# Nonlinear stability of self-gravitating massive fields.
A wave-Klein-Gordon model

Philippe G. LeFloch1 and and Yue Ma2 1 Laboratoire Jacques-Louis Lions and
Centre National de la Recherche Scientifique, Sorbonne Université, 4 Place
Jussieu, 75252 Paris, France.
2 School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an,
710049 Shaanxi, People’s Republic of China<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Significant advances were made in recent years on the global evolution problem
for self-gravitating massive matter in the small-perturbative regime close to
Minkowski spacetime. To study the coupling between a Klein-Gordon equation and
Einstein’s field equations, we introduced the “Euclidean-hyperboloidal
foliation method”, which is based on the construction of a spacetime foliation
adapted to the derivation of sharp decay estimates for wave and Klein-Gordon
equations in a curved spacetime. We give here an outline of this method,
together with a full proof for a wave-Klein-Gordon model which retains some
main challenges arising with the Einstein-matter system.

  * December 2022

###### Contents

  1. 1 Introduction
  2. 2 Geometric properties of Euclidean-Hyperboloidal foliations
  3. 3 Wave-Klein-Gordon equations in Euclidean-Hyperboloidal foliations
  4. 4 Nonlinear stability of self-gravitating massive fields
  5. 5 Nonlinear stability for the wave-Klein-Gordon model
  6. 6 Proof of stability

##  1 Introduction

###  1.1 Global evolution of self-gravitating massive field

#### Einstein-matter system.

We are interested in four-dimensional spacetimes $(\mathcal{M},\mathbf{g})$
where $\mathcal{M}$ is the manifold
$\mathcal{M}\simeq[0,+\infty)\times\mathbb{R}^{3}$, and $\mathbf{g}$ is a
Lorentzian metric with signature $(-,+,+,+)$. The Levi-Civita connection of
this metric is denoted by $\nabla$ from which we determine the Ricci curvature
tensor Ric and the scalar curvature $\mathbf{R}$, respectively. The components
of tensors such as $\mathbf{g}$ and Ric are denoted111Throughout, Greek
indices describe $0,1,2,3$ and we use the standard convention of 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
$g^{\alpha\beta}$. by $g_{\alpha\beta}$ and $R_{\alpha\beta}$, respectively.
In this notation, we impose Einstein’s field equations

$\mathbf{G}=8\pi\,\mathbf{T}\quad\mbox{ in }(\mathscr{M},\mathbf{g}),$ (1.1)

in which the left-hand side is Einstein’s curvature tensor
$\mathbf{G}:=\mathbf{R}-{1\over 2}\mathbf{R}\,\mathbf{g}$. In the right-hand
side, the energy-momentum tensor $\mathbf{T}$ with components
$T_{\alpha\beta}$ depends upon the nature of the matter under consideration
and, specifically, we are interested in real-valued, massive scalar fields
$\phi:\mathscr{M}\to\mathbb{R}$ described by the energy-momentum tensor

$T_{\alpha\beta}:=\nabla_{\alpha}\phi\nabla_{\beta}\phi-\Big{(}{1\over
2}\nabla_{\gamma}\phi\nabla^{\gamma}\phi+U(\phi)\Big{)}g_{\alpha\beta}.$ (1.2)

Here, the potential $U=U(\phi)$ is a prescribed real-valued function
satisfying

$U(\phi)={1\over 2}c^{2}\phi^{2}+\mathcal{O}(\phi^{3})$ (1.3)

for some constant $c>0$, referred to as the mass of the scalar field.

From the (twice contracted) Bianchi identities applied to (1.1) we deduce the
matter evolution equations

$\nabla^{\alpha}T_{\alpha\beta}=0\quad\mbox{ in }(\mathscr{M},\mathbf{g}).$
(1.4)

By denoting the wave operator by $\Box_{g}:=\nabla^{\alpha}\nabla_{\alpha}$,
it follows that the field $\phi$ satisfies a nonlinear Klein-Gordon equation
in a curved spacetime:

$\Box_{g}\phi-U^{\prime}(\phi)=0\quad\mbox{ in }(\mathscr{M},\mathbf{g}).$
(1.5)

For instance, with the choice $U(\phi)={1\over 2}c^{2}\phi^{2}$, (1.5) is
nothing but the linear Klein-Gordon equation $\Box_{g}\phi-c^{2}\phi=0$.

The initial value problem of interest here is formulated geometrically by
prescribing an initial data set that is close to data associated with a
asymptotically Euclidean, spacelike hypersurface of the (vacuum) Minkowski
spacetime. For suitably regular initial data, it is known that the Einstein
equations (1.1) together with (1.5) uniquely determines “locally in time” the
spacetime geometry and the evolution of the matter field. Our challenge is
precisely to the global evolution problem and to investigate the global
evolution of a massive matter field in the near-Minkowski regime.

#### Wave-Klein-Gordon formulation.

The equations under consideration are geometric in nature, and the degrees of
gauge freedom must be fixed before tackling the nonlinear stability problem of
interest by techniques of mathematical analysis. We assume the existence of
global coordinate functions $x^{\alpha}:\mathscr{M}\to\mathbb{R}$ satisfying
the wave gauge conditions ($\alpha=0,1,2,3$)

$\Box_{g}x^{\alpha}=0.$ (1.6)

In this gauge, the Einstein equations (1.1) take the form a nonlinear wave
system of second-order partial differential equations, supplemented with
second-order differential constraints. The main unknowns are then the metric
coefficients $g_{\alpha\beta}$ in the chosen coordinates, together with the
real-valued field $\phi$. It is well-known that the constraints are preserved
during the time evolution (cf., for instance, [3]) and therefore it is
sufficient to check them on the initial data set.

Specifically, by introducing the modified wave operator
$\widetilde{\Box}_{g}:=g^{\alpha^{\prime}\beta^{\prime}}\partial_{\alpha^{\prime}}\partial_{\beta^{\prime}}$
(which takes the wave gauge into account), (1.1) and (1.2) can be restated as
a nonlinear wave-Klein-Gordon system with unknowns $g_{\alpha\beta}$ and
$\phi$, namely

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

supplemented with the wave gauge conditions

$\displaystyle\Gamma^{\alpha}$
$\displaystyle:=g^{\alpha\beta}\Gamma_{\alpha\beta}^{\lambda}=0,\qquad\Gamma_{\alpha\beta}^{\lambda}:={1\over
2}\,g^{\lambda\lambda^{\prime}}\big{(}\partial_{\alpha}g_{\beta\lambda^{\prime}}+\partial_{\beta}g_{\alpha\lambda^{\prime}}-\partial_{\lambda^{\prime}}g_{\alpha\beta}\big{)},$
(1.8)

together with Einstein’s Hamiltonian and momentum constraints. We refer to [3]
for this standard formulation.

#### Nonlinear stability theory.

Our main result for the Einstein-massive field [24] is based on earlier
partial advances in [21]–[23] and establishes that initial data sets that are
sufficiently close to (vacuum) Minkowski data generates a global-in-time
solution to the Einstein-Klein-Gordon system in wave gauge (1.7)–(1.8). This
global existence result for the set of partial differential equations (PDEs)
translates into a geometric result, and the associated (globally hyperbolic)
Cauchy development is proven to be future causally geodesically
complete222That is, every affinely parameterized geodesic (of null or timelike
type) can be extended toward the future (for all values of its affine
parameter). and, in fact, to approach the Minkowski geometry in all timelike,
null, and spacelike directions. In other words, we prove that for a large
family of initial data sets satisfying smallness conditions in energy and
pointwise norms, the matter field disperses in the infinite future and the
formation of, for instance, black holes or gravitational singularities in the
future development is avoided.

We emphasize that, as our project came under completion, we learned that
Ionescu and Pausader [14]–[16] simultaneously solved the same problem by a
different method, which is based on the notion of spacetime resonances.

Let us recall that the nonlinear stability problem in the vacuum was solved by
Christodoulou and Klainerman via a gauge-invariant method [4]; see also Bieri
[1] for weaker decay conditions. Later on, Lindblad and Rodnianski discovered
a proof in wave coordinates [29]. For further contributions in the vacuum
regime we refer to Hintz and Vasy [10, 11]. On the other hand, the global
dynamics of self-gravitating massive matter fields has received far less
attention, even in the regime of small perturbations of Minkowski spacetime.
For other important contributions on various matter fields we refer to works
by Bigorgne, Fajman, Joudioux, Lindblad, Smulevici, Taylor, and Wang [2, 6, 8,
30, 33, 35].

###  1.2 Brief outlook on the method

#### Bootstrap strategy.

We rely on a bootstrap strategy which is based on the following arguments.

  * Blow-up criterion. A sufficiently regular, local-in-time solution cannot approach its maximal time of existence, say $s^{*}$, at a time at which the energy (at a sufficiently high order) remains bounded. Otherwise, we would be able to extend this solution beyond $s^{*}$ by applying a local-in-time existence argument and this would contradict the fact that $s^{*}$ is chosen to be maximal.

  * Continuity criterion. The functional norms under consideration, which determine the regularity of the initial data, depend continuously upon the time variable, as long as a local-in-time solution exists.

  * Formulating improved bound criterion. Suppose that on a time interval $[s_{0},s_{1}]$ the solution satisfies inequalities expressed in terms of an energy functional at a (sufficiently) high-order of differentiation and, possibly, some other functionals of the solution. (This later part is irrelevant for the model problem treated next, but important in the treatment of the Einstein-Klein-Gordon system.) Suppose that we can prove that these inequalities remain valid in a stronger form with strictly smaller constants. In these circumstances, we deduce that the solution extends beyond $s_{1}$.

  * Deriving the improved bounds. Indeed, if $[s_{0},s_{1}]$ is the maximal interval on which the set of inequalities holds, then at the “final” time $s_{1}$ thanks to the continuity criterion, at least one of the inequalities under consideration must become an equality. However, in the case when we can prove that stronger inequalities holds on the same time interval, it follows that none of the inequalities can be an equality at $s_{1}$. This leads us to the conclusion that all of our inequalities hold for $s<s^{*}$, which is impossible when $s^{*}<+\infty$ in view of the blow-up criterion above.

#### Euclidean-hyperboloidal framework.

The basic features of our method are as follows.

  * •

Foliation. Our framework is based on a foliation labelled by a parameter $s$,
which is asymptotically Euclidean in the vicinity of spacelike infinity while
timelike infinity is covered with slices that are asymptotically
hyperboloidal. The foliation (cf. (2.8), below) is described via the
introduction of a coefficient $\xi=\xi(s,r)$ (cf. (2.2), below) that
interpolates between the interior domain in which $\xi=1$ (for
$r<{\mathbf{r}}^{\mathcal{H}}(s)$, a time-dependent radius) and an exterior
domain in which $\xi=1$ (for $r>{\mathbf{r}}^{\mathcal{E}}(s)$, a larger
radius). Here, we work in in a global coordinate chart $(t,x^{a})$ with
$a=1,2,3$, and $r^{2}=\sum_{a}(x^{a})^{2}$. The two foliations are merged
across a transition region associated with the interval
$[{\mathbf{r}}^{\mathcal{H}}(s),{\mathbf{r}}^{\mathcal{E}}(s)]$. The time
variable $s$ is connected to the standard Cartesian time $t$ in such a way
that is coincides with the standard hyperbolic time $\sqrt{t^{2}-r^{2}}$ in
the interior domain while it is of the order of $\sqrt{t}$ in the exterior.

  * •

Calculus rules and hierarchy. To the proposed foliation we associated several
frames, including the semi-hyperboloidal frame $\partial^{\mathcal{H}}$ (in
(2.12), below) and the semi-null frame $\partial^{\mathcal{N}}$ (in (2.13),
below). These vector fields are used to define high-order operators and
differentiate with the evolution equations of interest, as well as to
decompose tensor fields such as the metric. The necessary calculus rules
enjoyed by the vector fields and the operators of interest are provided in our
theory [24] as a series of technical lemmas. For instance, ordering properties
allow us to work with ordered admissible operators
$Z=\partial^{I}L^{J}\Omega^{K}$, while commutator estimates are used to
commute vectors fields with differential operators. A key hierarchy structure
enjoyed by quasi-linear commutators was uncovered, as stated in Propositions
2.3 and 2.4. At this juncture, the notation $|u|_{p,k}$ (given in (2.17),
below) is very convenient to keep track of, both, the total order of
differentiation (which we call the order, denoted by $p$) and the total number
of boosts or rotations (which we call the rank, denoted by $k\leq p$).

  * •

Functional inequalities. New weighted Sobolev, Poincaré, and Hardy
inequalities are required which are adapted to our Euclidean–hyperboloidal
foliation. This includes a Sobolev inequality for the hyperboloidal domain in
Proposition 2.5 (which involves the boost vectors) and a Sobolev inequality
for the Euclidean-merging domain in Proposition 2.6 (which involves the
distance to the light cone). The weighted Hardy inequality in Proposition 2.7,
which will be necessary in order to control undifferentiated terms such as
metric coefficients. Further inequalities are required such as the Poincaré-
type Proposition 2.9.

  * •

Pointwise decay of wave fields. The pointwise behavior of wave fields and
their derivatives is based on an analysis of Kirchhoff’s formula and
establishes sharp estimates for solutions under assumptions on the source
term. In Proposition 3.1, we distinguish between sub-critical, critical, and
super-critical regimes, and we prove estimates with various decay behaviors in
terms of the radial distance and the distance to the light cone. We also
derive (cf. Case 0 therein) a property within the light cone. The control of
the Hessian of solutions to the wave equation must also be investigated at
arbitrary order, and in our analysis we find it useful to distinguish between
the near/far light cone regions and to rely on two different decompositions of
the wave operator.

  * •

Pointwise decay of Klein-Gordon fields on curved spacetimes. We also derive
sharp decay of solutions to Klein-Gordon fields and their derivatives.
Specifically, we establish Propositions 3.2 and 5.2, below. For a summary see
also Proposition 3.3, below.

###  1.3 Proposed wave-Klein-Gordon model

#### Model of interest.

We will illustrate some key features of our proof for the Einstein equation by
focusing here on a model. The system under consideration now is formally
derived from the Einstein-massive field system by suppressing null
nonlinearities and quasi-null nonlinearities, and by replacing the metric by a
scalar unknown. In turn, we obtain a wave equation and a Klein-Gordon equation
coupled through quadratic terms of zero, first, and second order, as follows:

$\displaystyle-\Box u$
$\displaystyle=P^{\alpha\beta}\partial_{\alpha}\phi\partial_{\beta}\phi+R\,\phi^{2},$
(1.9) $\displaystyle-\Box\phi+c^{2}\,\phi$
$\displaystyle=H^{\alpha\beta}u\,\partial_{\alpha}\partial_{\beta}\phi.$

Here, the unknown functions $u=u(t,x)$ and $\phi=\phi(t,x)$ are defined in the
future of a spacelike hypersurface o which initial data are prescribed. For
simplicity in our presentation, the coefficients of the nonlinearities
$P^{\alpha\beta}$, $R$, and $H^{\alpha\beta}$ are assumed to be constants, but
a generalization to non-constant coefficients would only require
straightforward conditions on the derivatives. The global existence theory for
(1.9) is presented in Section 4, below; cf. Theorem 5.1.

#### Outline of this paper.

We begin, in Section 2, with the definition of the proposed Euclidean-
Hyperboloidal foliation and we state various properties concerning the vector
frames of interest, commutator properties, and functional inequalities adapted
to this foliation. In Section 3, we present several pointwise estimates for
wave and Klein-Gordon equations posed on the Euclidean-Hyperboloidal
foliation. In Section 4, we give our main stability statement for the Einstein
equations and we outline its proof by including also a key analysis of the
null and quasi-null structure of the Einstein equations. In Section 5, we
state our global existence result for the model. Section 6 is devoted to the
proof of global existence for the model; we first derive energy and pointwise
estimates in the Euclidean-merging domain and in the hyperboloidal domain, and
we finally close the bootstrap by taking advantage of a hierarchy enjoyed by
our estimates.

Finally, let us summarize our main notation in the following table:

$\displaystyle\begin{aligned}
&\mathscr{M}_{s}=\mathscr{M}^{\mathcal{H}}_{s}\cup\mathscr{M}^{\mathcal{M}}_{s}\cup\mathscr{M}^{\mathcal{E}}_{s}&&\mbox{
spacelike slices}\\\ &\mathbf{T}(s,r)&&\mbox{ global time function}\\\
&\xi(s,r)&&\mbox{ foliation coefficient}\\\ &\zeta(s,r)&&\mbox{ energy
coefficient}\\\
&\partial^{\mathcal{H}}_{0}=\partial_{t},\quad\partial^{\mathcal{H}}_{a}=\frac{x^{a}}{t}\partial_{t}+\partial_{a}&&\mbox{
semi-hyperboloidal frame}\\\
&\partial^{\mathcal{N}}_{0}=\partial_{t},\quad\partial^{\mathcal{N}}_{a}=\not{\partial}^{\mathcal{N}}={x^{a}\over
r}\partial_{t}+\partial_{a}&&\mbox{ semi-null frame}\\\
&\partial^{\mathcal{E\hskip-1.70709ptH}}_{s}=(\partial_{s}T)\partial_{t},\quad\not{\partial}^{\mathcal{EH}}_{a}=\partial_{a}+(x^{a}/r)(\partial_{r}\mathbf{T})\partial_{t}&&\mbox{
Euclidean--hyperboloidal frame}\\\ &\mathbf{X}(s,r)&&\mbox{ energy weight }\\\
\end{aligned}$

##  2 Geometric properties of Euclidean-Hyperboloidal foliations

###  2.1 Geometry of the Euclidean-hyperboloidal slices

#### Euclidean–hyperboloidal time function.

We begin by describing the foliation of interest. It is defined over a
manifold
$\mathscr{M}\simeq\mathbb{R}^{1+3}_{+}=\mathbb{R}_{+}\times\mathbb{R}^{3}$
covered by a global coordinate chart denoted by
$(t,x):\mathscr{M}\to\mathbb{R}^{1+3}_{+}$. For convenience, we assume $t\geq
1$ and we will impose initial data on the hypersurface $t=1$ and solve a
Cauchy problem in the future of this initial hypersurface. We also set
$x=(x^{a})\in\mathbb{R}^{3}$ and $r^{2}=|x|^{2}=\sum_{a=1}^{3}|x^{a}|^{2}$.
The ground state of the gravity theory is given by the Minkowski metric
$g_{\textbf{Mink}}=-dt^{2}+\sum_{a}(dx^{a})^{2}$, and we are interested in
small perturbations of this metric.

Our first task is defining a suitable spacetime foliation. To label our
foliation, we use a (Euclidean-hyperboloidal) time parameter $s\geq s_{0}$ and
we distinguish between several domains on each hypersurface of constant $s$.
To this end, we introduce the hyperboloidal and Euclidean radii at any time
$s$

${\mathbf{r}}^{\mathcal{H}}(s):={1\over
2}(s^{2}-1),\qquad{\mathbf{r}}^{\mathcal{E}}(s):={1\over 2}(s^{2}+1).$ (2.1)

Let us consider any cut-off function $\chi:\mathbb{R}\to[0,1]$ satisfying
$\chi(x)=\begin{cases}0,&x\leq 0,\\\ 1,&x>1,\end{cases}$ and (for simplicity
in some of our arguments) $\chi^{(m)}(x)>0$ for all $x\in(0,1/2)$ and each
$m=0,1,2,3$. By definition, the foliation coefficient is the function

$\xi(s,r):=1-\chi(r-{\mathbf{r}}^{\mathcal{H}}(s))=\begin{cases}1,\quad&r<{\mathbf{r}}^{\mathcal{H}}(s),\\\
0,\quad&r>{\mathbf{r}}^{\mathcal{E}}(s),\end{cases}$ (2.2)

and will be applied to“select” the hyperboloidal domain.

