footnote

# Boundedness of the conformal hyperboloidal energy
for a wave-Klein-Gordon model

Philippe G. LeFlocha, Jesús Oliverb, and Yoshio Tsutsumic

(October 2022)

###### Abstract

We consider the global evolution problem for a model which couples together a
nonlinear wave equation and a nonlinear Klein-Gordon equation, and was
independently introduced by LeFloch and Y. Ma and by Q. Wang. By revisiting
the Hyperboloidal Foliation Method, we establish that a weighted energy of the
solutions remains (almost) bounded for all times. The new ingredient in the
proof is a hierarchy of fractional Morawetz energy estimates (for the wave
component of the system) which is defined from two conformal transformations.
The optimal case for these energy estimates corresponds to using the scaling
vector field as a multiplier for the wave component.

11footnotetext: Laboratoire Jacques-Louis Lions & Centre National de la
Recherche Scientifique, Sorbonne Université, 4 Place Jussieu, 75252 Paris,
France. Email<EMAIL_ADDRESS>
b Department of Mathematics, California State University, East Bay, 25800
Carlos Bee Blvd., Hayward, CA 94542, USA. Email<EMAIL_ADDRESS>
b Institute of Liberal Arts and Sciences, Kyoto University, Yoshida
Nihonmatsu-cho, Sakyo-ku, Kyoto 606–8501, Japan. Email:
<EMAIL_ADDRESS>
##  1 Introduction

#### Aim of this paper.

We are interested in the global-in-time existence problem for coupled systems
of wave-Klein-Gordon equations and we study a typical system that arises as a
simple model formally derived from the Einstein equations of general
relativity in presence of massive matter fields. We are interested in
spatially compactly supported solutions and in analyzing their decay in time,
and therefore our analysis focus on the interior of a light cone region.
Throughout, we work in Minkowski spacetime
$\mathbb{R}^{3+1}:=(\mathbb{R}^{4},\eta)$ where $\eta:=\text{diag}(-1,1,1,1)$
denotes the Minkowski metric in inertial coordinates denoted by
$(x^{0},x^{1},x^{2},x^{3})=(t,x)=(t,x^{i})\in\mathbb{R}^{4}$ ($i=1,2,3$). The
level sets of the time function $t:\mathbb{R}^{3+1}\mapsto\mathbb{R}$ are
uniformly spacelike and provide natural hypersurfaces on which the initial
value problem can be posed. On the other hand, an alternative standpoint is
provided by the slices (with $s\geq s_{0}>1$ for notational convenience)

$\mathcal{H}_{s}:=\big{\\{}(t,x)\,/\,t>0;\,t^{2}-r^{2}=s^{2}\big{\\}},$ (1.1)

in which we used the notation
$r^{2}:=|x|^{2}=(x^{1})^{2}+(x^{2})^{2}+(x^{3})^{2}$. These slices define a
foliation by spacelike hypersurfaces of (a subset of) the interior of the
future light cone from the origin.

In the present paper, we build upon pioneering works using the hyperbolic
foliation (1.1) together with earlier works on the global existence problems
for wave equations [8, 18] and Klein-Gordon equations [6, 14, 15, 17, 28, 29,
35, 36]. Specifically, we follow the presentation in the monograph [20] by
LeFloch and Ma who coined the word “hyperboloidal foliation method" to denote
a strategy which leads to global existence results for coupled wave-Klein-
Gordon systems possibly on a curved geometry, and combines several ingredients
motivated by the earlier works cited above:

  * (i)

a notion of weighted functional norm based on the Lorentz boosts,

  * (ii)

a notion of semi-hyperboloidal frame containing tangent vectors to the
hyperboloids,

  * (iii)

weighted Sobolev and Hardy inequalities on hyperboloids,

  * (iv)

sharp pointwise estimates in sup-norm for wave and Klein-Gordon equations, and

  * (v)

nonlinear coupling arguments based on a hierarchy enjoyed by the wave-Klein-
Gordon system.

The method in [20] overcomes the difficulty that solutions to wave and Klein-
Gordon equations have different decay in time, and that the scaling vector
fields does not commute with the Klein-Gordon operators, so that the natural
choice of weighted norms is too small for applying the standard vector field
techniques. This approach was later applied and generalized in [23] where the
spatial decay of solutions is investigated. This line of research was
motivated by the study of the Einstein equations in the regime near the
(vacuum) Minkowski spacetime.

#### The system of interest.

Specifically, we study here the following coupled system consisting of two
wave and Klein-Gordon equations, referred to as the LeFloch-Ma-Wang model [23,
39]:

$\displaystyle-\Box
u=P^{\alpha\beta}\partial_{\alpha}v\partial_{\beta}v+Rv^{2},\qquad-\Box
v+c^{2}v=uH^{\alpha\beta}\partial_{\alpha}\partial_{\beta}v,$ (1.2)

in which $\Box:=-\partial_{t}^{2}+\Delta_{x}$ denotes the wave operator on
$\mathbb{R}^{3+1}$ and $P^{\alpha\beta}$, $H^{\alpha\beta}$, $R$, and $c\neq
0$ are constants. The unknowns $u,v$ are scalar fields defined over Minkowski
space $\mathbb{R}^{3+1}$, and (1.2) is supplemented with initial data
$u_{0},u_{1},v_{0},v_{1}$ prescribed on the hyperboloid
$\mathcal{H}_{s_{0}}\subset\mathbb{R}^{3+1}$:

$\displaystyle u|_{\mathcal{H}_{s_{0}}}=u_{0},\quad$
$\displaystyle\partial_{t}u|_{\mathcal{H}_{s_{0}}}=u_{1},\qquad$
$\displaystyle v|_{\mathcal{H}_{s_{0}}}=v_{0},$
$\displaystyle\partial_{t}v|_{\mathcal{H}_{s_{0}}}=v_{1}.$ (1.3)

Provided these data are sufficiently small and regular, the existence of a
global-in-time solution $(u,v)$ to (1.2)- (1.3) was established in [23]. The
same problem was studied by Wang [39] and Ionescu and Pausader [11] in a
different functional framework.

