exterior domain. Throughout, we are given a smooth and non-decreasing function
$\aleph$ satisfying $\aleph(y)=0$ for $y\leq{-1}$ and $\aleph(y)=y+1$ for
$y\geq{0}$, and we set

$\mathbf{X}:=1+\aleph({r-t}).$ (2.6)

Consider the wave or Klein-Gordon equation (with $c\geq 0$) with unknown $u$,
namely $g^{\alpha\beta}\partial_{\alpha}\partial_{\beta}u-c^{2}u=f$ associated
with a metric
$g^{\alpha\beta}=:g_{\textbf{Mink}}^{\alpha\beta}+H^{\alpha\beta}$ and a
right-hand side $f$. The energy-flux vector

$V_{g,\eta,c}[u]:=-\mathbf{X}^{2\eta}\Big{(}g^{00}|\partial_{t}u|^{2}-g^{ab}\partial_{a}u\,\partial_{b}u-c^{2}u^{2},\
2g^{a\beta}\partial_{t}u\partial_{\beta}u\Big{)}$ (2.7a) depends upon the
metric $g$ as well as the weight $\mathbf{X}^{\eta}$. By setting
$\displaystyle\Omega_{g,\eta,c}[u]$
$\displaystyle:=-2\eta\mathbf{X}^{-1}\aleph^{\prime}({r-t})(-1,x^{a}/r)\cdot
V_{g,\eta,c}[u]$ (2.7b)
$\displaystyle=2\,\eta\mathbf{X}^{2\eta-1}\aleph^{\prime}({r-t})\,\big{(}g^{\mathcal{N}ab}\not{\partial}^{\mathcal{N}}_{a}u\not{\partial}^{\mathcal{N}}_{b}u-H^{\mathcal{N}00}|\partial_{t}u|^{2}+c^{2}u^{2}\big{)},$
$\displaystyle-G_{g,\eta}[u]$
$\displaystyle:=\partial_{t}H^{00}|\mathbf{X}^{\eta}\partial_{t}u|^{2}-\partial_{t}H^{ab}\mathbf{X}^{2\eta}\partial_{a}u\partial_{b}u+2\mathbf{X}^{2\eta}\partial_{a}H^{a\beta}\partial_{t}u\partial_{\beta}u,$
we find $\displaystyle-2\mathbf{X}^{2\eta}\,\partial_{t}uf=\text{div}\hskip
1.42271ptV_{g,\eta,c}[u]+\Omega_{g,\eta,c}[u]-G_{g,\eta}[u].$ (2.7c)

Next, we consider the integral $\mathcalboondox
E_{g,\eta,c}(s,u):=\int_{\mathscr{M}_{s}}V_{g,\eta,c}[u]\cdot
n_{s}\,d\sigma_{s}.$ (2.8a) By integrating the energy identity (2.7c) over the
domain limited by a slice of the foliation and the initial slice and using
Stokes’ formula, we arrive at the energy identity $\mathcalboondox
E_{g,\eta,c}(s_{1},u)-\mathcalboondox
E_{g,\eta,c}(s_{0},u)+\int_{\mathscr{M}_{[s_{0},s_{1}]}}\\!\\!\\!\\!\\!\\!\\!\\!\big{(}\Omega_{g,\eta,c}[u]-G_{g,\eta}[u]\big{)}\,dxdt=-2\int_{\mathscr{M}_{[s_{0},s_{1}]}}\\!\\!\\!\\!\\!\\!\\!\\!\partial_{t}uf\,\mathbf{X}^{2\eta}dxdt.$
(2.8b)

The Jacobian $J$ of our parameterization $(t,x)\mapsto(s,x)$, as established
in [30, Lemma LABEL:lem1-22-05-2020], satisfies

$J\leq\begin{cases}s/t\quad&\text{ in }\mathscr{M}^{\mathcal{H}}_{s},\\\ \xi
s\,(s^{2}+r^{2})^{-1/2}+(1-\xi)\,2s\quad&\text{ in
}\mathscr{M}^{\mathcal{M}}_{s},\\\ 2s\quad&\text{ in
}\mathscr{M}^{\mathcal{E}}_{s},\end{cases}$ (2.9)

and

$\zeta^{2}s\lesssim J\lesssim\,\zeta^{2}s\qquad\text{ in
}\mathscr{M}^{\mathcal{M}}.$ (2.10)

After a change of variable and differentiation we can rewrite (2.8b) in the
equivalent form

$\displaystyle\frac{d}{ds}\mathcalboondox
E_{g,\eta,c}(s,u)+2\eta\int_{\mathscr{M}_{s}}\big{(}g^{\mathcal{N}ab}\not{\partial}^{\mathcal{N}}_{a}u\not{\partial}^{\mathcal{N}}_{b}u+c^{2}u^{2}\big{)}\aleph^{\prime}({r-t})\mathbf{X}^{2\eta-1}\
Jdx$ (2.11a)
$\displaystyle=\int_{\mathscr{M}_{s}}\Big{(}G_{g,\eta}[u]+\eta\mathbf{X}^{2\eta-1}\aleph^{\prime}({r-t})H^{\mathcal{N}}{}^{00}|\partial_{t}u|^{2}\Big{)}\
Jdx+\int_{\mathscr{M}_{s}}|\partial_{t}uf|\,\mathbf{X}^{2\eta}\,Jdx,$ in which
the latter integral is controlled by
$\displaystyle\int_{\mathscr{M}_{s}}|\partial_{t}uf|\,\mathbf{X}^{2\eta}\,Jdx\lesssim\int_{s_{0}}^{s_{1}}\mathcalboondox
E_{\eta,c}(s,u)^{1/2}\|J\zeta^{-1}\mathbf{X}^{\eta}f\|_{L^{2}(\mathscr{M}_{s})}\,ds.$
(2.11b)

We will rely on the following weighted energy estimate in the Euclidean-
merging domain concerning any solution
$u:\mathscr{M}^{\mathcal{EM}}_{[s_{0},s_{1}]}\to\mathbb{R}$ to the wave or
Klein-Gordon equation
$g^{\alpha\beta}\partial_{\alpha}\partial_{\beta}u-c^{2}u=f$ with right-hand
side $f:\mathscr{M}^{\mathcal{EM}}_{[s_{0},s_{1}]}\to\mathbb{R}$:

$\displaystyle\frac{d}{ds}\mathcalboondox
E_{g,\eta,c}^{\mathcal{EM}}(s,u)+\frac{d}{ds}\mathcalboondox
E^{\mathcal{L}}_{g,c}(s,u;s_{0})+2\eta\int_{\mathscr{M}^{\mathcal{EM}}_{s}}\big{(}g^{\mathcal{N}ab}\not{\partial}^{\mathcal{N}}_{a}u\not{\partial}^{\mathcal{N}}_{b}u+c^{2}u^{2}\big{)}\mathbf{X}^{2\eta-1}\aleph^{\prime}({r-t})\,Jdx$
(2.12)
$\displaystyle=\int_{\mathscr{M}^{\mathcal{EM}}_{s}}\Big{(}G_{g,\eta}[u]+\eta\mathbf{X}^{2\eta-1}\aleph^{\prime}({r-t})H^{\mathcal{N}}{}^{00}|\partial_{t}u|^{2}\Big{)}\,Jdxds+\int_{\mathscr{M}^{\mathcal{EM}}_{s}}\mathbf{X}^{2\eta}\partial_{t}uf\,Jdxds,$

in which the latter integral is bounded by
$\int_{s_{0}}^{s_{1}}\,(\mathcalboondox
E_{\eta,c}^{\mathcal{EM}}(s,u))^{1/2}\,\big{\|}J\,\zeta^{-1}\mathbf{X}^{\eta}f\big{\|}_{L^{2}(\mathscr{M}^{\mathcal{EM}}_{s})}\,ds$,
while the second term in the left-hand side is defined as

$\frac{d}{ds}\mathcalboondox
E_{g,c}^{\mathcal{L}}(s,u;s_{0})=s\int\big{(}-H^{\mathcal{N}}{}^{00}|\partial_{t}u|^{2}+g^{\mathcal{N}ab}\not{\partial}^{\mathcal{N}}_{a}u\not{\partial}^{\mathcal{N}}_{b}u+c^{2}u^{2}\big{)}\,d\sigma,$
(2.13)

where in the domain of integration is defined by $t=r+1$ and
${\mathbf{r}}^{\mathcal{H}}(s_{0})\leq
r\leq{\mathbf{r}}^{\mathcal{H}}(s_{1})$.

#### Commutator estimates.

We also introduce the Japanese bracket $\langle y\rangle:=\sqrt{1+|y|^{2}}$
for all real $y$. Then, for any function $u=u(t,x)$ we define
$|u|_{N}:=\max_{\textbf{ord}{Z}\leq N}|Zu|$ and
$|u|_{p,k}:=\max_{\textbf{ord}{Z}\leq N\atop\textbf{rank}{Z}\leq k}|Zu|$,
where the first maximum is over all ordered admissible operators. Various
calculus rules based on this notation were established in [30], which we will
not repeat here. Let us only extract an important consequence of [30,
Proposition LABEL:prop1-12-02-2020]. We also decompose each slice
$\mathscr{M}^{\mathcal{EM}}_{s}$ into near and far regions so that

$\mathscr{M}^{\textbf{near}}_{s}:=\mathscr{M}^{\mathcal{EM}}_{s}\cap\big{\\{}t-1\leq
r\leq
2t\big{\\}},\qquad\qquad\mathscr{M}^{\textbf{far}}_{s}:=\mathscr{M}^{\mathcal{EM}}_{s}\cap\big{\\{}r\geq
2t\big{\\}}.$ (2.14)

######  Proposition 2.1 (Hierarchy property for quasi-linear commutators.
Euclidean-merging domain).

1\. Estimate in the near-light cone domain. For any function $u$ defined in
the near-light cone domain and for any operator $Z$ with $\textbf{ord}(Z)=p$
and $\textbf{rank}(Z)=k$, one has

$\displaystyle\,\big{|}[Z,H^{\alpha\beta}\partial_{\alpha}\partial_{\beta}]u\big{|}$
(2.15a)
$\displaystyle\lesssim\big{(}|H^{\mathcal{N}}{}^{00}|+t^{-1}|r-t||H|\big{)}\,|\partial\partial
u|_{p-1,k-1}+T_{p,k}^{\textbf{hier}}[H,u]+T_{p,k}^{\textbf{easy}}[H,u]+T_{p,k}^{\textbf{super}}[H,u]\quad\text{
in }\mathscr{M}^{\textbf{near}}_{[s_{0},s_{1}]},$ with $\displaystyle
T_{p,k}^{\textbf{hier}}[H,u]$ $\displaystyle:=\sum_{p_{1}+p_{2}=p\atop
p_{1}+k_{2}=k}\big{(}|LH^{\mathcal{N}}{}^{00}|_{p_{1}-1}+t^{-1}|r-t||LH|_{p_{1}-1}\big{)}\,|\partial\partial
u|_{p_{2},k_{2}},$ (2.15b) $\displaystyle T_{p,k}^{\textbf{easy}}[H,u]$
$\displaystyle:=\sum_{p_{1}+p_{2}=p\atop k_{1}+k_{2}=k}\big{(}|\partial
H^{\mathcal{N}}{}^{00}|_{p_{1}-1,k_{1}}+t^{-1}|r-t||\partial
H|_{p_{1}-1,k_{1}}\big{)}\,|\partial\partial u|_{p_{2},k_{2}},$ $\displaystyle
T_{p,k}^{\textbf{super}}[H,u]$ $\displaystyle:=t^{-1}|H||\partial
u|_{p}+t^{-1}\hskip-8.5359pt\sum_{0\leq p_{1}\leq p-1}|H|_{p_{1}+1}|\partial
u|_{p-p_{1}}.$

2\. Estimate in the Euclidean-merging domain. For any function $u$ defined in
the Euclidean-merging domain, one has (with $Y_{\text{rot}}=L_{a}$ or
$\Omega_{ab}$)

$\displaystyle|[Z,H^{\alpha\beta}\partial_{\alpha}\partial_{\beta}]u|$
$\displaystyle\lesssim|H|\,|\partial\partial u|_{p-1,k-1}$ (2.16)
$\displaystyle\quad+\hskip-17.07182pt\sum_{p_{1}+p_{2}=p\atop
p_{1}+k_{2}=k\text{ with
}k_{1}=p_{1}}\hskip-17.07182pt|Y_{\text{rot}}H|_{p_{1}-1}|\partial\partial
u|_{p_{2},k_{2}}+\hskip-8.5359pt\sum_{p_{1}+p_{2}=p\atop
k_{1}+k_{2}=k}\hskip-8.5359pt|\partial H|_{p_{1}-1,k_{1}}|\partial\partial
u|_{p_{2},k_{2}}\quad\text{ in }\mathscr{M}^{\mathcal{EM}}_{[s_{0},s_{1}]}.$

The above estimates will be applied to study the evolution of our high-order
energy functionals. In (2.15), the term $T_{p,k}^{\textbf{hier}}[H,u]$ is the
most challenging contribution to the change of the energy functional but
involves terms at a lower rank, that is, contains strictly fewer boosts or
rotations; this structure will allow us to formulate an induction argument on
the rank. The term $T^{\textbf{easy}}$ is easier since it contains the factor
$\partial H$ which enjoys (comparatively) good $L^{2}$ and pointwise decay,
while $T^{\textbf{super}}$ contains a favorable $1/t$ factor.

#### Functional inequalities.

It is necessary to revisit the standard Sobolev inequalities and formulate
them along the foliation of interest. We only select here two results. In the
inequalities below, recall that $\not{\partial}^{\mathcal{EM}}{}^{I}$ denotes
any $|I|$-order operator determined from the fields
$\\{\not{\partial}^{\mathcal{EM}}_{a}\\}_{a=1,2,3}$, while
$\not{\partial}^{\mathcal{E}}{}^{I}$ denotes any a $|I|$-order operator
determined from the fields $\\{\partial_{a}\\}_{a=1,2,3}$.

######  Proposition 2.2 (Sobolev inequality in the Euclidean-merging domain).

Fix an exponent $\eta\geq 0$ and set $C(\eta):=1+\eta+\eta^{2}$. For all
sufficiently regular functions defined in $\mathscr{M}_{[s_{0},s_{1}]}$ with
$2\leq s_{0}\leq s\leq s_{1}$, one has

$r\,\mathbf{X}^{\eta}|u(t,x)|\lesssim C(\eta)\sum_{|I|+|J|\leq
2}\|\mathbf{X}^{\eta}\not{\partial}^{\mathcal{EM}}{}^{I}\Omega^{J}u\|_{L^{2}(\mathscr{M}^{\mathcal{EM}}_{s})},\qquad(t,x)\in\mathscr{M}^{\mathcal{EM}}_{s},$
(2.17a) $r\,\mathbf{X}^{\eta}|u(t,x)|\lesssim C(\eta)\sum_{|I|+|J|\leq
2}\|\mathbf{X}^{\eta}\not{\partial}^{\mathcal{E}}{}^{I}\Omega^{J}u\|_{L^{2}(\mathscr{M}^{\mathcal{E}}_{s})},\qquad(t,x)\in\mathscr{M}^{\mathcal{E}}_{s}.$
(2.17b)

Next, we rely on the boosts $L_{a}=x^{a}\partial_{t}+t\,\partial_{a}$ which
are tangent to the hyperboloidal slices.

######  Proposition 2.3 (Sobolev inequality in the hyperboloidal domain).

For any function defined on a hypersurface $\mathscr{M}^{\mathcal{H}}_{s}$,
the following estimate holds (in which $t^{2}=s^{2}+|x|^{2}$):

$\sup_{\mathscr{M}^{\mathcal{H}}_{s}}t^{3/2}\,|u(t,x)|\lesssim\sum_{|J|\leq
2}\|L^{J}u\|_{L^{2}(\mathscr{M}^{\mathcal{H}}_{s})}\simeq\sum_{m=0,1,2}\|t^{m}(\not{\partial}^{\mathcal{H}})^{m}u\|_{L^{2}(\mathscr{M}^{\mathcal{H}}_{s})}.$
(2.18)

#### Estimates based on the energy functional.

We also have the following decay properties for the wave or Klein-Gordon
equation; cf. [30, Propositions LABEL:eq3-15-05-2020, LABEL:lem_2_d-e-I, and
LABEL:lem_1_d-KG-e], respectively, together with [30, Proposition
LABEL:17-08-2022-1] (namely (LABEL:eq1-18-08-2021) therein).

######  Proposition 2.4 (Hardy-Poincaré inequality for high-order
derivatives).

For any $\eta=1/2+\delta$ with $\delta>0$ and any sufficiently decaying
function $u$ defined in $\mathscr{M}_{[s_{0},s_{1}]}$ and for all
$s\in[s_{0},s_{1}]$ one has

$\|\mathbf{X}^{-1+\eta}|u|_{p,k}\|_{L^{2}(\mathscr{M}^{\mathcal{EM}}_{s})}\lesssim\big{(}1+\delta^{-1}\big{)}\,\mathcalboondox
F_{\eta}^{\mathcal{EM},p,k}(s,u)+\mathcalboondox F_{\eta}^{0}(s,u),$ (2.19)
$\|\mathbf{X}^{-1+\eta}\zeta|Y_{\text{rot}}u|_{k-1}\|_{L^{2}(\mathscr{M}^{\mathcal{EM}}_{s})}\lesssim(1+\delta^{-1})\,\mathcalboondox
F_{\eta}^{\mathcal{EM},k}(s,u).$ (2.20)

######  Proposition 2.5 (Sobolev decay for wave fields).

For all $\eta\in[0,1)$ and all functions $u$, one has (with $k\leq p$)

$\big{\|}r\,\mathbf{X}^{\eta}\,|\partial
u|_{p,k}\big{\|}_{L^{\infty}(\mathscr{M}^{\mathcal{EM}}_{s})}+\big{\|}r^{1+\eta}\,|\not{\partial}^{\mathcal{N}}u|_{p,k}\big{\|}_{L^{\infty}(\mathscr{M}^{\mathcal{EM}}_{s})}\lesssim(1-\eta)^{-1}\,\mathcalboondox
F_{\eta}^{\mathcal{EM},p+3,k+3}(s,u)$ (2.21a) and, for
$1/2<\eta=1/2+\delta<1$,
$\|r\,\mathbf{X}^{-1+\eta}|u|_{N-2}\|_{L^{\infty}(\mathscr{M}^{\mathcal{EM}}_{s})}\lesssim\delta^{-1}\,\mathcalboondox
F_{\eta}^{\mathcal{EM},N}(s,u)+\mathcalboondox F_{\eta}^{0}(s,u).$ (2.21b)

######  Proposition 2.6 (Pointwise decay of Klein-Gordon fields).

Given any exponents $\eta\in(0,1)$, for any solution $v$ to $-\Box
v+c^{2}\,v=f$ defined in $\mathscr{M}^{\mathcal{EM}}_{[s_{0},s_{1}]}$ one has

$c^{2}\,|v|_{p,k}\lesssim\begin{cases}r^{-2}\mathbf{X}^{1-\eta}\,\mathcalboondox
F_{\eta,c}^{\mathcal{EM},p+4,k+4}(s,v)+|f|_{p,k}&\text{in
}\mathscr{M}^{\textbf{near}}_{[s_{0},s_{1}]},\\\ r^{-1-\eta}\,\mathcalboondox
F_{\eta,c}^{\mathcal{EM},p+2,k+2}(s,v)\quad&\text{in
}\mathscr{M}^{\textbf{far}}_{[s_{0},s_{1}]}.\end{cases}$

##  3 Nonlinear stability of Einstein-Klein-Gordon spacetimes

###  3.1 The class of reference metrics

We introduce a class of reference spacetime metrics
$(\mathbb{R}^{3+1}_{+},g^{\star})$ which represent “approximate solutions” to
Einstein’s vacuum equations.