We define the function $t=\mathbf{T}(s,r)$ by solving the ordinary
differential equation

$\partial_{r}\mathbf{T}(s,r)=\frac{r\,\xi(s,r)}{(s^{2}+r^{2})^{1/2}},\qquad\mathbf{T}(s,0)=s.$
(2.3)

It can be checked that this time function enjoys the following properties:

$\mathbf{T}(s,r)=\begin{cases}(s^{2}+r^{2})^{1/2},\quad&r\leq{\mathbf{r}}^{\mathcal{H}}(s)\qquad\mbox{
(hyperboloidal domain),}\\\
r+1=(s^{2}+1)/2,\quad&r={\mathbf{r}}^{\mathcal{H}}(s),\\\
\mathbf{T}^{\mathcal{E}}(s),&r\geq{\mathbf{r}}^{\mathcal{E}}(s)\qquad\mbox{
(Euclidean domain),}\end{cases}$ (2.4)

in which $\mathbf{T}^{\mathcal{E}}=\mathbf{T}^{\mathcal{E}}(s)$ is independent
of $r$ and, for universal constants $K_{1},K_{2}>0$,

$K_{1}\,s^{2}\leq\mathbf{T}(s,r)\leq K_{2}\,s^{2}\qquad\text{ in
}\mathscr{M}^{\mathcal{EM}}_{[s_{0},+\infty)},$ (2.5)

together with

$\displaystyle 0\leq\partial_{r}\mathbf{T}(s,r)<1,\quad$ (slices of constant
$s$ are spacelike), (2.6) $\displaystyle
0<\partial_{r}\mathbf{T}(s,r)<1,\quad$ $\displaystyle\mbox{ when
$0<r\leq{\mathbf{r}}^{\mathcal{H}}(s)$},$
$\displaystyle|\partial_{r}\,\partial_{r}\mathbf{T}(s,r)|\lesssim 1.$

Figure 2.1: Spacetime foliation defined by merging
asymptotically Euclidean and asymptotically hyperboloidal slices

#### Foliation of interest.

A one-parameter family of spacelike, asymptotically Euclidean hypersurfaces is
defined as

$\mathscr{M}_{s}:=\big{\\{}(t,x^{a})\in\mathscr{M}\,\big{/}\,t=\mathbf{T}(s,r)\big{\\}}.$
(2.7)

In the future of the initial surface $\\{t=1\\}$, namely $\\{t\geq
1\\}=\mathscr{M}^{\textbf{init}}\cup\bigcup_{s\geq s_{0}}\mathscr{M}_{s}$, we
distinguish the initial domain (as we call it)
$\mathscr{M}^{\textbf{init}}=\\{(t,x)\,/\,1\leq t\leq T(s_{0},r)\\}$ within
which standard local-in-time existence arguments apply. Each slice
$\mathscr{M}_{s}=\mathscr{M}^{\mathcal{H}}_{s}\cup\mathscr{M}^{\mathcal{M}}_{s}\cup\mathscr{M}^{\mathcal{E}}_{s}$
is decomposed into three domains (with overlapping boundaries):

$\displaystyle\mathscr{M}^{\mathcal{H}}_{s}$
$\displaystyle:=\big{\\{}t=\mathbf{T}(s,|x|)\,\big{/}\,|x|\leq{\mathbf{r}}^{\mathcal{H}}(s)\big{\\}},$
asymptotically hyperboloidal, (2.8)
$\displaystyle\mathscr{M}^{\mathcal{M}}_{s}$
$\displaystyle:=\big{\\{}t=\mathbf{T}(s,|x|)\,\big{/}\,{\mathbf{r}}^{\mathcal{H}}(s)\leq|x|\leq{\mathbf{r}}^{\mathcal{E}}(s)\big{\\}},$
merging (or transition), $\displaystyle\mathscr{M}^{\mathcal{E}}_{s}$
$\displaystyle:=\big{\\{}t=\mathbf{T}(s)\,\big{/}\,{\mathbf{r}}^{\mathcal{E}}(s)\leq|x|\big{\\}},$
$\displaystyle\mbox{asymptotically Euclidean},$

with also
$\mathscr{M}^{\mathcal{EM}}_{s}:=\mathscr{M}^{\mathcal{E}}_{s}\cup\mathscr{M}^{\mathcal{M}}_{s}$.
Some additional notation is needed:

$\displaystyle\mathscr{M}_{[s_{0},s_{1}]}$
$\displaystyle:={\big{\\{}\mathbf{T}(s_{0},r)\leq
t\leq\mathbf{T}(s_{1},r)\big{\\}}=\bigcup_{s_{0}\leq s\leq
s_{1}}\mathscr{M}_{s}},\qquad\mathscr{M}_{[s_{0},+\infty)}$
$\displaystyle:=\bigcup_{s\geq s_{0}}\mathscr{M}_{s},$ (2.9)

and, similarly, we set $\mathscr{M}^{\mathcal{H}}_{[s_{0},s_{1}])}$,
$\mathscr{M}^{\mathcal{H}}_{[s_{0},+\infty)}$, etc. By construction, there
exists a function $c=c(s)\in(0,1)$ such that the radial variable $r$ in each
of the three domains satisfies

$r=|x|\in\,\begin{cases}\hskip
28.45274pt[0,t-1],&\mathscr{M}^{\mathcal{H}}_{[s_{0},+\infty)},\\\
[t-1,t-c(s)],\qquad&\mathscr{M}^{\mathcal{M}}_{[s_{0},+\infty)},\\\
[t-c(s),+\infty),&\mathscr{M}^{\mathcal{E}}_{[s_{0},+\infty)}.\end{cases}$
(2.10)

The future-oriented normal and the volume element associated with the
hypersurfaces are given by

$\displaystyle n_{s}$
$\displaystyle=\frac{\big{(}1,-(x^{a}/r)\partial_{r}\mathbf{T}\big{)}}{\sqrt{1+|\partial_{r}\mathbf{T}|^{2}}}=\nu^{-1}\Big{(}(s^{2}+r^{2})^{1/2},-x^{a}\xi(s,r)\Big{)},$
(2.11a) $\displaystyle\nu$
$\displaystyle=\big{(}s^{2}+r^{2}(1+\xi(s,r)^{2})\big{)}^{1/2},$ together with
$\displaystyle d\sigma_{s}$
$\displaystyle=(1+|\partial_{r}\mathbf{T}|^{2})\,dx=\nu\,(s^{2}+r^{2})^{-1/2}\,dx,$
(2.11b) $\displaystyle n_{s}d\sigma_{s}$
$\displaystyle=(1,-(x^{a}/r)\partial_{r}\mathbf{T})\,dx=(1,-\partial_{a}\mathbf{T})\,dx=\Big{(}1,\frac{-\xi(s,r)x^{a}}{(s^{2}+r^{2})^{1/2}}\Big{)}\,dx.$

#### Change of variables.

We will be relying on two parameterizations of the Euclidean-hyperboloidal
hypersurfaces, namely in terms of the variables $(t,x)$, or in terms of the
variables $(s,x)$ determined by the function $T$. This latter function by
construction is strictly increasing in $s$ and, in fact,
$(s,x)\mapsto(t,x)=(\mathbf{T}(s,x),x)$ is a smooth and global diffeomorphism.
The Jacobian matrix associated with the Euclidean–hyperboloidal foliation
reads $\left(\begin{array}[]{cc}\partial_{s}\mathbf{T}&\partial_{s}x\\\
\partial_{x}\mathbf{T}&\partial_{x}x\end{array}\right)=\left(\begin{array}[]{cc}\partial_{s}\mathbf{T}&0\\\
(x^{a}/r)\,\partial_{r}\mathbf{T}&I\end{array}\right)$ and the Jacobian is
$J(s,x)=\partial_{s}\mathbf{T}(s,x)$, leading to the corresponding volume
element $dtdx=J\,dsdx$. It can be established that

$J\leq\begin{cases}{s\over\mathbf{T}}=s\,(s^{2}+r^{2})^{-1/2},\\\ \xi
s\,(s^{2}+r^{2})^{-1/2}+(1-\xi)\,2s,\\\
2s,\end{cases}\,J\geq\begin{cases}{s\over\mathbf{T}}=s\,(s^{2}+r^{2})^{-1/2},\quad&\mathscr{M}^{\mathcal{H}}_{s},\\\
\xi\,s\,(s^{2}+r^{2})^{-1/2}+(1-\xi)3s/5,\quad&\mathscr{M}^{\mathcal{M}}_{s},\\\
3s/5\quad&\mathscr{M}^{\mathcal{E}}_{s}.\end{cases}$

###  2.2 Frames of interest and commutators

#### Frames of interest.

We will combine estimates involving different frames, as follows.

  * •

The semi-hyperboloidal frame

$\partial^{\mathcal{H}}_{0}=\partial_{t},\qquad\partial^{\mathcal{H}}_{a}=\not{\partial}^{\mathcal{H}}_{a}=\frac{x^{a}}{t}\partial_{t}+\partial_{a},$
(2.12)

and was already introduced by the authors in [20]. It is defined globally in
$\mathscr{M}_{s}$, relevant within the hyperboloidal domain and is relevant in
order to (1) exhibit the (quasi-)null form structure of the nonlinearities
and, in turn, (2) establish decay properties in timelike and null directions.
Some of our arguments also involve radial integration based on
$\not{\partial}^{\mathcal{H}}_{r}=(x^{a}/r)\not{\partial}^{\mathcal{H}}_{a}$.

  * •

The semi-null frame

$\partial^{\mathcal{N}}_{0}=\partial_{t},\qquad\partial^{\mathcal{N}}_{a}=\not{\partial}^{\mathcal{N}}_{a}={x^{a}\over
r}\partial_{t}+\partial_{a}$ (2.13)

is defined everywhere in $\mathscr{M}_{s}$ except on the center line $r=0$ and
is the appropriate frame within the Euclidean-merging domain in order to (1)
exhibit the structure of the (null, quasi-null) nonlinearities of the field
equations, and (2) establish decay properties in spatial and null directions.

  * •

The Euclidean–hyperboloidal frame

$\partial^{\mathcal{E\hskip-1.70709ptH}}_{0}=\partial_{t},\quad\partial^{\mathcal{E\hskip-1.70709ptH}}_{a}=\not{\partial}^{\mathcal{EH}}_{a}=\partial_{a}+(x^{a}/r)\partial_{r}\mathbf{T}\,\partial_{t}$
(2.14)

involves tangent vectors to the slices $\mathscr{M}_{s}$ and provides an
interpolation between
$\partial^{\mathcal{E\hskip-1.70709ptH}}_{a}=\partial^{\mathcal{H}}_{a}$ in
$\mathscr{M}^{\mathcal{H}}_{s}$, and
$\partial^{\mathcal{E\hskip-1.70709ptH}}_{a}=\partial_{a}$ in
$\mathscr{M}^{\mathcal{E}}_{s}$. Some of our arguments are also based on
radial integration based on
$\not{\partial}^{\mathcal{EH}}_{r}=(x^{a}/r)\not{\partial}^{\mathcal{EH}}_{a}$.

Moreover, various changes of frame formulas are useful such as
$\partial^{\mathcal{N}}_{\alpha}={\Phi^{\mathcal{N}}}{}{}_{\alpha}^{\beta}\,\partial_{\beta}$
and
$\partial_{\alpha}={\Psi^{\mathcal{N}}{}}_{\alpha}^{\beta}\partial^{\mathcal{N}}_{\beta}$
with

$\big{(}{\Phi^{\mathcal{N}}}{}_{\alpha}^{\beta}\big{)}=\left(\begin{array}[]{cccc}1&0&0&0\\\
x^{1}/r&1&0&0\\\ x^{2}/r&0&1&0\\\
x^{3}/r&0&0&1\end{array}\right),\qquad\big{(}{\Psi^{\mathcal{N}}{}}_{\alpha}^{\beta}\big{)}=\left(\begin{array}[]{cccc}1&0&0&0\\\
-x^{1}/r&1&0&0\\\ -x^{2}/r&0&1&0\\\ -x^{3}/r&0&0&1\end{array}\right).$ (2.15)

#### Commutator properties.

It is convenient to rely on the following definition. Let
$L_{a}=x^{a}\partial_{t}+t\partial_{a}$ and
$\Omega_{ab}=x^{a}\partial_{b}-x^{b}\partial_{a}$ be the Lorentzian and
Euclidean rotations. We use $L^{J},\Omega^{K}$ and $\partial^{I}$ for high-
order derivatives consisting of combinations of $L_{a},\Omega_{ab}$ and
$\partial_{\alpha}$ where $I,J,K$ denote ordered multi-indices.

######  Definition 2.1.

An operator $Z=\partial^{I}L^{J}\Omega^{K}$ is called an (ordered) admissible
operator and for such an operator one associates its order and rank by

$\textbf{ord}(Z)=|I|+|J|+|K|,\quad\textbf{rank}(Z)=|J|+|K|,\qquad
Z=\partial^{I}L^{J}\Omega^{K}.$ (2.16)

An operator $\Gamma=\partial^{I}L^{J}\Omega^{K}S^{l}$ is called an ordered
conformal operator. Its order and rank are defined similarly:

$\textbf{ord}(\Gamma)=|I|+|J|+|K|+l,\quad\textbf{rank}(\Gamma)=|J|+|K|+l,\qquad\Gamma=\partial^{I}L^{J}\Omega^{K}S^{l}.$

The proof of the following statement can be found in [24, Part 1]. To deal
with differential operators $\partial^{I}L^{J}\Omega^{K}$ we use the notation
$\textbf{ord}(Z)=|I|+|J|+|K|$ (the order) and $\textbf{rank}(Z)=|J|+|K|$ (the
rank). Given two integers $k\leq p$, it is convenient to introduce the
notation

$|u|_{p,k}:=\max_{\textbf{ord}(Z)\leq p,\textbf{rank}(Z)\leq
k}|Zu|,\qquad|u|_{p}:=\max_{\textbf{ord}(Z)\leq p}|Zu|.$ (2.17)

######  Lemma 2.2 (Estimates for linear commutators).

For any admissible field $Z$ satisfying $\textbf{ord}(Z)\leq p$ and
$\textbf{rank}(Z)\leq k$ one has

$|[Z,\partial]u|\lesssim|\partial u|_{p-1,k-1},$ (2.18a)
$|[Z,\partial\partial]u|\lesssim|\partial\partial
u|_{p-1,k-1}\lesssim|\partial u|_{p,k-1}.$ (2.18b)

In the following, it is convenient to denote by
$Y_{\text{rot}}\in\\{L_{a},\Omega_{ab}\\}$ the collection of boosts and
spatial rotations.

######  Proposition 2.3 (Hierarchy structure for quasi-linear commutators.
Euclidean-merging domain).

Let $Z$ be an admissible operator with ${\textbf{ord}(Z)=p}$ and
$\textbf{rank}(Z)=k$ and let $H,u$ be functions defined in the Euclidean-
merging domain $\mathscr{M}^{\mathcal{EM}}_{s}$. Then, one has

$|[Z,H]u|\lesssim\sum_{k_{1}+p_{2}=p\atop
k_{1}+k_{2}=k}|Y_{\text{rot}}H|_{k_{1}-1}|u|_{p_{2},k_{2}}+\sum_{p_{1}+p_{2}=p\atop
k_{1}+k_{2}=k}|\partial H|_{p_{1}-1,k_{1}}|u|_{p_{2},k_{2}},$ (2.19)
$\displaystyle|[Z,H\partial_{\alpha}\partial_{\beta}]u|$
$\displaystyle\lesssim\,|H|\,|\partial\partial
u|_{p-1,k-1}+\sum_{k_{1}+p_{2}=p\atop
k_{1}+k_{2}=k}|Y_{\text{rot}}H|_{k_{1}-1}|\partial\partial u|_{p_{2},k_{2}}$
(2.20) $\displaystyle+\sum_{p_{1}+p_{2}=p\atop k_{1}+k_{2}=k}|\partial
H|_{p_{1}-1,k_{1}}|\partial\partial u|_{p_{2},k_{2}},$

######  Proposition 2.4 (Hierarchy property for quasi-linear commutators.
Hyperboloidal domain).

For any function $u$ defined in $\mathscr{M}^{\mathcal{H}}_{[s_{0},s_{1}]}$
and for any admissible operator $Z$ with $\textbf{ord}(Z)=p$ and
$\textbf{rank}(Z)=k$ one has

$\displaystyle|[Z,H^{\alpha\beta}\partial_{\alpha}\partial_{\beta}]u|\lesssim
T^{\textbf{hier}}+T^{\textbf{easy}}+T^{\textbf{super}},$ (2.21a)
$\displaystyle T^{\textbf{hier}}$
$\displaystyle|H^{\mathcal{H}}{}^{00}|\,|\partial\partial
u|_{p-1,k-1}+\sum_{k_{1}+p_{2}=p\atop
k_{1}+k_{2}=k}|LH^{\mathcal{H}}{}^{00}|_{k_{1}-1,k_{1}-1}|\partial\partial
u|_{p_{2},k_{2}},$ (2.21b) $\displaystyle T^{\textbf{easy}}$
$\displaystyle:=\sum_{p_{1}+p_{2}=p\atop k_{1}+k_{2}=k}|\partial
H^{\mathcal{H}}{}^{00}|_{p_{1}-1,k_{1}}|\partial\partial
u|_{p_{2},k_{2}}+t^{-1}|H|\,|\partial u|_{p},$ $\displaystyle
T^{\textbf{super}}$
$\displaystyle:=\sum_{p_{1}+p_{2}=p}|\not{\partial}^{\mathcal{H}}H|_{p_{1}-1}|\partial
u|_{p_{2}+1}+t^{-1}\\!\\!\\!\\!\sum_{p_{1}+p_{2}=p}\\!\\!\\!\\!|\partial
H|_{p_{1}-1}|\partial u|_{p_{2}+1}.$

###  2.3 Energy functionals

#### Weight coefficients.

The fundamental energy functional (stated shortly below) involved another
geometric weight, denoted by $\zeta=\zeta(t,x)$ and defined by

$\zeta(s,r)^{2}=1-\frac{r^{2}\xi^{2}(s,r)}{s^{2}+r^{2}}={s^{2}\over
s^{2}+r^{2}}+(1-\xi^{2}(s,r)){r^{2}\over s^{2}+r^{2}}.$ (2.22)

This weight coincides with $s/t=s/(s^{2}+r^{2})^{1/2}$ in the hyperboloidal
domain, while it reduces to $1$ in the Euclidean domain. In fact, it provides
us with an interpolation between the energy density induced on hyperboloids
and the one induced on Euclidean slices. Various estimates on $\zeta$ can be
proven, for instance $\frac{|r-t\,|+1}{r}\lesssim\zeta^{2}\leq\zeta\leq 1$
valid within the Euclidean-merging domain, as well as $K_{1}\,\zeta^{2}s\leq
J\leq K_{2}\,\zeta^{2}s$ in the merging domain (for some universal constants
$K_{1},K_{2}$). Furthermore, we introduce a weight which measures the distance
to the light cone, and is defined from the prescription of a smooth and non-
decreasing function $\aleph$ satisfying

$\aleph(y)=\begin{cases}0,\quad&y\leq-2,\\\ y+2,\quad&y\geq{-1},\end{cases}$
(2.23)

and specifically we set $\mathbf{X}:=1+\aleph({r-t})$, which we refer to as
the energy weight. Recall that $\aleph^{\prime}$ is non-negative. This choice
is easier to work with, but an equivalent energy having a more geometric form
can be based on the unknown metric $g$.

#### Energy identity.

We multiply the wave-Klein-Gordon equation $\Box u-c^{2}u=F$ (with $c\geq 0$)
by $-2\,\mathbf{X}^{2\eta}\partial_{t}u$ with
$\Box=\Box_{g_{\textbf{Mink}}}=-\partial_{t}\,\partial_{t}+\sum_{a=1,2,3}\partial_{a}\partial_{a}$.
We treat simultaneously the wave and Klein-Gordon operators by assuming here
that $c\geq 0$. We find the divergence identity

$\displaystyle\partial_{t}\big{(}V^{0}_{\eta,c}[u]\big{)}+\partial_{a}\big{(}V^{a}_{\eta,c}[u]\big{)}$
$\displaystyle=2\eta\mathbf{X}^{-1}\aleph^{\prime}({r-t})(-1,x^{a}/r)\cdot
V_{\eta,c}[u]-2\,\mathbf{X}^{2\eta}\partial_{t}u\,F$ $\displaystyle
V_{\eta,c}[u]$
$\displaystyle=-\mathbf{X}^{2\eta}\big{(}-|\partial_{t}u|^{2}-\sum_{a}|\partial_{a}u|^{2}-c^{2}u^{2},2\partial_{t}u\partial_{a}u\big{)}.$

We define our energy functional on each Euclidean–hyperboloidal slice
$\mathscr{M}_{s}$ as

$\displaystyle\mathcalboondox E_{\eta,c}(s,u)$
$\displaystyle=\int_{\mathscr{M}_{s}}V_{\eta,c}[u]\cdot n_{s}d\sigma_{s}$
(2.24)
$\displaystyle=\int_{\mathscr{M}_{s}}\Big{(}|\partial_{t}u|^{2}+\sum_{a}|\partial_{a}u|^{2}+\frac{2x^{a}\xi(s,r)}{(s^{2}+r^{2})^{1/2}}\partial_{t}u\partial_{a}u+c^{2}u^{2}\Big{)}\,\mathbf{X}^{2\eta}\,dx$

or, equivalently,

$\displaystyle\mathcalboondox E_{\eta,c}(s,u)$
$\displaystyle=\int_{\mathscr{M}_{s}}\Big{(}\zeta^{2}|\partial_{t}u|^{2}+\sum_{a}|\not{\partial}^{\mathcal{EH}}_{a}u|^{2}+c^{2}u^{2}\Big{)}\,\mathbf{X}^{2\eta}\,dx,$
(2.25)
$\displaystyle=\int_{\mathscr{M}_{s}}\Big{(}\zeta^{2}\sum_{a}|\partial_{a}u|^{2}+\frac{\xi^{2}(s,r)}{s^{2}+r^{2}}\sum_{a<b}|\Omega_{ab}u|^{2}+|\not{\partial}^{\mathcal{EH}}_{r}u|^{2}+c^{2}\,u^{2}\Big{)}\,\mathbf{X}^{2\eta}\,dx.$

It involves the energy coefficient $\zeta$, which is non-trivial in the
merging and hyperboloidal domains, and depends upon our choice of coefficient
$\xi=\xi(s,r)$.