Our objective in the present paper is to revisit their result in a fractional
weighted energy norm, motivated from the Morawetz estimate and the conformal
energy. Namely, we will rely on two conformal transformations of the equations
under consideration and, in turn, in comparison to [23] we will establish an
improved time decay estimate, namely the (almost) boundedness of the energy
for all times. In other words, we improve upon the conclusion of LeFloch-Ma’s
theorem, but at the expense of imposing a stronger spatial decay on the
initial data. Our proof builds upon recent work by Oliver and Sterbenz [34]
and Oliver [33], which provided an extension of the standard vector field
method. The issue of the boundedness of the energy was also recently
established for a variety of models in [3, 4, 26].

#### Motivations from the Einstein equations.

Recall that, in [23], the model (1.2) was “extracted” from the Einstein the
equations of general relativity since it provided a simple, yet non-trivial,
example of coupling between a wave equation and a Klein-Gordon equation. For a
discussion of the Einstein equations, we refer to [22]–[25] as well as [2, 12,
39]. Furthermore, recall that a different strategy was proposed by Ionescu and
Pausader [12], who also successfully treated (1.2) by using Fourier resonance
techniques and also imposed a condition on the $1/2$-derivatives of the wave
component of the data. (Their regularity condition is reminiscent of our
condition with the exponent $a=1/2$; see below.) For further results
concerning the nonlinear coupling between wave and Klein-Gordon equations, we
refer to [19, 32] and the references cited therein. For other important
related developments, see [5, 7, 9, 10].

#### Notation and outline.

Throughout, we write $A\lesssim B$ (resp. “$A\ll B$"; “$A\approx B$") if $A\
\leqslant\ CB$ for some fixed $C>0$ which may change from line to line (resp.
$A\leqslant\epsilon B$ for a small $\epsilon$; both $A\lesssim B$ and
$B\lesssim A$). Our notation for the range of the indices is as follows:
$\alpha,\beta,\ldots=0,1,2,3$, while $i,j,\ldots=1,2,3$ and $b,c\ldots=1,2$.
We make use of Einstein summation convention throughout, and always rely on
the Minkowski metric $m$ to raise or lower indices. The notation
$m=m^{-1}=\text{diag}(-1,1,1,1)$ stands for the Minkowski metric in
$(t,x^{i})$ coordinates.

An outline of this paper is as follows. In Section 2, we present our global
energy bounds for the model, as well as an estimate for the wave equation
which is of interest for its own sake. We proceed by deriving, in Section 3, a
general conformal energy identity satisfied by solutions to the wave equation.
In Section 4, we then introduce Bondi coordinates and establish several
tensorial identities. This leads us, in Section 5, to our main fractional
Morawetz energy estimate. In Section 6, the framework is finally applied to
the model of interest.

##  2 Statement of the main results

###  2.1 The hyperboloidal foliation

#### Spacetime slices.

We need some additional notation and, by following [23], we introduce the
interior of the future light cone with vertex $(1,0,0,0)$, namely

$\mathcal{K}:=\big{\\{}(t,x^{i})\,/\,r<t-1\big{\\}},$ (2.1)

while we foliate the interior of the cone $\big{\\{}(t,x^{i})\,/r<t\big{\\}}$
from the origin $(0,0,0,0)$ by the spacelike hypersurfaces of constant
hyperboloidal time $s$

$\mathcal{H}_{s}:=\big{\\{}(t,x^{i})\,/\,t>0,\,t^{2}-r^{2}=s^{2}\big{\\}}.$
(2.2)

We have fixed some $s_{0}>1$ and for any pair $s_{0}\leq s_{1}<s_{2}$, we
introduce the following sub-domain of $\mathcal{K}$ bounded by two
hyperboloids:

$\mathcal{K}_{[s_{1},s_{2}]}:=\big{\\{}(t,x^{i})\,/\,s_{1}^{2}\leq
t^{2}-r^{2}\leq s_{2}^{2}\ ,r<t-1\big{\\}}\subset\mathcal{K}.$ (2.3)

Within $\mathcal{K}$ we single out the interior domain

$\mathcal{K}_{\text{int}}:=\big{\\{}(t,x^{i})\,/\,r<t/2\big{\\}}\
\subset\mathcal{K}.$ (2.4)

More precisely, the wave domain (i.e. essentially the support of the solutions
to the wave equation considered below) is defined to be the complement in
$\mathcal{K}$ of the domain $\mathcal{K}_{\text{int}}$, namely

$\mathcal{K}_{\text{W}}:=\big{\\{}(t,x^{i})\,/\,(t/2)\leq r\leq
t-1\big{\\}}\subset\mathcal{K}.$ (2.5)