  * •

The metric $g^{\star}=g_{\textbf{Mink}}+h^{\star}$ is asymptotically Minkowski
in the sense that

$|h^{\star}|_{N+2}+\langle r+t\rangle|\partial h^{\star}|_{N+1}+\langle
r+t\rangle^{2}|\partial\partial h^{\star}|_{N}\leq\epsilon_{\star}\langle
r+t\rangle^{-1},$ (3.1a)

  * •

This metric is almost Ricci flat in the sense that

$|\tensor[^{(w)}]{}{}R^{\star}|_{N}+\langle
r-t\rangle|\partial\tensor[^{(w)}]{}{}R^{\star}|_{N-1}\leq\begin{cases}\epsilon_{\star}^{2}\langle
r+t\rangle^{-4}\quad&\text{ in
}\mathscr{M}^{\mathcal{EM}}_{[s_{0},+\infty)},\\\ \epsilon_{\star}\langle
r+t\rangle^{-3}\quad&\text{ in
}\mathscr{M}^{\mathcal{H}}_{[s_{0},+\infty)},\end{cases}$ (3.1b)

where $\tensor[^{(w)}]{}{}R^{\star}$ denotes the “reduced” Ricci curvature
classically defined by removing suitable contractions of the Christoffel
symbols.

  * •

This metric satisfies the light-bending property, that is,

$\inf_{\mathscr{M}^{\textbf{near}}_{\ell}}\big{(}r\,h^{\star}(\mathbf{l},\mathbf{l})\big{)}\geq\epsilon_{\star},$
(3.1c)

in which $g^{\star}(\mathbf{l},\mathbf{l})$ is (linearly) equivalent to
$-h_{\star}^{\mathcal{N}00}$ and we recall that
$\mathbf{l}=\partial_{t}-(x^{a}/r)\partial_{a}$ (as stated earlier).

As a consequence of (3.1b) and (2.9) and after introducing a small parameter
$\delta>0$ and some decay exponent

$\kappa\in(1/2,1),$ (3.2)

the following rough bound follows by integration:

$\|\mathbf{X}^{\kappa}J{\zeta}^{-1}\,|\tensor[^{(w)}]{}{}R^{\star}|_{N}\|_{L^{2}(\mathscr{M}^{\mathcal{EM}}_{s})}\leq
R_{\textbf{err}}^{\star}(s)=C_{R^{\star}}\epsilon_{\star}^{2}\delta^{-1/2}\,s^{-1-\delta},$
(3.3)

in which and $C_{R^{\star}}>0$ is a constant determined by $N$ and

$\int_{s_{0}}^{s}R_{\textbf{err}}^{\star}(s^{\prime})\,ds^{\prime}\leq
C_{R^{\star}}\,\epsilon_{\star}^{2}\delta^{-3/2}s_{0}^{-\delta}.$ (3.4)

For further motivations and results we also refer the reader to the more
general theory in [30, Section LABEL:sec1-23-05-2021]. Furthermore, it is
straightforward to check that the Minkowski-Schwarzschild reference metric
$g^{\star}$ in the introduction satisfies all of the conditions above.

###  3.2 The class of initial data sets

In the following $\delta\in(0,1)$ denotes a fixed exponent which should be
small in comparison to a small multiple of $\kappa-1/2$, which arises as a
critical value for the decay of the metric perturbation. It will be
interesting to keep track, in our estimates, of the most relevant parameters
and constants. For the initial data we assume the following smallness
conditions.

  * •

In the hyperboloidal domain, we assume

$\mathcalboondox F^{\mathcal{H},N-5}(s_{0},u)+s_{0}^{-1/2}\,\mathcalboondox
F_{c}^{\mathcal{H},N-5}(s_{0},\phi)\leq C_{0}\epsilon s_{0}^{\delta},$ (3.5a)
$\mathcalboondox F^{\mathcal{H},N-7}(s_{0},u)+\mathcalboondox
F_{c}^{\mathcal{H},N-7}(s_{0},\phi)\leq C_{0}\epsilon s_{0}^{\delta}.$ (3.5b)

  * •

In the Euclidean-merging domain, we assume

$\mathcalboondox
F_{\kappa}^{\mathcal{EM},N}(s_{0},u)+s_{0}^{-1}\,\mathcalboondox
F_{\mu,c}^{\mathcal{EM},N}(s_{0},\phi)\leq C_{0}\epsilon s_{0}^{\delta},$
(3.6a) $\mathcalboondox F_{\kappa}^{\mathcal{EM},N-5}(s_{0},u)+\mathcalboondox
F_{\mu,c}^{\mathcal{EM},N-5}(s_{0},\phi)\leq C_{0}\epsilon s_{0}^{\delta}.$
(3.6b)

  * •

Finally, we impose the linear light-bending condition on the total initial
data (defined as the sum of the reference data and the perturbation) (with
$\ell\in(0,1/2]$ being fixed):

$\inf_{\mathscr{M}^{\textbf{near}}_{\ell,[s_{0},+\infty)}}\big{(}r\,g^{\star}(\mathbf{l},\mathbf{l})+r\,u_{\textbf{init}}(\mathbf{l},\mathbf{l})\big{)}\geq\epsilon_{\star}.$
(3.7)

Concerning (3.7), we point out that the contributions due to the
nonlinearities of the Einstein-matter equations on the key metric component
$h^{\mathcal{N}00}$ will turn out to be negligible in comparison with the
background and initial linear contributions which are of amplitude
$\epsilon_{\star}/r$. As a consequence of our energy bounds, recalling [30,
(LABEL:eq5-07-01-2022)] and applying the Klainerman-Sobolev inequality on the
initial slice, we obtain

$|\partial_{t}u(1,x)|+|\partial_{x}u(1,x)|\lesssim C_{0}\epsilon\langle
r\rangle^{-3/2-\kappa}.$

Integrating with the (spatial) radial derivative from spatial infinity, we
obtain

$\langle r\rangle|\partial_{t}u(1,x)|+|u(1,x)|\lesssim C_{0}\epsilon\langle
r\rangle^{-1/2-\kappa}.$

Recalling that $\kappa+1/2>1$, we apply [30, Proposition
LABEL:prop1-21-12-2021] (based on Kirchkoff formula) and obtain

$|u_{\textbf{init}}|_{N-4}\lesssim C_{0}\epsilon(t+r+1)^{-1}\quad\text{ in
}\mathscr{M}_{[s_{0},+\infty)}.$ (3.8)

Here, the linear development $u_{\textbf{init}}$ of the initial data
$(u_{0\alpha\beta},u_{1\alpha\beta})$ is the solution of the (free, linear)
wave equation with this initial data. (See also [30, Proposition
LABEL:prop1-12-01-2022].) The passage from geometric statements to statements
in coordinates was already discussed in an appendix of [30] to which the
reader is referred.

###  3.3 Nonlinear stability statement

We distinguish between estimates at low- or high-order of differentiation,
estimates within the hyperboloidal and Euclidean-merging domains, and a
positivity condition near the light cone. The following energy bounds will be
established at each time $s\geq s_{0}$.

  * •

In the hyperboloidal domain, we have

$\mathcalboondox F^{\mathcal{H},N-5}(s,u)+s^{-1/2}\,\mathcalboondox
F_{c}^{\mathcal{H},N-5}(s,\phi)\leq(\epsilon_{\star}+C_{1}\epsilon)\,s^{\delta},$
(3.9a) $\mathcalboondox F^{\mathcal{H},N-7}(s,u)+\mathcalboondox
F_{c}^{\mathcal{H},N-7}(s,\phi)\leq(\epsilon_{\star}+C_{1}\epsilon)\,s^{\delta}.$
(3.9b)

  * •

In the Euclidean-merging domain, we have

$\mathcalboondox F_{\kappa}^{\mathcal{EM},N}(s,u)+s^{-1}\,\mathcalboondox
F_{\mu,c}^{\mathcal{EM},N}(s,\phi)\leq(\epsilon_{\star}+C_{1}\epsilon)\,s^{\delta},$
(3.10a) $\mathcalboondox F_{\kappa}^{\mathcal{EM},N-5}(s,u)+\mathcalboondox
F_{\mu,c}^{\mathcal{EM},N-5}(s,\phi)\leq(\epsilon_{\star}+C_{1}\epsilon)\,s^{\delta}.$
(3.10b)

  * •

Near the light cone, we have the light-bending condition

$\inf_{\mathscr{M}^{\textbf{near}}_{\ell,s}}(-r\,h^{\mathcal{N}}{}^{00})\geq
0.$ (3.11)

######  Theorem 3.1 (Nonlinear stability of self-gravitating Klein-Gordon
fields. Perturbations of metrics with harmonic decay).

A constant $C_{\star}>0$ being fixed, the following result holds for all
sufficiently small $\epsilon,\epsilon_{\star}$ satisfying $\epsilon\leq
C_{\star}\epsilon_{\star}$. Consider a reference metric $g^{\star}$, defined
as a perturbation of the Minkowski metric satisfying (3.1) for some decay
exponent

$\kappa\in(1/2,1).$ (3.12)

Consider an initial data set $(u_{0},u_{1},\phi_{0},\phi_{1})$ satisfying the
decay and regularity conditions (3.5)–(3.7) at the initial time $s_{0}$ for
some parameters $(N,\mu,\epsilon)$ for a sufficiently large integer $N$ with

$\mu\in(3/4,1),\qquad\kappa\leq\mu.$ (3.13)

Then, the initial value problem for the Einstein-Klein-Gordon system in wave
gauge (1.12) admits a global in time solution $(g,\phi)$ defined for all
$t\geq 1$ and $x\in\mathbb{R}^{3}$. This solution satisfies the decay and
regularity conditions (3.9)–(3.11) for all times $s\geq s_{0}$ and remains
close to the reference spacetime $(\mathbb{R}_{+}^{3+1},g^{\star})$ (in the
norms under consideration) and enjoys the following decay estimate

$\inf_{\mathscr{M}^{\textbf{near}}_{\ell,s}}(-r\,h^{\mathcal{N}}{}^{00})\geq\epsilon_{\star}/2,\qquad\quad|h(\mathbf{l},\mathbf{l})|\lesssim{\epsilon_{\star}\over
t+r+1}.$ (3.14)

##  4 Consequences of the energy estimates

###  4.1 Bootstrap assumptions and basic estimates

#### Bootstrap assumptions.

This result will be established in the following sections by following the
strategy already outlined in the introduction. From now on we assume that the
local-in-time solution in fact extends over an interval $[s_{0},s_{1}]$, and
we assume that $[s_{0},s_{1}]$ is the maximal interval of time within which
(3.9), (3.10) and (3.11) hold for all $s\in[s_{0},s_{1}]$ so that, by
continuity, one of these conditions is an equality at the end time $s_{1}$.
Our objective is to establish the following improved estimates for all
$s\in[s_{0},s_{1}]$:

$\mathcalboondox F^{\mathcal{H},N-5}(s,u)+s^{-1/2}\,\mathcalboondox
F_{c}^{\mathcal{H},N-5}(s,\phi)\leq\frac{1}{2}(\epsilon_{\star}+C_{1}\epsilon)\,s^{\delta},$
(4.1a) $\mathcalboondox F^{\mathcal{H},N-7}(s,u)+\mathcalboondox
F_{c}^{\mathcal{H},N-7}(s,\phi)\leq\frac{1}{2}(\epsilon_{\star}+C_{1}\epsilon)\,s^{\delta},$
(4.1b) $\mathcalboondox
F_{\kappa}^{\mathcal{EM},N}(s,u)+s^{-1}\,\mathcalboondox
F_{\mu,c}^{\mathcal{EM},N}(s,\phi)\leq\frac{1}{2}(\epsilon_{\star}+C_{1}\epsilon)\,s^{\delta},$
(4.2a) $\mathcalboondox F_{\kappa}^{\mathcal{EM},N-5}(s,u)+\mathcalboondox
F_{\mu,c}^{\mathcal{EM},N-5}(s,\phi)\leq\frac{1}{2}(\epsilon_{\star}+C_{1}\epsilon)\,s^{\delta},$
(4.2b)
$\inf_{\mathscr{M}^{\textbf{near}}_{\ell,[s_{0},s_{1}]}}(-r\,h^{\mathcal{N}}{}^{00})\geq{1\over
2}\epsilon_{\star}.$ (4.3)

Observe that the factor $s^{-1/2}$ arises in (4.1a), while $s^{-1}$ arises in
(4.2a). In the following, our statements will be established under the
assumptions on the reference metric and initial data and the bootstrap
assumptions, as stated above.

#### Energy-based estimates.

The following $L^{2}$ estimates for the metric perturbation and the matter
field are immediate from the bootstrap assumption (3.10a) and (3.10b) in the
Euclidean-merging domain:

$\|\mathbf{X}^{\kappa}\zeta\,|\partial
u|_{N}\|_{L^{2}(\mathscr{M}^{\mathcal{EM}}_{s})}+\|\mathbf{X}^{\kappa}|\not{\partial}^{\mathcal{N}}u|_{N}\|_{L^{2}(\mathscr{M}^{\mathcal{EM}}_{s})}\lesssim(\epsilon_{\star}+C_{1}\epsilon)\,s^{\delta},$
(4.4a)
$\|\mathbf{X}^{\mu}\zeta\,|\partial\phi|_{p}\|_{L^{2}(\mathscr{M}^{\mathcal{EM}}_{s})}+\|\mathbf{X}^{\mu}|\not{\partial}^{\mathcal{N}}\phi|_{p}\|_{L^{2}(\mathscr{M}^{\mathcal{EM}}_{s})}+\|\mathbf{X}^{\mu}|\phi|_{p}\|_{L^{2}(\mathscr{M}^{\mathcal{EM}}_{s})}\lesssim(\epsilon_{\star}+C_{1}\epsilon)\,\begin{cases}s^{1+\delta},\,&p=N,\\\
s^{\delta},&p=N-5.\end{cases}$ (4.4b)

As a consequence of the weighted Poincaré inequality in Proposition 2.4 we
find

$\|\mathbf{X}^{-1+\kappa}|u|_{p,k}\|_{L^{2}(\mathscr{M}^{\mathcal{EM}}_{s})}\lesssim\delta^{-1}\,\mathcalboondox
F_{\kappa}^{\mathcal{EM},p,k}(s,u)+\mathcalboondox
F_{\kappa}^{0}(s,u)\lesssim\delta^{-1}(\epsilon_{\star}+C_{1}\epsilon)\,s^{\delta},$
(4.5)

which provides us with a control of the metric components possibly without
partial derivatives.

#### Pointwise decay of the metric.

Basic sup-norm estimates are then derived by applying the generalized Sobolev
estimate222We neglect a factor $1/(1-\kappa)$ since we have fixed $\kappa<1$.
in Proposition 2.5, and we thus control the metric perturbation $\partial u$
at order $N-3$, as follows:

$\|r\,\mathbf{X}^{\kappa}\,|\partial
u|_{N-3}\|_{L^{\infty}(\mathscr{M}^{\mathcal{EM}}_{s})}+\|r^{1+\kappa}|\not{\partial}^{\mathcal{N}}u|_{N-3}\|_{L^{\infty}(\mathscr{M}^{\mathcal{EM}}_{s})}\lesssim(\epsilon_{\star}+C_{1}\epsilon)\,s^{\delta}.$
(4.6)

In view of $h_{\alpha\beta}=h^{\star}_{\alpha\beta}+u_{\alpha\beta}$, by
combining this result with the asymptotically Minkowski condition (3.1a) on
the reference metric, for the metric unknown $\partial h$ we obtain

$\|r\,\mathbf{X}^{\kappa}|\partial
h|_{N-3}\|_{L^{\infty}(\mathscr{M}^{\mathcal{EM}}_{s})}+\|r^{1+\kappa}|\not{\partial}^{\mathcal{N}}h|_{N-3}\|_{L^{\infty}(\mathscr{M}^{\mathcal{EM}}_{s})}\lesssim(\epsilon_{\star}+C_{1}\epsilon)s^{\delta}.$
(4.7)

We apply the Sobolev decay for wave fields (2.21b), together with (3.9a) and
(3.10a), and obtain

$\|r\,\mathbf{X}^{\kappa-1}\,|u|_{N-2}\|_{L^{\infty}(\mathscr{M}^{\mathcal{EM}}_{s})}\lesssim\delta^{-1}\,(\epsilon_{\star}+C_{1}\epsilon)\,s^{\delta}$
(4.8)

and, using again (3.1a) enjoyed by the reference,

$\|r^{\kappa}|h|_{N-2}\|_{L^{\infty}(\mathscr{M}^{\mathcal{EM}}_{s})}\lesssim\delta^{-1}(\epsilon_{\star}+C_{1}\epsilon)s^{\delta}.$
(4.9)

#### Pointwise decay of the matter field.

The Sobolev decay inequality in [30, Proposition LABEL:lem_2_d-e-I] and the
bootstrap assumptions (3.10) provide us with sup-norm estimates

$\|r\,\mathbf{X}^{\mu}\,|\partial\phi|_{p-3}\|_{L^{\infty}(\mathscr{M}^{\mathcal{EM}}_{s})}+\|r^{1+\mu}|\not{\partial}^{\mathcal{N}}\phi|_{p-3}\|_{L^{\infty}(\mathscr{M}^{\mathcal{EM}}_{s})}\lesssim(\epsilon_{\star}+C_{1}\epsilon)\,\begin{cases}s^{1+\delta},\,&p=N,\\\
s^{\delta},\,&p=N-5,\end{cases}$ (4.10)

and, thanks to the consequence in [30, (LABEL:eq_decay-v-repeat000-two)] of
our generalized Sobolev inequality,

$\|r\,\mathbf{X}^{\mu}\,|\phi|_{p-2}\|_{L^{\infty}(\mathscr{M}^{\mathcal{EM}}_{s})}\lesssim(\epsilon_{\star}+C_{1}\epsilon)\,\begin{cases}s^{1+\delta},\quad&p=N,\\\
s^{\delta},&p=N-5.\end{cases}$ (4.11)

However, within $\mathscr{M}^{\textbf{near}}_{s}$ this is not sufficient for
our purpose below and we establish a stronger decay, as follows.