The energy identity in curved spacetime associated with the wave or Klein-
Gordon equation $g^{\alpha\beta}\partial_{\alpha}\partial_{\beta}u-c^{2}u=F$
is expressed by decomposing the curved metric $g$ as
$g^{\alpha\beta}=g_{\textbf{Mink}}^{\alpha\beta}+H^{\alpha\beta}.$

After defining the energy-flux vector (with $a=1,2,3$)
$V_{g,\eta,c}[u]=-\mathbf{X}^{2\eta}\Big{(}g^{00}|\partial_{t}u|^{2}-g^{ab}\partial_{a}u\,\partial_{b}u-c^{2}u^{2},\
2g^{a\beta}\partial_{t}u\partial_{\beta}u\Big{)},$ (2.26a) which depends upon
$g$ as well as the weight $\mathbf{X}^{\eta}$, we easily find the energy
identity $\displaystyle\text{div}\hskip
1.42271ptV_{g,\eta,c}[u]=-\Omega_{g,\eta,c}[u]+G_{g,\eta}[u]-2\mathbf{X}^{2\eta}\,\partial_{t}u\,F,$
(2.26b) in which $\displaystyle\Omega_{g,\eta,c}[u]$
$\displaystyle=-2\eta\,\mathbf{X}^{-1}\aleph^{\prime}({r-t})(-1,x^{a}/r)\cdot
V_{g,\eta,c}[u]$ (2.26c)
$\displaystyle=2\,\eta\,\mathbf{X}^{2\eta-1}\aleph^{\prime}({r-t})\,\Big{(}g^{\mathcal{N}ab}\not{\partial}^{\mathcal{N}}_{a}u\not{\partial}^{\mathcal{N}}_{b}u-H^{\mathcal{N}00}|\partial_{t}u|^{2}+c^{2}u^{2}\Big{)},$
$\displaystyle G_{g,\eta}[u]$
$\displaystyle=-\partial_{t}H^{00}|\mathbf{X}^{\eta}\partial_{t}u|^{2}+\partial_{t}H^{ab}\mathbf{X}^{2\eta}\partial_{a}u\partial_{b}u-2\mathbf{X}^{2\eta}\partial_{a}H^{a\beta}\partial_{t}u\partial_{\beta}u.$

In turn we arrive at a weighted energy estimate associated with the Euclidean-
hyperboloidal foliation:

$\displaystyle\frac{d}{ds}\mathcalboondox
E_{g,\eta,c}(s,u)+2\eta\int_{\mathscr{M}_{s}}\big{(}g^{\mathcal{N}ab}\not{\partial}^{\mathcal{N}}_{a}u\not{\partial}^{\mathcal{N}}_{b}u+c^{2}u^{2}\big{)}\aleph^{\prime}({r-t})\mathbf{X}^{2\eta-1}\
Jdx$ (2.27)
$\displaystyle=\int_{\mathscr{M}_{s}}\Big{(}G_{g,\eta}[u]+\eta\mathbf{X}^{2\eta-1}\aleph^{\prime}({r-t})H^{\mathcal{N}}{}^{00}|\partial_{t}u|^{2}\Big{)}\
Jdx+\int_{\mathscr{M}_{s}}|\partial_{t}uF|\,\mathbf{X}^{2\eta}\,Jdx,$

in which the latter integral is controlled by

$\displaystyle\int_{s_{0}}^{s_{1}}\mathcalboondox
E_{\eta,c}(s,u)^{1/2}\,\big{(}\|F\|_{L^{2}(\mathscr{M}^{\mathcal{H}}_{s})}+\big{\|}s\zeta
F\big{\|}_{L^{2}(\mathscr{M}^{\mathcal{M}}_{s})}+\|s\,\mathbf{X}^{\eta}f\|_{L^{2}(\mathscr{M}^{\mathcal{E}}_{s})}\big{)}\,ds.$
(2.28)

It will be convenient to also use the notation $\mathcalboondox
F_{\eta,c}(s,u)=\big{(}\mathcalboondox E_{\eta,c}(s,u)\big{)}^{1/2}$ and
similarly with the subscript $c$ omitted.

###  2.4 Functional inequalities on Euclidean-hyperboloidal slices

#### Sobolev inequalities.

The proof of the statements in this section can be found in [24, Part 1].

######  Proposition 2.5 (Sup-norm Sobolev inequality. Hyperboloidal domain).

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

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

######  Proposition 2.6 (Weighted sup-norm Sobolev inequality. 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

$\textstyle r\mathbf{X}^{\eta}|u(t,x)|\lesssim C(\eta)\sum_{|I|+|J|\leq
2}\|\mathbf{X}^{\eta}\not{\partial}^{\mathcal{EM}}{}^{I}\Omega^{J}u\|_{L^{2}(\mathscr{M}^{\mathcal{EM}}_{s})},\qquad(t,x)\in\mathscr{M}^{\mathcal{EM}}_{s},$
(2.29a) $\textstyle r\mathbf{X}^{\eta}|u(t,x)|\lesssim
C(\eta)\sum_{|I|+|J|\leq
2}\|\mathbf{X}^{\eta}\not{\partial}^{\mathcal{E}}{}^{I}\Omega^{J}u\|_{L^{2}(\mathscr{M}^{\mathcal{E}}_{s})},\qquad(t,x)\in\mathscr{M}^{\mathcal{E}}_{s}.$
(2.29b)

#### Hardy inequalities.

Several functional inequalities will be used to control undifferentiated
functions from the energy functional.

######  Proposition 2.7 (Weighted Hardy inequality on the Euclidean-
hyperboloidal foliation).

Fix some exponent $\eta\geq 0$. For any function $u$ defined in
$\mathscr{M}_{[s_{0},s_{1}]}$ and sufficiently decaying at infinity, one has

$\|r^{-1}\mathbf{X}^{\eta}u\|_{L^{2}(\mathscr{M}_{s})}\lesssim\|\mathbf{X}^{\eta}\not{\partial}^{\mathcal{EH}}u\|_{L^{2}(\mathscr{M}_{s})}\lesssim\mathcalboondox
F_{\eta}(s,u).$

######  Proposition 2.8 (Hardy-type inequalities in the hyperboloidal domain).

For all functions $u$ defined on a hyperboloid $\mathcal{H}_{s}$:

$\Big{\|}{u\over
r}\Big{\|}_{L^{2}(\mathcal{H}_{s})}\lesssim\sum_{a}\|\partial^{\mathcal{H}}_{a}u\|_{L^{2}(\mathcal{H}_{s})}.$
(2.30a) On the other hand, for all functions defined within a slab of the
hyperboloidal foliation $[s_{0},s_{1}]$ $\displaystyle\Big{\|}{u\over
s_{1}}\Big{\|}_{L^{2}(\mathcal{H}_{s_{1}})}\lesssim$
$\displaystyle\textstyle\Big{\|}{u\over
s_{0}}\Big{\|}_{L^{2}(\mathcal{H}_{s_{0}})}+\sum_{a}\|\partial^{\mathcal{H}}_{a}u\|_{L^{2}(\mathcal{H}_{s_{1}})}$
(2.30b)
$\displaystyle\textstyle\quad+\sum_{a}\int_{s_{0}}^{s_{1}}\Big{(}\|\partial^{\mathcal{H}}_{a}u\|_{L^{2}(\mathcal{H}_{s})}+\|(s/t)\partial_{a}u\|_{L^{2}(\mathcal{H}_{s})}\Big{)}\,{ds\over
s}.$

We point out that while the proof of (2.30a) is analogue to the proof of the
standard Hardy inequality, in contrast the proof of (2.30b) is more involved
and is based on computing the divergence of the vector field

$\chi\Big{(}{r\over t}\Big{)}^{2}\,\Big{(}0,{t\,x^{a}\over
1+r^{2}}\,{u^{2}\over s^{2}}\Big{)},\qquad\chi\Big{(}{r\over
t}\Big{)}=\begin{cases}0,\,&\displaystyle r/t\leq 1/3,\\\
1,\,&\displaystyle\hskip 0.0ptr/t\geq 2/3,\end{cases}$ (2.31)

a smooth cut-off function being introduced here in order to removes a wedge
containing the center of coordinates.

#### Poincaré inequalities.

In our analysis the following functional inequalities will also be useful.

######  Proposition 2.9 (Poincaré-type inequalities in the Euclidean-merging
domain).

Fix an exponent $\eta=1/2+\delta$ with $\delta>0$. For any function $u$
defined in
$\mathscr{M}^{\mathcal{EM}}_{s}=\\{(t,x)\in\mathscr{M}_{s}\,/\,|x|\geq{\mathbf{r}}^{\mathcal{H}}(s)\\}$,
one has
$\|\mathbf{X}^{-1+\eta}u\|_{L^{2}(\mathscr{M}^{\mathcal{EM}}_{s})}\lesssim\big{(}1+\delta^{-1}\big{)}\|\mathbf{X}^{\eta}\not{\partial}^{\mathcal{EM}}u\|_{L^{2}(\mathscr{M}^{\mathcal{EM}}_{s})}+\|r^{-1}\mathbf{X}^{\eta}u\|_{L^{2}(\mathscr{M}^{\mathcal{EM}}_{s})},$
(2.32a) $\|\mathbf{X}^{-1+\eta}\zeta
u\|_{L^{2}(\mathscr{M}^{\mathcal{EM}}_{s})}\lesssim(1+\delta^{-1})\|\mathbf{X}^{\eta}\zeta\not{\partial}^{\mathcal{EM}}u\|_{L^{2}(\mathscr{M}^{\mathcal{EM}}_{s})}+\|r^{-1}\mathbf{X}^{\eta}\zeta
u\|_{L^{2}(\mathscr{M}^{\mathcal{EM}}_{s})}.$ (2.32b)

##  3 Wave-Klein-Gordon equations in Euclidean-Hyperboloidal foliations

###  3.1 Pointwise estimates for wave equations

We now investigate the decay of solutions to wave equations and state
$L^{\infty}$–$L^{\infty}$ estimates. Given some data $f,u_{0},u_{1}$ with
sufficient regularity and decay (so that Kirchhoff’s formula below makes
sense), we consider the solution $u=u(t,x)$ to the initial value problem

$\Box u=f,\quad u(1,x)=u_{0}(x),\quad\partial_{t}u(1,x)=u_{1}(x),\qquad
x\in\mathbb{R}^{3}.$ (3.1)

In order to separate the contributions from the initial data and from the
source, we find it convenient to use the short-hand notation
$u=\Box^{-1}[u_{0},u_{1},f]$,
$\Box^{-1}_{\textbf{init}}[u_{0},u_{1}]:=\Box^{-1}[u_{0},u_{1},0]$, and
$\Box^{-1}_{\textbf{sour}}[f]:=\Box^{-1}[0,0,f]$. While referring to [24] for
the analysis of initial data, let us here consider the effect of a source,
namely the operator $\Box^{-1}_{\textbf{sour}}$. An earlier result was
established in [23] where the source was supported in the interior of a light
cone — a restriction we overcome in the statement below. Throughout,
$\Lambda_{t,x}:=\big{\\{}(\tau,y)\big{/}\,t-\tau=|x-y|,\,1\leq\tau\leq
t\big{\\}}$ is the truncated cone from a point $(t,x)$.

######  Proposition 3.1 (Wave equation. Contribution from the source).

Consider the wave operator $\Box^{-1}_{\textbf{sour}}$ acting on a source
function $f$ satisfying the decay conditions (with
$\alpha_{1},\alpha_{2},\alpha_{3}$)

$|f(\tau,y)|\lesssim
C_{1}\,\tau^{\alpha_{1}}(\tau+|y|)^{\alpha_{2}}\big{(}1+|\tau-|y||\big{)}^{\alpha_{3}},\qquad(\tau,y)\in\Lambda_{t,x}.$
(3.2)

Then the solution to the wave equation enjoys the following properties.

– Case 0 (interior). When the support of $f$ is contained in $\\{r\leq
t-1\\}$, and $\alpha_{1}=-2-\upsilon,\alpha_{2}=0,\alpha_{3}=-1+\mu$ for
$0<|\upsilon|\leq 1/2$, $0<\nu\leq 1/2$, one has

$|\Box^{-1}_{\textbf{sour}}[f](t,x)|\lesssim
C_{1}\begin{cases}\frac{1}{\nu\mu}(t-r)^{\mu-\upsilon}t^{-1},\qquad&0<\nu\leq
1/2,\\\
\frac{1}{|\nu|\mu}(t-r)^{\mu}t^{-1-\upsilon},&-1/2\leq\nu<0.\end{cases}$
(3.3a)

– 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

$|\Box^{-1}_{\textbf{sour}}[f](t,x)|\lesssim
C_{1}\,\big{(}\upsilon^{-1}+\mu^{-1}+|\mu-\nu|^{-1}\big{)}\,|\upsilon+\mu-\nu|^{-1}(t+r)^{-1}.$
(3.3b)

– 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

$|\Box^{-1}_{\textbf{sour}}[f](t,x)|\lesssim
C_{1}\,\begin{cases}\mu^{-1}|\mu-\nu|^{-1}(t+r)^{-1}t^{\mu-\nu},\qquad&\mu>\nu,\\\
\mu^{-1}(t+r)^{-1}\ln(t+1),\quad&\mu=\nu,\\\
\mu^{-1}|\mu-\nu|^{-1}(t+r)^{-1},&\mu<\nu.\end{cases}$ (3.3c)

– 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

$|\Box^{-1}_{\textbf{sour}}[f](t,x)|\lesssim
C_{1}\,\mu^{-1}(t+r)^{-1}\Big{(}1+\mathbf{X}^{-\mu}\ln\big{(}t/\mathbf{X}\big{)}\Big{)}.$
(3.3d)

– 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

$|\Box^{-1}_{\textbf{sour}}[f](t,x)|\lesssim
C_{1}\,\big{(}|\mu-\nu|^{-1}+\mu^{-1}\nu^{-1}\mathbf{X}^{-\mu}t^{\nu}\big{)}(t+r)^{-1}.$
(3.3e)

(In the last two cases, observe that $\mathbf{X}\equiv 1$ when $r\leq t-1$).

###  3.2 Pointwise estimates for Klein-Gordon equations

The method below was introduced in Klainerman [18] and the derivation below
was proposed in LeFloch and Ma [22]. Here, we present yet another version of
the argument which takes into account the contribution from the boundary
(namely the light cone).

We focus here on the hyperboloidal domain and rely on the decomposition

$g^{\alpha\beta}\partial_{\alpha}\partial_{\beta}=s^{-3/2}(t/s)^{2}g^{\mathcal{H}}{}^{00}\mathcal{L}^{2}(s^{3/2}\phi)+\mathbb{R}_{g}[\phi],$
(3.4)

where $g^{\alpha\beta}=g_{\textbf{Mink}}^{\alpha\beta}+H^{\alpha\beta}$ and
the following field (which is nothing but the unit normal to the hyperboloids
for the Minkowski metric)

$\mathcal{L}:=(s/t)\partial_{t}+(x^{a}/s)\not{\partial}^{\mathcal{H}}_{a}=(t/s)\partial_{t}+(x^{a}/s)\partial_{a},$
(3.5)

while

$\displaystyle\mathbb{R}_{g}[\phi]=$
$\displaystyle-(3/4)s^{-2}\phi-3s^{-1}(x^{a}/s)\not{\partial}^{\mathcal{H}}_{a}\phi-(x^{a}x^{b}/s^{2})\not{\partial}^{\mathcal{H}}_{a}\not{\partial}^{\mathcal{H}}_{b}\phi-\sum_{a}\not{\partial}^{\mathcal{H}}_{a}\not{\partial}^{\mathcal{H}}_{a}\phi$
(3.6)
$\displaystyle-(t/s)^{2}H^{\mathcal{H}}{}^{00}\Big{(}(3/4)s^{-2}\phi+\big{(}3+(r/t)^{2}\big{)}t^{-1}\partial_{t}\phi+3s^{-1}(x^{a}/s)\not{\partial}^{\mathcal{H}}_{a}\phi\Big{)}$
$\displaystyle-(t/s)^{2}H^{\mathcal{H}}{}^{00}\Big{(}(2x^{a}/t)\partial_{t}\not{\partial}^{\mathcal{H}}_{a}\phi+(x^{a}x^{b}/s^{2})\not{\partial}^{\mathcal{H}}_{a}\not{\partial}^{\mathcal{H}}_{b}\phi\Big{)}$
$\displaystyle\quad+H^{\mathcal{H}}{}^{a0}\not{\partial}^{\mathcal{H}}_{a}\partial_{t}\phi+H^{\mathcal{H}}{}^{0a}\partial_{t}\not{\partial}^{\mathcal{H}}_{a}\phi+H^{\mathcal{H}}{}^{ab}\not{\partial}^{\mathcal{H}}_{a}\not{\partial}^{\mathcal{H}}_{b}\phi+H^{\alpha\beta}\partial_{\alpha}\big{(}{\Psi^{\mathcal{H}}}{}_{\beta}^{\beta^{\prime}}\big{)}\partial^{\mathcal{H}}_{\beta^{\prime}}\phi.$

For a Klein-Gordon equation $\widetilde{\Box}_{g}\phi-c^{2}\phi=f,$ thanks to
$g^{\mathcal{H}}{}^{00}=-(s/t)^{2}+H^{\mathcal{H}}{}^{00}$, under the
assumption $(s/t)^{2}|H^{\mathcal{H}}{}^{00}|\leq 1/3,$ the above identity
leads us to

$\mathcal{L}^{2}(s^{3/2}\phi)+\frac{c^{2}}{1-(t/s)^{2}H^{\mathcal{H}}{}^{00}}(s^{3/2}\phi)=\frac{s^{3/2}(f+\mathbb{R}_{g})}{1-(t/s)^{2}H^{\mathcal{H}}{}^{00}}.$
(3.7)

For any function $\phi$ defined in $\mathscr{M}^{\mathcal{H}}_{[s_{0},s_{1}]}$
and at each point $(t,x)\in\mathscr{M}^{\mathcal{H}}_{[s_{0},s_{1}]}$, we use
the notation $\Phi_{t,x}(\lambda):=\lambda^{3/2}\phi(\lambda t/s,\lambda
x/s)$. Since $\mathcal{L}(s^{3/2}\phi)|_{(\lambda t/s,\lambda
x/s)}=\Phi_{t,x}^{\prime}(\lambda)$, we find

$\Phi_{t,x}^{\prime\prime}(\lambda)+\frac{c^{2}}{1-\overline{H}_{t,x}}\Phi_{t,x}=\frac{\lambda^{3/2}}{1-\overline{H}_{t,x}}(f+\mathbb{R}_{g})|_{(\lambda
t/s,\lambda x/s)},$ (3.8)

where $\overline{H}_{t,x}=(t/s)^{2}H^{\mathcal{H}}{}^{00}|_{(\lambda
t/s,\lambda x/s)}$.