#### Vector fields.

We define the semi-hyperboloidal frame
$\underline{\partial}=(\underline{\partial}_{\alpha})=(\underline{\partial}_{0},\underline{\partial}_{1},\underline{\partial}_{2},\underline{\partial}_{3})$
to be

$\underline{\partial}_{0}:=\partial_{t},\qquad\underline{\partial}_{i}:=\partial_{i}+\frac{x^{i}}{t}\partial_{t},$
(2.6)

and we also set
$\underline{\partial}_{x}:=(\underline{\partial}_{1},\underline{\partial}_{2},\underline{\partial}_{3})$.
Observe that the vector fields $\underline{\partial}_{i}$ generate the tangent
space to the hyperboloids of constant $s$. We also further decompose
Span$\\{\underline{\partial}_{i}\\}_{i=1,2,3}$ into radial and angular
components $(\underline{\partial}_{r},{\,{/\\!\\!\\!\nabla}})$ with

$\underline{\partial}_{r}:=\partial_{r}+{r\over t}\partial_{t},$ (2.7)

while ${\,{/\\!\\!\\!\nabla}}$ denotes derivatives tangential to the surfaces
with constant $t-r$ and $r$.

###  2.2 Functional spaces of interest

#### Standard energy norms.

On any spacelike hypersurface $\mathcal{H}_{s}$ of constant hyperboloidal
radius $s=\sqrt{t^{2}-r^{2}}$, we use the $L^{2}$ norm

$\|\,\phi(s)\,\|_{L^{2}(\mathcal{H}_{s})}:=\Big{(}\int_{\mathcal{H}_{s}}\phi^{2}\,dx\Big{)}^{1/2},\qquad
s\geq s_{0},$ (2.8)

where $dx=dx^{1}dx^{2}dx^{3}$ is the standard Euclidean volume form. Observe
that $dx$ differs from the natural volume form on hyperboloids (which is
$dV=(s/t)dx$); the above notation will turn out to be convenient in our
analysis. On any given hypersurface $\mathcal{H}_{s}$, the standard
hyperboloidal energy of wave and Klein-Gordon fields, respectively,
read111after normalization of the Klein-Gordon mass.

$\displaystyle\|\,\phi\,\|_{E_{\text{W}}}:=\|\,\frac{s}{t}\partial_{t}\phi\,\|_{L^{2}(\mathcal{H}_{s})}+\|\,\underline{\partial}_{x}\phi\,\|_{L^{2}(\mathcal{H}_{s})},$
$\displaystyle\phi=\phi(s),$ (2.9)
$\displaystyle\|\,\psi\,\|_{E_{\text{KG}}}:=\|\,\frac{s}{t}\partial_{t}\psi\,\|_{L^{2}(\mathcal{H}_{s})}+\|\,\underline{\partial}_{x}\psi\,\|_{L^{2}(\mathcal{H}_{s})}+\|\,\psi\,\|_{L^{2}(\mathcal{H}_{s})},$
$\displaystyle\psi=\psi(s).$ (2.10)

#### Weighted energy norms.

Next, let us define the so-called optical functions222It will be clear from
the context whether $u$ denotes the Minkowski optical function $t-r$ or the
wave component of (1.2). $u:=t-r$ and $\underline{u}:=t+r,$ which is now used
in defining our weights as follows:

$\tau_{-}:={\langle
u\rangle},\qquad\tau_{+}:={\langle\underline{u}\rangle},\qquad\tau_{0}:=\tau_{-}/\tau_{+}.$
(2.11)

Observe that $\tau_{-}$ can be regarded as a distance to the light cone, while
$\tau_{+}$ provides a decay weight in, both, timelike and spacelike
directions. Here, for any function $f$ we use the notation ${\langle
f\rangle}=(1+f^{2})^{1/2}$ for the so-called Japanese bracket. The, given an
arbitrary exponent $a\in[0,1]$, we intrduce the weighted hyperboloidal energy
of the wave component $\phi=\phi(s)$ by

$\|\,\phi\,\|_{E_{\text{W}}^{a}}:=\|\,\big{(}1+a\tau_{+}^{a}\tau_{0}^{\max(a,1/2)}\big{)}\frac{s}{t}\partial_{t}\phi\,\|_{L^{2}(\mathcal{H}_{s})}+\|\,s^{a}\underline{\partial}_{x}\phi\,\|_{L^{2}(\mathcal{H}_{s})}+a\,\|\,\frac{s^{a}}{t}\phi\,\|_{L^{2}(\mathcal{H}_{s})},$
(2.12)

where, by definition, $E_{W}^{0}=E_{W}$. In addition, to deal with a source
term $F$ in the wave equation, we will use a spacetime weighted norm denoted
by the subscript $N_{W}^{a}$, given by

$\|\,F\,\|_{N_{W}^{a}[s_{0},s_{1}]}:=\int_{s_{0}}^{s_{1}}\|\,\tau_{+}^{a}\frac{s}{t}\,F(s)\,\|_{L^{2}(\mathcal{H}_{s})}\,ds.$
(2.13)

#### High-order norms.

We now define versions of the norms which incorporate additional decay based
on the translations and Lorentz boosts of Minkowski space. In the $(t,x^{i})$
coordinates, we define the Lie algebra of translations and boosts as follows
(with $\alpha=0,1,2,3$ and $i=1,2,3$):

$\mathbb{L}:=\big{\\{}\partial_{\alpha}\big{\\}}\cup\big{\\{}L_{i}=t\partial_{i}+x^{i}\partial_{t}\big{\\}}.$
(2.14)