######  Lemma 4.1.

The matter field satisfies the pointwise bound

$\|r^{2}\,\mathbf{X}^{\mu-1}\,|\phi|_{p-4}\|_{L^{\infty}(\mathscr{M}^{\mathcal{EM}}_{s})}\lesssim(\epsilon_{\star}+C_{1}\epsilon)\,\begin{cases}s^{1+2\delta},\quad&p=N,\\\
s^{2\delta},\quad&p=N-5.\end{cases}$

###### Proof.

In view of the inequality (4.11), we want to “trade” a factor $\mathbf{X}$ for
a factor $r$, and we only need to deal with the domain
$\mathscr{M}^{\textbf{near}}_{s}$. We need the decay property near the light
cone stated in Proposition 2.6. We consider the Klein-Gordon equation
$g^{\alpha\beta}\partial_{\alpha}\partial_{\beta}\phi-c^{2}\phi=0$ and with
the notation therein, we set

$f=h^{\mu\nu}\partial_{\mu}\partial_{\nu}\phi=h^{\star\mu\nu}\partial_{\mu}\partial_{\nu}\phi+u^{\mu\nu}\partial_{\mu}\partial_{\nu}\phi.$

For the first term $h^{\star\mu\nu}\partial_{\mu}\partial_{\nu}\phi$ above,
the decay condition on $h^{\star}$ in (3.1a) yields us

$|h^{\star\mu\nu}\partial_{\mu}\partial_{\nu}\phi|_{p-4}\lesssim\epsilon_{\star}(\epsilon_{\star}+C_{1}\epsilon)\,r^{-2}\mathbf{X}^{-\mu}\begin{cases}s^{1+\delta},\quad&p=N,\\\
s^{\delta},\quad&p=N-5.\end{cases}$

For the second term $u^{\mu\nu}\partial_{\mu}\partial_{\nu}\phi$, by [30,
Lemma LABEL:lem-small] we have $|u^{\mu\nu}|_{p}\lesssim|u|_{p}$ and, by
recalling the Sobolev decay (4.8) and (4.10), we find

$|u^{\mu\nu}\partial_{\mu}\partial_{\nu}\phi|_{p-4}\lesssim|u|_{p-4}|\partial\phi|_{p-3}\lesssim\delta^{-1}(\epsilon_{\star}+C_{1}\epsilon)^{2}r^{-2}\mathbf{X}^{1-\kappa-\mu}\begin{cases}s^{1+2\delta},&\quad
p=N,\\\ s^{2\delta},&\quad p=N-5.\end{cases}$

On the other hand, recalling (3.10) we have

$r^{-2}\mathbf{X}^{1-\mu}\,\mathcalboondox
F_{\mu,c}^{\mathcal{EM},p,k}(s,\phi)\lesssim(\epsilon_{\star}+C_{1}\epsilon)\,r^{-2}\mathbf{X}^{1-\mu}\begin{cases}s^{1+\delta},\quad&p=N,\\\
s^{\delta},\quad&p=N-5.\end{cases}$

We are thus in a position to use the general pointwise decay enjoyed by Klein-
Gordon fields, as stated [30, Proposition LABEL:lem_1_d-KG-e] and we arrive at
the desired conclusion. ∎

###  4.2 Basic estimates for nonlinearities: energy norm

#### Improving the energy estimates.

We differentiate the metric and matter evolution equations (1.12) with respect
to $Z=\partial^{I}L^{J}\Omega^{K}$ (with $\textbf{ord}(Z)=|I|+|J|+|K|\leq N$
or $\leq N-5$) and obtain

$\displaystyle\widetilde{\Box}_{g}Zu$
$\displaystyle=-[Z,h^{\mu\nu}\partial_{\mu}\partial_{\nu}]u_{\alpha\beta}+Z\big{(}\mathbb{P}_{\alpha\beta}^{\star}[u]\big{)}+Z\big{(}\mathbb{Q}_{\alpha\beta}^{\star}[u]\big{)}$
(4.12a)
$\displaystyle\quad+Z\Big{(}\mathbb{I}^{\star}_{\alpha\beta}[u]+2\,\tensor[^{(w)}]{R}{{}^{\star}}_{\alpha\beta}-u^{\mu\nu}\partial_{\mu}\partial_{\nu}g^{\star}_{\alpha\beta}\Big{)}-8\pi\,Z\big{(}2\,T_{\alpha\beta}-Tg_{\alpha\beta}\big{)},$
and
$\widetilde{\Box}_{g}Z\phi-c^{2}Z\phi=-[Z,h^{\mu\nu}\partial_{\mu}\partial_{\nu}]\phi,$
(4.12b)

in which $\mathbb{I}^{\star}[u]$ represent the reference-perturbation
interaction terms presented in [30, (LABEL:eq1-06-02-2022)]. Our main task is
to control
$\|J\,\zeta^{-1}\mathbf{X}^{\kappa}|T|_{N}\|_{L^{2}(\mathscr{M}^{\mathcal{EM}}_{s})},$
where $T$ represents any of the terms in the right-hand sides of (4.12).
Thanks to (2.10) we have

$\|J\,\zeta^{-1}\mathbf{X}^{\kappa}|T|_{N}\|_{L^{2}(\mathscr{M}^{\mathcal{EM}}_{s})}\lesssim\|s\,\mathbf{X}^{\kappa}\zeta|T|_{N}\|_{L^{2}(\mathscr{M}^{\mathcal{EM}}_{s})}.$
(4.13)

#### Linear-critical and super-critical nonlinearities.

We treat first the comparatively easier terms, that is, the reference-
perturbation interaction terms $\mathbb{I}^{\star}[u]$, the term
$u^{\mu\nu}\partial_{\mu}\partial_{\nu}h^{\star}_{\alpha\beta}$, and the
source terms associated with the scalar field. On the other hand, the null
terms, the quasi-null term and the commutators require different arguments and
will be the subject of later sections. We set

$\displaystyle W^{\textbf{linear}}$
$\displaystyle:=\mathbb{F}_{\alpha\beta}(g^{\star},g^{\star};\partial
g^{\star},\partial u)+\mathbb{F}_{\alpha\beta}(g^{\star},g^{\star};\partial
u,\partial g^{\star}),$ (4.14) $\displaystyle W^{\textbf{super}}$
$\displaystyle:=\mathbb{F}_{\alpha\beta}(u,g^{\star};\partial
g^{\star},\partial g^{\star})+\mathbb{F}_{\alpha\beta}(g^{\star},u;\partial
g^{\star},\partial g^{\star})$
$\displaystyle\quad+\mathbb{B}^{\star}_{\alpha\beta}[u]+\mathbb{C}^{\star}_{\alpha\beta}[u]-8\pi\,(2\,T_{\alpha\beta}-Tg_{\alpha\beta})+2\,\tensor[^{(w)}]{R}{{}^{\star}}_{\alpha\beta}.$

It is a simple matter to check that

$\|J\,\zeta^{-1}\mathbf{X}^{\kappa}|W^{\textbf{linear}}|_{p,k}\|_{L^{2}(\mathscr{M}^{\mathcal{EM}}_{s})}\lesssim\delta^{-1}(\epsilon_{\star}+C_{1}\epsilon)^{2}s^{-1-\delta},$
(4.15)

while

$\|J\,\zeta^{-1}\mathbf{X}^{\kappa}|W^{\textbf{super}}|_{p,k}\|_{L^{2}(\mathscr{M}^{\mathcal{EM}}_{s})}\lesssim\delta^{-1}(\epsilon_{\star}+C_{1}\epsilon)^{2}s^{-1-\delta}.$
(4.16)

On the other hand, for super-critical terms involving the reference metric we
find

$\displaystyle\|s\mathbf{X}^{\kappa}\zeta|\mathbb{F}_{\alpha\beta}(u,g^{\star};\partial
g^{\star},\partial
g^{\star})|_{N}\|_{L^{2}(\mathscr{M}^{\mathcal{EM}}_{s})}+\|s\mathbf{X}^{\kappa}\zeta|\mathbb{B}^{\star}_{\alpha\beta}[u]|_{N}\|_{L^{2}(\mathscr{M}^{\mathcal{EM}}_{s})}+\|s\mathbf{X}^{\kappa}\zeta|\mathbb{C}^{\star}_{\alpha\beta}[u]|_{N}\|_{L^{2}(\mathscr{M}^{\mathcal{EM}}_{s})}$
(4.17)
$\displaystyle\lesssim(\epsilon_{\star}+C_{1}\epsilon)^{2}s^{-1-\delta},$

as well as for super-critical terms involving the matter field

$\|s\,\mathbf{X}^{\kappa}\zeta|2\,T_{\alpha\beta}-Tg_{\alpha\beta}|_{N}\|_{L^{2}(\mathscr{M}^{\mathcal{EM}}_{s})}\lesssim(\epsilon_{\star}+C_{1}\epsilon)^{2}s^{-1-\delta}.$

We can now apply the Hardy-Poincaré inequality in Proposition 2.4 and obtain

$\|s\mathbf{X}^{\kappa}\zeta\,|u\,\partial\partial
h^{\star}|_{N}\|_{L^{2}(\mathscr{M}^{\mathcal{EM}}_{s})}\lesssim\delta^{-1}\epsilon_{\star}(\epsilon_{\star}+C_{1}\epsilon)s^{-3+\delta}\lesssim\delta^{-1}(\epsilon_{\star}+C_{1}\epsilon)^{2}s^{-2}.$
(4.18)

###  4.3 Basic estimates for nonlinearities: pointwise norm

#### Estimates for the metric.

From [30, (LABEL:eq1-04-12-2020) and (LABEL:eq11-04-06-2020)] we also recall

$|\mathbb{I}^{\star}_{\alpha\beta}[u]|_{N-4}+|u^{\mu\nu}\partial_{\mu}\partial_{\nu}g^{\star}|_{N-4}\lesssim{\delta^{-1}}(\epsilon_{\star}+C_{1}\epsilon)^{2}r^{-3}\mathbf{X}^{-\kappa}s^{2\delta}$
(4.19)

and

$|u^{\mu\nu}\partial_{\mu}\partial_{\nu}g^{\star}_{\alpha\beta}|_{N-3}\lesssim{\delta^{-1}}\epsilon_{\star}(\epsilon_{\star}+C_{1}\epsilon)\,r^{-4}\mathbf{X}^{1-\kappa}s^{\delta}.$
(4.20)

#### Estimates for the matter field.

For the matter interaction terms, thanks to (4.10), (4.11), and Lemma 4.1 we
have

$\sum_{\alpha,\beta}|2\,T_{\alpha\beta}-(T_{\gamma\gamma}g^{\gamma\gamma})\,g_{\alpha\beta}|_{N-3}=:|\mathbb{T}(\phi)|_{N-3}\lesssim(\epsilon_{\star}+C_{1}\epsilon)^{2}r^{-3}\mathbf{X}^{1-2\mu}s^{1+3\delta}.$
(4.21)

Here and from now on, we use the short-hand notation $\mathbb{T}(\phi)$ for
the matter term contributions
$2\,T_{\alpha\beta}-(T_{\gamma\gamma}g^{\gamma\gamma})\,g_{\alpha\beta}$. For
the proof, we recall the expression of $T_{\alpha\beta}$ and, using the
pointwise decay of Klein-Gordon fields in Proposition 2.6 obtain (see [30,
(LABEL:eq11a-02-05-2020) and Lemma LABEL:lemma-111])

$|\partial\phi\partial\phi|_{N-3}\lesssim|\partial\phi|_{N-3}|\partial\phi|_{[(N-3)/2]}\lesssim(\epsilon_{\star}+C_{1}\epsilon)^{2}\,r^{-3}\mathbf{X}^{1-2\mu}s^{1+3\delta}.$
(4.22)

The bound for $|\phi^{2}|_{N-3}$ is similar and we omit the details. Using the
fact that $|h^{\alpha\beta}|_{N-3}\lesssim 1$, we obtain (4.21).

#### Bounds on null metric component.

The null component $g^{\mathcal{N}00}$ of the metric plays a special role in
our analysis, and the decay of its gradient considered now. Thanks to the wave
gauge condition as derived in [30, Lemma LABEL:lemma-12-04-2020], we have the
decay property

$|\partial
g^{\mathcal{N}00}|_{N-3}\lesssim{\delta^{-1}}\big{(}\epsilon_{\star}+C_{1}\epsilon\big{)}r^{-1-\kappa}s^{\delta}\qquad\text{
in }\mathscr{M}^{\mathcal{EM}}_{[s_{0},s_{1}]}.$ (4.23)

Indeed, in the right-hand side the inequality in [30, Lemma
LABEL:lemma-12-04-2020] with $p=N-3$, namely

$\displaystyle|\partial g^{\mathcal{N}00}|_{N-3}\lesssim$
$\displaystyle|\not{\partial}^{\mathcal{N}}h|_{N-3}+r^{-1}|h|_{N-3}+\sum_{p_{1}+p_{2}=N-3}|h|_{p_{1}}|\partial
h|_{p_{2}}.$

We need to substitute the Sobolev bounds (4.7) on $|\partial h|_{N-3}$ and
$|\not{\partial}^{\mathcal{N}}h|_{N-3}$, together with the bound for
$|h|_{N-2}$ from (4.9).

##  5 Estimates based on the structure of the Einstein equations

###  5.1 Commutator and Hessian estimates for the metric

#### Basic pointwise estimates.

Throughout, the bootstrap assumptions (3.10) and (3.11) are assumed, and
$(\epsilon_{\star}+C_{1}\epsilon)$ is taken to be sufficiently small. So far
all the estimates are similar to the ones we had derived in the general regime
(but by taking $\lambda=1$) in [30]. Now, we focus on the harmonic decay
estimates. From [30, Sections LABEL:sectionN-7 and LABEL:section-8-added], we
recall the following result (extracted from Propositions
LABEL:prop1-12-02-2020 and LABEL:prop1-22-05-2020 therein). Observe that the
inequalities now are simpler in comparison to the general $\lambda$ regime,
due to the fact that now we have a “good” control of the component
$H^{\mathcal{N}00}$. We first write [30, Proposition LABEL:prop1-12-02-2020]
in a different form that will be more appropriate in the present paper, and
next we state our improved estimate for the Hessian.

######  Proposition 5.1 (Hessian for the wave equation near the light cone).

Consider the wave operator
$\widetilde{\Box}_{g}u=g^{\alpha\beta}\partial_{\alpha}\partial_{\beta}u$ in
which $g^{\alpha\beta}=g_{\textbf{Mink}}^{\alpha\beta}+H^{\alpha\beta}$, and
assume that, for some $\epsilon_{1}\ll 1$,

$|H|\leq\epsilon_{1}\quad\text{ in
}\mathscr{M}^{\textbf{near}}_{\ell,[s_{0},s_{1}]},$ (5.1)
$\big{|}H^{\mathcal{N}}{}^{00}\,\big{|}\leq\epsilon_{1}\frac{1+|r-t|}{r}\quad\text{
in }\mathscr{M}^{\textbf{near}}_{\ell,[s_{0},s_{1}]}.$ (5.2)

Then, for any function $u$ one has

$\frac{1+|r-t|}{r}|\partial\partial
u|\lesssim|\widetilde{\Box}_{g}u|+t^{-1}|\partial u|_{1,1}\quad\text{ in
}\mathscr{M}^{\textbf{near}}_{\ell,[s_{0},s_{1}]}$ (5.3)

and, more generally at arbitrary order $(p,k)$,

$\frac{1+|r-t|}{r}|\partial\partial
u|_{p,k}\lesssim|\widetilde{\Box}_{g}u|_{p,k}+t^{-1}|\partial
u|_{p+1,k+1}+T_{p,k}^{\textbf{hier}}[H,u]+T_{p,k}^{\textbf{easy}}[H,u]+T_{p,k}^{\textbf{super}}[H,u]\quad\text{
in }\mathscr{M}^{\textbf{near}}_{\ell,[s_{0},s_{1}]},$ (5.4)

where the notation in (2.15) is used.

###### Sketch of the proof of Proposition 5.1.

From [30, Proposition LABEL:prop1-22-05-2020] (and by recalling
(LABEL:eq3-28-12-2020) therein) we find

$\frac{1+|r-t|}{r}|\partial\partial
u|_{p,k}\lesssim|\widetilde{\Box}_{g}u|_{p,k}+t^{-1}|\partial
u|_{p+1,k+1}+\sum_{\textbf{ord}(Z)\leq p\atop\textbf{rank}(Z)\leq
k}\big{|}[Z,H^{\alpha\beta}\partial_{\alpha}\partial_{\beta}]u\big{|},$ (5.5)

in which we control the last term thanks to the hierarchy property for
commutators (2.15), namely

$\displaystyle\frac{1+|r-t|}{r}|\partial\partial u|_{p,k}$
$\displaystyle\lesssim|\widetilde{\Box}_{g}u|_{p,k}+t^{-1}|\partial
u|_{p+1,k+1}+\big{(}|H^{\mathcal{N}}{}^{00}|+t^{-1}|r-t||H|\big{)}\,|\partial\partial
u|_{p-1,k-1}$
$\displaystyle\quad+T_{p,k}^{\textbf{hier}}[H,u]+T_{p,k}^{\textbf{easy}}[H,u]+T_{p,k}^{\textbf{super}}[H,u].$

Thanks to our smallness assumptions (5.1) and (5.2) (for a sufficiently small
$\epsilon_{1}$), the third term in the right-hand side above is absorbed in
the left-hand side, and (5.4) is established. ∎

We will also use the second part of [30, Proposition LABEL:prop1-22-05-2020],
restated in the following form.

######  Proposition 5.2 (Hessian for the wave equation away from the light
cone).

Suppose that, for some $\epsilon_{1}\ll\ell$,

$|H|\leq\epsilon_{1}\quad\text{ in
}\mathscr{M}^{\textbf{far}}_{\ell,[s_{0},s_{1}]}.$ (5.6)

Then, for any function $u$ the following pointwise inequality holds (where the
commutator is bounded by (2.16)):

$\displaystyle|\partial\partial u|_{p,k}$
$\displaystyle\lesssim(1+t\,\mathbf{X}^{-1})\big{(}|\widetilde{\Box}_{g}u|_{p,k}+t^{-1}|\partial
u|_{p+1,k+1}\big{)}+\sum_{\textbf{ord}(Z)\leq p\atop\textbf{rank}(Z)\leq
k}|[Z,H^{\alpha\beta}\partial_{\alpha}\partial_{\beta}]u|\qquad\text{ in
}\mathscr{M}^{\textbf{far}}_{\ell,[s_{0},s_{1}]}.$

#### Estimates on the Hessian of the metric perturbation.

The arguments in [30, Section LABEL:section---12] somewhat simplify since the
coupling between the commutator and the Hessian contributions is no longer
required at the leading order. For clarity in the presentation, we treat first
the remaining terms in (5.4).

######  Lemma 5.3.

Using the notation (2.15b), for all $p\leq N-4$ the metric perturbation
satisfies

$\displaystyle|T^{\textbf{hier}}[H,u]|_{p,k}+|T^{\textbf{easy}}[H,u]|_{p,k}+|T^{\textbf{super}}[H,u]|_{p,k}$
(5.7)
$\displaystyle\lesssim\big{(}(\epsilon_{\star}+C_{1}\epsilon)+r\mathbf{X}^{-1}|Lh^{\mathcal{N}00}|_{k-1}\big{)}\frac{\mathbf{X}}{r}|\partial\partial
u|_{p-1,k-1}+\delta^{-1}(\epsilon_{\star}+C_{1}\epsilon)^{2}\langle
r\rangle^{-2}\mathbf{X}^{-\kappa}s^{2\delta}\quad\text{ in
}\mathscr{M}^{\textbf{near}}_{\ell,[s_{0},s_{1}]},$

while

$|[Z,H^{\alpha\beta}\partial_{\alpha}\partial_{\beta}]u|\lesssim(\epsilon_{\star}+C_{1}\epsilon)|\partial\partial
u|_{p,k},\qquad\textbf{ord}(Z)\leq p,\ \textbf{rank}(Z)\leq k\quad\text{ in
}\mathscr{M}^{\textbf{far}}_{\ell,[s_{0},s_{1}]}.$ (5.8)

###### Proof.

The result is analogous to the one in [30, Proposition LABEL:Proposition12.1].
In view of (3.1a), (4.7), (4.8) and (4.23), we have

$\displaystyle|H^{\mathcal{N}}{}^{00}|_{k}$
$\displaystyle\lesssim|h^{\mathcal{N}00}|_{k},$ $\displaystyle|\partial
H^{\mathcal{N}}{}^{00}|_{N-3}\lesssim\delta^{-1}\big{(}\epsilon_{\star}+C_{1}\epsilon\big{)}r^{-1-\kappa}s^{\delta},$
(5.9) $\displaystyle|H|_{N-3}$
$\displaystyle\lesssim{\delta^{-1}}(\epsilon_{\star}+C_{1}\epsilon)r^{-\kappa}s^{\delta},\qquad\qquad$
$\displaystyle|\partial
H|_{N-3}\lesssim(\epsilon_{\star}+C_{1}\epsilon)r^{-1}\mathbf{X}^{-\kappa}s^{\delta}.$

Observing that

$|\partial\partial u|_{N-4}\lesssim|\partial
u|_{N-3}\lesssim(\epsilon_{\star}+C_{1}\epsilon)r^{-1}\mathbf{X}^{-\kappa}s^{\delta}$

and substituting these bounds in (2.15b), we obtain the desired result (5.7).
The estimate (5.8) is established in the same manner and we omit the details.
∎

######  Proposition 5.4.

There exists a constant $\epsilon_{s}>0$, determined by $N$, such that
provided

$|h^{\mathcal{N}00}|_{N-4}\leq\epsilon_{s}r^{-1}\quad\text{ in
}\mathscr{M}^{\mathcal{EM}}_{[s_{0},s_{1}]},$ (5.10)

then the metric perturbation satisfies

$\frac{\mathbf{X}}{r}|\partial\partial
u|_{N-4}+|\partial\not{\partial}^{\mathcal{N}}u|_{N-4}\lesssim(\epsilon_{\star}+C_{1}\epsilon)r^{-2}\mathbf{X}^{-\kappa}s^{2\delta}\quad\text{
in }\mathscr{M}^{\textbf{near}}_{\ell,[s_{0},s_{1}]},$ (5.11)

together with

$|\partial\partial
u|_{N-4}\lesssim\ell^{-1}(\epsilon_{\star}+C_{1}\epsilon)t^{-1}r^{-1}\mathbf{X}^{-\kappa}s^{2\delta}\quad\text{
in }\mathscr{M}^{\textbf{far}}_{\ell,[s_{0},s_{1}]}.$ (5.12)

###### Proof.

We recall [30, Lemma LABEL:lemma-123], which we apply with the choice
$\lambda$ replaced by $\kappa$ which is possible since $\kappa<1$. We obtain

$|\widetilde{\Box}_{g}u|_{p,k}\lesssim(\epsilon_{\star}+C_{1}\epsilon)\,r^{-2}\mathbf{X}^{-\kappa}s^{2\delta}+\epsilon_{\star}r^{-3}\qquad\text{
in }\mathscr{M}^{\mathcal{EM}}_{[s_{0},s_{1}]},\quad p\leq N-4.$ (5.13)

Then we substitute (5.7) and (4.6) (applied in $t^{-1}|\partial u|_{p+1,k+1}$)
into (5.4), and we obtain the bound for $|\partial\partial u|_{p,k}$, provided
$(\epsilon_{\star}+C_{1}\epsilon)+r\mathbf{X}^{-1}|Lh^{\mathcal{N}00}|_{k-1}$
is sufficiently small. The bound for
$|\partial\not{\partial}^{\mathcal{N}}u|_{p,k}$ is obtained thanks to [30,
(LABEL:eq1-06-02-2020)]. For (5.12), we apply (5.8) and Proposition 5.2 with
$(\epsilon_{\star}+C_{1}\epsilon)$ sufficiently small. ∎

#### Hessian of the null component.

We also need the following pointwise estimate which is a direct application of
the wave gauge condition (1.11).