We observe that $(s/t)$ is constant along a given path $\gamma_{t,x}$, and we
multiply the above equation by $(s/t)^{\eta}$ with $\eta\in\mathbb{R}$. By an
elementary ODE lemma, we then arrive at the following result.

######  Proposition 3.2 (Sharp decay of Klein-Gordon solutions in the
hyperboloidal domain).

Suppose that for all $(t,x)\in\mathscr{M}^{\mathcal{H}}_{[s_{0},s_{1}]}$ and
for all $\lambda_{0}\leq\lambda\leq s_{1}$, one has

$|\overline{H}_{t,x}|\leq
1/3,\qquad\int_{\lambda_{0}}^{s_{1}}|\overline{H}_{t,x}^{\prime}(\lambda)|d\lambda\lesssim
1.$ (3.9)

Then for any $\eta\in\mathbb{R}$, any solution $\phi$ to the Klein-Gordon
equation $\widetilde{\Box}_{g}\phi-c^{2}\phi=f$ satisfies

$\displaystyle(s/t)^{\eta}s^{3/2}\,\Big{(}|\phi(t,x)|+(s/t)\,|\partial\phi(t,x)|\Big{)}$
(3.10)
$\displaystyle\lesssim(s/t)^{\eta}s^{1/2}|\phi|_{1}(t,x)+\sup_{\mathscr{M}^{\mathcal{H}}_{s_{0}}\cup\mathscr{L}_{[s_{0},s]}}(s/t)^{\eta}\Big{(}t^{1/4}(t^{1/2}|\phi|+|\partial\phi|+t\,|\not{\partial}^{\mathcal{N}}\phi|)\Big{)}$
$\displaystyle\quad+(s/t)^{\eta}\int_{\lambda_{0}}^{s}\lambda^{3/2}\big{(}|f|+|\mathbb{R}_{g}[\phi]\big{)}\big{|}_{(\lambda
t/s,\lambda x/s)}d\lambda,$

in which

$\lambda_{0}=\begin{cases}s_{0},\quad&0\leq
r/t\leq\frac{s_{0}^{2}-1}{s_{0}^{2}+1},\\\
\sqrt{\frac{t+r}{t-r}},&\frac{s_{0}^{2}-1}{s_{0}^{2}+1}\leq r/t<1\end{cases}$
(3.11)

and

$\displaystyle|\mathbb{R}_{g}[\phi]|_{p,k}$ $\displaystyle\lesssim
s^{-2}|\phi|_{p+2}+(t/s)^{2}\sum_{p_{1}+p_{2}=p}|H^{\mathcal{H}}{}^{00}|_{p_{1}}\big{(}s^{-2}|\phi|_{p_{2}+2}+t^{-1}|\partial\phi|_{p_{2}+1}\big{)}$
(3.12)
$\displaystyle\quad+\sum_{p_{1}+p_{2}=p}|H|_{p_{1}}\big{(}t^{-1}|\partial\phi|_{p_{2}+1}+t^{-2}|\phi|_{p_{2}+2}\big{)}.$

###### Proof.

Observe that the integral curve of $\mathcal{L}$ is $\gamma_{t,x}=\\{(\lambda
t/s,\lambda x/s)\\}$. When $0\leq r/t\leq\frac{s_{0}^{2}-1}{s_{0}^{2}+1}$, the
segment $\big{\\{}(\lambda t/s,\lambda x/s)|s_{0}\leq\lambda\leq s\big{\\}}$
is contained in $\mathscr{M}^{\mathcal{H}}_{[s_{0},s]}$ and $\gamma_{t,x}$
meets $\partial\mathscr{M}^{\mathcal{H}}_{[s_{0},s]}$ at
$(s_{0}/s)(t,x)\in\mathscr{M}^{\mathcal{H}}_{s_{0}}$. When
$\frac{s_{0}^{2}-1}{s_{0}^{2}+1}\leq r/t<1$, the segment

$\Big{\\{}(\lambda t/s,\lambda x/s)|\sqrt{\frac{t+r}{t-r}}\leq\lambda\leq
s\Big{\\}}$

is contained in $\mathscr{M}^{\mathcal{H}}_{[s_{0},s_{1}]}$ and meets
$\partial\mathscr{M}^{\mathcal{H}}_{[s_{0},s_{1}]}$ at the point
$\frac{1}{t-r}(t,r)\in\mathscr{M}^{\mathcal{H}}_{[s_{0},s_{1}]}\cap\mathscr{M}^{\mathcal{EM}}_{[s_{0},s_{1}]}$.
Here we emphasize that $\sqrt{\frac{t+r}{t-r}}\simeq t/s.$ We apply the
multiplier $(s/t)^{2\eta}\Phi^{\prime}_{t,x}$ to (3.8) and obtain

$\frac{d}{d\lambda}\Big{(}|\Phi_{t,x,\eta}^{\prime}|^{2}+\frac{|\Phi_{t,x,\eta}|^{2}}{1-\overline{H}_{t,x}}\Big{)}-\frac{\overline{H}_{t,x}^{\prime}}{(1-\overline{H}_{t,x})^{2}}|\Phi_{t,x,\eta}|^{2}=\Phi_{t,x,\eta}^{\prime}\frac{(s/t)^{\eta}(f+\mathbb{R}_{g})|_{(\lambda
t/s,\lambda x/s)}}{1-\overline{H}_{t,x}}$

with $\Phi_{t,x,\eta}:=(s/t)^{\eta}\Phi_{t,x}$. Then by Gronwall’s inequality
applied on the interval $[\lambda_{0},s]$, we find

$\displaystyle|\Phi_{t,x,\eta}^{\prime}(s)|+|\Phi_{t,x,\eta}(s)|$
$\displaystyle\lesssim\big{(}|\Phi_{t,x,\eta}^{\prime}(\lambda_{0})|+|\Phi_{t,x,\eta}(\lambda_{0})|\big{)}e^{\int_{\lambda_{0}}^{s}|\overline{H}_{t,x}(\lambda)|d\lambda}$
$\displaystyle\qquad+(s/t)^{\eta}\int_{\lambda_{0}}^{s}|f(\lambda t/s,\lambda
x/s)|e^{\int_{\lambda}^{s}|\overline{H}_{t,x}(\tau)|d\tau}d\lambda.$

On the other hand, when $\lambda_{0}\geq s_{0}\geq 2$ we have

$|\Phi_{t,x}^{\prime}(\lambda)|+|\Phi_{t,x}(\lambda)|\simeq\lambda^{3/2}\big{(}|\phi(\lambda
t/s,\lambda x/s)|+|\mathcal{L}\phi(\lambda t/s,\lambda x/s)|\big{)}$

in which
$\mathcal{L}=(t/s)\partial_{t}+(x^{a}/s)\partial_{a}=(t/s)\partial_{t}+(x^{a}/s)\partial_{a}$.
We also observe that

$\mathcal{L}=(s/t)\partial_{t}+s^{-1}(x^{a}/t)L_{a},\quad\partial_{a}=t^{-1}L_{a}-(x^{a}/t)\partial_{t},$

thus

$\displaystyle(s/t)^{\eta}\lambda^{3/2}\big{(}|\phi(\lambda t/s,\lambda
x/s)|+(s/t)\,|\partial\phi(\lambda t/s,\lambda x/s)|\big{)}$
$\displaystyle\lesssim(s/t)^{\eta}\lambda^{-1}|\phi|_{1}(\lambda t/s,\lambda
x/s)+|\Phi_{t,x,\eta}^{\prime}(\lambda)|+|\Phi_{t,x,\eta}(\lambda)|,$

and

$|\Phi_{t,x,\eta}^{\prime}(\lambda)|+|\Phi_{t,x,\eta}(\lambda)|\lesssim(s/t)^{\eta}\lambda^{3/2}\,\big{(}|\phi|_{1}+(s/t)\,|\partial\phi|+(t/s)|\not{\partial}^{\mathcal{N}}\phi|\big{)}\big{|}_{(\lambda
t/s,\lambda x/s)}.$

This gives the desired result. ∎

We continue to rely on the linear structure of the Klein-Gordon equation. We
use here the notation
$\mathscr{M}^{\textbf{near}}_{[s_{0},s_{1}]}:=\mathscr{M}^{\mathcal{EM}}_{[s_{0},s_{1}]}\cap\big{\\{}t-1\leq
r\leq 2t\big{\\}}$, while the complement is denoted by
$\mathscr{M}^{\textbf{far}}_{[s_{0},s_{1}]}$. In the near light cone region we
take advantage of the Klein-Gordon structure and control the mass term by the
wave operator and a source term, while Sobolev decay is available in the far
region.

######  Proposition 3.3 (Pointwise decay of Klein-Gordon fields).

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

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

##  4 Nonlinear stability of self-gravitating massive fields

###  4.1 Nonlinear stability statement

#### Merging the Minkowski and Schwarzschild solutions.

We present now our stability theory in the form established in [25]
(corresponding to $\lambda=1$ in [24]) and refer the reader to [24] for the
treatment of weaker spatial decay conditions. In wave coordinates the
Schwarzschild metric $g_{\textbf{Sch}}$ reads

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

with $\omega_{a}:=x_{a}/r$. Let $\chi^{\star}(r)$ be (regular) cut-off
function vanishing for all $r\leq 1/2$ and which is identically $1$ for all
$r\geq 3/4$. Given a mass coefficient $m>0$, the reference metric of interest
here is (by restricting attention to $t\geq 2$) for convenience in the
discussion)

$g^{\star}=g_{\textbf{Mink}}+\chi^{\star}(r)\,\chi^{\star}(r/(t-1))(g_{\textbf{Sch}}-g_{\textbf{Mink}}),\qquad
t\geq 2,$ (4.1)

which coincides with $g_{\textbf{Mink}}$ in the cone
$\big{\\{}r/(t-1)<1/2\big{\\}}$ and with $g_{\textbf{Sch}}$ in the exterior
$\big{\\{}r/(t-1)\geq 3/4\big{\\}}$ (containing the light cone). This metric
satisfies the light-bending property in the sense that the coefficient

$r\,g^{\star}(\mathbf{l},\mathbf{l})=4m+\mathcal{O}(1/r)\qquad\text{ for the
metric }g^{\star},$ (4.2)

is positive —the light cone direction being
$\mathbf{l}:=\partial_{t}-(x^{a}/r)\partial_{a}.$

#### Class of initial data sets.

The initial metric $g_{0}$ is assumed to be close to the Euclidean metric
while the initial second fundamental form $k_{0}$ is small. We assume the
following decomposition ($a,b=1,2,3$)

$g_{0ab}=g^{\star}_{0ab}+u_{0ab}=\delta_{ab}+h^{\star}_{0ab}+u_{0ab},\qquad
k_{0ab}=k^{\star}_{0ab}+l_{0ab},$ (4.3)

and we propose the following terminology.

  * •

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

  * •

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

An example of a such decomposition is provided by the construction in Lindblad
and Rodnianski [28], 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 [24], the two parts are treated differently. Indeed, $h_{0}^{\star}$ is
the initial trace of $h^{\star}$ while $u_{0}$ is propagated.

Let us fix some exponents $\kappa\in(1/2,1)$ and $\mu\in(3/4,1)$. For the
metric perturbation and the matter field, we introduce the energy norms

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

Given an initial data set we decompose it according to (4.3) 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 can be checked
that (using here that $\kappa>1/2$)

$|u_{\textbf{init}}|\lesssim C_{0}\epsilon(t+r+1)^{-1}.$ (4.5)

Our main assumption beyond the smallness on the norms (4.4) is the following
light-bending condition:

$\inf_{\mathscr{M}^{\textbf{near}}_{\ell}}\big{(}4m+r\,u_{\textbf{init}}(\mathbf{l},\mathbf{l})\big{)}\geq
m.$ (4.6)

Here, the parameter $\ell\in(0,1/2]$ is fixed and we focus on the near-light
cone domain $\mathscr{M}^{\textbf{near}}_{\ell}=\Big{\\{}t\geq 2,\quad t-1\leq
r\leq\frac{t}{1-\ell}\Big{\\}}$.

  * •

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

  * •

or $u_{\textbf{init}}(\mathbf{l},\mathbf{l})$ is non-negative (which can
follow from positivity assumptions on the initial data and the fact that the
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 for the Einstein equations.

We are in a position to state our main result. In fact, a slightly more
general statement concerning perturbations of reference metrics with harmonic
decay is actually established [25] (while much weaker decay is proven to be
sufficient for nonlinear stability in [24]. As explained earlier on in this
text, similar results (in a rather different functional framework) was
simultaneously and independently established by Ionescu and Pausader [16].

######  Theorem 4.1 (Nonlinear stability of self-gravitating Klein-Gordon
fields. Near Schwarzschild decay).

A constant $C_{\star}>0$ being fixed, the following result holds for all
sufficiently small $\epsilon,m$ satisfying $\epsilon\leq C_{\star}m$. Consider
the reference metric $g^{\star}$ defined in (4.1) by merging together the
Minkowski and Schwarzschild metrics. Consider constraint-satisfying initial
data $(g_{0},k_{0},\phi_{0},\phi_{1})$, a large integer $N$, and exponents
$(\kappa,\mu,\epsilon)$ satisfying

$\kappa\in(1/2,1),\qquad\mu\in(3/4,1),\qquad\kappa\leq\mu.$ (4.7)

Then provided the initial data satisfies the light-bending condition (4.6)
together with the smallness condition

$\displaystyle\mathbf{E}^{\text{metric}}_{\kappa,N}(g_{0},k_{0})+\mathbf{E}^{\text{matter}}_{\mu,N}(\phi_{0},\phi_{1})\leq\epsilon,$
(4.8)

the maximal globally hyperbolic Cauchy development of
$(g_{0},k_{0},\phi_{0},\phi_{1})$ associated with the Einstein-massive field
system 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

$|g(\mathbf{l},\mathbf{l})|\lesssim{m+\epsilon\over
t+r+1},\qquad\quad\inf_{\mathscr{M}^{\textbf{near}}_{\ell}}r\,g(\mathbf{l},\mathbf{l})\geq
m/2.$ (4.9)

In the rest of this paper, we will provide a full proof of nonlinear stability
for a simplified model, which retains some of the main challenges arising with
the Einstein-massive matter system; cf. Theorem 5.1.

###  4.2 Analysis of the Einstein equations in the Euclidean-hyperboloidal
foliation

Next, having presented all of our technical tools, we turn our attention to
the Einstein equations. The existence theory is established in wave gauge, and
one of our tasks is to connect geometric components and PDEs components.

  * •

Structure of Einstein’s field equations. We decompose the Einstein-massive
matter system in a form that is adapted to the Euclidean-hyperboloidal
foliation and we analyze the nonlinear structure of these equations. In
particular, we exploit the wave gauge conditions and distinguish between
different components of the metric. One major challenge comes from the fact
that the nonlinearities arising in the Einstein equations do not obey the null
condition. The wave gauge conditions play a central role in several instances,
in the derivation of, both, energy and pointwise estimates. Cf. the statements
in Lemmas 4.3 and 4.4.

  * •

Consequences of the energy estimates. We proceed by postulating certain
bootstrap assumptions which distinguish between low- and high-order
derivatives of the metric and matter fields, and involve the translations, the
boosts, and the spatial rotations. Our estimates involve the geometric weight
denoted by $\zeta$, which allows us to distinguish between the interior and
exterior domains of our foliation. Decay in space is incorporated by
subtracting a reference metric and adding the weight $\mathbf{X}$ in terms of
the distance from the light cone. A broad range of metric and matter exponents
are allowed by our method. By applying the energy estimate, we derive directly
several bounds and, in turn, we write direct consequences of the weighted
Poincaré inequality in Proposition 2.9 (see for instance Proposition 5.4,
below) and of the generalized Sobolev inequality in Proposition 5.2.

  * •

Commutator and Hessian estimates for the metric. We then focus on the metric
perturbation and establish estimates that are localized near the light cone as
well as estimates away from it. We use various calculus rules enjoyed by our
frame of vector fields and we uncover the boost-rotation hierarchy enjoyed by
quasi-linear commutators, as stated earlier in Propositions 2.3 and 2.4.

  * •

Near-Schwarzschild decay of the null metric component. A key contribution in
our method is proving that a certain component of the metric, referred to as
the null metric component, has a ‘near-Schwarzschild’ decay; see (4.9). In
addition, a related argument allows us to prove a so-called light-bending
condition: see (4.9). We find it useful to decompose the spacetime domain into
two sub-domains, referred to as the “bad” and “good” regions: in the bad
region, which is a (thick) neighborhood of the light cone (covering points up
to a distance $\sqrt{t}$) we integrate toward the light cone from the good
region; in the good region we apply Kirchhoff’s formula and integrate the
effect of the initial data, by taking the properties of the source terms into
account. Here, we make use of the assumed decay of the reference metric and
the contribution of the initial perturbation.

  * •

Sharp decay for good metric components. Our next task is to estimate the
gradient and Hessian of the “good metric components” and derive suitably
weighted pointwise estimate. In the wave equations satisfied by the metric
perturbation, the source-terms contain the quasi-null terms $\mathbb{P}$ which
may not enjoy integrable decay. By virtue of the tensorial structure, the
quasi-null terms in the evolution equations of the good components of the
metric are actually null terms and, consequently, enjoy sufficient decay. This
allows us to uncover a hierarchy between the Einstein equations.

  * •

Pointwise estimate for metric components at low order. We next control general
components of the metric at low order of differentiation, and we derive a
near-Schwarzschild decay which is essential in order to deal with massive
matter fields. On the other hand, for massless fields a weaker estimate would
be sufficient to close the bootstrap argument.

  * •

Improved energy estimates. In turn we can close the bootstrap argument by
establishing improved energy estimates at the highest-order of
differentiation, both, first for general metric components and then for the
Klein-Gordon field. The boost-rotation hierarchy made evident in our earlier
estimates is also here the key ingredient of this final step of the prof. For
this argument applied to the model, cf. Section 6.7.

  * •

Asymptotically hyperboloidal domain. Estimates are required also within the
asymptotically hyperboloidal domain. For this region, a proof of global
existence was given first in our monograph [23] in which the emphasis was on
spacetime coincide exactly with the Schwarzschild spacetime outside a (large,
say) light cone. In the light cone region, we also investigated the structure
of the nonlinearities of the Einstein equations coupled to a Klein-Gordon
fields, and uncovered the boost-rotation hierarchy, as well.