Given any multi-indices $I=(I_{\alpha})$ and $J=(J_{i})$, we use the standard
notation
$\partial^{I}=\partial^{I_{0}}_{0}\partial^{I_{1}}_{1}\partial^{I_{2}}_{2}\partial^{I_{3}}_{3}$
and $L^{J}=L^{J_{1}}_{1}L^{J_{2}}_{2}L^{J_{3}}_{3}$, and we also set
$|I|:=I_{0}+I_{1}+I_{2}+I_{3}$ and $|J|:=J_{1}+J_{2}+J_{3}$. We define the
high-order energy norms of interest on any given hyperboloid $\mathcal{H}_{s}$
as

$\|\,\phi\,\|_{E_{\text{W}}^{a,k}}:=\sum_{|I|+|J|\leqslant
k}\|\,\partial^{I}L^{J}\phi\,\|_{E_{\text{W}}^{a}},\quad\qquad\phi=\phi(s)$
(2.15)

as well as, for a source term in the wave equation, the spacetime norm

$\|\,F\,\|_{N^{a,k}[s_{0},s_{1}]}:=\sum_{|I|+|J|\leqslant
k}\|\,\partial^{I}L^{J}F\,\|_{N_{W}^{a}[s_{0},s_{1}]}.$ (2.16)

Similar definitions are used the remaining norms.

#### Conformal norms.

Finally, let us introduce the Bondi-like derivative operators
$\partial^{B}_{u}:=\partial_{t}$ and
$\partial^{B}_{r}:=\partial_{t}+\partial_{r}$, which are well-defined except
at the center $r=0$. We will work with two choices of conformal weight denoted
by $\Omega$, namely

$\Omega_{r}:=r,\qquad\Omega_{s}:=s^{2}=u\underline{u}.$ (2.17)

For each parameter value $a\in[0,1]$, we associate the modified weighted
energy of a function $\phi=\phi(s)$ in Bondi coordinates defined by

$\|\,r\phi\,\|_{E_{r}^{a}}:=\|\,\big{(}1+a\tau_{+}^{a}\tau_{0}^{\max(a,1/2)}\big{)}\Big{(}\frac{s}{t}\frac{\partial^{B}_{u}(r\phi)}{r},{\,{/\\!\\!\\!\nabla}}\phi\Big{)}\,\|_{L^{2}(\mathcal{H}_{s})}+\|\,\tau_{+}^{a}\big{(}\frac{\partial^{B}_{r}(r\phi)}{r},\frac{s}{t}{\,{/\\!\\!\\!\nabla}}\phi\big{)}\,\|_{L^{2}(\mathcal{H}_{s})}$
(2.18)

for the choice of weight $\Omega_{s}$, while for the choice $\Omega_{r}$ we
set

$\|\,s^{2}\phi\,\|_{E_{s}^{a}}:=\|\,\big{(}1+s^{a-1}\tau_{-}\big{)}\Big{(}\frac{s}{t}\frac{\partial^{B}_{u}(s^{2}\phi)}{s^{2}},{\,{/\\!\\!\\!\nabla}}\phi\Big{)}\,\|_{L^{2}(\mathcal{H}_{s})}+\|\,s^{a-1}\tau_{+}\big{(}\frac{\partial^{B}_{r}(s^{2}\phi)}{s^{2}},\frac{s}{t}{\,{/\\!\\!\\!\nabla}}\phi\big{)}\,\|_{L^{2}(\mathcal{H}_{s})}.$
(2.19)

Observe that a computation using Hardy’s inequality shows the inequalities

$\displaystyle\|\,\phi(s)\,\|_{E_{\text{W}}}\lesssim\|\,r\phi(s)\,\|_{E_{r}^{a}},\qquad\|\,\phi(s)\,\|_{E_{\text{W}}}\lesssim\|\,s^{2}\phi(s)\,\|_{E_{s}^{a}},$
(2.20)
$\displaystyle\|\,\phi(s)\,\|_{E_{\text{W}}}\approx\|\,r\phi(s)\,\|_{E^{0}_{r}},\qquad\|\,\phi(s)\,\|_{E_{\text{W}}}\approx\|\,s^{2}\phi(s)\,\|_{E^{0}_{s}}.$

###  2.3 Existence theory for the wave-Klein-Gordon model

#### Main statement.

We are now in a position to state our main result.

######  Theorem 2.1 (Global energy hierarchy for the wave-Klein-Gordon model).

Consider the coupled nonlinear wave and Klein-Gordon system (1.2) for some
given constants $P^{\alpha\beta},R,H^{\alpha\beta}$, and $c>0$. Then for any
integer $N\geq 8$, there exist positive constants
$\epsilon=\epsilon(N)\in(0,1)$, $\delta=\delta(N,\epsilon)\in(0,1)$, and
$C_{1}>0$ such that the initial value problem (1.2)- (1.3) enjoys the
following property. For all initial data satisfying, for some $a\in[0,1]$, the
smallness and decay conditions

$\displaystyle\|u(s_{0})\|_{E_{W}^{N}}+{a\|\,ru(s_{0})\,\|_{E_{r}^{a,N}}+a\,\|\,s^{2}u(s_{0})\,\|_{E_{s}^{a,N}}}+\|v(s_{0})\|_{E_{KG}^{N}}\leq\epsilon,$
(2.21)

the Cauchy problem (1.2)- (1.3) admits a global-in-time solution $(u,v)$
defined in the future of $\mathcal{H}_{s_{0}}$ and satisfying the energy
bounds for all $s\geq s_{0}$ and any $k=|J|=0,\ldots,N$ and $|I|+|J|\leq N$