######  Lemma 5.5.

As a consequence of (5.11) hold and the wave gauge condition one has

$|\partial\partial
h^{\mathcal{N}00}|_{N-4}\lesssim(\epsilon_{\star}+C_{1}\epsilon)r^{-1-\kappa}\mathbf{X}^{-\kappa}s^{2\delta}\qquad\text{
in }\mathscr{M}^{\textbf{near}}_{[s_{0},s_{1}]}.$

###### Proof.

We give only an outline the argument. Recalling [30, (LABEL:lem2-08-12-2020)],
we have

$\displaystyle|\partial_{t}\partial_{t}h^{\mathcal{N}00}|_{p,k}$
$\displaystyle\lesssim|\partial\not{\partial}^{\mathcal{N}}h|_{p,k}+r^{-1}|\partial
h|_{p,k}+\hskip-8.5359pt\sum_{p_{1}+p_{2}=p\atop
k_{1}+k_{2}=k}\hskip-2.84544pt\big{(}|h|_{p_{1},k_{1}}|\partial\partial
h|_{p_{2},k_{2}}+|\partial h|_{p_{1},k_{1}}|\partial
h|_{p_{2},k_{2}}\big{)}+r^{-1}\hskip-8.5359pt\sum_{p_{1}+p_{2}=p\atop
k_{1}+k_{2}=k}|\partial h|_{p_{1},k_{1}}|h|_{p_{2},k_{2}}.$

By substituting (4.6), (4.9), and (5.11), the desired result holds under the
condition $\delta^{-1}(\epsilon_{\star}+C_{1}\epsilon)\lesssim 1$. ∎

###  5.2 Asymptotically harmonic decay of the null metric component

#### Objective.

We now establish the following result which is similar to [30, Proposition
LABEL:prop1-14-08-2021]. Importantly, our bound here is sharper than the one
derived in [30], which was only concerned with a sub-harmonic decay.

######  Proposition 5.6.

Under the bootstrap assumptions within $[s_{0},s_{1}]$, one has

$|h^{\mathcal{N}00}|_{N-4}\lesssim(\epsilon_{\star}+C_{1}\epsilon)\langle
r\rangle^{-1}\quad\text{ in }\mathscr{M}^{\mathcal{EM}}_{[s_{0},s_{1}]},$
(5.14)
$\inf_{\mathscr{M}^{\textbf{near}}_{\ell,[s_{0},s_{1}]}}(-r\,h^{\mathcal{N}}{}^{00})\geq{1\over
2}\epsilon_{\star}.$ (5.15)

#### Pointwise estimates.

An important ingredient of the method is the derivation of pointwise
estimates, as now stated. Given a triple of data $(f,u_{0},u_{1})$ we consider
the solution $u=u(t,x)$ to

$\Box u=f,\qquad u(1,x)=u_{0}(x),\qquad\partial_{t}u(1,x)=u_{1}(x),\qquad
x\in\mathbb{R}^{3},$ (5.16)

and we use the notation

$\displaystyle
u=\Box^{-1}[u_{0},u_{1},f],\qquad\Box^{-1}_{\textbf{init}}[u_{0},u_{1}]:=\Box^{-1}[u_{0},u_{1},0],\qquad\Box^{-1}_{\textbf{sour}}[f]:=\Box^{-1}[0,0,f].$
(5.17)

We consider the effect of a decaying source, represented by the operator
$\Box^{-1}_{\textbf{sour}}$. We denote by
$\Lambda_{t,x}:=\big{\\{}(\tau,y)\big{/}\,t-\tau=|x-y|,\,1\leq\tau\leq
t\big{\\}}$ the past light cone associated with a point $(t,x)$. In the
present paper, we will rely especially on the critical case below (and refer
to [30] for additional details).

######  Proposition 5.7 (Sup-norm estimates for the wave equation).

Consider a function $f$ satisfying, for some
$\alpha_{1},\alpha_{2},\alpha_{3}$,

$|f(\tau,y)|\lesssim
C_{1}\,\tau^{\alpha_{1}}(\tau+|y|)^{\alpha_{2}}\big{(}1+|\tau-|y||\big{)}^{\alpha_{3}},\qquad(\tau,y)\in\Lambda_{t,x}.$
(5.18)

Case 1 (typical). When $\alpha_{1}=-1+\upsilon$ and $\alpha_{2}=-1-\nu$ and
$\alpha_{3}=-1+\mu$ for some $\upsilon+\mu<\nu$ and $0<\mu,\nu,\upsilon\leq
1/2$, one has

$|\Box^{-1}_{\textbf{sour}}[f](t,x)|\lesssim
C_{1}\,\big{(}\upsilon^{-1}+\mu^{-1}+|\mu-\nu|^{-1}\big{)}\,|\upsilon+\mu-\nu|^{-1}(t+r)^{-1}.$
(5.19a)

Case 2 (sub-critical). When $\alpha_{1}=0$ and $\alpha_{2}=-2-\nu$ and
$\alpha_{3}=-1+\mu$ for some $0<\nu,\mu\leq 1/2$, one has

$|\Box^{-1}_{\textbf{sour}}[f](t,x)|\lesssim
C_{1}\,\begin{cases}\mu^{-1}|\mu-\nu|^{-1}(t+r)^{-1}t^{\mu-\nu},\qquad&\mu>\nu,\\\
\mu^{-1}(t+r)^{-1}\ln(t+1),\quad&\mu=\nu,\\\
\mu^{-1}|\mu-\nu|^{-1}(t+r)^{-1},&\mu<\nu.\end{cases}$ (5.19b)

Case 3 (critical). When $\alpha_{1}=0$ and $\alpha_{2}=-2$ and
$\alpha_{3}=-1-\mu$ for some $\mu\in(0,1/2)$, one has

$|\Box^{-1}_{\textbf{sour}}[f](t,x)|\lesssim
C_{1}\,\mu^{-1}(t+r)^{-1}\Big{(}1+\mathbf{X}^{-\mu}\ln\Big{(}\frac{t}{\mathbf{X}}\Big{)}\Big{)}.$
(5.19c)

Case 4 (super-critical). When $\alpha_{1}=0$ and $\alpha_{2}=-2+\nu$ and
$\alpha_{3}=-1-\mu$ for some $0<\nu<\mu<1/2$, one has

$|\Box^{-1}_{\textbf{sour}}[f](t,x)|\lesssim
C_{1}\,\big{(}|\mu-\nu|^{-1}+\mu^{-1}\nu^{-1}\mathbf{X}^{-\mu}t^{\nu}\big{)}(t+r)^{-1}.$
(5.19d)

#### Controlling $|\Box u|_{p,k}$.

We recall the following decomposition introduced in [30, Section LABEL:section
---13]:

$\displaystyle\mathscr{M}^{\textbf{good}}_{[s_{0},s_{1}]}$
$\displaystyle:=\big{\\{}r\geq
t-1+(\epsilon_{\star}+C_{1}\epsilon)t^{1/2}\big{\\}}\cap\mathscr{M}^{\mathcal{EM}}_{[s_{0},s_{1}]},\quad$
(5.20) $\displaystyle\mathscr{M}^{\textbf{bad}}_{[s_{0},s_{1}]}$
$\displaystyle:=\big{\\{}t-1\leq r\leq
t-1+(\epsilon_{\star}+C_{1}\epsilon)t^{1/2}\big{\\}}\cap\mathscr{M}^{\mathcal{EM}}_{[s_{0},s_{1}]}.$

The formulation [30, (LABEL:eq1-19-04-2021)–(LABEL:eq3-19-04-2021)], and
specifically [30, (LABEL:eq4-19-04-2021)], can be restated as

$|\mathbb{A}^{\alpha\beta}[h]|_{N-2}\lesssim\delta^{-2}(\epsilon_{\star}+C_{1}\epsilon)^{2}r^{-2\kappa}s^{2\delta}.$
(5.21)

Then the statement in [30, (LABEL:eq3-19-04-2021)] still holds. Furthermore,
thanks to (3.1a) and (3.8), we have

$r|\Xi^{\mathcal{N}00}|_{N-4}+r|u_{\textbf{init}}^{\mathcal{N}00}|_{N-4}\lesssim(\epsilon_{\star}+C_{1}\epsilon).$

Then we find

$(1+r+t)\,|h^{\mathcal{N}00}|_{p,k}\lesssim\langle
r\rangle\,|u_{\textbf{sour}}|_{p,k}+(\epsilon_{\star}+C_{1}\epsilon),\qquad
p\leq N-4.$ (5.22)

The only issue is to estimate $r|u_{\textbf{sour}}|_{p,k}$ and, for this
purpose, we need to bound $|\Box u|_{p,k}$. The following result is analogous
to [30, Lemma LABEL:lem1-25-04-2021].

######  Lemma 5.8.

Under the condition (5.10) so that (5.11) and (5.12) hold, one has

$\displaystyle|\Box u|_{N-4}$
$\displaystyle\lesssim\ell^{-1}\delta^{-1}(\epsilon_{\star}+C_{1}\epsilon)^{2}t^{-1+(3/2)\delta}r^{-1-\kappa}\mathbf{X}^{-1+(1-\kappa)}+\delta^{-1}(\epsilon_{\star}+C_{1}\epsilon)^{2}r^{-2-3\delta}\mathbf{X}^{-1+\delta}\quad\text{
in }\mathscr{M}^{\textbf{far}}_{\ell,[s_{0},s_{1}]}$

and

$\displaystyle|\Box u|_{N-4}$
$\displaystyle\lesssim(\epsilon_{\star}+C_{1}\epsilon)r^{-2+\delta}\mathbf{X}^{-1-\kappa}(r\,|h^{\mathcal{N}00}|_{N-4})+\delta^{-1}(\epsilon+C_{1}\epsilon)^{2}r^{-2-3\delta}\mathbf{X}^{-1+\delta}\quad\text{
in }\mathscr{M}^{\textbf{near}}_{\ell,[s_{0},s_{1}]}.$

###### Proof.

As in the proof of [30, Lemma LABEL:lem1-25-04-2021], we need to establish
pointwise bounds on the following quantities:

$\displaystyle|h^{\mu\nu}\partial_{\mu}\partial_{\nu}u|_{k},\qquad|u^{\mu\nu}\partial_{\mu}\partial_{\nu}h^{\star}|_{k},\qquad|\mathbb{P}^{\star}[u]|_{k},\qquad|\mathbb{Q}^{\star}[u]|_{k},$
(5.23)
$\displaystyle\qquad|\mathbb{I}^{\star}[u]|_{k},\qquad|\mathbb{T}[\phi]|_{k},\qquad|^{(w)}R^{\star}_{\alpha\beta}|_{k}.$

The estimates established in the general theory are sufficient for all but the
first term and, for convenience, we recall the previous bounds as follows:

$\displaystyle|\mathbb{P}^{\star}[u]|_{N-4}+|\mathbb{Q}^{\star}[u]|_{N-4}\lesssim(\epsilon_{\star}+C_{1}\epsilon)^{2-4\delta}r^{-2-3\delta}\mathbf{X}^{-1+\delta},$
(5.24)
$\displaystyle|\mathbb{I}^{\star}[u]|_{N-4}+|u^{\mu\nu}\partial_{\mu}\partial_{\nu}h^{\star}|_{N-4}\lesssim\delta^{-1}(\epsilon_{\star}+C_{1}\epsilon)^{2}r^{-2-3\delta}\mathbf{X}^{-1+\delta},$
$\displaystyle|\mathbb{T}[\phi]|_{N-4}\lesssim(\epsilon_{\star}+C_{1}\epsilon)^{2}\,r^{-2-3\delta}\mathbf{X}^{-1+\delta},$

and $|^{(w)}R^{\star}_{\alpha\beta}|_{k}$ is bounded by (3.1b), which is
stronger than the inequality [30, (LABEL:eq5-25-04-2021)] applied in the proof
of [30, Lemma LABEL:lem1-25-04-2021]. We thus focus on the first term
$|h^{\mu\nu}\partial_{\mu}\partial_{\nu}u|_{k}$ and, first of all, we have

$|h^{\mu\nu}\partial_{\mu}\partial_{\nu}u_{\alpha\beta}|_{k}\lesssim|h^{\mathcal{N}00}\partial_{t}\partial_{t}u|_{k}+|H\,\partial\not{\partial}^{\mathcal{N}}u|_{k}+r^{-1}|H\,\partial
u|_{N-4}.$ (5.25)

The second term is bounded as, thanks to (4.8) and (5.11) and provided
$\kappa-1/2\geq(3/2)\delta$,

$|H\partial\not{\partial}^{\mathcal{N}}u|_{N-4}\lesssim\delta^{-1}(\epsilon_{\star}+C_{1}\epsilon)^{2}r^{-3}\mathbf{X}^{1-2\kappa}s^{3\delta}\lesssim\delta^{-1}(\epsilon_{\star}+C_{1}\epsilon)^{2}r^{-2-3\delta}\mathbf{X}^{-1+\delta}\qquad\text{in
}\mathscr{M}^{\textbf{near}}_{[s_{0},s_{1}]}.$

On the other hand, to handle the “far” region we rely on (5.12) in combination
with (5.9) which leads us to the relevant estimate that is stated in Lemma
5.8.

The last term in the right-hand side of (5.25) is trivial thanks to the
decreasing factor $r^{-1}$:

$r^{-1}|H\,\partial
u|_{N-4}\lesssim\delta^{-1}(\epsilon_{\star}+C_{1}\epsilon)^{2}r^{-2-\kappa}\mathbf{X}^{-\kappa}s^{\delta}\lesssim\delta^{-1}(\epsilon_{\star}+C_{1}\epsilon)^{2}r^{-2-3\delta}\mathbf{X}^{-1+\delta},$

provided $\kappa-1/2\geq(5/4)\delta$. For the first term in the right-hand
side (5.25), we first apply [30, (LABEL:eq1-15-08-2021)] and (4.9)

$|h^{\mathcal{N}00}\partial_{t}\partial_{t}u|_{N-4}\lesssim\ell^{-1}\delta^{-1}(\epsilon_{\star}+C_{1}\epsilon)^{2}t^{-1+(3/2)\delta}r^{-1-\kappa}\mathbf{X}^{-1+(1-\kappa)}\qquad\text{
in }\mathscr{M}^{\textbf{far}}_{[s_{0},s_{1}]}.$

We have arrived at the first inequality. On the other hand, we rely on (5.10)
and apply (5.11) and obtain

$|h^{\mathcal{N}00}\partial_{t}\partial_{t}u|_{N-4}\lesssim(\epsilon_{\star}+C_{1}\epsilon)r^{-2+\delta}\mathbf{X}^{-1-\kappa}\big{(}r\,|h^{\mathcal{N}00}|_{N-4}\big{)}\qquad\text{in
}\mathscr{M}^{\textbf{near}}_{[s_{0},s_{1}]}.$

and, in combination with the above inequalities, we arrive at the second
inequality. ∎

###### Proof of Proposition 5.6.

Step 1. We consider (5.14) first. We first establish the bound in
$\mathscr{M}^{\textbf{far}}_{\ell,[s_{0},s_{1}]}$, which is achieved by a
direct application of Proposition 5.7 in combination with Lemma 5.8 and
(5.22). We observe that $\mathscr{M}^{\textbf{far}}_{\ell,[s_{0},s_{1}]}$ is
past complete, in the sense that for any
$(t,x)\in\mathscr{M}^{\textbf{far}}_{\ell,[s_{0},s_{1}]}$ the past cone
$\Lambda_{(t,x)}:=\\{(\tau,y)\in\mathscr{M}_{[s_{0},s_{1}]}||y-x|\leq
t-\tau\\}$ is entirely contained in
$\mathscr{M}^{\textbf{far}}_{\ell,[s_{0},s_{1}]}$. Then by Proposition 5.7
Case 1 ($\upsilon=(3/2)\delta$, $\mu=1-\kappa$, $\nu=\kappa$ ) and Case 2
($\mu=\delta$, $\nu=3\delta$), we obtain (under the condition
$\kappa-1/2\geq(5/4)\delta$)

$|u_{\textbf{sour}}|_{N-4}\lesssim\ell^{-1}\delta^{-2}(\epsilon_{\star}+C_{1}\epsilon)^{2}(t+r)^{-1}+\delta^{-1}(\epsilon_{\star}+C_{1}\epsilon)^{2}(t+r)^{-1},$

which leads us to

$|u_{\textbf{sour}}|_{N-4}\lesssim\ell^{-1}\delta^{-2}(\epsilon_{\star}+C_{1}\epsilon)^{2}\langle
r\rangle^{-1}.$ (5.26)

Then recalling (5.22), we obtain (5.14) in
$\mathscr{M}^{\textbf{far}}_{\ell,[s_{0},s_{1}]}$.