###  4.3 Null and quasi-null structures in the Euclidean-hyperboloidal
foliation

The Einstein equations can be decomposed in the frames that are relevant in
the Euclidean–hyperboloidal foliation framework. In the global coordinate
chart $(x^{\alpha})=(t,x^{a})$, we introduce
$\Gamma^{\gamma}:=g^{\alpha\beta}\Gamma_{\alpha\beta}^{\gamma}$ and
$\Gamma_{\alpha}:=g_{\alpha\beta}\Gamma^{\beta}$ determined from the
corresponding Christoffel symbols. The Ricci curvature depends upon (up to)
second-order derivatives of the metric $g$ and specifically ([23])

$2\,R_{\alpha\beta}=-g^{\mu\nu}\partial_{\mu}\partial_{\nu}g_{\alpha\beta}+\mathbb{F}_{\alpha\beta}(g,g;\partial
g,\partial
g)+\big{(}\partial_{\alpha}\Gamma_{\beta}+\partial_{\beta}\Gamma_{\alpha}\big{)}+W_{\alpha\beta},$
(4.10)

where
$W_{\alpha\beta}:=g^{\delta\delta^{\prime}}\partial_{\delta}g_{\alpha\beta}\Gamma_{\delta^{\prime}}-\Gamma_{\alpha}\Gamma_{\beta}$
and
$\mathbb{F}_{\alpha\beta}=\mathbb{P}_{\alpha\beta}+\mathbb{Q}_{\alpha\beta}$
which involves the quasi-null quadratic forms

$\mathbb{P}_{\alpha\beta}(g,g;\partial g,\partial
g):=-\frac{1}{2}g^{\mu\mu^{\prime}}g^{\nu\nu^{\prime}}\partial_{\alpha}g_{\mu\nu}\partial_{\beta}g_{\mu^{\prime}\nu^{\prime}}+\frac{1}{4}g^{\mu\mu^{\prime}}g^{\nu\nu^{\prime}}\partial_{\alpha}g_{\mu\mu^{\prime}}\partial_{\beta}g_{\nu\nu^{\prime}}$
(4.11)

and null quadratic forms

$\displaystyle\mathbb{Q}_{\alpha\beta}(g,g;\partial g,\partial
g):=g^{\mu\mu^{\prime}}g^{\nu\nu^{\prime}}\partial_{\mu}g_{\alpha\nu}\partial_{\mu^{\prime}}g_{\beta\nu^{\prime}}-g^{\mu\mu^{\prime}}g^{\nu\nu^{\prime}}\big{(}\partial_{\mu}g_{\alpha\nu^{\prime}}\partial_{\nu}g_{\beta\mu^{\prime}}-\partial_{\mu}g_{\beta\mu^{\prime}}\partial_{\nu}g_{\alpha\nu^{\prime}}\big{)}$
(4.12)
$\displaystyle+g^{\mu\mu^{\prime}}g^{\nu\nu^{\prime}}\big{(}\partial_{\alpha}g_{\mu\nu}\partial_{\nu^{\prime}}g_{\mu^{\prime}\beta}-\partial_{\alpha}g_{\mu^{\prime}\beta}\partial_{\nu^{\prime}}g_{\mu\nu}\big{)}+\frac{1}{2}g^{\mu\mu^{\prime}}g^{\nu\nu^{\prime}}\big{(}\partial_{\alpha}g_{\mu\beta}\partial_{\mu^{\prime}}g_{\nu\nu^{\prime}}-\partial_{\alpha}g_{\nu\nu^{\prime}}\partial_{\mu^{\prime}}g_{\mu\beta}\big{)}$
$\displaystyle+g^{\mu\mu^{\prime}}g^{\nu\nu^{\prime}}\big{(}\partial_{\beta}g_{\mu\nu}\partial_{\nu^{\prime}}g_{\mu^{\prime}\alpha}-\partial_{\beta}g_{\mu^{\prime}\alpha}\partial_{\nu^{\prime}}g_{\mu\nu}\big{)}+\frac{1}{2}g^{\mu\mu^{\prime}}g^{\nu\nu^{\prime}}\big{(}\partial_{\beta}g_{\mu\alpha}\partial_{\mu^{\prime}}g_{\nu\nu^{\prime}}-\partial_{\beta}g_{\nu\nu^{\prime}}\partial_{\mu^{\prime}}g_{\mu\alpha}\big{)}.$

We focus on the quadratic terms in the perturbation, namely

$\mathbb{P}^{\star}_{\alpha\beta}[u,u]:=\mathbb{P}_{\alpha\beta}(g^{\star},g^{\star};\partial
u,\partial
u),\quad\mathbb{Q}^{\star}_{\alpha\beta}[u,u]:=\mathbb{Q}_{\alpha\beta}(g^{\star},g^{\star};\partial
u,\partial u).$ (4.13)

Since these expressions are quadratic in $(g,g)$ as well as in $(\partial
g,\partial g)$, we apply a polarization argument and define the corresponding
symmetric bilinear forms. We consider
$|\mathbb{Q}^{\star}[u,v]|_{p,k}:=\max_{\alpha,\beta}|\mathbb{Q}_{\alpha\beta}^{\star}[u,v]|_{p,k}$.

######  Lemma 4.2 (Null interaction terms at arbitrary order).

Null forms are controlled by good derivatives and a contribution depending
upon the reference metric and, specifically, n the Euclidean-merging domain
one has

$\displaystyle|\mathbb{Q}^{\star}[u,v]|_{p,k}$
$\displaystyle\lesssim\sum_{p_{1}+p_{2}=p\atop k_{1}+k_{2}=k}\Big{(}|\partial
u|_{p_{1},k_{1}}|\not{\partial}^{\mathcal{N}}v|_{p_{2},k_{2}}+|\partial
v|_{p_{1},k_{1}}|\not{\partial}^{\mathcal{N}}u|_{p_{2},k_{2}}\Big{)}$ (4.14)
$\displaystyle\textstyle\quad+|h^{\star}|_{p}\sum_{p_{1}+p_{2}=p\atop
k_{1}+k_{2}=k}|\partial u|_{p_{1},k_{1}}|\partial v|_{p_{2},k_{2}},$

while, in the hyperboloidal domain $\mathscr{M}^{\mathcal{H}}$,

$|\mathbb{Q}^{\star}[u]|_{p}\lesssim\sum_{p_{1}+p_{2}=p}|\partial
u|_{p_{1}}\big{(}|\not{\partial}^{\mathcal{H}}u|_{p_{2}}+(s/t)^{2}|\partial
u|_{p_{2}}\big{)}+|h^{\star}|_{p}\sum_{p_{1}+p_{2}=p}|\partial
u|_{p_{1}}|\partial u|_{p_{2}}.$ (4.15)

Next, in the Euclidean-merging domain let us introduce

$w[v]_{\gamma}:=(g_{\textbf{Mink}}+v)^{\alpha\beta}\partial_{\alpha}v_{\beta\gamma}-\frac{1}{2}(g_{\textbf{Mink}}+v)^{\alpha\beta}\partial_{\gamma}v_{\alpha\beta},$
(4.16) $\displaystyle\mathbb{W}^{\mathcal{EM}}_{p,k}[v]$
$\displaystyle:={\sum_{\gamma}|w[v]_{\gamma}|_{p,k}}+\sum_{p_{1}+p_{2}=p\atop
k_{1}+k_{2}=k}|\partial v|_{p_{1},k_{1}}|v|_{p_{2},k_{2}}.$ (4.17)

######  Lemma 4.3 (Quasi-null interaction terms at arbitrary order. Euclidean-
merging domain).

In the Euclidean-merging domain $\mathscr{M}^{\mathcal{EM}}$, under the
smallness condition $|h^{\star}|_{p}+|u|_{[p/2]}\ll 1$ and $|v|_{[p/2]}\ll 1$
one has

$\displaystyle|\not{\mathbb{P}}^{\star\mathcal{N}}[u,v]|_{p,k}$
$\displaystyle\lesssim\hskip-5.69046pt\sum_{p_{1}+p_{2}=p\atop
k_{1}+k_{2}=k}\big{(}|\partial
u|_{p_{1},k_{1}}|\not{\partial}^{\mathcal{N}}v|_{p_{2},k_{2}}+|\partial\phi|_{p_{1}}|\not{\partial}^{\mathcal{N}}u|_{p_{2}}\big{)}+\hskip-11.38092pt\sum_{p_{1}+p_{2}+p_{3}=p}\hskip-8.5359pt|h^{\star}|_{p_{3}}|\partial
u|_{p_{1}}|\partial\phi|_{p_{2}},$
$\displaystyle|\mathbb{P}_{00}^{\star\mathcal{N}}[u,v]|_{p,k}$
$\displaystyle\lesssim\sum_{p_{1}+p_{2}=p\atop
k_{1}+k_{2}=k}|\partial{\not{u}}^{\mathcal{N}}|_{p_{1},k_{1}}|\partial\not{v}^{\mathcal{N}}|_{p_{2},k_{2}}+\hskip-8.5359pt{\sum_{p_{1}+p_{2}=p}|\not{\partial}^{\mathcal{N}}u|_{p_{1}}|\partial\phi|_{p_{2}}}+\hskip-8.5359pt\sum_{p_{1}+p_{2}=p}\hskip-8.5359pt|\not{\partial}^{\mathcal{N}}v|_{p_{1}}|\partial
u|_{p_{2}}$ $\displaystyle\quad+\hskip-8.5359pt\sum_{p_{1}+p_{2}=p\atop
k_{1}+k_{2}=k}\mathbb{W}^{\mathcal{EM}}_{p_{1},k_{1}}[v]\,|\partial
u|_{p_{2},k_{2}}+\hskip-8.5359pt\sum_{p_{1}+p_{2}=p}\hskip-8.5359pt\mathbb{S}^{\mathcal{EM}}_{p_{1}}[u]|\partial\phi|_{p_{2}}+\hskip-14.22636pt\sum_{p_{1}+p_{2}+p_{3}=p}\hskip-14.22636pt|h^{\star}|_{p_{3}}|\partial
u|_{p_{1}}|\partial\phi|_{p_{2}}.$

Finally, in the hyperboloidal domain we introduce

$\displaystyle\mathbb{S}_{p}^{\mathcal{H}}[u]$
$\displaystyle:=t^{-1}|u|_{p_{1}}+\big{(}|\partial
h^{\star}|_{p_{1}}+t^{-1}|h^{\star}|_{p_{1}}\big{)}+\sum_{p_{1}+p_{2}=p_{1}}\big{(}|\partial
u|_{p_{1}}|u|_{p_{2}}+|u|_{p_{1}}|u|_{p_{2}}\big{)}$ (4.18)
$\displaystyle\quad+\sum_{p_{1}+p_{2}=p_{1}}\big{(}|h^{\star}|_{p_{1}}|\partial
u|_{p_{2}}+|u|_{p_{1}}|\partial
h^{\star}|_{p_{2}}+|h^{\star}|_{p_{1}}|\partial h^{\star}|_{p_{2}}\big{)}$
$\displaystyle\quad+\sum_{p_{1}+p_{2}=p_{1}}\big{(}|h^{\star}|_{p_{1}}|u|_{p_{2}}+|u|_{p_{1}}|h^{\star}|_{p_{2}}+|h^{\star}|_{p_{1}}|h^{\star}|_{p_{2}}\big{)}.$

######  Lemma 4.4 (Quasi-null interaction terms at arbitrary order.
Hyperboloidal domain).

In the hyperboloidal domain $\mathcal{M}^{\mathcal{H}}$ and under the
smallness condition $|h^{\star}|_{p}+|u|_{[p/2]}\ll 1$, the quasi-null terms
satisfy

$\displaystyle|\not{\mathbb{P}}^{\star\mathcal{H}}[u]|_{p}$
$\displaystyle\lesssim\sum_{p_{1}+p_{2}=p}|\partial
u|_{p_{1}}\big{(}|\not{\partial}^{\mathcal{H}}u|_{p_{2}}+(s/t)^{2}|\partial
u|_{p_{2}}\big{)}+\sum_{p_{1}+p_{2}+p_{3}=p}|h^{\star}|_{p_{3}}|\partial
u|_{p_{1}}|\partial u|_{p_{2}},$
$\displaystyle|\mathbb{P}_{00}^{\star\mathcal{H}}[u]|_{p,k}$
$\displaystyle\lesssim\hskip-8.5359pt\sum_{p_{1}+p_{2}=p}\hskip-8.5359pt\Big{(}|\partial\not{u}^{\mathcal{H}}|_{p_{1}}|\partial\not{u}^{\mathcal{H}}|_{p_{2}}+\big{(}|\not{\partial}^{\mathcal{H}}u|_{p_{1}}+(s/t)^{2}|\partial
u|_{p_{1}}\big{)}|\partial u|_{p_{2}}\Big{)}$
$\displaystyle\qquad+\hskip-8.5359pt\sum_{p_{1}+p_{2}=p}\hskip-8.5359pt|\mathbb{S}_{p}^{\mathcal{H}}[u]|_{p_{1}}|\partial
u|_{p_{2}}+\hskip-8.5359pt\sum_{p_{1}+p_{2}+p_{3}=p}\hskip-8.5359pt|h^{\star}|_{p_{3}}|\partial
u|_{p_{1}}|\partial u|_{p_{2}}.$

##  5 Nonlinear stability for the wave-Klein-Gordon model

We work with the proposed Euclidean–hyperboloidal foliation $\bigcup_{s\geq
1}\mathscr{M}_{s}$ of Minkowski spacetime. For any integer $N\geq 1$ and any
values $s\geq 1$ of the foliation parameter, we consider the following
weighted energy functional for the wave component $u$ on the hypersurface
$\mathscr{M}_{s}$:

$\mathcalboondox E^{N}_{\kappa}(s,u):=\sum_{\textbf{ord}(Z)\leq
N}\mathcalboondox E_{\kappa}(s,Zu),\qquad\mathcalboondox
F_{\kappa}^{N}(s,u):=\big{(}\mathcalboondox E^{N}_{\kappa}(s,u)\big{)}^{1/2},$
(5.1)

while for the Klein-Gordon field with mass coefficient $c>0$ we set

$\mathcalboondox E^{N}_{c,\kappa}(s,\phi):=\sum_{\textbf{ord}(Z)\leq
N}\mathcalboondox E_{c,\kappa}(s,Z\phi),\qquad\mathcalboondox
F^{N}_{c,\kappa}(s,\phi):=\big{(}\mathcalboondox
E^{N}_{c,\kappa}(s,\phi)\big{)}^{1/2}.$ (5.2)

The summations above are over all ordered admissible operators of order $\leq
N$. Later on, we will also need the notation $\mathcalboondox
F_{\kappa}^{p,k}(s,u)$ for the energy defined by restricting the summation to
fields with $\textbf{ord}(Z)\leq p$ and $\textbf{rank}(Z)\leq k$. We also
write $\mathcalboondox F_{\kappa}^{\mathcal{EM},p,k}(s,u)$ and
$\mathcalboondox F^{\mathcal{H},p,k}(s,u)$ when the integrals defining the
energy is restricted to the domains $\mathscr{M}^{\mathcal{EM}}$ or
$\mathscr{M}^{\mathcal{H}}$, respectively. (The subscript $\kappa$ is
irrelevant and omitted for the energy in the hyperboloidal domain.)

We now state our main result for the model.

######  Theorem 5.1 (Global existence theory for the wave-Klein-Gordon model).

Consider the nonlinear wave-Klein-Gordon model (1.9) with given real constants
$P^{\alpha\beta},R,H^{\alpha\beta}$, and a mass coefficient $c>0$. For
$\kappa>3/4$ and any integer $N\geq 11$, there exists a sufficiently small
$\epsilon>0$ such that the initial value problem associated with the system
(1.9) admits a global-in-time solution $(u,v)$, provided the data set on the
initial hypersurface has sufficiently small energy in the sense that (for a
fixed $C_{0}>0$)

$\mathcalboondox F^{N}_{\kappa}(s_{0},u)+\mathcalboondox
F^{N}_{c,\kappa}(s_{0},\phi)\leq C_{0}\,\epsilon,$ (5.3) $\mathcalboondox
F_{\kappa}^{N-4}(s_{0},u)+\mathcalboondox F^{N-4}_{c,\kappa}(s_{0},\phi)\leq
C_{0}\,\epsilon$ (5.4)

and, for every admissible field $Z$ with $\textbf{ord}(Z)\leq N-4$, the
solution $w$ to the following free wave problem

$\displaystyle\Box w=0,\quad\qquad w|_{\mathscr{M}_{s_{0}}}$
$\displaystyle=w_{0},\quad\qquad$
$\displaystyle\partial_{t}w|_{\mathscr{M}_{s_{0}}}$ $\displaystyle=w_{1},$
(5.5) $\displaystyle w_{0}$ $\displaystyle=Zu|_{\mathscr{M}_{s_{0}}},\quad$
$\displaystyle w_{1}$ $\displaystyle=\partial_{t}Zu|_{\mathscr{M}_{s_{0}}},$

satisfies the decay property

$|w(t,x)|\lesssim\epsilon\,(r+t)^{-1}.$ (5.6)

On the other hand, similarly as we did for the Einstein equations, we could
also introduce a notion of reference solution and establish the above
existence theory under milder decay conditions. On the other hand, we point
out that (5.6) is satisfied not only by compactly supported initial data, but
also by a class of non-compact initial data described in [24, Section 10]. To
proceed, we fix some exponents

$3/4<\kappa<1,\qquad 0<\delta\ll\kappa-3/4,$ (5.7)

where the exponent $\delta$ is used to allow a (mild) growth of our energy
norms. (For instance, it is possible to take $\delta=(\kappa-3/4)/20$.) Our
bootstrap energy assumptions are stated on a time interval $s\in[s_{0},s_{1}]$
and are based on a sufficient large constant $C_{1}>0$ (in comparison to
$C_{0}$ in (5.3) and (5.4)) which will be chosen later on:

$s^{-\delta}\,\mathcalboondox
F^{N}_{\kappa}(s,u)+s^{-1/2-\delta}\,\mathcalboondox
F^{N}_{c,\kappa}(s,\phi)\leq C_{1}\epsilon,$ (5.8) $\mathcalboondox
F_{\kappa}^{N-4}(s,u)+s^{-\delta}\,\mathcalboondox
F^{N-4}_{c,\kappa}(s,\phi)\leq C_{1}\epsilon.$ (5.9)

We emphasize that, in (5.8), the high-order energy norm of the wave component
may grow at the mild rate $s^{\delta}$, while the high-order energy norm of
the Klein-Gordon component may grow at the rate $s^{1/2+\delta}$. On the other
hand, in (5.9) a uniform control is required on the low-order energy norm of
the wave component, while the Klein-Gordon component may suffer a mild growth
$s^{\delta}$.

###  5.1 High-order estimates

Functional inequalities together with calculus rules allow us to derive
estimates at arbitrary orders and for instance we arrive at the following
statement.

######  Proposition 5.2 (Sobolev decay for wave fields in the Euclidean-
merging domain).

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

$\displaystyle\big{\|}r\,\mathbf{X}^{\eta}\,|\partial
u|_{p,k}\big{\|}_{L^{\infty}(\mathscr{M}^{\mathcal{EM}}_{s})}+\big{\|}r^{1+\eta}\,|\not{\partial}^{\mathcal{N}}u|_{p,k}\big{\|}_{L^{\infty}(\mathscr{M}^{\mathcal{EM}}_{s})}$
(5.10a) $\displaystyle\lesssim(1-\eta)^{-1}\,\mathcalboondox
F_{\eta}^{\mathcal{EM},p+3,k+3}(s,u)$ and, for $\eta\in(1/2,1)$,
$\|r\,\mathbf{X}^{-1+\eta}|u|_{N-2}\|_{L^{\infty}(\mathscr{M}^{\mathcal{EM}}_{s})}\lesssim(2\eta-1)^{-1}\,\mathcalboondox
F_{\eta}^{\mathcal{EM},N}(s,u)+\mathcalboondox F_{\eta}^{0}(s,u).$ (5.10b)

######  Proposition 5.3 (Sobolev decay for wave fields in the hyperboloidal
domain).

For all function $u$ defined in $\mathscr{M}^{\mathcal{H}}_{[s_{0},s_{1}]}$,
one has (for $k\leq p$)

$\|t^{3/2}(s/t)\,|\partial
u|_{p,k}\|_{L^{\infty}(\mathscr{M}^{\mathcal{H}}_{s})}+\|t^{3/2}|\not{\partial}^{\mathcal{H}}u|_{p,k}\|_{L^{\infty}(\mathscr{M}^{\mathcal{H}}_{s})}\lesssim\mathcalboondox
F^{\mathcal{H},p+2,k+2}(s,u),$ (5.11)
$\displaystyle\|t^{3/2}(s/t)\,|\partial\phi|_{p,k}\|_{L^{\infty}(\mathscr{M}^{\mathcal{H}}_{s})}+\|t^{3/2}|\not{\partial}^{\mathcal{H}}\phi|_{p,k}\|_{L^{\infty}(\mathscr{M}^{\mathcal{H}}_{s})}+c\|t^{3/2}|\phi|_{p,k}\|_{L^{\infty}(\mathscr{M}^{\mathcal{H}}_{s})}$
(5.12) $\displaystyle\lesssim\mathcalboondox
F_{c}^{\mathcal{H},p+2,k+2}(s,u).$

######  Proposition 5.4 (Hardy-Poincaré inequality for high-order
derivatives).

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

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

##  6 Proof of stability

###  6.1 Direct consequences in the Euclidean-merging domain

#### Hardy-Poincaré inequality.

Throughout the following derivation, the notation $A\lesssim B$ stands for
$A\leq C_{N,\delta}B$ where $C_{N,\delta}$ a constant depending upon
$N,\delta$. In view of the bootstrap conditions (5.8)-(5.9) and Proposition
5.4, we have

$\|\mathbf{X}^{-1+\kappa}|u|_{p}\|_{L^{2}(\mathscr{M}^{\mathcal{EM}}_{s})}\lesssim\begin{cases}C_{1}\epsilon,\quad&p=N-4,\\\
C_{1}\epsilon s^{\delta},&p=N.\end{cases}$ (6.1)

#### Sobolev decay.

In view of (5.8)-(5.9) together with (6.1), we can apply Proposition 5.2 to
both the wave field $u$ and the Klein-Gordon field $\phi$, and we obtain