$\displaystyle\|\partial^{I}L^{J}u(s)\|_{E_{\text{W}}^{a}}\lesssim\epsilon
s^{\max(a-1/2,0)\delta{+{\max(k,1)\delta}}},$ (2.22)
$\displaystyle\|\partial^{I}L^{J}v(s)\|_{E_{\text{KG}}}\lesssim\epsilon
s^{\max(-a+1/2,0)+k\delta}.$

Furthermore, the following pointwise decay properties hold (for all $s\geq
s_{0}$ and some constants $C_{k}$)

$\displaystyle\sup_{|I|\leq
k}\|\tau^{1/2}_{+}\partial^{I}u(s)\|_{L^{\infty}(\mathcal{H}_{s})}\lesssim\epsilon
s^{k\delta-1/2},$ (2.23) $\displaystyle\sup_{|I|\leq
k}\|\tau^{3/2}_{+}\partial^{I}v(s)\|_{L^{\infty}(\mathcal{H}_{s})}\lesssim\epsilon
s^{\max(-a+1/2,0)\delta+k\delta}.$

The conclusion of main interest is obtained by choosing $a=1/2$ in the above
theorem, since in this case the exponent $-a+1/2$ cancels out and we arrive at
the almost sharp growth $s^{k\delta}$ for both the wave and the Klein-Gordon
components. Compare this (almost) energy boundedness statement with the
previous result in [23] which had a rate of growth of $s^{1/2+k\delta}$ for
the energy of the Klein-Gordon component. Hence, under stronger decay
conditions in comparison to [23] we arrive at a stronger conclusion as far as
the decay of solutions is concerned.

#### Main energy and pointwise estimates.

In the course of our analysis, relying on the weighted spaces defined earlier
in this section we will study solution to the wave equation

$\displaystyle\Box\phi=F,\qquad(\phi,\partial_{t}\phi)|_{\mathcal{H}_{s_{0}}}=(\phi_{0},\phi_{1}),$
(2.24)

where the data $(\phi_{0},\phi_{1})$ are prescribed on the initial hyperboloid
$\mathcal{H}_{s_{0}}$.

######  Theorem 2.2 (Fractional Morawetz estimate in Bondi coordinates).

Given any $a\in[0,1]$, any sufficiently regular and compactly supported
solution $\phi$ to the Cauchy problem (2.24) satisfies ($k\geq 0$ being an
arbitrary integer) satisfies

$\sup_{s_{0}\leq s\leq
s_{1}}\big{(}\|\,\phi(s)\,\|_{E_{r}^{a,k}}+\|\,\phi(s)\,\|_{E_{s}^{a,k}}\big{)}\lesssim\|\,\phi(s_{0})\,\|_{E_{r}^{a,k}}+\|\,\phi(s_{0})\,\|_{E_{s}^{a,k}}+\|\,F\,\|_{N_{\text{W}}^{a,k}[s_{0},s_{1}]}.$
(2.25)

As a consequence, we have the following energy bounds in terms of the semi-
hyperboloidal frame. These, in turn, yield a pointwise decay estimate.

######  Corollary 2.3.

Given any $a\in[0,1]$, any sufficiently regular and compactly supported
solution $\phi$ to the Cauchy problem (2.24) satisfies the following
properties ($k\geq 0$ being an arbitrary integer).

  * •

Weighted energy estimate:

$\sup_{s_{0}\leq s\leq
s_{1}}\|\,\phi(s)\,\|_{E_{\text{W}}^{a,k}}\lesssim\|\,\phi(s_{0})\,\|_{E_{r}^{a,k}}+\|\,\phi(s_{0})\,\|_{E_{s}^{a,k}}+\|\,\phi(s_{0})\,\|_{E_{\text{W}}^{k}}+\|\,F\,\|_{N_{\text{W}}^{a,k}[s_{0},s_{1}]}.$
(2.26)

  * •

Weighted pointwise decay:

$\displaystyle\sup_{s_{0}\leq s\leq s_{1}}\sum_{|I|+|J|\leq
k}\|\,s^{a}\tau_{+}^{1/2}\partial^{I}L^{J}\phi\,\|_{L^{\infty}(\mathcal{H}_{s})}\lesssim\|\,\phi(s_{0})\,\|_{E_{r}^{a,k+2}}+\|\,\phi(s_{0})\,\|_{E_{s}^{a,k+2}}+\|\,\phi(s_{0})\,\|_{E_{W}^{k+2}}+\|\,F\,\|_{N_{\text{W}}^{a,k+2}[s_{0},s_{1}]}.$
(2.27)

For $a=1/2$ we obtain the relevant estimate for our purpose in the present
paper and this is the key in proving the (essential) and the weighted energy
boundedness stated in Theorem 2.1 follows by following the same lines as in
[20]. Hence, our main task for the rest of this paper is giving a proof of
Theorem 2.2. Note in passing that plugging in the extremal value $a=1$ above
allows us to recover (based on the standard vector field arguments) the
$L^{\infty}$ decay for the linear wave equation in Minkowski space, namely
$t^{-3/2}$ in timelike regions and $t^{-1}$ in the vicinity of the light cone
$t=r$.