Then we consider the region $\mathscr{M}^{\textbf{near}}_{\ell,[s_{0},s_{1}]}$
and define

$s^{*}=\sup_{s^{\prime}\in[s_{0},s_{1}]}\big{\\{}\text{\eqref{eq1-30-05-2022}
is valid in }\mathscr{M}^{\textbf{near}}_{\ell,[s_{0},s^{\prime}]}\big{\\}}.$
(5.27)

Provided $(\epsilon_{\star}+C_{1}\epsilon)$ is sufficiently small, we have
$s^{*}>s_{0}$ and for all
$(t,x)\in\mathscr{M}^{\textbf{good}}_{\ell,[s_{0},s^{*}]}$

$\Lambda_{(t,x)}=\\{(\tau,y)\in\mathscr{M}_{[s_{0},s_{1}]}||y-x|\leq
t-\tau\\}\subset\mathscr{M}^{\textbf{good}}_{[s_{0},s^{*}]}.$

We also have

$\mathbf{X}^{-8\delta}\lesssim(\epsilon_{\star}+C_{1}\epsilon)^{-8\delta}r^{-4\delta}\qquad\text{
in
}\mathscr{M}^{\textbf{good}}_{[s_{0},s_{1}]}\cap\mathscr{M}^{\textbf{near}}_{\ell,[s_{0},s_{1}]}.$

Thus by Lemma 5.8, in $\mathscr{M}^{\textbf{good}}_{[s_{0},s^{*}]}$,

$\displaystyle|\Box u|_{N-4}$
$\displaystyle\lesssim\ell^{-1}\delta^{-1}(\epsilon_{\star}+C_{1}\epsilon)^{2}t^{-1+(3/2)\delta}r^{-1-\kappa}\mathbf{X}^{-1+(1-\kappa)}$
(5.28)
$\displaystyle\quad+(\delta^{-1}(\epsilon_{\star}+C_{1}\epsilon)^{2}+(\epsilon_{\star}+C_{1}\epsilon)^{1-8\delta}r\,|h^{\mathcal{N}00}|_{N-4})r^{-2-3\delta}\mathbf{X}^{-1+\delta}$

provided $7\delta\leq\kappa$. (Here $C$ is a constant determined by $N$.) Then
by Proposition 5.7 Case 1 and Case 2, we obtain

$|u_{\textbf{sour}}|_{N-4}\lesssim\ell^{-1}\delta^{-2}(\epsilon_{\star}+C_{1}\epsilon)^{2}\langle
r\rangle^{-1}+\delta^{-1}(\epsilon_{\star}+C_{1}\epsilon)^{1-8\delta}\langle
r\rangle^{-1}\sup_{\mathscr{M}^{\textbf{good}}_{[s_{0},s^{*}]}}\big{(}r\,|h^{\mathcal{N}00}|_{N-4}\big{)}.$

Recalling (5.22), we obtain

$\langle
r\rangle|h^{\mathcal{N}00}|_{N-4}\lesssim(\epsilon_{\star}+C_{1}\epsilon)+\ell^{-1}\delta^{-1}(\epsilon_{\star}+C_{1}\epsilon)^{2}+\delta^{-1}(\epsilon_{\star}+C_{1}\epsilon)^{1-8\delta}\sup_{\mathscr{M}^{\textbf{good}}_{[s_{0},s^{*}]}}\Big{(}r\,|h^{\mathcal{N}00}|_{N-4}\Big{)},$

which leads us to (provided $(\epsilon_{\star}+C_{1}\epsilon)$ is sufficiently
small, as determined by $N$)

$\langle
r\rangle|h^{\mathcal{N}00}|_{N-4}\lesssim(\epsilon_{\star}+C_{1}\epsilon)+\ell^{-1}\delta^{-2}(\epsilon_{\star}+C_{1}\epsilon)^{2}\quad\text{in
}\mathscr{M}^{\textbf{good}}_{[s_{0},s^{*}]}.$ (5.29)

Next we consider the region
$\mathscr{M}^{\textbf{bad}}_{[s_{0},s^{*}]}\cap\mathscr{M}^{\textbf{near}}_{\ell,[s_{0},s_{1}]}$
and we rely on a technique of integration toward the light cone. We observe
that for $(t,x)\in\mathscr{M}^{\textbf{bad}}_{[s_{0},s^{*}]}$, we denote by
$(t,\bar{x})\in\mathscr{M}^{\textbf{good}}_{[s_{0},s^{*}]}\cap\mathscr{M}^{\textbf{bad}}_{[s_{0},s^{*}]}$
with $x/|x|=\bar{x}/|\bar{x}|$. Then, for $\textbf{ord}(Z)\leq N-4$ we have

$Zh^{\mathcal{N}00}(t,x)=Zh^{\mathcal{N}00}(t,\bar{x})-\int_{|x|}^{|\bar{x}|}\partial^{\mathcal{E\hskip-1.70709ptH}}_{r}Zh^{\mathcal{N}00}(t,\rho
x/|x|)d\rho,$

where
$\partial^{\mathcal{E\hskip-1.70709ptH}}_{r}=(x^{a}/r)\partial^{\mathcal{E\hskip-1.70709ptH}}_{a}$.
Then by (5.29) and (4.23),

$\displaystyle\langle r\rangle|Zh^{\mathcal{N}00}|$
$\displaystyle\lesssim(\epsilon_{\star}+C_{1}\epsilon)+\ell^{-1}\delta^{-2}(\epsilon_{\star}+C_{1}\epsilon)^{2}+\langle
t\rangle\big{|}|\bar{x}|-|x|\big{|}\sup_{\mathscr{M}^{\mathcal{EM}}_{[s_{0},s^{*}]}}\\{|\partial
Zh^{\mathcal{N}00}|_{N-4}\\}$
$\displaystyle\lesssim(\epsilon_{\star}+C_{1}\epsilon)+\ell^{-1}\delta^{-2}(\epsilon_{\star}+C_{1}\epsilon)^{2}+(\epsilon_{\star}+C_{1}\epsilon)^{2}\langle
t\rangle^{1/2-\kappa+\delta/2}.$

Then, combining with (5.29) we have

$\sup_{\mathscr{M}^{\mathcal{EM}}_{[s_{0},s^{*}]}}\big{\\{}\langle
r\rangle|h^{\mathcal{N}00}|_{N-4}\big{\\}}\lesssim(\epsilon_{\star}+C_{1}\epsilon)+\ell^{-1}\delta^{-2}(\epsilon_{\star}+C_{1}\epsilon)^{2}.$
(5.30)

Recalling (5.27), we observe that when $(\epsilon_{\star}+C_{1}\epsilon)$ is
sufficiently small, the above bound leads us to

$r\,|h^{\mathcal{N}00}|_{N-4}<\frac{1}{2}\epsilon_{s}\quad\text{in
}\mathscr{M}^{\textbf{near}}_{\ell,[s_{0},s^{*}]}.$ (5.31)

By continuity, we conclude that $s^{*}=s_{1}$ since, otherwise,
$\sup_{\mathscr{M}^{\textbf{good}}_{s^{*}}}\Big{(}r\,|h^{\mathcal{N}00}|_{N-4}\Big{)}=\epsilon_{s}$
would contradict (5.31). Next, (5.14) follows from (5.30), provided
$\ell^{-1}\delta^{-1}(\epsilon_{\star}+C_{1}\epsilon)\lesssim 1$. This
completes the proof of (5.14).

Step 2. We now turn to (5.15), which, in some sense, a special case of (5.14)
but also provides us with a key sign. We recall the following identity
(established in [30, (LABEL:eq2-19-04-2021)])

$\displaystyle rh^{\mathcal{N}00}$
$\displaystyle=rg^{\star}(\mathbf{l},\mathbf{l})+ru_{\textbf{init}}(\mathbf{l},\mathbf{l})+r\,h^{\mathcal{N}00}_{\textbf{pertur}}$
$\displaystyle=r\big{(}g^{\star}(\mathbf{l},\mathbf{l})+u_{\textbf{init}}(\mathbf{l},\mathbf{l})\big{)}-r\sum_{\alpha,\beta}{\Psi^{\mathcal{N}}{}}_{\alpha}^{0}{\Psi^{\mathcal{N}}{}}_{\beta}^{0}u_{\textbf{sour},\alpha\beta}+r{\Psi^{\mathcal{N}}{}}_{\alpha}^{0}{\Psi^{\mathcal{N}}{}}_{\beta}^{0}\mathbb{A}^{\alpha\beta}[h].$

Thanks to (3.7) and (5.21), we find
$rh^{\mathcal{N}00}\leq-\epsilon_{\star}+Cr|u_{\textbf{sour}}|+\delta^{-2}(\epsilon_{\star}+C_{1}\epsilon)^{2}r^{1-2\kappa+\delta}.$
Substituting (5.14) in (5.28) we obtain

$|\Box
u|\lesssim\ell^{-1}\delta^{-1}(\epsilon_{\star}+C_{1}\epsilon)^{2}t^{-1+(3/2)\delta}r^{-1-\kappa}\mathbf{X}^{-1+(1-\kappa)}+\delta^{-1}(\epsilon_{\star}+C_{1}\epsilon)^{2-8\delta}r^{-2-3\delta}\mathbf{X}^{-1+\delta}\quad\text{in
}\mathscr{M}^{\textbf{good}}_{[s_{0},s_{1}]}.$ (5.32)

By Proposition 5.7 (Case 1 and Case 2), we obtain

$\langle
r\rangle|u_{\textbf{sour}}|\lesssim\ell^{-1}\delta^{-2}(\epsilon_{\star}+C_{1}\epsilon)^{2-8\delta}\qquad\text{in
}\mathscr{M}^{\textbf{good}}_{[s_{0},s_{1}]},$

which leads us to (for some constant $C>0$)

$rh^{\mathcal{N}00}\leq-\epsilon_{\star}+C\ell^{-1}\delta^{-2}(\epsilon_{\star}+C_{1}\epsilon)^{2-8\delta}\qquad\text{in
}\mathscr{M}^{\textbf{good}}_{[s_{0},s_{1}]}.$ (5.33)

Again, we perform an integration towards the light-cone in order to cover
$\mathscr{M}^{\textbf{bad}}_{[s_{0},s_{1}]}$:

$\displaystyle h^{\mathcal{N}00}(t,x)$
$\displaystyle=h^{\mathcal{N}00}(t,\bar{x})-\int_{|x|}^{|\bar{x}|}\partial^{\mathcal{E\hskip-1.70709ptH}}_{r}h^{N00}(t,\rho
x/|x|)\,d\rho\leq
h^{\mathcal{N}00}(t,\bar{x})+C(\epsilon_{\star}+C_{1}\epsilon)^{2}t^{-1}$
$\displaystyle\leq-\epsilon_{\star}|\bar{x}|^{-1}+C\ell^{-1}\delta^{-2}(\epsilon_{\star}+C_{1}\epsilon)^{2-8\delta}|\bar{x}|^{-1}+C(\epsilon_{\star}+C_{1}\epsilon)^{2}t^{-1}$
$\displaystyle\leq-\epsilon_{\star}r^{-1}+C\big{(}(\epsilon_{\star}+C_{1}\epsilon)^{2}+\ell^{-1}\delta^{-2}(\epsilon_{\star}+C_{1}\epsilon)^{2-8\delta}\big{)}r^{-1}\hskip
56.9055pt\text{ in }\mathscr{M}^{\textbf{bad}}_{[s_{0},s_{1}]}.$

We recall that $\epsilon\leq C_{\star}\epsilon_{\star}$ with $C_{\star}$ a
given constant, thus when $\epsilon_{\star}$ sufficiently small, the above
inequality together with (5.33) leads us to (5.15). ∎

###  5.3 Sharp decay for the gradient of good metric components

#### Objective and strategy.

We now turn our attention to the estimates for $\partial\not{u}$.

######  Proposition 5.9.

Under the bootstrap assumption in $[s_{0},s_{1}]$ and as a consequence of
(5.14), one has in $\mathscr{M}^{\textbf{near}}_{\ell,[s_{0},s_{1}]}$

$|\partial\not{u}|_{N-4,k}\lesssim\big{(}\ell^{-\delta/2}+\delta^{-2}\big{)}(\epsilon_{\star}+C_{1}\epsilon)\langle
r\rangle^{-1}\mathbf{X}^{-1/2-\delta/2}\big{(}1+(\epsilon_{\star}+C_{1}\epsilon)(\ln\langle
r\rangle)^{k}\big{)},\qquad 0\leq k\leq N-4,$ (5.34)
$|\partial\partial\not{u}|_{N-5,k}\lesssim(\ell^{-\delta}+\delta^{-2})(\epsilon_{\star}+C_{1}\epsilon)\langle
r\rangle^{-1}\mathbf{X}^{-1-\delta}\big{(}1+(\epsilon_{\star}+C_{1}\epsilon)(\ln\langle
r\rangle)^{k}\big{)},\qquad 0\leq k\leq N-5.$ (5.35)

Our proof relies on the following result, which was pointed out first in [35]
and formulated as follows in [30].

######  Proposition 5.10 (Weighted pointwise estimate in the Euclidean-merging
domain).

Consider a metric
$g^{\alpha\beta}=g_{\textbf{Mink}}^{\alpha\beta}+H^{\alpha\beta}$ defined in
$\mathscr{M}^{\textbf{near}}_{\ell,[s_{0},s_{1}]}$ and satisfying

$|H^{\mathcal{N}}{}^{00}|\ll 1,\qquad H^{\mathcal{N}}{}^{00}\leq 0\quad\text{
in }\mathscr{M}^{\textbf{near}}_{\ell,[s_{0},s_{1}]}.$ (5.36)

Given any $\rho\geq 0$, for any function $u$ defined
$\mathscr{M}^{\textbf{near}}_{\ell,[s_{0},s_{1}]}$ one has

$\displaystyle\mathbf{X}^{\rho}|(\partial_{t}-\partial_{r})(ru)|(t,x)$
$\displaystyle\lesssim\sup_{\Omega^{\ell}_{s_{0},s_{1}}}(r-t+2)^{\rho}\big{(}r\,|\partial
u|+|u|\big{)}+\int_{t_{0}}^{t}\mathbf{X}(\tau,r)^{\rho}\,r\Big{(}r^{-1}|\not{\partial}^{\mathcal{N}}u|_{1,1}+|H|\,|\partial\not{\partial}^{\mathcal{N}}u|+r^{-1}|H|\,|\partial
u|+|\widetilde{\Box}_{g}u|\Big{)}\big{|}_{\varphi_{t,x}(\tau)}d\tau,$

in which the supremum is taken over the set
$\Omega_{s_{0},s_{1}}^{\ell}=\mathscr{L}_{\ell,[s_{0},s_{1}]}\cup\mathscr{M}^{\textbf{near}}_{\ell,s_{0}}$.

Our strategy is to apply the above estimate to the wave equations satisfied by
the components $Z\not{u}^{\mathcal{N}}_{\alpha\beta}$. More precisely, we
observe that

$\widetilde{\Box}_{g}u^{\mathcal{N}}_{\alpha\beta}=L_{1\alpha\beta}+L_{2\alpha\beta}+S_{1\alpha\beta},$
(5.37)

where
$u^{\mathcal{N}}_{\alpha\beta}:={\Psi^{\mathcal{N}}{}}^{\alpha^{\prime}}_{\alpha}{\Psi^{\mathcal{N}}{}}^{\beta^{\prime}}_{\beta}u_{\alpha^{\prime}\beta^{\prime}}$
and

$\displaystyle
L_{1\alpha\beta}:=u_{\alpha^{\prime}\beta^{\prime}}\widetilde{\Box}_{g}\big{(}{\Phi^{\mathcal{N}}}{}_{\alpha}^{\alpha^{\prime}}{\Phi^{\mathcal{N}}}{}_{\beta}^{\beta^{\prime}}\big{)},$
$\displaystyle
L_{2\alpha\beta}:=g^{\mu\nu}\partial_{\mu}\big{(}{\Phi^{\mathcal{N}}}{}_{\alpha}^{\alpha^{\prime}}{\Phi^{\mathcal{N}}}{}_{\beta}^{\beta^{\prime}}\big{)}\partial_{\nu}u_{\alpha^{\prime}\beta^{\prime}},\qquad$
$\displaystyle
S_{1\alpha\beta}:={\Phi^{\mathcal{N}}}{}_{\alpha}^{\alpha^{\prime}}{\Phi^{\mathcal{N}}}{}_{\beta}^{\beta^{\prime}}\widetilde{\Box}_{g}u_{\alpha^{\prime}\beta^{\prime}}.$

For any $Z=\partial^{I}L^{J}\Omega^{K}$ with $\textbf{ord}(Z)\leq N-4$, we
then write

$\displaystyle\widetilde{\Box}_{g}\big{(}Zu^{\mathcal{N}}_{\alpha\beta}\big{)}=ZL_{1\alpha\beta}+ZL_{2\alpha\beta}+ZS_{1\alpha\beta}+S_{2\alpha\beta},\qquad$
$\displaystyle
S_{2\alpha\beta}=S_{2\alpha\beta}[u]:=-[Z,h^{\mu\nu}\partial_{\mu}\partial_{\nu}]u^{\mathcal{N}}_{\alpha\beta},$
(5.38)
$\widetilde{\Box}_{g}(Z\partial_{t}u^{\mathcal{N}}_{\alpha\beta})=Z\partial_{t}\big{(}L_{1\alpha\beta}+L_{2\alpha\beta}+S_{1\alpha\beta}\big{)}+S^{\prime}_{2\alpha\beta},\qquad
S^{\prime}_{2\alpha\beta}=S^{\prime}_{2\alpha\beta}[u]:=-[Z\partial_{t},h^{\mu\nu}\partial_{\mu}\partial_{\nu}]u^{\mathcal{N}}_{\alpha\beta}.$
(5.39)

In order to eventually control $Z\not{u}^{\mathcal{N}}_{\alpha\beta}$, we need
to estimate the source terms in the right-hand side, which is our main task in
the rest of this Section 5.3.

#### Estimates on source terms.

We are in a position to establish the following result.