$\langle r\rangle\mathbf{X}^{\kappa}|\partial u|_{p}+\langle
r\rangle^{1+\kappa}|\not{\partial}^{\mathcal{N}}u|_{p}\lesssim\begin{cases}C_{1}\epsilon,\quad&p=N-7,\\\
C_{1}\epsilon s^{\delta},\quad&p=N-3,\end{cases}\quad\text{ in
}\mathscr{M}^{\mathcal{EM}}_{[s_{0},s_{1}]},$ (6.2) $\langle
r\rangle\mathbf{X}^{-1+\kappa}|u|_{p}\lesssim\begin{cases}C_{1}\epsilon,\quad&p=N-6,\\\
C_{1}\epsilon s^{\delta}\quad&p=N-2,\end{cases}\quad\text{ in
}\mathscr{M}^{\mathcal{EM}}_{[s_{0},s_{1}]},$ (6.3) $\langle
r\rangle\mathbf{X}^{\kappa}|\partial\phi|_{p}+\langle
r\rangle^{1+\kappa}|\not{\partial}^{\mathcal{N}}\phi|_{p}+\langle
r\rangle\mathbf{X}^{\kappa}|\phi|_{p+1}\lesssim\begin{cases}C_{1}\epsilon
s^{\delta},\,&p=N-7,\\\ C_{1}\epsilon
s^{1/2+\delta},\,&p=N-3,\end{cases}\text{ in
}\mathscr{M}^{\mathcal{EM}}_{[s_{0},s_{1}]}.$ (6.4)

#### Bounds on the light cone.

Along the light cone
$\mathscr{L}_{[s_{0},s]}=\mathscr{M}^{\mathcal{H}}_{[s_{0},s_{1}]}\cap\mathscr{M}^{\mathcal{EM}}_{[s_{0},s_{1}]}$,
we have
$\not{\partial}^{\mathcal{H}}_{a}=\not{\partial}^{\mathcal{N}}_{a}-(x^{a}/r)t^{-1}\partial_{t}$.
By restricting (6.2) to $\mathscr{L}_{[s_{0},s]}$ and by observing that
$t\simeq s^{2}$, we find

$t\,|\partial
u|_{p}+t^{1+\kappa}|\not{\partial}^{\mathcal{H}}u|_{p}\lesssim\begin{cases}C_{1}\epsilon,\quad&p=N-7,\\\
C_{1}\epsilon\,t^{\delta/2},&p=N-3,\end{cases}\qquad\text{on
}\mathscr{L}_{[s_{0},s]},$ (6.5)

and

$t\,|\partial\phi|_{p}+t^{1+\kappa}|\not{\partial}^{\mathcal{H}}\phi|_{p}+t\,|\phi|_{p+1}\lesssim\begin{cases}C_{1}\epsilon\,t^{\delta/2},\quad&p=N-7,\\\
C_{1}\epsilon\,t^{1/4+\delta/2},&p=N-3,\end{cases}\qquad\text{on
}\mathscr{L}_{[s_{0},s]}.$ (6.6)

In the same manner, from (6.3) we deduce

$t\,|u|_{p}\lesssim\begin{cases}C_{1}\epsilon,\quad&p=N-6,\\\
C_{1}\epsilon\,t^{\delta/2},\quad&p=N-2,\end{cases}\qquad\text{on
}\mathscr{L}_{[s_{0},s]}$ (6.7)

and, in particular, this leads us to the bound (since
$\not{\partial}^{\mathcal{H}}_{a}=t^{-1}L_{a}$):

$t^{2}|\not{\partial}^{\mathcal{H}}u|_{p-1}\lesssim\begin{cases}C_{1}\epsilon,\quad&p=N-6,\\\
C_{1}\epsilon\,t^{\delta/2},\quad&p=N-2,\end{cases}\qquad\text{on
}\mathscr{L}_{[s_{0},s]}.$ (6.8)

###  6.2 Direct consequences in the hyperboloidal domain

#### Sobolev decay.

In view of the Sobolev decay stated in Proposition 5.3 in combination with the
bootstrap bounds (5.8)-(5.9), we have

$t^{3/2}(s/t)\,|\partial
u|_{p}+t^{3/2}|\not{\partial}^{\mathcal{H}}u|_{p}\lesssim\begin{cases}C_{1}\epsilon,\quad&p=N-6,\\\
C_{1}\epsilon s^{\delta},&p=N-2.\end{cases}\quad\text{ in
}\mathscr{M}^{\mathcal{H}}_{[s_{0},s_{1}]},$ (6.9)
$t^{3/2}(s/t)\,|\partial\phi|_{p}+t^{3/2}|\not{\partial}^{\mathcal{H}}\phi|_{p}+t^{3/2}|\phi|_{p}\lesssim\begin{cases}C_{1}\epsilon
s^{\delta},\quad&p=N-6,\\\ C_{1}\epsilon
s^{1/2+\delta},&p=N-2,\end{cases}\quad\text{ in
}\mathscr{M}^{\mathcal{H}}_{[s_{0},s_{1}]}.$ (6.10)

Furthermore, this implies that

$(s/t)^{-2+\delta}s^{1/2}|\phi|_{N-3}\lesssim C_{1}\epsilon\quad\text{ in
}\mathscr{M}^{\mathcal{H}}_{[s_{0},s_{1}]}$ (6.11)

and, in particular,

$\displaystyle|\partial_{r}(Z\not{\partial}^{\mathcal{H}}_{a}u)|$
$\displaystyle\lesssim\begin{cases}C_{1}\epsilon\,t^{-3/2}s^{-1},\quad&\textbf{ord}Z\leq
N-7,\\\ C_{1}\epsilon\,t^{-3/2}s^{-1+\delta},&\textbf{ord}Z\leq
N-3,\end{cases}$ (6.12)
$\displaystyle\simeq\begin{cases}C_{1}\epsilon\,t^{-2}(t-r)^{-1/2},\quad&\textbf{ord}Z\leq
N-7,\\\ C_{1}\epsilon\,t^{-2+\delta/2}(t-r)^{-1/2+\delta/2},&\textbf{ord}Z\leq
N-3,\end{cases}\quad\text{ in }\mathscr{M}^{\mathcal{H}}_{[s_{0},s_{1}]}.$

#### Integration from the light cone.

Recalling (6.8) and performing an integration —along a constant-time slice—
from the light cone
$\mathscr{L}_{[s_{0},s]}=\mathscr{M}^{\mathcal{H}}_{[s_{0},s_{1}]}\cap\mathscr{M}^{\mathcal{EM}}_{[s_{0},s_{1}]}$
towards the center, we deduce that

$|\not{\partial}^{\mathcal{H}}u|_{p}\lesssim\begin{cases}C_{1}\epsilon\,t^{-3/2}(s/t),\quad&p{=N-7,}\\\
C_{1}\epsilon\,t^{-3/2}(s/t)s^{\delta},&p{=N-3,}\end{cases}\quad\text{ in
}\mathscr{M}^{\mathcal{H}}_{[s_{0},s_{1}]}.$ (6.13)

In the same manner, we can integrate $\partial_{r}u=(x^{a}/r)\partial_{a}u$
from $\mathscr{L}_{[s_{0},s_{1}]}$ towards the center on a constant-time slice
and, in view of (6.7) and (6.9), we find

$|u|_{p}\lesssim\begin{cases}C_{1}\epsilon\,t^{-1}(t-r)^{1/2},\quad&p=N-6,\\\
C_{1}\epsilon\,t^{-1+\delta/2}(t-r)^{1/2+\delta/2},&p=N-2,\end{cases}\quad\text{
in }\mathscr{M}^{\mathcal{H}}_{[s_{0},s_{1}]}.$ (6.14)

#### Further estimate.

Thanks to (6.9), (6.13) and the relation
$\overline{H}_{t,x}^{\prime}(\lambda)=(t/s)^{2}H^{\mathcal{H}}{}^{00}\mathcal{L}u$,
we have

$|\overline{H}_{t,x}^{\prime}(\lambda)|\lesssim
C_{1}\epsilon(s/t)^{-1/2}s^{-3/2}\quad\text{ in
}\mathscr{M}^{\mathcal{H}}_{[s_{0},s_{1}]}$ (6.15)

(provided $N\geq 7$). This guarantees the assumption (3.9) in Proposition 3.2
(by observing that $(t/s)\lesssim\lambda_{0}$) and this observation will be
required in Section 6.4, below.

###  6.3 Commutators and source terms in the Euclidean-merging domain

#### Pointwise decay for the Klein-Gordon component.

By a direct application of the bounds (6.3)-(6.4) we have

$\displaystyle|H^{\alpha\beta}u\partial_{\alpha}\partial_{\beta}\phi|_{p-4}$
$\displaystyle\lesssim(C_{1}\epsilon)^{2}r^{-2}\mathbf{X}^{1-2\kappa}\begin{cases}s^{1/2+2\delta},\quad&p=N,\\\
s^{2\delta},\quad&p=N-4,\end{cases}$ $\displaystyle\lesssim
C_{1}\epsilon\langle
r\rangle^{-2}\mathbf{X}^{1-\kappa}\begin{cases}s^{1/2+2\delta},\quad&p=N,\\\
s^{2\delta},\quad&p=N-4,\end{cases}\quad\text{ in
}\mathscr{M}^{\mathcal{EM}}_{[s_{0},s_{1}]}.$

We emphasize that, for convenience in the discussion, the left-hand side is
stated here in terms of a norm at the order $(p-4)$. Thanks to the pointwise
decay property stated in Proposition 3.3 in combination with the bootstrap
bounds (5.8)-(5.9), we find

$|\phi|_{p-4}\lesssim\begin{cases}C_{1}\epsilon\langle
r\rangle^{-2}\mathbf{X}^{1-\kappa}s^{1/2+2\delta},\quad&p=N,\\\
C_{1}\epsilon\langle
r\rangle^{-2}\mathbf{X}^{1-\kappa}s^{2\delta},\quad&p=N-4,\end{cases}\quad\text{
in }\mathscr{M}^{\mathcal{EM}}_{[s_{0},s_{1}]}$ (6.16)

and, consequently, on the light cone

$|\phi|_{p}\lesssim\begin{cases}C_{1}\epsilon\,t^{-7/4+\delta},\quad&p=N-4,\\\
C_{1}\epsilon\,t^{-2+\delta},\quad&p=N-8,\end{cases}\quad\text{on
}\mathscr{L}_{[s_{0},s]}.$ (6.17)

#### Pointwise decay for the wave component.

We now apply Kirchhoff formula. We have

$|\partial_{\alpha}\phi\partial_{\beta}\phi|_{N-4}+|\phi^{2}|_{N-4}\lesssim(C_{1}\epsilon)^{2}\langle
r\rangle^{-2-(1/2-(3/2)\delta)}\mathbf{X}^{-1+2(1-\kappa)}\quad\text{ in
}\mathscr{M}^{\mathcal{EM}}_{[s_{0},s_{1}]},$

where we used (provided $N\geq 5$)

$|\partial\phi|_{N-5}\lesssim C_{1}\epsilon\langle
r\rangle^{-2}\mathbf{X}^{1-\kappa}s^{1/2+2\delta},\quad|\partial\phi|_{N-3}\lesssim
C_{1}\epsilon\langle
r\rangle^{-1}\mathbf{X}^{-\kappa}s^{1/2+\delta}\quad\text{ in
}\mathscr{M}^{\mathcal{EM}}_{[s_{0},s_{1}]}.$

Therefore, we have the following low-order, pointwise bound on the source-term
of the wave equation:

$|\Box u|_{N-4}\lesssim(C_{1}\epsilon)^{2}\langle
r\rangle^{-2-(1/2-(3/2)\delta)}\mathbf{X}^{-1+2(1-\kappa)}\quad\text{ in
}\mathscr{M}^{\mathcal{EM}}_{[s_{0},s_{1}]}.$ (6.18)

In turn, we can apply Proposition 3.1 (Case 2 therein) within the domain
$\mathscr{M}^{\mathcal{EM}}_{[s_{0},s_{1}]}$, by observing that this region is
“past complete” in the sense that for any
$(t,x)\in\mathscr{M}^{\mathcal{EM}}_{[s_{0},s_{1}]}$ we have
$\Lambda_{t,x}\subset\mathscr{M}^{\mathcal{EM}}_{[s_{0},s_{1}]}$. Therefore,
by recalling our assumption (5.6) concerning the sharp pointwise decay of the
initial data, we arrive at the following key statement.

######  Lemma 6.1.

The wave field satisfies the low-order, sharp pointwise estimate

$|u|_{N-4}\lesssim(C_{1}\epsilon)^{2}\langle
r\rangle^{-1}+C_{0}\epsilon\,t^{-1}\quad\text{ in
}\mathscr{M}^{\mathcal{EM}}_{[s_{0},s_{1}]}.$ (6.19)

For points $(t,x)\in\mathscr{M}^{\mathcal{H}}_{[s_{0},s_{1}]}$, the bound
(6.18) and Case 2 of Proposition 3.1 are also relevant to control the
contribution of the source due to the Euclidean-merging domain, but of course
we still need to control the contribution due to the hyperboloidal domain, and
we are going to apply Case 0 of Proposition 3.1; cf. (6.24) below.

#### Estimates for source-terms.

We now derive $L^{2}$ estimates for the source terms (of the wave equation)
and the commutators (of the Klein-Gordon equation). To this end, we want to
apply the commutator estimate (2.20) with $H=H^{\alpha\beta}u$ and the
function $u$ therein chosen to be the Klein-Gordon field $\phi$. In (2.20),
the operator $Z$ is of order $p=\textbf{ord}(Z)\leq N$ and
$k=\textbf{rank}(Z)\leq p$. In view of the Sobolev bound (6.2) (low-order case
therein) and the pointwise Klein-Gordon decay (6.16), and the sharp pointwise
estimate (6.19), we find $|u||\partial\partial\phi|_{p-1,k-1}\lesssim
C_{1}\epsilon\,t^{-1}|\partial\phi|_{p,k-1}$ and

$|Y_{\text{rot}}u|_{k_{1}-1}|\partial\partial\phi|_{p_{2},k_{2}}\lesssim\begin{cases}C_{1}\epsilon\,t^{-1}|\partial\phi|_{p,k-1},\quad&k_{1}\leq
N-4,\\\ C_{1}\epsilon\langle
r\rangle^{-1}\mathbf{X}^{-\kappa}s^{\delta}|u|_{k_{1}}\quad&k_{1}\geq
N-3,\end{cases}$

where the notation in (2.20) is used. Here we applied the Sobolev bound (6.10)
to deal with the Klein-Gordon term $|\partial\partial\phi|_{p_{2},k_{2}}$ and
we used the conditions $p_{2}\leq N-8$ and $N\geq 11$. For the last term in
the right-hand side of (2.20), we have

$|\partial
u|_{p_{1}-1,k_{1}}|\partial\partial\phi|_{p_{2},k_{2}}\lesssim\begin{cases}C_{1}\epsilon\,t^{-1}|\partial\phi|_{p,k},\quad&p_{1}\leq
N-4,\\\ C_{1}\epsilon\langle
r\rangle^{-2}\mathbf{X}^{1-\kappa}s^{1/2+2\delta}|\partial
u|_{p,k},\quad&p_{2}\leq N-6.\end{cases}$

Collecting the bounds above together and integrating over the Euclidean-
merging domain, we obtain

$\|J\zeta^{-1}\mathbf{X}^{\kappa}[Z,H^{\alpha\beta}\partial_{\alpha}\partial_{\beta}]\phi\|_{L^{2}(\mathscr{M}^{\mathcal{EM}}_{s})}\lesssim
C_{1}\epsilon s^{-1}\mathcalboondox
F_{\kappa,c}^{p,k}(s,\phi)+\begin{cases}(C_{1}\epsilon)^{2}s^{-5/4},\quad&k\leq
N-4,\\\ (C_{1}\epsilon)^{2}s^{-1/2},&k\geq N-3.\end{cases}$ (6.20)

On the other hand, for the wave equation we have

$\displaystyle|\Box u|_{p,k}$
$\displaystyle\lesssim|\partial_{\alpha}\phi\partial_{\beta}\phi|_{p,k}+|\phi^{2}|_{p,k}$
(6.21) $\displaystyle\lesssim C_{1}\epsilon\langle
r\rangle^{-2}\mathbf{X}^{1-\kappa}s^{1/2+2\delta}(|\partial\phi|_{p,k}+|\phi|_{p,k})\quad\text{
in }\mathscr{M}^{\mathcal{EM}}_{[s_{0},s_{1}]}$

(provided $N\geq 9$) and, after integration, with $p=\textbf{ord}(Z)\leq N$
and $k=\textbf{rank}(Z)\leq p$

$\|J\zeta^{-1}\mathbf{X}^{\kappa}|\Box
u|_{p,k}\|_{L^{2}(\mathscr{M}^{\mathcal{EM}}_{s})}\lesssim(C_{1}\epsilon)^{2}s^{-3/2+3\delta}.$
(6.22)

###  6.4 Sharp decay in the hyperboloidal domain: hierarchy inequalities

#### Sharp decay of the wave component.

We now rely on Kirchhoff formula, that is, on Proposition 3.1 and we thus need
first to control $|\Box u|$. At this stage, (6.18) already gives us the
required control in $\mathscr{M}^{\mathcal{EM}}_{[s_{0},s_{1}]}$, and we can
focus on the domain $\mathscr{M}^{\mathcal{H}}_{[s_{0},s_{1}]}$. The Sobolev
decay bound (6.10) gives us

$|\Box
u|_{N-6}\lesssim(C_{1}\epsilon)^{2}t^{-2-(1/2-\delta)}(t-r)^{-1+(1/2+\delta)}\quad\text{
in }\mathscr{M}^{\mathcal{H}}_{[s_{0},s_{1}]},$ (6.23)

where we used that
$|\partial\phi\partial\phi|_{N-6}\lesssim|\partial\phi|_{N-6}|\phi|_{N-6}$
(provided $N\geq 7$). In view of the estimate in Case 0 of Proposition 3.1 and
thanks to the sharp initial decay condition (5.6), we obtain the following
almost sharp pointwise decay of the wave component at a low order:

$|u|_{N-6}\lesssim C_{1}\epsilon\,t^{-1+2\delta}\quad\text{ in
}\mathscr{M}^{\mathcal{H}}_{[s_{0},s_{1}]}.$ (6.24)

#### Sharp decay of the Klein-Gordon component.

On the other hand, we can apply the integration technique in Proposition 3.2
to the Klein-Gordon equation
$-\widetilde{\Box}_{g}Z\phi+c^{2}\phi=[Z,H^{\alpha\beta}\partial_{\alpha}\partial_{\beta}]\phi,$
in which $g^{\alpha\beta}=\eta^{\alpha\beta}+H^{\alpha\beta}$ and
$\widetilde{\Box}_{g}=g^{\alpha\beta}\partial_{\alpha}\partial_{\beta}$. We
pick up any operator $Z$ such that $\textbf{ord}(Z)=p\leq N-4$ and
$\textbf{rank}(Z)=k\leq p$. With the notation $|\mathbb{R}_{g}[Z\phi]|$ given
in (3.12) and thanks to the Sobolev decay estimates, we claim that

$\displaystyle|\mathbb{R}_{g}[Z\phi]|$ $\displaystyle\lesssim
s^{-2}|\phi|_{N-2}+(t/s)^{2}s^{-2}|u|_{N-4}|\phi|_{N-2}+(t/s^{2})\sum_{p_{1}+p_{2}=N-4}|u|_{p_{1}}|\partial\phi|_{p_{2}+1}$
(6.25) $\displaystyle\lesssim(C_{1}\epsilon)(s/t)s^{-3+3\delta}\qquad\text{ in
}\mathscr{M}^{\mathcal{H}}_{[s_{0},s_{1}]}.$

Indeed, this is checked as follow. For the first two terms in (3.12) we apply
the Sobolev decay (6.10) and (6.14), and we obtain the upper bound
$(C_{1}\epsilon)^{2}(s/t)^{3/2}s^{-3+2\delta}$. On the other hand, dealing
with the last term in (3.12) is more involved, and we observe that $p_{1}\leq
N-6$ or $p_{2}\leq N-7$ (with $N\geq 9$) and we write

$\displaystyle(t/s^{2})|u|_{N-4}|\partial\phi|_{N-6}\lesssim(C_{1}\epsilon)^{2}(s/t)s^{-2+2\delta}\quad$
$\displaystyle\text{ in }\mathscr{M}^{\mathcal{H}}_{[s_{0},s_{1}]},$
$\displaystyle(t/s^{2})|u|_{N-6}|\partial\phi|_{N-3}\lesssim(C_{1}\epsilon)^{2}(s/t)^{3/2-2\delta}s^{-3+3\delta}\quad$
$\displaystyle\text{ in }\mathscr{M}^{\mathcal{H}}_{[s_{0},s_{1}]},$

where for the second case we use the decay (6.24).