##  3 A conformal multiplier identity for the hyperboloidal foliation

###  3.1 Energy formalism for the wave equation

#### Divergence identity.

To any solution $\phi$ to the equation (2.24), we associate the energy-
momentum tensor associated with the Minkowski metric $\eta$ and an arbitrary
function $\phi$, namely

$Q_{\alpha\beta}[\phi]=\partial_{\alpha}\phi\partial_{\beta}\phi-\frac{1}{2}\eta_{\alpha\beta}\partial^{\gamma}\phi\partial_{\gamma}\phi.$
(3.1)

It is well-known that this symmetric two-tensor is related to the wave
operator by the identities

$\partial^{\gamma}\big{(}Q_{\gamma\beta}[\phi]\big{)}=F\,\partial_{\beta}\phi.$
(3.2)

To each vector field $X=X^{\gamma}\partial_{\gamma}$, we can then associate
the one-form field

${}^{(X)}\\!P_{\alpha}[\phi]:=Q_{\alpha\gamma}[\phi]\,X^{\gamma}.$ (3.3)

which, sometimes, is simply denoted by $P_{\alpha}$ when the choice made for
$X$ is obvious.

We also define the symmetric two-tensor ${}^{(X)}\pi=\mathcal{L}_{X}\eta$ as
being the Lie derivative of the metric $\eta$ with respect to $X$. It is
called the deformation tensor of $X$ and, in local coordinates, one has

${}^{(X)}\pi_{\alpha\beta}:=(\mathcal{L}_{X}\eta)_{\alpha\beta}=\partial_{\alpha}X_{\beta}+\partial_{\beta}X_{\alpha}.$
(3.4)

When the choice of the vector field is obvious from context we simply write
$\pi$ instead of ${}^{(X)}\pi$. Taking the divergence in (3.3) and using the
identities (3.2), we find

$\partial^{\gamma}\big{(}P_{\gamma}[\phi]\big{)}=FX\phi+\frac{1}{2}{}^{(X)}\pi^{\alpha\beta}Q_{\alpha\beta}[\phi].$
(3.5)

#### Integrating between two hyperboloids.

For any $s_{1}\geq s_{0}>1$, integrating (3.5) over the spacetime domain
$\mathcal{K}_{[s_{0},s_{1}]}$ and using Stokes’ theorem we get the multiplier
identity associated with the vector field $X$

$\int_{s=s_{0}}{}^{(X)}P_{\alpha}[\phi]\,N^{\alpha}dV-\int_{s=s_{1}}\\!{}^{(X)}\\!P_{\alpha}[\phi]\,N^{\alpha}dV=\int_{s_{0}}^{s_{1}}\\!\\!\\!\int_{s=s^{\prime}}\Big{(}FX\phi+\frac{1}{2}{}^{(X)}\pi^{\alpha\beta}Q_{\alpha\beta}[\phi]\big{)}dVds^{\prime},$
(3.6)

where $dV$ denotes the volume element on hyperboloids: $dV=\frac{s}{t}dx$.
Here, $N$ denotes the future-directed unit normal vector to the level sets
$s=const$

$N^{\alpha}=\frac{-m^{\alpha\beta}\partial_{\beta}s}{(-m^{\alpha\beta}\partial_{\alpha}s\partial_{\beta}s)^{1/2}}=-m^{\alpha
0}\frac{t}{s}+m^{\alpha r}\frac{r}{s}.$ (3.7)

We also define the rescaled normal vector $N^{\prime}$ via

$N{{}^{\prime}}^{\alpha}=\frac{s}{t}N^{\alpha}=-m^{\alpha 0}+m^{\alpha
r}{r\over t}.$ (3.8)

In terms of $N{{}^{\prime}}$ we have the identity

${}^{(X)}P_{\alpha}N^{\alpha}\,dV={}^{(X)}P_{\alpha}N{{}^{\prime}}^{\alpha}\,dx.$
(3.9)

Last, we refer to the integrand on the left hand side of the equation (3.6) as
the energy density associated with $X$ with respect to the foliation by
spacelike hypersurfaces $s=const$.

###  3.2 Multipliers and commutators

The following setup will be useful for constructing weighted energies. Given a
vector field $X$, we define the traceless deformation tensor of $X$ to be
${}^{(X)}\widehat{\pi}:={}^{(X)}\pi-\frac{1}{2}\eta(\text{Tr }{}^{(X)}\pi).$

######  Lemma 3.1 (Basic formulas involving ${}^{(X)}\widehat{\pi}$).

Let $\phi$ be a solution to the equation (2.24) and $X$ be a smooth vector
field. The following identities hold:

${}^{(X)}\widehat{\pi}^{\alpha\beta}=-X(\eta^{\alpha\beta})-\eta^{\alpha\beta}\partial_{\gamma}X^{\gamma}+\eta^{\alpha\gamma}\partial_{\gamma}X^{\beta}+\eta^{\beta\gamma}\partial_{\gamma}X^{\alpha},$
(3.10)
$\partial^{\gamma}P_{\gamma}=FX\phi+\frac{1}{2}\widehat{\pi}^{\alpha\beta}\partial_{\alpha}\phi\partial_{\beta}\phi=FX\phi+\frac{1}{2}\pi^{\alpha\beta}Q_{\alpha\beta},$
(3.11)

where (3.10) is expressed in local coordinates.