######  Proposition 5.11.

As a consequence of (5.14), for all $\textbf{ord}(Z)\leq N-4$ and
$\textbf{rank}(Z)=k\leq N-4$ one has

$\displaystyle\mathbf{X}^{1/2+\delta/2}r|\widetilde{\Box}_{g}Z\not{u}^{\mathcal{N}}|\lesssim(\epsilon_{\star}+C_{1}\epsilon)r^{-1}\big{(}r\mathbf{X}^{1/2+\delta/2}|\partial\not{u}^{\mathcal{N}}|_{N-4,k-1}\big{)}+\delta^{-1}(\epsilon_{\star}+C_{1}\epsilon)r^{-1-\delta},$
(5.40a) while for all $\textbf{ord}(Z)\leq N-5$ and $\textbf{rank}(Z)=k\leq
N-5$ one has
$r\,\mathbf{X}^{1+\delta}|\widetilde{\Box}_{g}Z\partial_{t}\not{u}^{\mathcal{N}}|\lesssim(\epsilon_{\star}+C_{1}\epsilon)r^{-1}\big{(}r\mathbf{X}^{\kappa}|\partial\partial\not{u}^{\mathcal{N}}|_{N-5,k-1}\big{)}+\delta^{-1}(\epsilon_{\star}+C_{1}\epsilon)r^{-1-\delta}.$
(5.40b)

###### Proof.

The proof is similar to that in [30, Proposition LABEL:proposition-label-144].
The only difference concerns the estimates for $S_{2\mu\nu}[u]$ and
$S^{\prime}_{2\mu\nu}[u]$, while the remaining terms are bounded by
$\delta^{-1}(\epsilon_{\star}+C_{1}\epsilon)r^{-1-\delta}$. Recalling
Proposition 2.1 together with (5.11) and (5.14) (which guarantees (5.10)), for
$(\alpha,\beta)\neq(0,0)$ we find

$\displaystyle|S_{2\alpha\beta}|_{p,k}$
$\displaystyle\lesssim(\epsilon_{\star}+C_{1}\epsilon)\langle
r\rangle^{-1}|\partial\partial\not{u}^{\mathcal{N}}|_{p-1,k-1}+\delta^{-1}(\epsilon_{\star}+C_{1}\epsilon)^{2}\langle
r\rangle^{-2-\kappa}\mathbf{X}^{-\kappa}s^{3\delta},\quad$ $\displaystyle
k\leq p\leq N-4$ (5.41) $\displaystyle|S^{\prime}_{2\alpha\beta}|_{p,k}$
$\displaystyle\lesssim(\epsilon_{\star}+C_{1}\epsilon)\langle
r\rangle^{-1}|\partial\partial\partial_{t}\not{u}^{\mathcal{N}}|_{p-1,k-1}+\delta^{-1}(\epsilon_{\star}+C_{1}\epsilon)^{2}\langle
r\rangle^{-2-\kappa}\mathbf{X}^{-\kappa}s^{3\delta},\quad$ $\displaystyle
k\leq p\leq N-5.$

Hence, (5.40a) and (5.40b) are established. ∎

#### Proof of Proposition 5.9.

Relying on the notation in Proposition 5.10, we observe that by (4.6) and
(5.11) expressed on the relevant cone $r=t(1-\ell)$ or $r-t\simeq\ell\,r$

$\displaystyle\sup_{\Omega_{s_{0},s_{1}}^{\ell}}\mathbf{X}^{1/2+\delta/2}\big{(}r\,|\partial
Z\not{u}^{\mathcal{N}}|+|Z\not{u}^{\mathcal{N}}|\big{)}$
$\displaystyle\lesssim\big{(}\ell^{-\delta/2}+\delta^{-1}\big{)}(\epsilon_{\star}+C_{1}\epsilon),$
(5.42)
$\displaystyle\sup_{\Omega^{\ell}_{s_{0},s_{1}}}\mathbf{X}^{1+\delta}\big{(}r\,|\partial
Z\partial_{t}\not{u}^{\mathcal{N}}|+|Z\partial_{t}\not{u}^{\mathcal{N}}|\big{)}$
$\displaystyle\lesssim\ell^{-\delta}(\epsilon_{\star}+C_{1}\epsilon).$

Then for $(t,x)\in\mathscr{M}^{\textbf{near}}_{\ell,[s_{0},s_{1}]}$ and
$\textbf{ord}(Z)=N-4,\textbf{rank}(Z)=k$, after observing that (5.36) are
guaranteed by (3.11) and (5.9), we obtain

$\displaystyle\mathbf{X}^{1/2+\delta/2}|(\partial_{t}-\partial_{r})(rZ\not{u}^{\mathcal{N}})(t,x)|\lesssim(\ell^{-\delta/2}+\delta^{-1})(\epsilon_{\star}+C_{1}\epsilon)+\int_{t_{0}}^{t}r\langle
r-\tau\rangle^{1/2+\delta/2}|\widetilde{\Box}_{g}Z\not{u}^{\mathcal{N}}|_{\varphi_{t,x}(\tau)}d\tau$
$\displaystyle\hskip 142.26378pt+\int_{t_{0}}^{t}\langle
r-\tau\rangle^{1/2+\delta/2}\Big{(}|\not{\partial}^{\mathcal{N}}\not{u}^{\mathcal{N}}|_{N-3}+r|H||\partial\not{\partial}^{\mathcal{N}}Z\not{u}^{\mathcal{N}}|+|H||\partial\not{u}^{\mathcal{N}}|_{N-4}\Big{)}\Big{|}_{\varphi_{t,x}(\tau)}d\tau$
$\displaystyle\lesssim(\ell^{-\delta/2}+\delta^{-1})(\epsilon_{\star}+C_{1}\epsilon)+\int_{t_{0}}^{t}(\epsilon_{\star}+C_{1}\epsilon)r^{-1}\mathbf{X}^{1/2+\delta/2}r|\partial\not{u}^{\mathcal{N}}|_{N-4,k-1}\Big{|}_{\varphi_{t,x}(\tau)}d\tau+\delta^{-1}(\epsilon_{\star}+C_{1}\epsilon)\int_{t_{0}}^{t}\tau^{-1-\delta}d\tau$
$\displaystyle\lesssim\big{(}\ell^{-\delta/2}+\delta^{-2}\big{)}(\epsilon_{\star}+C_{1}\epsilon)+(\epsilon_{\star}+C_{1}\epsilon)\int_{t_{0}}^{t}\tau^{-1}\mathbf{X}^{1/2+\delta/2}r|\partial\not{u}^{\mathcal{N}}|_{N-4,k-1}\Big{|}_{\varphi_{t,x}(\tau)}d\tau,$

where we used333$\not{u}^{\mathcal{N}}$ is a finite linear combination of $u$
with homogeneous coefficients of degree zero, thus the bounds can be applied.
(5.41), (4.6), and (4.4a). On the other hand, observe that
$|Zu|\lesssim\delta^{-1}(\epsilon_{\star}+C_{1}\epsilon)r^{-1}\mathbf{X}^{1-\kappa}s^{\delta}\lesssim\delta^{-1}(\epsilon_{\star}+C_{1}\epsilon)r^{\delta/2-\kappa}\lesssim(\epsilon_{\star}+C_{1}\epsilon),$
which we apply with $u$ replaced by $\not{u}^{\mathcal{N}}$. This leads us to

$\displaystyle
r\,\big{|}(\partial_{t}-\partial_{r})Z\not{u}^{\mathcal{N}}_{\alpha\beta}\,\big{|}\lesssim(\epsilon_{\star}+C_{1}\epsilon)+|(\partial_{t}-\partial_{r})(rZ\not{u}^{\mathcal{N}})|.$
(5.43)

Now recalling (4.6), for all $\textbf{ord}(Z)\leq N-4$ we have
$r\,|\not{\partial}^{\mathcal{N}}Zu|\lesssim
C_{1}\epsilon\,r^{-\kappa+\delta/2}\lesssim(\epsilon_{\star}+C_{1}\epsilon)$.
Recalling the identities
$2\partial_{t}=(x^{a}/r)\not{\partial}^{\mathcal{N}}_{a}+(\partial_{t}-\partial_{r})$
and $\partial_{a}=\not{\partial}^{\mathcal{N}}_{a}-(x^{a}/r)\partial_{t}$,
together with the ordering lemma [30, Lemma LABEL:lem_2_high-order], we obtain
the following bound in $\mathscr{M}^{\textbf{near}}_{\ell,[s_{0},s_{1}]}$:

$\displaystyle
r\,\mathbf{X}^{1/2+\delta/2}|\partial\not{u}^{\mathcal{N}}|_{N-4,k}$
$\displaystyle\lesssim\big{(}\ell^{-\delta/2}+\delta^{-2}\big{)}(\epsilon_{\star}+C_{1}\epsilon)+(\epsilon_{\star}+C_{1}\epsilon)\int_{t_{0}}^{t}\tau^{-1}r\,\mathbf{X}^{1/2+\delta/2}|\partial\not{u}^{\mathcal{N}}|_{N-4,k-1}\Big{|}_{\varphi_{t,x}(\tau)}d\tau.$

Finally, with the notation
$\mathbf{B}_{k}(t):=\sup_{\mathscr{M}^{\textbf{near}}_{\ell,[s_{0},s]}}\big{(}r\,\mathbf{X}^{1/2+\delta/2}|\partial\not{u}^{\mathcal{N}}|_{N-4,k}\big{)}$
(with $t=T^{\mathcal{E}}(s)$), the above estimate reads

$\mathbf{B}_{k}(t)\lesssim\big{(}\ell^{-\delta/2}+\delta^{-2}\big{)}(\epsilon_{\star}+C_{1}\epsilon)+(\epsilon_{\star}+C_{1}\epsilon)\int_{t_{0}}^{t}\tau^{-1}\mathbf{B}_{k-1}(\tau)d\tau,$
(5.44)

in which the last term does not exist when $k=0$. Next, by induction on $k$
varying from $k=0$ to $k=N-4$, we conclude and arrive at (5.34). The estimate
(5.35) is established in a similar way, and we omit the details.

#### Useful inequalities.

For $k\geq 1$, we can relax (5.44) in the form

$\mathbf{B}_{k}(t)\lesssim\big{(}\ell^{-\delta/2}+\delta^{-2}\big{)}(\epsilon_{\star}+C_{1}\epsilon)+(\epsilon_{\star}+C_{1}\epsilon)\int_{t_{0}}^{t}\tau^{-1}\mathbf{B}_{k}(\tau)d\tau.$
(5.45)

By Gronwall’s inequality, we obtain the slightly weaker version

$|\partial\not{u}|_{N-4,k}\lesssim\begin{cases}\big{(}\ell^{-\delta/2}+\delta^{-2}\big{)}(\epsilon_{\star}+C_{1}\epsilon)\langle
r\rangle^{-1+C(\epsilon_{\star}+C_{1}\epsilon)}\mathbf{X}^{-1/2-\delta/2},\quad&1\leq
k\leq N-4,\\\
\big{(}\ell^{-\delta/2}+\delta^{-2}\big{)}(\epsilon_{\star}+C_{1}\epsilon)\langle
r\rangle^{-1}\mathbf{X}^{-1/2-\delta/2},\quad&k=0;\end{cases}$ (5.46a)
$|\partial\partial\not{u}|_{N-5,k}\lesssim\begin{cases}(\ell^{-\delta}+\delta^{-2})(\epsilon_{\star}+C_{1}\epsilon)\langle
r\rangle^{-1+C(\epsilon_{\star}+C_{1}\epsilon)}\mathbf{X}^{-1-\delta},\quad&1\leq
k\leq N-5,\\\
(\ell^{-\delta}+\delta^{-2})(\epsilon_{\star}+C_{1}\epsilon)\langle
r\rangle^{-1}\mathbf{X}^{-1-\delta},\quad&k=0,\end{cases}$ (5.46b)

###  5.4 Pointwise estimates for metric components at low order

#### Objective.

We now collect the pointwise estimates obtained in the previous sections for
the different metric components. The main result is stated as follows.

######  Proposition 5.12.

In $\mathscr{M}^{\textbf{near}}_{\ell,[s_{0},s_{1}]}$, the metric components
satisfy

$|\partial
h|_{N-4,k}\lesssim\begin{cases}\big{(}\ell^{-\delta/2}+\delta^{-2}\big{)}(\epsilon_{\star}+C_{1}\epsilon)\mathbf{X}^{-1/2-\delta/2}\langle
r\rangle^{-1},\quad&k=0,\\\
\big{(}\ell^{-\delta/2}+\delta^{-2}\big{)}(\epsilon_{\star}+C_{1}\epsilon)\mathbf{X}^{-1/2-\delta/2}\langle
r\rangle^{-1+C(\epsilon_{\star}+C_{1}\epsilon)},\quad&1\leq k\leq
N-4,\end{cases}$ (5.47) $|\partial\partial
h|_{N-5,k}\lesssim\begin{cases}\big{(}\ell^{-\delta}+\delta^{-2}\big{)}(\epsilon_{\star}+C_{1}\epsilon)\mathbf{X}^{-1-\delta}\langle
r\rangle^{-1},\quad&k=0,\\\
\big{(}\ell^{-\delta}+\delta^{-2}\big{)}(\epsilon_{\star}+C_{1}\epsilon)\mathbf{X}^{-1-\delta}\langle
r\rangle^{-1+C(\epsilon_{\star}+C_{1}\epsilon)},\quad&1\leq k\leq
N-5.\end{cases}$ (5.48)

Furthermore, the source terms $u_{\textbf{sour},\alpha\beta}$ defined in [30,
(LABEL:eq1-19-04-2021)] enjoys the near-harmonic decay in the whole domain
$\mathscr{M}^{\mathcal{EM}}_{[s_{0},s_{1}]}$:

$|u_{\textbf{sour}}|_{k}\lesssim(\epsilon_{\star}+C_{1}\epsilon)r^{-1+C(\epsilon_{\star}+C_{1}\epsilon)^{1/2}},\qquad
0\leq k\leq N-5.$ (5.49)

We also have estimates for the metric with upper indices, that is, $|\partial
H|:=\max_{\alpha,\beta}|\partial h^{\alpha\beta}|$.

######  Corollary 5.13.

In $\mathscr{M}^{\textbf{near}}_{\ell,[s_{0},s_{1}]}$, the metric perturbation
satisfies

$\displaystyle|\partial u|_{N-4,k}+|\partial H|_{N-4,k}$
$\displaystyle\lesssim\begin{cases}\big{(}\ell^{-\delta/2}+\delta^{-2}\big{)}(\epsilon_{\star}+C_{1}\epsilon)\mathbf{X}^{-1/2-\delta/2}\langle
r\rangle^{-1},\quad&k=0,\\\
\big{(}\ell^{-\delta/2}+\delta^{-2}\big{)}(\epsilon_{\star}+C_{1}\epsilon)\mathbf{X}^{-1/2-\delta/2}\langle
r\rangle^{-1+C(\epsilon_{\star}+C_{1}\epsilon)},\quad&1\leq k\leq
N-4,\end{cases}$ (5.50) $\displaystyle|\partial\partial
u|_{N-5,k}+|\partial\partial H|_{N-5,k}$
$\displaystyle\lesssim\begin{cases}\big{(}\ell^{-\delta}+\delta^{-2}\big{)}(\epsilon_{\star}+C_{1}\epsilon)\mathbf{X}^{-1-\delta}\langle
r\rangle^{-1},\quad&k=0,\\\
\big{(}\ell^{-\delta}+\delta^{-2}\big{)}(\epsilon_{\star}+C_{1}\epsilon)\mathbf{X}^{-1-\delta}\langle
r\rangle^{-1+C(\epsilon_{\star}+C_{1}\epsilon)},\quad&1\leq k\leq
N-4.\end{cases}$

Furthermore, as a consequence of (3.1a), in
$\mathscr{M}^{\mathcal{EM}}_{[s_{0},s_{1}]}$, one has

$|u|_{N-5}+|h_{\alpha\beta}|_{N-5}+|h^{\alpha\beta}|_{N-5}\lesssim(\epsilon_{\star}+C_{1}\epsilon)\langle
r\rangle^{-1+C(\epsilon_{\star}+C_{1}\epsilon)^{1/2}}.$ (5.51)

#### Sharp decay bounds on gradient and Hessian.

The following identities will be used.

######  Lemma 5.14.

When $|h|_{[p/2]}\ll 1$, one has
$h^{\mathcal{N}}_{00}=h_{00}=-\frac{1}{4}h^{\mathcal{N}00}+\mathbb{B}[h]$ with

$\big{|}\partial_{t}\mathbb{B}[h]\big{|}_{p,k}\lesssim|\partial_{t}\not{h}^{\mathcal{N}}|_{p,k}+|\partial\mathbb{A}[h]|_{p,k},\qquad\big{|}\partial_{t}\partial_{t}\mathbb{B}[h]\big{|}_{p,k}\lesssim|\partial_{t}\partial_{t}\not{h}^{\mathcal{N}}|_{p,k}+|\partial\partial\mathbb{A}[h]|_{p,k}$
(5.52)

with

$\displaystyle|\partial\mathbb{A}[h]|_{p,k}\lesssim\sum_{p_{1}+p_{2}=p\atop
k_{1}+k_{2}=k}|\partial h|_{p_{1},k_{1}}|h|_{p_{2},k_{2}},$
$\displaystyle|\partial\partial\mathbb{A}[h]|_{p,k}\lesssim\sum_{p_{1}+p_{2}=p\atop
k_{1}+k_{2}=k}\Big{(}|\partial\partial
h|_{p_{1},k_{1}}|h|_{p_{2},k_{2}}+|\partial h|_{p_{1},k_{1}}|\partial
h|_{p_{2},k_{2}}\Big{)}+\sum_{p_{1}+p_{2}+p_{3}=p\atop
k_{1}+k_{2}+k_{3}=k}|\partial h|_{p_{1},k_{1}}|\partial
h|_{p_{2},k_{2}}|h|_{p_{3},k_{3}}.$

###### Proof.

We only outline the proof and refer to [30] where we derived the algebraic
identity

$-h^{\mathcal{N}00}=h_{00}-\frac{2x^{a}}{r}h_{a0}+\frac{x^{a}x^{b}}{r^{2}}h_{ab}-{\Psi^{\mathcal{N}}{}}_{\alpha}^{0}{\Psi^{\mathcal{N}}{}}_{\beta}^{0}\mathbb{A}^{\alpha\beta}[h],$
(5.53a) with $\displaystyle h_{00}=h^{\mathcal{N}}_{00},\quad\qquad
h_{a0}=h^{\mathcal{N}}_{\alpha\beta}{\Psi^{\mathcal{N}}{}}_{a}^{\alpha}{\Psi^{\mathcal{N}}{}}_{0}^{\beta}=-(x^{a}/r)h^{\mathcal{N}}_{00}+\sum_{(\alpha,\beta)\neq(0,0)}{\Psi^{\mathcal{N}}{}}_{a}^{\alpha}{\Psi^{\mathcal{N}}{}}_{0}^{\beta}\not{h}^{\mathcal{N}}_{\alpha\beta},$
(5.53b) $\displaystyle
h_{ab}=\frac{x^{a}x^{b}}{r^{2}}h^{\mathcal{N}}_{00}+\sum_{(\alpha,\beta)\neq(0,0)}{\Psi^{\mathcal{N}}{}}_{a}^{\alpha}{\Psi^{\mathcal{N}}{}}_{b}^{\beta}\not{h}^{\mathcal{N}}_{\alpha\beta}.$

Substituting (5.53b) into (5.53a) and separating the linear terms from the
nonlinear ones, we obtain the result. ∎

#### Proof of (5.47).

Combining Lemma 5.14 with (4.7) and (4.9), we obtain

$|\partial\mathbb{A}|_{N-4}+|\partial\partial\mathbb{A}|_{N-5}\lesssim\delta^{-1}(\epsilon_{\star}+C_{1}\epsilon)^{2}r^{-1-\kappa}\mathbf{X}^{-\kappa}s^{2\delta}.$
(5.54)

Combining (4.23), Lemma 5.5, and the condition
$\delta^{-1}(\epsilon_{\star}+C_{1}\epsilon)\lesssim 1$, we thus find

$\displaystyle|\partial_{t}h^{\mathcal{N}}_{00}|_{N-4,k}$
$\displaystyle\lesssim|\partial_{t}\not{h}^{\mathcal{N}}|_{N-4,k}+\delta^{-1}(\epsilon_{\star}+C_{1}\epsilon)r^{-1-\kappa}s^{2\delta},$
$\displaystyle|\partial_{t}\partial_{t}h^{\mathcal{N}}_{00}|_{N-5,k}$
$\displaystyle\lesssim|\partial_{t}\partial_{t}\not{h}^{\mathcal{N}}|_{N-5,k}+(\epsilon_{\star}+C_{1}\epsilon)r^{-1-\kappa}\mathbf{X}^{-\kappa}s^{2\delta},$

together with

$\displaystyle|\partial_{t}^{m}h|_{p,k}$
$\displaystyle\lesssim|\partial^{m}\not{h}^{\mathcal{N}}|_{p,k}+|\partial_{t}^{m}h^{\mathcal{N}}_{00}|_{p,k}\lesssim|\partial^{m}\not{u}^{\mathcal{N}}|_{p,k}+|\partial^{m}h^{\star}|_{p,k}+|\partial_{t}^{m}h^{\mathcal{N}}_{00}|_{p,k}.$

In view of Proposition 5.9 and (3.1a), we arrive at (5.47) and (5.48).

#### Proof of (5.50).

We only outline the argument. For the bounds on $|\partial u|$ and
$|\partial\partial u|$, we only need to observe that
$h_{\alpha\beta}=h^{\star}_{\alpha\beta}+u_{\alpha\beta}$ together with
(3.1a). For the bounds on $|\partial H|$ and $|\partial\partial H|$, namely
$|\partial h^{\alpha\beta}|$ and $|\partial\partial h^{\alpha\beta}|$, we
recall that $-h^{\alpha\beta}=h_{\alpha\beta}+\mathbb{A}^{\alpha\beta}[h]$ and
apply (5.54).