Next let us consider the boundary “sup” term in (3.10). In view of the light
cone estimates (6.6) and (6.17) and recalling $\kappa-3/4\geq\delta$, we find
the following estimate on the boundary of the hyperboloidal domain:

$t^{1/4}\big{(}t^{1/2}|\phi|_{p}+|\partial\phi|_{p}+t\,|\not{\partial}^{\mathcal{N}}\phi|_{p}\big{)}\lesssim\begin{cases}C_{1}\epsilon(s/t)^{1-\delta},\quad&p=N-4,\\\
C_{1}\epsilon(s/t)^{2-2\delta},\quad&p=N-5.\end{cases}\quad\text{on
}\mathscr{M}^{\mathcal{H}}_{s_{0}}\cup\mathcal{L}_{[s_{0},s_{1}]}.$ (6.26)

We now recall (3.10) together with (6.26), (6.11) and (6.15), since we still
need to bound the source term of the Klein-Gordon equation together with the
additional term $\mathbb{R}_{g}[\phi]$. We observe that, for a high-order
operator $Z$ with $\textbf{ord}(Z)=p\leq N-5$ and $\textbf{rank}(Z)=k$ and
thanks to (2.20),

$\displaystyle\big{|}[Z,H^{\alpha\beta}u\partial_{\alpha}\partial_{\beta}]\phi\big{|}$
$\displaystyle\lesssim|u||\partial\partial\phi|_{p-1,k-1}+\sum_{k_{1}+p_{2}=p\atop
k_{1}+k_{2}=k}|Y_{\text{rot}}u|_{k_{1}-1}|\partial\partial\phi|_{p_{2},k_{2}}+\sum_{p_{1}+p_{2}=p\atop
k_{1}+k_{2}=k}|\partial
u|_{p_{1}-1,k_{1}}|\partial\partial\phi|_{p_{2},k_{2}}$
$\displaystyle\lesssim(s/t)^{2-2\delta}s^{-5/2}\mathbf{A}^{\flat}_{k-1}(s)\mathbf{B}_{0}(s)+(s/t)^{2-2\delta}s^{-5/2}\sum_{k_{1}=1}^{k}\mathbf{A}^{\flat}_{k-k_{1}}(s)\mathbf{B}_{k_{1}}(s)$
$\displaystyle\quad+C_{1}\epsilon(s/t)^{3/2-2\delta}s^{-3+\delta}\mathbf{A}^{\flat}_{k}(s),$

where we have introduced

$\displaystyle\mathbf{A}^{\flat}_{k}(s):=\sup_{\mathscr{M}^{\mathcal{H}}_{[s_{0},s_{1}]}}\Big{(}(s/t)^{-2+2\delta}s^{3/2}\big{(}|\phi|_{N-5,k}+(s/t)\,|\partial\phi|_{N-5,k}\big{)}\Big{)},$
$\displaystyle\mathbf{B}_{k}(s):=\sup_{\mathscr{M}^{\mathcal{H}}_{[s_{0},s_{1}]}}\big{(}t\,|u|_{N-4,k}\big{)}.$

Importantly, in the above estimate for
$[Z,H^{\alpha\beta}u\partial_{\alpha}\partial_{\beta}]\phi$ the first two
terms in the right-hand side do not exist when $k=0$. Consequently, by
collecting the previous estimates together we are in a position to apply
Proposition 3.2 and obtain our inductive inequality (at the order $N-5$)
concerning the Klein-Gordon component

$\displaystyle\mathbf{A}^{\flat}_{k}(s)\lesssim
C_{1}\epsilon+C_{1}\epsilon(s/t)^{3/2-2\delta}\int_{\lambda_{0}}^{s}\lambda^{-3/2+\delta}\mathbf{A}^{\flat}_{k}(\lambda)d\lambda$
(6.27)
$\displaystyle+(s/t)^{2-2\delta}\int_{\lambda_{0}}^{s}\lambda^{-1}\mathbf{A}^{\flat}_{k-1}(\lambda)\mathbf{B}_{0}(\lambda)d\lambda+(s/t)^{2-2\delta}\sum_{k_{1}=1}^{N-5}\int_{\lambda_{0}}^{s}\lambda^{-1}\mathbf{A}^{\flat}_{k-k_{1}}(\lambda)\mathbf{B}_{k_{1}}(\lambda)d\lambda,$

where the last two terms do not exist when $k=0$.

Following the same lines and setting

$\mathbf{A}^{\sharp}_{k}:=\sup_{\mathscr{M}^{\mathcal{H}}_{[s_{0},s_{1}]}}\big{(}(s/t)^{-1+\delta}s^{3/2}\big{(}|\phi|_{N-4,k}+(s/t)\,|\partial\phi|_{N-4,k}\big{)}\big{)},$
(6.28)

we proceed with similar quantities with $N-5$ replaced by $N-4$ and, by now
including a defavorable factor $(s/t)$, we find our inductive inequality (at
the order $N-4$)

$\displaystyle\mathbf{A}^{\sharp}_{k}(s)\lesssim
C_{1}\epsilon+(s/t)^{1/2-\delta}\int_{\lambda_{0}}^{s}\lambda^{-3/2+\delta}\mathbf{A}^{\sharp}_{k}(\lambda)d\lambda$
(6.29)
$\displaystyle+(s/t)^{1-\delta}\int_{\lambda_{0}}^{s}\lambda^{-1}\mathbf{A}^{\sharp}_{k-1}(\lambda)\mathbf{B}_{0}(\lambda)d\lambda+(s/t)^{1-\delta}\sum_{k_{1}=1}^{N-4}\int_{\lambda_{0}}^{s}\lambda^{-1}\mathbf{A}^{\sharp}_{k-k_{1}}(\lambda)\mathbf{B}_{k_{1}}(\lambda)d\lambda,$

where the last two terms do not exist when $k=0$.

Finally, considering the case $k=0$ and applying Gronwall inequality to
(6.27)-(6.29), we arrive at the following statement. Larger values of $k$ will
be treated next.

######  Lemma 6.2 (Version without boosts and rotations).

In the hyperboloidal domain, the Klein-Gordon field satisfies the following
sharp pointwise estimates at low-order:

$\displaystyle|\phi|_{N-5,0}+(s/t)\,|\partial\phi|_{N-5,0}\lesssim
C_{1}\epsilon(s/t)^{2-2\delta}s^{-3/2}\quad$ $\displaystyle\text{ in
}\mathscr{M}^{\mathcal{H}}_{[s_{0},s_{1}]},$ (6.30)
$\displaystyle|\phi|_{N-4,0}+(s/t)\,|\partial\phi|_{N-4,0}\lesssim
C_{1}\epsilon(s/t)^{1-\delta}s^{3/2}\quad$ $\displaystyle\text{ in
}\mathscr{M}^{\mathcal{H}}_{[s_{0},s_{1}]}.$

#### Sharp decay of the wave component.

In order to now apply Proposition 3.1, we estimate $|\Box u|_{k}$ for $k\leq
N-5$. In fact, in $\mathscr{M}^{\mathcal{H}}_{[s_{0},s]}$ we have

$|\partial\phi\partial\phi|_{k}\lesssim\sum_{p_{1}+p_{2}=k\atop
k_{1}+k_{2}=k}|\partial\phi|_{p_{1},k_{2}}|\partial\phi|_{p_{2},k_{2}}\lesssim(s/t)^{2-4\delta}s^{-3}\sum_{k_{1}=0}^{N-5}\mathbf{A}^{\flat}_{k_{1}}(s)\mathbf{A}^{\flat}_{N-5-k_{1}}(s).$

Similar estimate holds for $|\phi^{2}|_{N-5,k}$. We observe (6.18) provides us
with a sufficient estimate on $|\Box u|_{k}$ in
$\mathscr{M}^{\mathcal{EM}}_{[s_{0},s]}$. Thus by Proposition 3.1 (Case 0 and
Case 1 therein) together with the pointwise decay on the initial data (5.6),
we find

$\textstyle\mathbf{B}_{k}(s)\lesssim
C_{0}\epsilon+\sum_{k_{1}=0}^{N-5}\mathbf{A}^{\flat}_{k_{1}}(s)\mathbf{A}^{\flat}_{N-5-k_{1}}(s).$
(6.31)

When $k=0$, thanks to (6.30) this leads us to $\mathbf{B}_{0}(s)\lesssim
C_{1}\epsilon.$

###  6.5 Sharp decay in the hyperboloidal domain: induction argument

We consider (6.27) together with (6.31). By induction on $k$ and Gronwall
inequality, provided $C_{1}\epsilon\lesssim\theta\leq\delta/(10N)$ we obtain

$\mathbf{A}^{\flat}_{k}(s)\lesssim C_{1}\epsilon
s^{2k\theta},\qquad\mathbf{B}_{k}(s)\lesssim C_{1}\epsilon s^{2k\theta},\qquad
0\leq k\leq N-5.$ (6.32)

Then we turn to the control of $\mathbf{A}^{\sharp}_{k}$ and
$\mathbf{B}_{N-4}$. To this end we need a more detailed analysis of the
structure of commutators. Recalling the hierarchy structure (2.20), in the
hyperboloidal domain and for all $\textbf{ord}(Z)=p=N-4$ and
$\textbf{rank}(Z)=k\leq N-4$ we find

$\displaystyle\,\big{|}[Z,H^{\alpha\beta}u\partial_{\alpha}\partial_{\beta}]\phi\big{|}$
(6.33)
$\displaystyle\lesssim|u||\partial\partial\phi|_{p-1,k-1}+\sum_{k_{1}+p_{2}=p\atop
k_{1}+k_{2}=k}|Y_{\text{rot}}u|_{k_{1}-1}|\partial\partial\phi|_{p_{2},k_{2}}+\sum_{p_{1}+p_{2}=p\atop
k_{1}+k_{2}=k}|\partial
u|_{p_{1}-1,k_{1}}|\partial\partial\phi|_{p_{2},k_{2}}$
$\displaystyle\lesssim(s/t)^{1-\delta}s^{-3/2}\mathbf{A}^{\sharp}_{k-1}(s)\mathbf{B}_{0}(s)+|Y_{\text{rot}}u||\partial\partial\phi|_{p-1,k-1}+\hskip-8.5359pt\sum_{k_{1}+p_{2}=p,k_{1}\geq
2\atop
k_{1}+k_{2}=k}\hskip-8.5359pt|Y_{\text{rot}}u|_{k_{1}-1}|\partial\partial\phi|_{p_{2},k_{2}}$
$\displaystyle\quad+C_{1}\epsilon(s/t)^{1/2-\delta}s^{-3+\delta}\mathbf{A}^{\sharp}_{k}(s)$
$\displaystyle\lesssim(s/t)^{1-\delta}s^{-3/2}\mathbf{A}^{\sharp}_{k-1}(s)\big{(}\mathbf{B}_{0}(s)+\mathbf{B}_{1}(s)\big{)}+(s/t)^{2-2\delta}s^{-5/2}\sum_{k_{1}=2}^{k}\mathbf{A}^{\flat}_{k-k_{1}}(s)\mathbf{B}_{k_{1}}(s)$
$\displaystyle\quad+C_{1}\epsilon(s/t)^{1/2-\delta}s^{-3+\delta}\mathbf{A}^{\sharp}_{k}(s).$

Then, thanks to (6.32), for all $k\leq N-5$ we have

$\displaystyle\big{|}[Z,H^{\alpha\beta}\partial_{\alpha}\partial_{\beta}]\phi\big{|}$
$\displaystyle\lesssim
C_{1}\epsilon(s/t)^{1/2-\delta}s^{-3+\delta}\mathbf{A}^{\sharp}_{k}(s)+C_{1}\epsilon(s/t)^{1-\delta}s^{-5/2+2\theta}\mathbf{A}^{\sharp}_{k-1}(s)$
(6.34)
$\displaystyle\quad+(C_{1}\epsilon)^{2}(s/t)^{2-2\delta}s^{-5/2+2k\theta}.$

When $k=N-4$, the second sum in the right-hand side of (6.33) can be
decomposed and controlled as follows:

$\displaystyle\textstyle(s/t)^{2-2\delta}s^{-5/2}\sum_{k_{1}=2}^{N-4}\mathbf{A}^{\flat}_{N-4-k_{1}}(s)\mathbf{B}_{k_{1}}(s)$
$\displaystyle\lesssim(s/t)^{2-2\delta}s^{-5/2}\mathbf{A}^{\flat}_{0}(s)\mathbf{B}_{N-4}(s)+(C_{1}\epsilon)^{2}(s/t)^{2-2\delta}s^{-5/2}s^{2(N-4)\theta}$
$\displaystyle\lesssim
C_{1}\epsilon(s/t)^{2-2\delta}s^{-5/2}\mathbf{B}_{N-4}(s)+(C_{1}\epsilon)^{2}(s/t)^{2-2\delta}s^{-5/2}s^{2(N-4)\theta}.$

We are now ready to apply Proposition 3.2 concerning the sharp decay of Klein-
Gordon solutions. Thanks to (6.26), (6.11), and (6.15), for all $k\leq N-5$ we
find

$\displaystyle\mathbf{A}^{\sharp}_{k}(s)$ $\displaystyle\lesssim C_{1}\epsilon
s^{2k\theta}+C_{1}\epsilon(s/t)^{1/2-\delta}\int_{\lambda_{0}}^{s}\lambda^{-3/2}\mathbf{A}^{\sharp}_{k}(\lambda)d\lambda$
(6.35)
$\displaystyle\quad+C_{1}\epsilon(s/t)^{1-\delta}\int_{\lambda_{0}}^{s}\lambda^{-1+2\theta}\mathbf{A}^{\sharp}_{k-1}(\lambda)d\lambda.$

By induction on $k$ and Gronwall’s inequality, provided
$C_{1}\epsilon\theta\lesssim 1$ we obtain

$\mathbf{A}^{\sharp}_{k}(s)\lesssim C_{1}\epsilon s^{2k\theta},\quad k\leq
N-5.$ (6.36)

On the other hand, we have

$\displaystyle\mathbf{A}^{\sharp}_{N-4}(s)$ $\displaystyle\lesssim
C_{1}\epsilon
s^{2(N-4)\theta}+C_{1}\epsilon(s/t)^{1/2-\delta}\int_{\lambda_{0}}^{s}\lambda^{-3/2}\mathbf{A}^{\sharp}_{N-4}(\lambda)d\lambda$
(6.37)
$\displaystyle\quad+C_{1}\epsilon(s/t)^{2-2\delta}\int_{\lambda_{0}}^{s}\lambda^{-1}\mathbf{B}_{N-4}(\lambda)d\lambda.$

A final estimate on $\mathbf{B}_{N-4}$ is required and we write

$\displaystyle|\partial\phi\partial\phi|_{N-4}$
$\displaystyle\textstyle\lesssim|\partial\phi||\partial\phi|_{N-4}+\sum_{k=1}^{N-5}|\partial\phi|_{k}|\partial\phi|_{N-4-k}$
$\displaystyle\textstyle\lesssim(s/t)^{-\delta}s^{-3/2}|\phi|_{1,0}\mathbf{A}^{\sharp}_{N-4}(s)+\sum_{k=1}^{N-5}\mathbf{A}^{\flat}_{k}(s)\mathbf{A}^{\flat}_{N-4-k}(s)$
$\displaystyle\lesssim(s/t)^{-\delta}s^{-3/2}|\phi|_{N-5,0}\mathbf{A}^{\sharp}_{N-4}(s)+(C_{1}\epsilon)^{2}(s/t)^{2-4\delta}s^{-3+2(N-4)\theta}$
$\displaystyle\lesssim
C_{1}\epsilon(s/t)^{2-3\delta}s^{-3}\mathbf{A}^{\sharp}_{N-4}(s)+(C_{1}\epsilon)^{2}(s/t)^{2-4\delta}s^{-3+2(N-4)\theta}$

(provided $N\geq 6$). A similar (but simpler) estimate holds for $|\phi|^{2}$.
Then from Proposition 3.1 concerning the wave equation (Case 0 and Case 1
therein), we deduce that

$\mathbf{B}_{N-4}(s)\lesssim
C_{0}\epsilon+C_{1}\epsilon\mathbf{A}^{\sharp}_{N-4}(s)+(C_{1}\epsilon)^{2}s^{2(N-4)\theta}.$
(6.38)

Combining this inequality (6.38) together with (6.37), we apply Gronwall’s
inequality and obtain

$\mathbf{A}^{\sharp}_{N-4}(s)\lesssim C_{1}\epsilon
s^{(N-4)\theta},\qquad\mathbf{B}_{N-4}(s)=C_{1}\epsilon s^{2(N-4)\theta}.$
(6.39)

We have arrived at the following main conclusion.

######  Lemma 6.3 (Version with a hierarchy).

In the hyperboloidal domain $\mathscr{M}^{\mathcal{H}}_{[s_{0},s_{1}]}$, the
wave and Klein-Gordon fields satisfy the following sharp pointwise estimates
at low-order (provided $C_{1}\epsilon\lesssim\theta\leq\delta/(10N)$):

$t\,|u|_{k}\lesssim C_{1}\epsilon s^{2k\theta},\qquad k\leq N-4,$ (6.40)
$|\phi|_{N-5,k}+(s/t)\,|\partial\phi|_{N-5,k}\lesssim
C_{1}\epsilon(s/t)^{2-2\delta}s^{-3/2+2k\theta},\qquad k\leq N-5,$ (6.41a)
$|\phi|_{N-4,k}+(s/t)\,|\partial\phi|_{N-4,k}\lesssim
C_{1}\epsilon(s/t)^{1-\delta}s^{-3/2+2k\theta},\qquad k\leq N-4.$ (6.41b)

###  6.6 Commutators and source terms in the hyperboloidal domain

We now turn our attention to the $L^{2}$-type estimates of the commutators and
source terms. First of all, to deal with the Klein-Gordon commutators, we
apply the commutator estimate (2.20). For the first term in the right-hand
side therein, we have

$|u||\partial\partial\phi|_{p-1,k-1}\lesssim C_{1}\epsilon
s^{-1}(s/t)\,|\partial\phi|_{p,k-1}.$

For the second term, when $k_{1}\leq N-4$ we apply (6.40) and obtain

$|Y_{\text{rot}}u|_{k_{1}-1}|\partial\partial\phi|_{p_{2},k_{2}}\lesssim
C_{1}\epsilon s^{-1+2k_{1}\theta}(s/t)\,|\partial\phi|_{p,k-k_{1}}.$

The case $N-3\leq k_{1}\leq k$ occurs only when $p\geq k\geq N-3$. In this
case $p_{2}=p-k_{1}\leq 3$ and

$\displaystyle|Y_{\text{rot}}u|_{k_{1}-1}|\partial\partial\phi|_{p_{2},k_{2}}$
$\displaystyle\lesssim
C_{1}\epsilon(s/t)^{2-2\delta}s^{-3/2+2k_{2}\theta}|Y_{\text{rot}}u|_{k_{1}-1}$
$\displaystyle\lesssim
C_{1}\epsilon(s/t)^{1-2\delta}s^{-1/2+2k_{2}\theta}|\not{\partial}^{\mathcal{H}}u|_{k_{1}-1},$

where we applied (6.41a) (provided $3\leq N-7$, namely $N\geq 10$).