###### Proof.

To derive the identity (3.10), we write
${}^{(X)}\pi^{\alpha\beta}=-X(\eta^{\alpha\beta})+\eta^{\alpha\gamma}\partial_{\gamma}X^{\beta}+\eta^{\beta\gamma}\partial_{\gamma}X^{\alpha}$
in local coordinates. Subtracting the expression
$\frac{1}{2}\eta^{\alpha\beta}(\text{Tr
}\pi)=\eta^{\alpha\beta}\partial_{\gamma}X^{\gamma}$ from both sides completes
the proof. The identity (3.11) follows directly from the definition of
$Q_{\alpha\beta}[\phi]$. ∎

###  3.3 Conformal transformations for the wave equation

Let $g=(g_{\alpha\beta})$ be a Lorentzian metric on an $(3+1)$-dimensional
spacetime. Given any positive weight
$\Omega:\mathbb{R}^{3+1}\to\mathbb{R}_{+}$ (assumed to be smooth throughout
this paper), we consider the conformal metric $\widetilde{g}_{\alpha\beta}$
defined by $\widetilde{g}:=\Omega^{-2}g,$ and we denote by
$\widetilde{\nabla}$ and
$\Box_{\widetilde{g}}=\widetilde{\nabla}^{\alpha}\widetilde{\nabla}_{\alpha}$
the corresponding covariant derivative and wave operator, respectively. A
standard calculation (see [33]) shows that the function
$\widetilde{\phi}:=\Omega\phi$ satisfies the Klein-Gordon equation

$\Box_{\widetilde{g}}\widetilde{\phi}+V\widetilde{\phi}=\Omega^{3}F,\qquad
V:=\Omega^{3}\Box_{g}\Omega^{-1},\qquad F:=\Box_{g}\phi.$ (3.12)

Furthermore, given any smooth and positive weight
$\Lambda:\mathbb{R}^{3+1}\to\mathbb{R}_{+}$ and in view of the equation
(3.12), it is natural to introduce the conformal energy-momentum tensor
associated with the function $\widetilde{\phi}$ defined as

$\widetilde{Q}_{\alpha\beta}[\widetilde{\phi},\Lambda]=\Big{(}\partial_{\alpha}\widetilde{\phi}\partial_{\beta}\widetilde{\phi}-\frac{1}{2}\widetilde{g}_{\alpha\beta}(\widetilde{g}^{\mu\nu}\partial_{\mu}\widetilde{\phi}\partial_{\nu}\widetilde{\phi}-V\widetilde{\phi}^{2})\Big{)}\Lambda.$
(3.13)

This tensor satisfies the divergence identity

$\widetilde{\nabla}^{\alpha}\widetilde{Q}_{\alpha\beta}[\widetilde{\phi},\Lambda]=\Lambda\Omega^{3}F\partial_{\beta}\widetilde{\phi}+\frac{\Lambda}{2}(\partial_{\beta}V)\widetilde{\phi}^{2}+\frac{\Omega^{2}}{\Lambda}(\nabla^{\alpha}\Lambda)\widetilde{Q}_{\alpha\beta}.$
(3.14)

In fact, our weights may vanish at the center $r=0$, and in such a case the
above identities are only valid for $r>0$. From the identity (3.3) we have

$\displaystyle\widetilde{\nabla}^{\alpha}X^{\beta}(\widetilde{Q}_{\alpha\beta}[\widetilde{\phi},\Lambda])-(\widetilde{\nabla}^{\alpha}X^{\beta})\widetilde{Q}_{\alpha\beta}[\widetilde{\phi},\Lambda]$
$\displaystyle=\Lambda\Omega^{3}FX^{\beta}\partial_{\beta}\widetilde{\phi}+\frac{\Lambda}{2}(X^{\beta}\partial_{\beta}V)\widetilde{\phi}^{2}+\frac{\Omega^{2}}{\Lambda}(\nabla^{\alpha}\Lambda)X^{\beta}\widetilde{Q}_{\alpha\beta},$

and we have arrived at the following result.

######  Proposition 3.2.

Let $X$ be a vector field associated with the energy flux
${}^{(X)}\widetilde{P}_{\alpha}=\widetilde{Q}_{\alpha\beta}X^{\beta}$. Then
the one-form field ${}^{(X)}\widetilde{P}_{\alpha}$ satisfies the following
conformal multiplier identity associated with the vector field $X$ and the
weight $\Omega$

$\int_{s=s_{0}}\\!\\!{}^{(X)}\widetilde{P}_{\alpha}N^{\alpha}\Omega^{-2}\
dV-\int_{s=s_{1}}\\!\\!{}^{(X)}\widetilde{P}_{\alpha}N^{\alpha}\Omega^{-2}\,dV=\int_{s_{0}}^{s_{1}}\\!\\!\\!\int_{s=s^{\prime}}\big{(}\widetilde{\nabla}^{\alpha}{}^{(X)}\widetilde{P}_{\alpha}\big{)}\Omega^{-4}\,dVds^{\prime},$
(3.15)

in which the normal vector field $N$ is given by (3.7) and the divergence term
on the right-hand side reads