#### Proof of (5.49).

We rely on Proposition 5.7, and the proof is quite similar to that of the
source in [30, Proposition LABEL:section-15-1 ], while only difference is that
here we have better pointwise bounds (5.47), (5.48) and (5.50). More
precisely, by (4.19), (4.20), (4.21) and (3.1b), the terms
$\mathbb{I}^{\star}[u]$, ${u^{\mu\nu}\partial_{\mu}\partial_{\nu}h^{\star}}$,
$\mathbb{T}[\phi]$ and ${{}^{(w)}R^{\star}_{\alpha\beta}}$ are bounded
(provided $\kappa\geq 5\delta$ and $\mu\geq 3/4+(7/4)\delta$) by

$\delta^{-1}(\epsilon_{\star}+C_{1}\epsilon)^{2}r^{-2-3\delta}\mathbf{X}^{-1+\delta}.$

We only need to improve the estimate of $|\Box u_{\alpha\beta}|_{N-5}$ in
$\mathscr{M}^{\textbf{near}}_{\ell,[s_{0},s_{1}]}$. More precisely, we have

$|\mathbb{P}^{\star}[u]|_{k}+|\mathbb{Q}^{\star}[u]|_{k}\lesssim\begin{cases}(\ell^{-\delta}+\delta^{-4})(\epsilon_{\star}+C_{1}\epsilon)^{2}\mathbf{X}^{-1-\delta}r^{-2},\quad&k=0,\\\
(\ell^{-\delta}+\delta^{-4})(\epsilon_{\star}+C_{1}\epsilon)^{2}\mathbf{X}^{-1-\delta}r^{-2+C(\epsilon_{\star}+C_{1}\epsilon)},\quad&1\leq
k\leq N-4.\end{cases}$ (5.55)

Similarly, considering the most challenging component
$h^{\mathcal{N}00}\partial_{t}\partial_{t}u$ we improve the estimate of quasi-
linear terms as

$|h^{\mu\nu}\partial_{\mu}\partial_{\nu}u|_{k}\lesssim\begin{cases}(\ell^{-\delta}+\delta^{-2})(\epsilon_{\star}+C_{1}\epsilon)^{2}\mathbf{X}^{-1-\delta}r^{-2},\quad&k=0,\\\
(\ell^{-\delta}+\delta^{-2})(\epsilon_{\star}+C_{1}\epsilon)^{2}\mathbf{X}^{-1-\delta}r^{-2+C(\epsilon_{\star}+C_{1}\epsilon)},\quad&1\leq
k\leq N-5,\end{cases}$ (5.56)

provided $\kappa\geq 1/2+(5/4)\delta$. We thus conclude that, in
$\mathscr{M}^{\textbf{near}}_{[s_{0},s_{1}]}$,

$\displaystyle|\Box u_{\alpha\beta}|_{k}$
$\displaystyle\lesssim\delta^{-1}(\epsilon_{\star}+C_{1}\epsilon)^{2}s^{-2-3\delta}\mathbf{X}^{-1+\delta}$
(5.57)
$\displaystyle\quad+\begin{cases}(\ell^{-\delta}+\delta^{-4})(\epsilon_{\star}+C_{1}\epsilon)^{2}\mathbf{X}^{-1-\delta}r^{-2},\quad&k=0,\\\
(\ell^{-\delta}+\delta^{-4})(\epsilon_{\star}+C_{1}\epsilon)^{2}\mathbf{X}^{-1-\delta}r^{-2+C(\epsilon_{\star}+C_{1}\epsilon)},\quad&1\leq
k\leq N-4.\end{cases}$

In $\mathscr{M}^{\textbf{far}}_{\ell,[s_{0},s_{1}]}$ and for all $k\leq N-5$,
we use $\mathbf{X}^{-1}\lesssim\ell^{-1}\langle r\rangle^{-1}$ together with
(4.6), (4.9) and (5.12), and arrive at

$\displaystyle|\partial u\partial u|_{N-5}$
$\displaystyle\lesssim\ell^{-4\delta}(\epsilon_{\star}+C_{1}\epsilon)^{2}r^{-2-3\delta}\mathbf{X}^{-1+\delta},$
(5.58)
$\displaystyle|h^{\alpha\beta}\partial_{\alpha}\partial_{\beta}u|_{N-5}$
$\displaystyle\lesssim\ell^{-1}{\delta^{-1}}(\epsilon_{\star}+C_{1}\epsilon)^{2}t^{-1+{(3/2)}\delta}r^{-1-\kappa}\mathbf{X}^{-1+\kappa}.$

In conclusion, we have a control of the wave operator

$\displaystyle|\Box u_{\alpha\beta}|_{k}$
$\displaystyle\lesssim\big{(}\ell^{-4\delta}+\delta^{-1}\big{)}(\epsilon_{\star}+C_{1}\epsilon)^{2}r^{-2-3\delta}\mathbf{X}^{-1+\delta}+\ell^{-1}{\delta^{-1}}(\epsilon_{\star}+C_{1}\epsilon)^{2}t^{-1+{(3/2)}\delta}r^{-1-\kappa}\mathbf{X}^{-1+\kappa}$
(5.59)
$\displaystyle\quad+(\ell^{-\delta}+\delta^{-4})(\epsilon_{\star}+C_{1}\epsilon)^{2}\mathbf{X}^{-1-\delta}r^{-2+C(\epsilon_{\star}+C_{1}\epsilon)^{1/2}}.$

Here for simplicity we have replaced the decay factors $r^{-1}$ and
$r^{-2+C(\epsilon_{\star}+C_{1}\epsilon)}$ by
$r^{-2+C(\epsilon_{\star}+C_{1}\epsilon)^{1/2}}$. Applying Proposition 5.7 for
the case $k\geq 1$, we obtain

$\displaystyle|u_{\textbf{sour}}|_{k}$
$\displaystyle\lesssim\underbrace{\big{(}\ell^{-4\delta}+\delta^{-1}\big{)}\delta^{-2}(\epsilon_{\star}+C_{1}\epsilon)^{2}r^{-1}}_{\text{Case
2 with
}\mu=\delta,\nu=3\delta}+\underbrace{\ell^{-1}\delta^{-3}(\epsilon_{\star}+C_{1}\epsilon)^{2}r^{-1}}_{\text{Case
1 with
}\upsilon=(3/2)\delta\atop\mu=1-\min(\lambda,\kappa),\nu=\min(\lambda,\kappa),\nu-\mu-\upsilon\geq\delta}$
$\displaystyle\quad+\underbrace{(\ell^{-\delta}+\delta^{-4})\delta^{-1}(\epsilon_{\star}+C_{1}\epsilon)^{3/2}r^{-1+C(\epsilon_{\star}+C_{1}\epsilon)^{1/2}}}_{\text{Case
4 with
}\mu=\delta,\nu=C(\epsilon_{\star}+C_{1}\epsilon)^{1/2},C(\epsilon_{\star}+C_{1}\epsilon)^{1/2}\leq\delta/2}.$

Provided
$(\ell^{-\delta}+\delta^{-4})\delta^{-1}(\epsilon_{\star}+C_{1}\epsilon)^{1/2}\lesssim
1$, we arrive at (5.49).

#### Proof of (5.51).

We rely on the decomposition
$h_{\alpha\beta}=h^{\star}_{\alpha\beta}+u_{\alpha\beta}=h^{\star}_{\alpha\beta}+u_{\textbf{init},\alpha\beta}+u_{\textbf{sour},\alpha\beta}$
and, by recalling (3.1a) and (3.8) together with (5.49), we can control
$|h_{\alpha\beta}|$. For $|h^{\alpha\beta}|$, we only need to observe that
$\max_{\alpha,\beta}|h^{\alpha\beta}|\lesssim\max_{\alpha,\beta}|h_{\alpha\beta}|$,
provided that $\max_{\alpha,\beta}|h_{\alpha\beta}|$ is sufficiently small.

##  6 Closing the bootstrap estimates

###  6.1 Improved energy estimate for general metric components

#### Objective.

This section is devoted to the following result. The proof begins with the
energy estimate (2.12) applied to (4.12). Sufficient integrable $L^{2}$ decay
must be checked for the terms arising in the right-hand side of the energy
estimate.

######  Proposition 6.1 (Improved energy estimates for the matter field).

Under the bootstrap assumptions, the matter field satisfies

$\mathcalboondox F_{\kappa}^{\mathcal{EM},N}(s,u)\leq{C_{1}\over
2}\,\epsilon\,s^{\delta},\qquad s\in[s_{0},s_{1}].$ (6.1)

#### Nonlinearities.

######  Proposition 6.2 (Sharp energy estimates for nonlinearities).

Under the bootstrap assumptions and as a consequence of the pointwise metric
estimates (5.50), for all $s\in[s_{0},s_{1}]$ one has

$\displaystyle\|\mathbf{X}^{\kappa}J\zeta^{-1}|\widetilde{\Box}_{g}u|_{p,k}\|_{L^{2}(\mathscr{M}^{\mathcal{EM}}_{s})}$
$\displaystyle\lesssim(\ell^{-\delta/2}+\delta^{-2})(\epsilon_{\star}+C_{1}\epsilon)\big{(}s^{-1}\,\mathcalboondox
F_{\kappa}^{\mathcal{EM},p,k}(s,u)+s^{-1+C(\epsilon_{\star}+C_{1}\epsilon)}\mathcalboondox
F_{\kappa}^{\mathcal{EM},p,k-1}(s,u)\big{)}$ (6.2)
$\displaystyle\quad+(\epsilon_{\star}+C_{1}\epsilon)^{2}s^{-1-\delta}+R_{\textbf{err}}^{\star}(s),$

where $R_{\textbf{err}}^{\star}(s)$ denotes the Ricci upper bound in (3.3).

###### Proof.

Null semi-linear terms near the light cone. Recalling the structure of the
null terms, we find

$|\mathbb{Q}^{\star}[u]|_{p,k}:=\max_{\alpha,\beta}|\mathbb{Q}_{\alpha\beta}^{\star}[u]|_{p,k}\lesssim\sum_{p_{1}+p_{2}=p\atop
k_{1}+k_{2}=k}|\partial
u|_{p_{1},k_{1}}|\not{\partial}^{\mathcal{N}}u|_{p_{2},k_{2}}+|h^{\star}|_{p}\sum_{p_{1}+p_{2}=p\atop
k_{1}+k_{2}=k}|\partial u|_{p_{1},k_{1}}|\partial u|_{p_{2},k_{2}}$ (6.3)

and, in combination with (5.50),

$\displaystyle|\mathbb{Q}^{\star}[u]|_{p,k}$
$\displaystyle\lesssim\sum_{p_{1}+p_{2}=p\atop k_{1}+k_{2}=k}|\partial
u|_{p_{1},k_{1}}|\not{\partial}^{\mathcal{N}}u|_{p_{2},k_{2}}+\epsilon_{\star}r^{-1}|\partial
u|_{p,k}|\partial u|_{p_{1},k_{1}}$ (6.4)
$\displaystyle\lesssim(\ell^{-\delta/2}+\delta^{-2})(\epsilon_{\star}+C_{1}\epsilon)\big{(}\langle
r\rangle^{-1}|\not{\partial}^{\mathcal{N}}u|_{p,k}+\langle
r\rangle^{-1+C(\epsilon_{\star}+C_{1}\epsilon)}|\not{\partial}^{\mathcal{N}}u|_{p-1,k-1}\big{)}$
$\displaystyle\quad+(\epsilon_{\star}+C_{1}\epsilon)\,s^{\delta}r^{-1-\kappa}|\partial
u|_{p}=:G_{1}+G_{2}.$

We integrate the above inequality so that, thanks to (2.10),

$\|\mathbf{X}^{\kappa}J\zeta^{-1}\,G_{2}\|_{L^{2}(\mathscr{M}^{\textbf{near}}_{\ell,s})}\lesssim(\epsilon_{\star}+C_{1}\epsilon)s^{-1-2\kappa+\delta}\|\mathbf{X}^{\kappa}\zeta|\partial
u|_{p}\|_{L^{2}(\mathscr{M}^{\textbf{near}}_{\ell,s})}\lesssim(\epsilon_{\star}+C_{1}\epsilon)^{2}s^{-1-\delta}.$
(6.5)

On the other hand, we have

$\displaystyle\|\mathbf{X}^{\kappa}J\zeta^{-1}\,G_{1}\|_{L^{2}(\mathscr{M}^{\textbf{near}}_{\ell,s})}$
$\displaystyle\lesssim(\ell^{-\delta/2}+\delta^{-2})(\epsilon_{\star}+C_{1}\epsilon)s^{-1}\|\mathbf{X}^{\kappa}\zeta|\not{\partial}^{\mathcal{N}}u|_{p,k}\|_{L^{2}(\mathscr{M}^{\textbf{near}}_{\ell,s})}$
$\displaystyle\quad+(\ell^{-\delta/2}+\delta^{-2})(\epsilon_{\star}+C_{1}\epsilon)s^{-1+C(\epsilon_{\star}+C_{1}\epsilon)^{1/2}}\|\mathbf{X}^{\kappa}\zeta|\not{\partial}^{\mathcal{N}}u|_{p-1,k-1}\|_{L^{2}(\mathscr{M}^{\textbf{near}}_{\ell,s})}$
$\displaystyle\lesssim(\ell^{-\delta/2}+\delta^{-2})(\epsilon_{\star}+C_{1}\epsilon)\big{(}s^{-1}\,\mathcalboondox
F_{\kappa}^{\mathcal{EM},p,k}(s,u)+s^{-1+C(\epsilon_{\star}+C_{1}\epsilon)}\mathcalboondox
F_{\kappa}^{\mathcal{EM},p-1,k-1}(s,u)\big{)}.$

Quasi-null terms near the light cone. Recalling of [30, Lemma
LABEL:lem1-31-01-2021], we have

$\displaystyle|\mathbb{P}^{\star}[u]|_{p,k}\lesssim|\mathbb{P}_{00}^{\star}[u]|_{p,k}+|\not{\mathbb{P}}^{\star}[u]|_{p,k}$
$\displaystyle\lesssim\underbrace{\sum_{p_{1}+p_{2}=p\atop
k_{1}+k_{2}=k}|\partial\not{u}^{\mathcal{N}}|_{p_{1},k_{1}}|\partial\not{u}^{\mathcal{N}}|_{p_{2},k_{2}}}_{G_{3}}+\underbrace{\sum_{p_{1}+p_{2}=p}\Big{(}|\not{\partial}^{\mathcal{N}}u|_{p_{1}}|\partial
u|_{p_{2}}+|\mathbb{S}^{\mathcal{EM}}_{p_{1}}[u]||\partial
u|_{p_{2}}\Big{)}+\sum_{p_{1}+p_{2}+p_{3}=p}|h^{\star}|_{p_{3}}|\partial
u|_{p_{1}}|\partial u|_{p_{2}}}_{G_{4}}$

The terms $G_{4}$ are null terms and high-order terms, bounded similarly as
$G_{2}$, so we can focus our attention on $G_{3}$. By applying (5.46a), we
find

$\|s\,\mathbf{X}^{\kappa}\zeta\,G_{3}\|_{L^{2}(\mathscr{M}^{\textbf{near}}_{\ell,s})}\lesssim(\ell^{-\delta/2}+\delta^{-2})(\epsilon_{\star}+C_{1}\epsilon)\big{(}s^{-1}\,\mathcalboondox
F_{\kappa}^{\mathcal{EM},p,k}(s,u)+s^{-1+C(\epsilon_{\star}+C_{1}\epsilon)}\mathcalboondox
F_{\kappa}^{\mathcal{EM},p-1,k-1}(s,u)\big{)}.$

Quadratic semi-linear terms away from the light cone. It remains to derive the
desired bound in $\mathscr{M}^{\textbf{far}}_{\ell,s}$. We observe that
$\mathbf{X}^{-1}\lesssim\ell^{-1}s^{-2}$ so, thanks to (6.3) and our bound on
$|\mathbb{P}^{\star}[u]|_{p,k}$

$\displaystyle|\mathbb{Q}^{\star}[u]|_{p,k}+|\mathbb{P}^{\star}[u]|_{p,k}\lesssim(1+|h^{\star}|_{N})|\partial
u\partial u|_{N}\lesssim\sum_{p_{1}+p_{2}=p\atop k_{1}+k_{2}=k}|\partial
u|_{p_{1},k_{1}}|\partial u|_{p_{2},k_{2}}$ $\displaystyle\lesssim|\partial
u|_{p,k}|\partial
u|_{[N/2]}\lesssim(\epsilon_{\star}+C_{1}\epsilon)r^{-1}\mathbf{X}^{-\kappa}s^{\delta}|\partial
u|_{p,k}\lesssim\ell^{-\delta/2}(\epsilon_{\star}+C_{1}\epsilon)r^{-1}\mathbf{X}^{-\kappa+\delta/2}|\partial
u|_{p,k}.$

We thus find

$\begin{aligned}
\|\mathbf{X}^{\kappa}J\zeta^{-1}\,G_{3}\|_{L^{2}(\mathscr{M}^{\textbf{far}}_{\ell,s})}&\lesssim\ell^{-\delta/2}(\epsilon_{\star}+C_{1}\epsilon)\|r^{-1}s\,\mathbf{X}^{\kappa}\zeta|\partial
u|_{p,k}\|_{L^{2}(\mathscr{M}^{\textbf{far}}_{\ell,s})}\lesssim\ell^{-\delta/2}(\epsilon_{\star}+C_{1}\epsilon)s^{-1}\,\mathcalboondox
F_{\kappa}^{\mathcal{EM},p,k}(s,u).\end{aligned}\qed$

#### Commutators.

######  Proposition 6.3 (Sharp energy estimates for commutators).

As a consequence of Lemma 5.13, for all $\textbf{ord}(Z)=p$ and
$\textbf{rank}(Z)=k$ one has

$\displaystyle\|\mathbf{X}^{\kappa}J\zeta^{-1}[Z,h^{\alpha\beta}\partial_{\alpha}\partial_{\beta}]u\|_{L^{2}(\mathscr{M}^{\mathcal{EM}}_{s})}$
$\displaystyle\lesssim\delta^{-1}(\ell^{-1}+\delta^{-2})(\epsilon_{\star}+C_{1}\epsilon)s^{-1}\,\mathcalboondox
F_{\kappa}^{\mathcal{EM},p,k}(s,u)$
$\displaystyle\quad+\delta^{-1}(\ell^{-1}+\delta^{-2})(\epsilon_{\star}+C_{1}\epsilon)s^{-1+C(\epsilon_{\star}+C_{1}\epsilon)}\,\mathcalboondox
F_{\kappa}^{\mathcal{EM},p-1,k-1}(s,u)$
$\displaystyle\quad+\|\mathbf{X}^{\kappa}J\zeta^{-1}|\widetilde{\Box}_{g}u|_{p-1,k-1}\|_{L^{2}(\mathscr{M}^{\mathcal{EM}}_{s})}+\ell^{-1}\delta^{-1}(\epsilon_{\star}+C_{1}\epsilon)^{2}s^{-2}.$

###### Proof.

Estimates away from the light cone. For simplicity in the notation, we define

$\displaystyle\mathbf{A}_{p,k}(s)$ $\displaystyle:=\sum_{\textbf{ord}(Z)\leq
p\atop\textbf{rank}(z)\leq
k}\|\mathbf{X}^{\kappa}J\zeta^{-1}[Z,h^{\alpha\beta}\partial_{\alpha}\partial_{\beta}]u\|_{L^{2}(\mathscr{M}^{\textbf{far}}_{\ell,s})},\quad\mathbf{B}_{p,k}(s):=\|\mathbf{X}^{\kappa}J\zeta^{-1}\,|\partial\partial
u|_{p,k}\|_{L^{2}(\mathscr{M}^{\textbf{far}}_{\ell,s})}.$

We recall Proposition 5.2, which gives us

$\displaystyle\mathbf{B}_{p,k}(s)$
$\displaystyle\lesssim\mathbf{A}_{p,k}(s)+\ell^{-1}\|\mathbf{X}^{\kappa}J\zeta^{-1}\,|\widetilde{\Box}_{g}u|_{p,k}\|_{L^{2}(\mathscr{M}^{\textbf{far}}_{\ell,s})}+\ell^{-1}s^{-1}\,\mathcalboondox
F_{\kappa}^{\mathcal{EM},p+1,k+1}(s,u).$ (6.6)

We then rely on (2.16), and control each term in this inequality as follows.