For the last term in the right-hand side of (2.20), we observe that when
$p_{2}\leq N-7$ we can apply (6.41a) on
$|\partial\partial\phi|_{p_{2},k_{2}}\leq|\phi|_{N-5,k_{2}}\lesssim
C_{1}\epsilon(s/t)^{2-\delta}s^{-3/2+(N-5)\theta}$:

$|\partial u|_{p_{1}-1}|\partial\partial\phi|_{p_{2},k_{2}}\lesssim
C_{1}\epsilon s^{-5/4}(s/t)\,|\partial u|_{p_{1}-1}.$

When $N-2\geq p_{2}\geq N-6$, in this case $p_{1}\leq 6\leq N-5$. We apply
(6.9) to the term $|\partial u|_{p_{1}-1}$ and we find

$|\partial u|_{p_{1}-1}|\partial\partial\phi|_{p_{2},k_{2}}\lesssim
C_{1}\epsilon s^{-1}(s/t)\,|\partial\phi|_{p,k}.$

In the hyperboloidal domain, for all $\textbf{ord}(Z)=p$ and
$\textbf{rank}(Z)=k$ we conclude that

$\displaystyle\,\big{|}[Z,H^{\alpha\beta}\partial_{\alpha}\partial_{\beta}]\phi\big{|}$
$\displaystyle\lesssim C_{1}\epsilon s^{-1}(s/t)\,|\partial
u|_{p,k}+C_{1}\epsilon(s/t)^{-3/2+2(N-5)\theta}|\partial u|_{p,k}$
$\displaystyle\quad+\begin{cases}C_{1}\epsilon\sum_{k_{1}=1}^{k}s^{-1+2k_{1}\theta}(s/t)\,|\partial\phi|_{p,k-k_{1}},\quad&k\leq
N-4,\\\ {\begin{aligned}
&C_{1}\epsilon\sum_{k_{1}=1}^{N-4}s^{-1+2k_{1}\theta}(s/t)\,|\partial\phi|_{p,k-k_{1}}\\\
&+\sum_{k_{1}=0}^{k-N+3}s^{-1/2+2k_{1}\theta}|\not{\partial}^{\mathcal{H}}u|_{k-k_{1}-1},\end{aligned}}&k\geq
N-3.\end{cases}$

Provided that $2(N-5)\theta+\delta\leq 1/4$, for the $L^{2}$ norm this
pointwise inequality leads us to

$\displaystyle\|[Z,H^{\alpha\beta}\partial_{\alpha}\partial_{\beta}]\phi\|_{L^{2}(\mathscr{M}^{\mathcal{H}}_{s})}\lesssim(C_{1}\epsilon)^{2}s^{-5/4}$
(6.42) $\displaystyle\quad+\begin{cases}C_{1}\epsilon
s^{-1+2k_{1}\theta}\sum_{k_{1}=0}^{k}\mathcalboondox
F_{\kappa,c}^{p,k-k_{1}}(s,\phi),\quad&k\leq N-4,\\\ \\\ {\begin{aligned}
&\textstyle C_{1}\epsilon
s^{-1+2k_{1}\theta}\sum_{k_{1}=0}^{N-4}\mathcalboondox
F_{\kappa,c}^{p,k-k_{1}}(s,\phi)\\\ \\\
&\textstyle\quad+\sum_{k_{1}=0}^{k-N+3}s^{-1/2+2k_{1}\theta}\mathcalboondox
F_{\kappa,c}^{k-k_{1}-1}(s,u),\end{aligned}}&k\geq N-3\end{cases}$

On the other hand, dealing with the source term for the wave equation is
simpler and we write directly

$\||\Box
u|_{p,k}\|_{L^{2}(\mathscr{M}^{\mathcal{H}}_{s})}\lesssim\begin{cases}(C_{1}\epsilon)^{2}s^{-3/2+2\delta},\quad&p\leq
N-4,\\\ (C_{1}\epsilon)\sum_{k_{1}=0}^{k}s^{-3/2+2k_{1}\theta}\mathcalboondox
F_{\kappa,c}^{p,k-k_{1}}(s,\phi),&p\geq N-3.\end{cases}$ (6.43)

###  6.7 Improved energy estimate and conclusion

#### Preliminaries.

Now we are ready to apply the energy estimate (2.27) and establish the
following improved energy estimates:

$s^{-\delta}\,\mathcalboondox
F^{N}_{\kappa}(s,u)+s^{-1/2-\delta}\,\mathcalboondox
F^{N}_{c,\kappa}(s,\phi)\leq(C_{1}/2)\epsilon,$ (6.44) $\mathcalboondox
F_{\kappa}^{N-4}(s,u)+s^{-\delta}\mathcalboondox
F^{N-4}_{c,\kappa}(s,\phi)\leq(C_{1}/2)\epsilon.$ (6.45)

We need to distinguish between high-order case $p\geq N-3$ and low-order case
$p\leq N-4$. But before these, we establish the common estimates needed for
both.

######  Lemma 6.4.

In the spacetime slab $\mathscr{M}_{[s_{0},s_{1}]}$, the following estimates
hold for $\textbf{ord}(Z)=p$ and $\textbf{rank}(Z)=k$:

$\int_{\mathscr{M}_{s}}|G_{g,\kappa}[Z\phi]|\ Jdxds\lesssim C_{1}\epsilon
s^{-1}\mathcalboondox E_{\kappa}^{p,k}(s,\phi)ds,$ (6.46)
$\int_{\mathscr{M}_{s}}\mathbf{X}^{2\kappa-1}\aleph^{\prime}(r-t)H^{\mathcal{N}00}|\partial_{t}Z\phi|\
Jdxds\lesssim C_{1}\epsilon s^{-1}\mathcalboondox E_{\kappa}^{p,k}(s,\phi)ds.$
(6.47)

Here we have $g^{\alpha\beta}=\eta^{\alpha\beta}+H^{\alpha\beta}u$, while the
notation $G_{g,\kappa}$ was introduced in (2.26c). Furthermore, when
$C_{1}\epsilon$ sufficiently small one also has

$\frac{1}{4}\,\mathcalboondox E_{g,\kappa,c}(s,\phi)\leq\mathcalboondox
E_{\kappa,c}(s,\phi)\leq 4\,\mathcalboondox E_{g,\kappa,c}(s,\phi).$ (6.48)

###### Proof.

We rely here on direct consequences of the pointwise decay derived earlier.
Thanks to (6.2) and (6.9), we have

$\displaystyle|G_{g,\kappa}[Z\phi]|J\lesssim|\partial
u|\mathbf{X}^{2\kappa}|\partial Z\phi|$
$\displaystyle\lesssim\begin{cases}C_{1}\epsilon\,t^{-1/2}s^{-1}(s/t)^{-1}|(s/t)\partial
Z\phi|^{2}\quad&\text{ in }\mathscr{M}^{\mathcal{H}}_{[s_{0},s_{1}]},\\\
C_{1}\epsilon\,t^{-1}\mathbf{X}^{-\kappa}(J\zeta^{-2})|\zeta\mathbf{X}^{\kappa}\partial
Z\phi|^{2}\quad&\text{ in
}\mathscr{M}^{\mathcal{EM}}_{[s_{0},s_{1}]},\end{cases}$
$\displaystyle\lesssim\begin{cases}C_{1}\epsilon s^{-1}|(s/t)\partial
Z\phi|^{2}\quad&\text{ in }\mathscr{M}^{\mathcal{H}}_{[s_{0},s_{1}]},\\\
C_{1}\epsilon s^{-1}|\zeta\mathbf{X}^{\kappa}\partial Z\phi|^{2}\quad&\text{
in }\mathscr{M}^{\mathcal{EM}}_{[s_{0},s_{1}]},\end{cases}$

which, by integration, implies (6.46). On the other hand, the proof of (6.47)
is similar and we apply (6.19) (in the Euclidean-merging domain) and (6.40)
(in the hyperboloidal domain). Concerning the inequalities (6.48), we observe
that $|\mathcalboondox E_{g,\kappa,c}(s,Z\phi)-\mathcalboondox
E_{\kappa,c}(s,Z\phi)|\lesssim\int_{\mathscr{M}_{s}}\zeta^{-2}|u|\,|\zeta\mathbf{X}^{\kappa}\partial
Z\phi|^{2}\,dx$ and, in view of (6.19) and (6.40), we find (since
$t^{-1}\lesssim\zeta^{2}$)

$|\mathcalboondox E_{g,\kappa,c}(s,Z\phi)-\mathcalboondox
E_{\kappa,c}(s,Z\phi)|\lesssim
C_{1}\epsilon\int_{\mathscr{M}_{s}}\mathcalboondox E_{\kappa,c}(s,Z\phi).\qed$

#### Energy inequalities.

Now we apply the energy equation (2.27). For the wave operator, the metric
under consideration in the model is flat, thus for all $\textbf{ord}(Z)=p$ and
$\textbf{rank}(Z)=k$

$\frac{d}{ds}\mathcalboondox
E_{\kappa}(s,Zu)=\int_{\mathscr{M}_{s}}\hskip-8.5359pt\zeta\mathbf{X}^{2\kappa}\partial_{t}Zu\,\mathbf{X}^{\kappa}J\zeta^{-1}\Box
Zu\ dx\lesssim\mathcalboondox
F_{\kappa}(s,Zu)^{1/2}\|J\zeta^{-1}\,\mathbf{X}^{\kappa}|\Box
u|_{p,k}\|_{L^{2}(\mathscr{M}_{s})}.$

Thus we have $\frac{d}{ds}\mathcalboondox
F_{\kappa}(s,Zu)\lesssim\|(s/t)\,|\Box
u|_{p,k}\|_{L^{2}(\mathscr{M}^{\mathcal{H}}_{s})}+\|J\zeta^{-1}\,\mathbf{X}^{\kappa}|\Box
u|_{p,k}\|_{L^{2}(\mathscr{M}^{\mathcal{EM}}_{s})}.$ Thanks to the $L^{2}$
bound (6.22) and (6.43) concerning the source-terms, and after summation over
$\textbf{ord}(Z)\leq p$ and $\textbf{rank}(Z)\leq k$, we find

$\displaystyle\frac{d}{ds}\mathcalboondox F_{\kappa}^{p,k}(s,u)$ (6.49)
$\displaystyle\leq\begin{cases}C_{N,\delta}(C_{1}\epsilon)^{2}s^{-5/4},&p\leq
N-4,\\\
C_{N,\delta}(C_{1}\epsilon)^{2}s^{-5/4}+C_{N,\delta}C_{1}\epsilon\sum_{k_{1}=0}^{k}s^{-3/2+2k_{1}\theta}\mathcalboondox
F_{\kappa,c}^{p,k-k_{1}}(s,\phi),&p\geq N-3,\end{cases}$

where we the notation $C_{N,\delta}$ we now make explicit the implicit
constant corresponding by the symbol “$\lesssim$” in all previous estimates.
This constant is determined by $N,\delta$ and the model itself.

Now we turn our attention to the Klein-Gordon component. Recall (2.27)
together with Lemma 6.4, for all $\textbf{ord}(Z)\leq p$ and
$\textbf{rank}(Z)\leq k$ we have

$\frac{d}{ds}\mathcalboondox E_{g,\kappa,c}(s,Z\phi)\lesssim C_{1}\epsilon
s^{-1}\mathcalboondox E_{\kappa}^{p,k}(s,\phi)+\mathcalboondox
E_{g,\kappa,c}(s,Z\phi)^{1/2}\big{\|}[Z,H^{\alpha\beta}\partial_{\alpha}\partial_{\beta}]\phi\big{\|}_{L^{2}(\mathscr{M}_{s})}.$

Summing over all $\textbf{ord}(Z)\leq p$ and $\textbf{rank}(Z)\leq k$, we find

$\frac{d}{ds}\mathcalboondox E_{g,\kappa,c}^{p,k}(s,\phi)\lesssim
C_{1}\epsilon s^{-1}\mathcalboondox E_{\kappa}^{p,k}(s,\phi)+\mathcalboondox
E_{g,\kappa,c}(s,Z\phi)^{1/2}\big{\|}[Z,H^{\alpha\beta}\partial_{\alpha}\partial_{\beta}]\phi\big{\|}_{L^{2}(\mathscr{M}_{s})},$

which, thanks to (6.48), leads us to

$\frac{d}{ds}\mathcalboondox F_{\kappa,c}^{p,k}(s,\phi)\lesssim C_{1}\epsilon
s^{-1}\mathcalboondox F_{\kappa,c}^{p,k}(s,\phi)+\sum_{\textbf{ord}(Z)\leq
p\atop\textbf{rank}(Z)\leq
k}\|[Z,H^{\alpha\beta}\partial_{\alpha}\partial_{\beta}]\phi\|_{L^{2}(\mathscr{M}_{s})}.$
(6.50)

Taking into account the commutator estimates in (6.20) and (6.42), we thus
have

$\frac{d}{ds}\mathcalboondox
F_{\kappa,c}^{p,k}(s,\phi)\leq\begin{cases}\displaystyle
C_{N,\delta}(C_{1}\epsilon)^{2}s^{-5/4}+C_{N,\delta}C_{1}\epsilon\sum_{k_{1}=0}^{k}s^{-1+2k_{1}\theta}\mathcalboondox
F_{\kappa,c}^{p,k-k_{1}}(s,\phi),&k\leq N-4,\\\ {\begin{aligned}
&C_{N,\delta}(C_{1}\epsilon)^{2}s^{-1/2}+C_{N,\delta}C_{1}\epsilon\sum_{k_{1}=0}^{N-4}s^{-1+2k_{1}\theta}\mathcalboondox
F_{\kappa,c}^{p,k-k_{1}}(s,\phi)\\\
&\quad+C_{N,\delta}C_{N,\delta}\sum_{k_{1}=0}^{k-N+3}s^{-1/2+2k_{1}\theta}\mathcalboondox
F_{\kappa,c}^{k-k_{1}-1}(s,u),\end{aligned}}\quad&\hskip 0.0ptk\geq
N-3.\end{cases}$ (6.51)

#### Low-order estimates.

Consider first the range $p\leq N-4$, therefore $k\leq p\leq N-4$. Thus in
view of the energy inequality (6.49) and (6.51), we find

$\displaystyle\mathcalboondox F_{\kappa}^{N-4}(s,u)$ $\displaystyle\leq
C_{0}\epsilon+C_{N,\delta}(C_{1}\epsilon)^{2},$ $\displaystyle k\leq p\leq
N-4,$ (6.52) $\displaystyle\mathcalboondox F_{\kappa,c}^{N-4,k}(s\phi)$
$\displaystyle\leq(C_{0}\epsilon+5C_{N,\delta}(C_{1}\epsilon)^{2})s^{(2k+1)\theta},$
$\displaystyle k\leq p\leq N-4,$

provided

$C_{N}C_{1}\epsilon\leq\theta,\quad C_{0}/C_{1}+5C_{N}C_{1}\epsilon\leq
2\theta.$ (6.53)

#### High-order estimates.

We consider next the range $p\geq N-3$, and also rely on (6.49) and (6.51).
However, the system of inequalities is more involved. We first treat the case
$k\leq N-4$, as follows: the system (6.49) and (6.51) reduces to

$\displaystyle\frac{d}{ds}\mathcalboondox F_{\kappa}^{p,k}(s,u)$
$\displaystyle\leq
C_{N,\delta}(C_{1}\epsilon)^{2}s^{-5/4}+C_{N,\delta}C_{1}\epsilon
s^{-3/2}\mathcalboondox{A}^{p,k}(s,\phi),$
$\displaystyle\frac{d}{ds}\mathcalboondox F_{\kappa,c}^{p,k}(s,\phi)\leq$
$\displaystyle
C_{N,\delta}(C_{1}\epsilon)^{2}s^{-5/4}+C_{N,\delta}C_{1}\epsilon
s^{-1}\mathcalboondox{A}^{p,k}(s,\phi),$

in which

$\mathcalboondox{A}^{p,k}(s,\phi):=\sum_{k_{1}=0}^{k}s^{2k_{1}\theta}\mathcalboondox
F_{\kappa,c}^{p,k-k_{1}}(s,\phi).$ (6.54)

By induction and Gronwall’s inequality, we obtain

$\displaystyle\mathcalboondox F_{\kappa}^{N,k}(s,u)$ $\displaystyle\leq
C_{0}\epsilon+5C_{N,\delta}(C_{1}\epsilon)^{2},$ $\displaystyle p\geq N-3,$
(6.55) $\displaystyle\mathcalboondox F_{\kappa,c}^{N,k}(s,\phi)$
$\displaystyle\leq\big{(}C_{0}\epsilon+5C_{N,\delta}(C_{1}\epsilon)^{2}\big{)}s^{(2k+1)\theta},$
$\displaystyle p\geq N-3,$

provided

$(C_{0}/C_{1})+5C_{N,\delta}C_{1}\epsilon\leq\min\\{1/4N,2\theta\\},\quad
C_{N,\delta}C_{1}\epsilon\leq\theta.$ (6.56)

On the other hand, for the range $k\geq N-3$ we write

$\displaystyle\mathcalboondox{A}^{p,k}(s,\phi)=$
$\displaystyle\sum_{k_{1}=0}^{N-4}s^{2k_{1}\theta}\mathcalboondox
F_{\kappa,c}^{N,k}(s,\phi)+\sum_{k_{1}=N-3}^{k}s^{2k_{1}\theta}\mathcalboondox
F_{\kappa,c}^{N,k-k_{1}}(s,\phi)$
$\displaystyle\lesssim(N-4)(C_{0}\epsilon+C_{N}(C_{1}\epsilon)^{2})s^{(2N+1)\theta}+\mathcalboondox{B}^{k}(s,\phi),$

in which

$\mathcalboondox{B}^{k}(s,\phi):=\sum_{k_{1}=N-3}^{k}s^{2k_{1}\theta}\mathcalboondox
F_{\kappa,c}^{N,k-k_{1}}(s,\phi).$ (6.57)

Substituting this result into (6.49) and (6.51), provided $(2N+1)\theta\leq
1/4$ we obtain

$\displaystyle\frac{d}{ds}\mathcalboondox F_{\kappa}^{N,k}(s,u)$
$\displaystyle\leq
C_{N,\delta}^{\prime}(C_{1}\epsilon)^{2}s^{-5/4}+C_{N,\delta}^{\prime}C_{1}\epsilon
s^{-3/2}\mathcalboondox{B}^{k}(s,\phi),$
$\displaystyle\frac{d}{ds}\mathcalboondox F_{\kappa,c}^{N,k}(s,\phi)$
$\displaystyle\leq
C_{N,\delta}^{\prime}(C_{1}\epsilon)^{2}s^{-1/2}+C_{N,\delta}^{\prime}C_{1}\epsilon
s^{-1/2}\mathcalboondox{C}^{k}(s,u).$

where

$\mathcalboondox{C}^{k}(s,u):=\sum_{k_{1}=N-3}^{k}s^{2k_{1}\theta}\mathcalboondox
F_{\kappa}^{k-k_{1}-1}(s,u).$ (6.58)

We especially observe that, by definition, $\mathcalboondox{C}^{N-3}(s,u)=0$.
Proceeding again by induction, we arrive at

$\displaystyle\mathcalboondox F_{\kappa}^{N,k}(s,u)$
$\displaystyle\leq(C_{0}\epsilon+5C_{N,\delta}^{\prime}(C_{1}\epsilon)^{2})s^{2k\theta},\quad$
$\displaystyle\mathcalboondox F_{\kappa,c}^{N,k}(s,\phi)$
$\displaystyle\leq(C_{0}\epsilon+5C_{N,\delta}^{\prime}(C_{1}\epsilon)^{2})s^{1/2+2k\theta},$

so that

$\displaystyle\mathcalboondox F_{\kappa}^{N,k}(s,u)$
$\displaystyle\leq(C_{0}\epsilon+5C_{N,\delta}^{\prime}(C_{1}\epsilon)^{2})s^{2(k-(N-3)+1)\theta},$
$\displaystyle k\geq N-3,$ (6.59) $\displaystyle\mathcalboondox
F_{\kappa}^{N,k}(s,\phi)$
$\displaystyle\leq(C_{0}\epsilon+5C_{N,\delta}^{\prime}(C_{1}\epsilon)^{2})s^{2(k-(N-3))\theta},$
$\displaystyle k\geq N-3,$

provided

$C_{0}/C_{1}+5C_{N,\delta}^{\prime}C_{1}\epsilon\leq\theta/2,\qquad
C_{N,\delta}^{\prime}C_{1}\epsilon\leq\theta.$ (6.60)

If we now impose that the constants satisfy

$2N\theta\leq\delta,\quad C_{0}/C_{1}\leq
1/4\theta,\quad\epsilon\leq\frac{2C_{1}-C_{0}}{10\max\big{(}C_{N,\delta},C_{N,\delta^{\prime}}\big{)}},$
(6.61)

together with the previous inequalities (6.53), (6.56) and (6.60), then
(6.52), (6.55) and (6.59) lead to the conclusion (6.44)-(6.45). This completes
the bootstrap argument.

#### Acknowledgments.

This work was done in parts when PLF was a visiting research fellow at the
Courant Institute for Mathematical Sciences, New York University, and a
visiting professor at the School of Mathematical Sciences, Fudan University,
Shanghai. The work of YM was supported by a Special Financial Grant from the
China Postdoctoral Science Foundation under the grant number NSFC 11601414.

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