$\bullet$ For $|Y_{\text{rot}}h|_{p_{1}-1}|\partial\partial u|_{p_{2},k_{2}}$
we find
$|Y_{\text{rot}}h|_{p}\lesssim|Y_{\text{rot}}h^{\star}|_{p}+|Y_{\text{rot}}u|_{p}\lesssim\epsilon_{\star}\langle
r\rangle^{-1}+|Y_{\text{rot}}u|_{p},$ which leads us to

$\|\mathbf{X}^{\kappa}J\zeta^{-1}\,|Y_{\text{rot}}h|_{p_{1}-1}|\partial\partial
u|_{p_{2},k_{2}}\|_{L^{2}(\mathscr{M}^{\textbf{far}}_{\ell,s})}\lesssim\begin{cases}\delta^{-1}(\epsilon_{\star}+C_{1}\epsilon)\mathbf{B}_{p-1,k-1}(s),&p_{1}\leq
N-3,\\\
\epsilon_{\star}\mathbf{B}_{p-1,k-1}(s)+\ell^{-1}\delta^{-1}(\epsilon_{\star}+C_{1}\epsilon)^{2}s^{-2},&p_{1}\geq
N-2,\end{cases}$ (6.7)

where for the first case we have applied the third bound of (5.9), while for
the second bound we have applied (4.5) and (5.12), provided $\kappa\geq
1/2+(3/2)\delta$.

$\bullet$ The term $|H|\,|\partial\partial u|_{p-1,k-1}$ is also bounded by
the first case of (6.7).

$\bullet$ The term $|\partial h|_{p_{1}-1,k_{1}}|\partial\partial
u|_{p_{2},k_{2}}$ is easier since a partial derivative is acting on $h$. We
observe that

$|\partial h|_{p}\lesssim|\partial h^{\star}|_{p}+|\partial
u|_{p}\lesssim\epsilon_{\star}\langle r\rangle^{-2}+|\partial u|_{p}$ (6.8)

and, using the inequality $\mathbf{X}^{-1}\lesssim\ell^{-1}\langle
r\rangle^{-1}$ valid in $\mathscr{M}^{\textbf{far}}_{\ell,[s_{0},s_{1}]}$,

$\|\mathbf{X}^{\kappa}J\zeta^{-1}\,|\partial
h|_{p_{1}-1,k_{1}}|\partial\partial
u|_{p_{2},k_{2}}\|_{L^{2}(\mathscr{M}^{\textbf{far}}_{\ell,s})}\lesssim\ell^{-\delta/2}(\epsilon_{\star}+C_{1}\epsilon)s^{-1}\,\mathcalboondox
F_{\kappa}^{\mathcal{EM},p,k}(s,u).$ (6.9)

We then deduce that

$\mathbf{A}_{p,k}(s)\lesssim\delta^{-1}(\epsilon_{\star}+C_{1}\epsilon)\mathbf{B}_{p-1,k-1}(s)+\ell^{-\delta/2}(\epsilon_{\star}+C_{1}\epsilon)s^{-1}\,\mathcalboondox
F_{\kappa}^{\mathcal{EM},p,k}(s,u)+\ell^{-1}\delta^{-1}(\epsilon_{\star}+C_{1}\epsilon)^{2}s^{-2}.$
(6.10)

Combining (6.10) and (6.6) and observing that
$\delta^{-1}(\epsilon_{\star}+C_{1}\epsilon)\ll 1$, we obtain

$\displaystyle\sum_{\textbf{ord}(Z)\leq p\atop\textbf{rank}(z)\leq
k}\|\mathbf{X}^{\kappa}J\zeta^{-1}[Z,h^{\alpha\beta}\partial_{\alpha}\partial_{\beta}]u\|_{L^{2}(\mathscr{M}^{\textbf{far}}_{\ell,s})}$
(6.11)
$\displaystyle\lesssim\|\mathbf{X}^{\kappa}J\zeta^{-1}\,|\widetilde{\Box}_{g}u|_{p-1,k-1}\|_{L^{2}(\mathscr{M}^{\textbf{far}}_{\ell,s})}+\ell^{-1}\delta^{-1}(\epsilon_{\star}+C_{1}\epsilon)s^{-1}\mathcalboondox
F_{\kappa}^{\mathcal{EM},p,k}(s,u)+\ell^{-1}\delta^{-1}(\epsilon_{\star}+C_{1}\epsilon)^{2}s^{-2},$

which completes the proof of Proposition 6.3 in
$\mathscr{M}^{\textbf{far}}_{\ell,[s_{0},s_{1}]}$.

Estimates near the light cone. We also introduce the notation

$\displaystyle\mathbf{A}_{p,k}(s)$ $\displaystyle:=\sum_{\textbf{ord}(Z)\leq
p\atop\textbf{rank}(z)\leq
k}\|\mathbf{X}^{\kappa}J\zeta^{-1}[Z,h^{\alpha\beta}\partial_{\alpha}\partial_{\beta}]u\|_{L^{2}(\mathscr{M}^{\textbf{near}}_{\ell,s})},\qquad\mathbf{B}_{p,k}(s):=\|t^{-1}\mathbf{X}^{1+\kappa}J\zeta^{-1}\,|\partial\partial
u|_{p,k}\|_{L^{2}(\mathscr{M}^{\textbf{near}}_{\ell,s})}.$

Then, in view of (5.5) we get

$\mathbf{B}_{p,k}(s)\lesssim\mathbf{A}_{p,k}(s)+\|\mathbf{X}^{\kappa}J\zeta^{-1}|\widetilde{\Box}_{g}u|_{p,k}\|_{L^{2}(\mathscr{M}^{\textbf{near}}_{\ell,s})}+s^{-1}\mathcalboondox
F_{\kappa}^{\mathcal{EM},p+,k+1}(s,u).$ (6.12)

On the other hand, we recall (2.15) and we need to control each term in its
right-hand side.

$\bullet$ The term
$\big{(}|H^{\mathcal{N}}{}^{00}|+t^{-1}|r-t||H|\big{)}\,|\partial\partial
u|_{p-1,k-1}$, thanks to (5.14) and (5.9) (the third bound), is directly
controlled by

$\big{\|}\mathbf{X}^{\kappa}J\zeta^{-1}(|H^{\mathcal{N}}{}^{00}|+t^{-1}|r-t||H|\big{)}\,|\partial\partial
u|_{p-1,k-1})\big{\|}\lesssim(\epsilon_{\star}+C_{1}\epsilon)\mathbf{B}_{p-1,k-1}(s).$
(6.13)

$\bullet$ Thanks to (5.14) and (5.9) (i.e. the third bound therein), the term
$T_{p,k}^{\textbf{hier}}[H,u]$ is controlled as follows. First of all, we have

$|LH^{\mathcal{N}00}|_{N-5}+t^{-1}|r-t||H|_{N-5}\lesssim(\epsilon_{\star}+C_{1}\epsilon)\mathbf{X}r^{-1}$

and

$|LH^{\mathcal{N}00}|_{k-1}+t^{-1}|r-t||H|_{k-1}\lesssim|h^{\star}|_{k}+|Lu|_{k-1}+t^{-1}\mathbf{X}|u|_{k}\lesssim\epsilon_{\star}\langle
r\rangle^{-1}+|Y_{\text{rot}}u|_{k-1}+t^{-1}\mathbf{X}|u|_{k}.$

We thus deduce that

$\displaystyle\|\mathbf{X}^{\kappa}J\zeta^{-1}T_{p,k}^{\textbf{hier}}[H,u]\|_{L^{2}(\mathscr{M}^{\textbf{near}}_{\ell,s})}$
$\displaystyle\lesssim(\epsilon_{\star}+C_{1}\epsilon)\|\mathbf{X}^{\kappa}J\zeta^{-1}\mathbf{X}r^{-1}|\partial\partial
u|_{p-1,k-1}\|_{L^{2}(\mathscr{M}^{\textbf{near}}_{\ell,s})}+\sum_{k_{1}=N-4}^{k}\||\mathbf{X}^{\kappa}J\zeta^{-1}|\partial\partial
u|_{p-k_{1},k-k_{1}}|Y_{\text{rot}}u|_{k_{1}-1}\|_{L^{2}(\mathscr{M}^{\textbf{near}}_{\ell,s})}$
$\displaystyle\quad+(\epsilon_{\star}+C_{1}\epsilon)s^{2\delta}\|\mathbf{X}^{\kappa}J\zeta^{-1}\,r^{-2}\mathbf{X}^{-\kappa}|u|_{N}\|_{L^{2}(\mathscr{M}^{\textbf{near}}_{\ell,s})},$

where (4.23) and (5.11) were used. For the latter term, we apply (4.5) which
is seen to be bounded by $\delta^{-1}(\epsilon_{\star}+C_{1}\epsilon)s^{-2}$
(provided $\kappa\geq 1/2+(3/2)\delta$). For the second term we recall (2.20)
together with (5.50) and obtain

$\displaystyle\||\mathbf{X}^{\kappa}J\zeta^{-1}|\partial\partial
u|_{p-k_{1},k-k_{1}}|Y_{\text{rot}}u|_{k_{1}-1}|\|_{L^{2}(\mathscr{M}^{\textbf{near}}_{\ell,s})}$
$\displaystyle\lesssim\begin{cases}(\ell^{-\delta}+\delta^{-2})(\epsilon_{\star}+C_{1}\epsilon)s^{-1}\|\zeta\mathbf{X}^{-1+\kappa}|Y_{\text{rot}}u|_{k-1}\|_{L^{2}(\mathscr{M}^{\textbf{near}}_{\ell,s})}\quad&\text{
(when $k_{1}=k$)}\\\
(\ell^{-\delta}+\delta^{-2})(\epsilon_{\star}+C_{1}\epsilon)s^{-1+C(\epsilon_{\star}+C_{1}\epsilon)}\|\zeta\mathbf{X}^{-1+\kappa}|Y_{\text{rot}}u|_{k-2}\|_{L^{2}(\mathscr{M}^{\textbf{near}}_{\ell,s})}\quad&\text{
(when $k_{1}\leq k-1$)}\end{cases}$
$\displaystyle\lesssim(\ell^{-\delta}+\delta^{-2})(\epsilon_{\star}+C_{1}\epsilon)s^{-1}\mathcalboondox
F_{\kappa}^{\mathcal{EM},p,k}(s,u)+(\ell^{-\delta}+\delta^{-2})(\epsilon_{\star}+C_{1}\epsilon)s^{-1+C(\epsilon_{\star}+C_{1}\epsilon)}\mathcalboondox
F_{\kappa}^{\mathcal{EM},p-1,k-1}(s,u).$

Hence, it follows that

$\displaystyle\|\mathbf{X}^{\kappa}J\zeta^{-1}T_{p,k}^{\textbf{hier}}[H,u]\|_{L^{2}(\mathscr{M}^{\textbf{near}}_{\ell,s})}$
(6.14)
$\displaystyle\lesssim(\epsilon_{\star}+C_{1}\epsilon)\mathbf{B}_{p-1,k-1}(s)+(\ell^{-\delta}+\delta^{-2})(\epsilon_{\star}+C_{1}\epsilon)s^{-1}\mathcalboondox
F_{\kappa}^{\mathcal{EM},p,k}(s,u)$
$\displaystyle\quad+(\ell^{-\delta}+\delta^{-2})(\epsilon_{\star}+C_{1}\epsilon)s^{-1+C(\epsilon_{\star}+C_{1}\epsilon)}\mathcalboondox
F_{\kappa}^{\mathcal{EM},p-1,k-1}(s,u)+\delta^{-1}(\epsilon_{\star}+C_{1}\epsilon)s^{-2}.$

$\bullet$ We consider next the terms $T^{\bf easy}$ and $T^{\bf super}$, and
we observe that $T^{\textbf{super}}$ contains a favorable factor $t^{-1}$
while the second term of $T^{\textbf{easy}}$ contains the factor
$t^{-1}|r-t|$, supplying sufficient decay in $\langle r\rangle^{-1}$. Thus
these terms are relatively easier to handle and we only discuss now the most
challenging term, namely $|h^{\mathcal{N}00}|_{p_{1}-1,k_{1}}|\partial\partial
u|_{p_{2},k_{2}}$. When $p_{1}-1\leq N-3$, we apply (4.23) and obtain

$\displaystyle\|\mathbf{X}^{\kappa}J\zeta^{-1}|\partial
h^{\mathcal{N}00}|_{p_{1}-1,k_{1}}|\partial\partial
u|_{p_{2},k_{2}}\|_{L^{2}(\mathscr{M}^{\textbf{near}}_{\ell,s})}$
$\displaystyle\lesssim\delta^{-1}(\epsilon_{\star}+C_{1}\epsilon)s^{-1-2\kappa+\delta}\|\mathbf{X}^{\kappa}\zeta|\partial\partial
u|_{p,k}\|_{L^{2}(\mathscr{M}^{\textbf{near}}_{\ell,s})}$
$\displaystyle\lesssim\delta^{-1}(\epsilon_{\star}+C_{1}\epsilon)^{2}s^{-2}.$

When $p_{1}-1\geq N-2$, we have $p_{2}\leq 1\leq N-5$ and thus

$\displaystyle\|\mathbf{X}^{\kappa}J\zeta^{-1}|\partial
h^{\mathcal{N}00}|_{p_{1}-1,k_{1}}|\partial\partial
u|_{p_{2},k_{2}}\|_{L^{2}(\mathscr{M}^{\textbf{near}}_{\ell,s})}$
$\displaystyle\lesssim s\|\mathbf{X}^{\kappa}\zeta|\partial
h^{\star}|_{p_{1}-1,k_{1}}|\partial\partial
u|_{p_{2},k_{2}}\|_{L^{2}(\mathscr{M}^{\textbf{near}}_{\ell,s})}+s\|\mathbf{X}^{\kappa}\zeta|\partial
u|_{p_{1}-1,k_{1}}|\partial\partial
u|_{p_{2},k_{2}}\|_{L^{2}(\mathscr{M}^{\textbf{near}}_{\ell,s})}$
$\displaystyle\lesssim(\ell^{-\delta}+\delta^{-2})(\epsilon_{\star}+C_{1}\epsilon)s^{-1}\mathcalboondox
F_{\kappa}^{\mathcal{EM},p,k}(s,u)+(\ell^{-\delta}+\delta^{-2})(\epsilon_{\star}+C_{1}\epsilon)s^{-1+C(\epsilon_{\star}+C_{1}\epsilon)}\mathcalboondox
F_{\kappa}^{\mathcal{EM},p-1,k-1}(s,u).$

The weighted $L^{2}$ norm of the remaining terms is bounded by
$\delta^{-1}(\epsilon_{\star}+C_{1}\epsilon)^{2}s^{-2}$.

Now we conclude with (6.13), (6.14), and the bounds established on $T^{\bf
easy}$ and $T^{\bf super}$:

$\displaystyle\mathbf{A}_{p,k}$
$\displaystyle\lesssim(\epsilon_{\star}+C_{1}\epsilon)\mathbf{B}_{p-1,k-1}(s)+(\ell^{-\delta}+\delta^{-2})(\epsilon_{\star}+C_{1}\epsilon)s^{-1}\mathcalboondox
F_{\kappa}^{\mathcal{EM},p,k}(s,u)$
$\displaystyle\quad+(\ell^{-\delta}+\delta^{-2})(\epsilon_{\star}+C_{1}\epsilon)s^{-1+C(\epsilon_{\star}+C_{1}\epsilon)}\mathcalboondox
F_{\kappa}^{\mathcal{EM},p-1,k-1}(s,u)+\delta^{-1}(\epsilon_{\star}+C_{1}\epsilon)^{2}s^{-2}.$

Substituting (6.12) in the above estimate and recalling
$(\epsilon_{\star}+C_{1}\epsilon)\ll 1$, we obtain

$\displaystyle\sum_{\textbf{ord}(Z)\leq p\atop\textbf{rank}(z)\leq
k}\|\mathbf{X}^{\kappa}J\zeta^{-1}[Z,h^{\alpha\beta}\partial_{\alpha}\partial_{\beta}]u\|_{L^{2}(\mathscr{M}^{\textbf{near}}_{\ell,s})}$
(6.15)
$\displaystyle\lesssim\|\mathbf{X}^{\kappa}J\zeta^{-1}|\widetilde{\Box}_{g}u|_{p,k}\|_{L^{2}(\mathscr{M}^{\textbf{near}}_{\ell,s})}+(\ell^{-\delta}+\delta^{-2})(\epsilon_{\star}+C_{1}\epsilon)s^{-1}\mathcalboondox
F_{\kappa}^{\mathcal{EM},p,k}(s,u)$
$\displaystyle\quad+(\ell^{-\delta}+\delta^{-2})(\epsilon_{\star}+C_{1}\epsilon)s^{-1+C(\epsilon_{\star}+C_{1}\epsilon)}\mathcalboondox
F_{\kappa}^{\mathcal{EM},p-1,k-1}(s,u)+\delta^{-1}(\epsilon_{\star}+C_{1}\epsilon)^{2}s^{-2}.$

In conclusion, by combining (6.11) and (6.15) we arrive at the desired result.
∎

#### Conclusion.

We summarize our result so far as follows.

######  Proposition 6.4.

Under the conclusions of Propositions 6.2 and 6.3, the following estimate
holds for all $\textbf{ord}(Z)\leq p$ and $\textbf{rank}(Z)\leq k$:

$\displaystyle\|\mathbf{X}^{\kappa}J\zeta^{-1}\,\widetilde{\Box}_{g}Zu_{\alpha\beta}\|_{L^{2}(\mathscr{M}^{\mathcal{EM}}_{s})}$
(6.16)
$\displaystyle\lesssim\delta^{-1}(\ell^{-1}+\delta^{-2})(\epsilon_{\star}+C_{1}\epsilon)\big{(}s^{-1}\,\mathcalboondox
F_{\kappa}^{\mathcal{EM},p,k}(s,u)+s^{-1+C(\epsilon_{\star}+C_{1}\epsilon)}\,\mathcalboondox
F_{\kappa}^{\mathcal{EM},p-1,k-1}(s,u)\big{)}$
$\displaystyle\quad+\ell^{-1}\delta^{-1}(\epsilon_{\star}+C_{1}\epsilon)^{2}s^{-1-\delta}+R_{\textbf{err}}^{\star}(s).$

At this juncture it is useful to state the following elementary property.

######  Lemma 6.5 (Comparison of energy functionals).

Let $(\eta,c^{*},w)=(\kappa,0,u)$ or $(\eta,c^{*},w)=(\mu,c,\phi)$ one has

$\mathcalboondox
E_{g,\eta,c^{*}}^{\mathcal{EM}}(s,u)\geq(1/4)\,\mathcalboondox
E_{\eta,c^{*}}^{\mathcal{EM}}(s,u).$

###### Proof.

All of the assumptions required in [30, Section LABEL:section-label-11-1] are
valid in the present context, especially the inequality $h^{\mathcal{N}00}\leq
0$ in $\mathscr{M}^{\textbf{near}}_{\ell,[s_{0},s_{1}]}$ and the decay
$|h|\lesssim(\epsilon_{\star}+C_{1}\epsilon)r^{-\kappa}s^{\delta}\leq(\epsilon_{\star}+C_{1}\epsilon)r^{-1/2},$
and we can follow the same arguments as in [30]. ∎

In order to apply (2.12), we still need the following two bounds for
($\textbf{ord}(Z)\leq p$ and $\textbf{rank}(Z)\leq k$ and)