$\big{(}\widetilde{\nabla}^{\alpha}{}^{(X)}\\!\widetilde{P}_{\alpha}\big{)}\Omega^{-4}=\Lambda
F\Omega^{-1}X\widetilde{\phi}+\Lambda\Omega^{-2}\mathcal{A}^{\alpha\beta}\partial_{\alpha}\widetilde{\phi}\partial_{\beta}\widetilde{\phi}+\Lambda\mathcal{B}\widetilde{\phi}^{2}+\Omega^{-2}\Lambda^{-1}\mathcal{C}^{\alpha}{}^{(X)}\widetilde{P}_{\alpha}\
,$ (3.16)

with $\mathcal{A},\mathcal{B},\mathcal{C}$ given by

$\mathcal{A}^{\alpha\beta}:=\frac{1}{2}\big{(}{}^{(X)}\widehat{\pi}+2X\ln(\Omega)g^{-1}\big{)},\qquad\mathcal{B}:=\frac{1}{2\Omega^{4}}\big{(}X(V)-(\text{Tr
}\mathcal{A})V\big{)},\qquad\mathcal{C}^{\alpha}:=\nabla^{\alpha}\Lambda.$
(3.17)

##  4 Tensor identities in Bondi coordinates

###  4.1 Basic definitions

As shown in Oliver [33] and Oliver and Sterbenz [34], the Bondi coordinate
system provides one with a very convenient way to establish energy estimates
for the wave equation. In establishing our multiplier estimates, while we work
here with a different, hyperboloidal, foliation, we can still follow the
outline of the arguments in [33, 34]. Recall that $u=t-r$ is the Minkowski
optical function whose level sets are outgoing null hypersurfaces and that
$x^{i}$, $1\leq i\leq 3$ are the standard Cartesian coordinates. The set of
coordinates $(u,x^{i})$ is referred to as the Bondi coordinate system for
Minkowski space. We also define the polar Bondi coordinates by $(u,r,x^{A})$,
where locally we choose $x^{A}$ to be two members of
$\widehat{x}^{i}=r^{-1}x^{i}$. We also let
$|{\,{/\\!\\!\\!\nabla}}^{B}\phi|^{2}$ denote the angular gradient of $\phi$
with respect to the spheres $u=const$ and $r=const$.

We denote by $\eta$ the Minkowski metric in Bondi coordinates. In (polar)
Bondi coordinates, the components of $\eta$ have the form

$\eta_{uu}=-1,\qquad\eta_{ur}=-1,\qquad\eta_{rr}=0,\qquad\eta_{bc}=r^{2}\delta_{bc},$
(4.1)

with $\delta_{bc}$ denoting the standard metric on 2-spheres. We denote by
$\eta^{-1}$ the inverse Minkowski metric in polar Bondi coordinates, with
$\eta^{uu}=0$, $\eta^{ui}=-\omega^{i},$ and $\eta^{ij}=\delta^{ij}.$ In
particular $\det(\eta^{-1})=-1$. Thus, the wave operator $\Box_{\eta}$ in
Bondi coordinates takes the form

$\Box_{\eta}=-2\partial^{B}_{u}\partial^{B}_{r}+(\partial^{B}_{r})^{2}-2r^{-1}\partial^{B}_{u}+2r^{-1}\partial^{B}_{r}+r^{-2}\sum_{i<j}(\Omega_{ij})^{2}.$
(4.2)

Henceforth we let $\partial=(\partial_{t},\partial_{x})$ to denote the $(t,x)$
coordinate derivatives, and likewise
$\partial^{B}=({\partial_{u}^{B}},\partial_{x}^{B})$ denote the
$(u,x)$-coordinate derivatives. By the chain rule, we have immediately

$\displaystyle\partial^{B}_{u}=\partial_{t},\qquad\partial^{B}_{i}=\partial_{i}+\omega^{i}\partial_{t},\qquad\partial^{B}_{r}=\partial_{r}+\partial_{t}.$
(4.3)

###  4.2 Various identities and estimates in Bondi coordinates

Next we compute the key quantities from the equations (3.16), (3.17). The
proof of the following two lemmas is postponed to Appendix A.

######  Lemma 4.1 (Identity for deformation tensor).

Let $\Omega\geq 0$ be a weight and $X=X^{\alpha}\partial_{\alpha}$ be a vector
field in Bondi coordinates. The contravariant tensor $\mathcal{A}$ in (3.17)
satisfies (at points where $\Omega>0$)

$\displaystyle\mathcal{A}(X,\Omega,\eta)$
$\displaystyle=\frac{1}{2}\big{(}{}^{(X)}\widehat{\pi}+2X\ln(\Omega)\eta^{-1}\big{)}$
(4.4)
$\displaystyle=-\frac{1}{2}\big{(}\mathcal{L}_{X}\eta^{-1}+\Big{(}\partial^{B}_{u}X^{u}+\partial^{B}_{r}X^{r}+\partial^{B}_{i}\overline{X}^{i}+2(\frac{X^{r}}{r}-X\ln\Omega)\Big{)}\eta^{-1}\big{)},$

where $\overline{X}^{i}=X^{i}-\omega^{i}\omega_{j}X^{j}$ denotes the angular
part and $X^{r}=\omega_{i}X^{i}$ the radial part of $X$. Furthermore, if $X$
has no angular part, that is
$X=X^{u}(u,r)\partial^{B}_{u}+X^{r}(u,r)\partial^{B}_{r}$, then the components
of the tensor $\mathcal{A}$ in polar Bondi coordinates are

$\displaystyle\mathcal{A}^{uu}$ $\displaystyle=-\partial_{r}^{B}X^{u},\,$
(4.5a) $\displaystyle\mathcal{A}^{rr}$
$\displaystyle=\frac{1}{2}\partial_{r}^{B}X^{r}-\frac{1}{2}{\partial_{u}^{B}}X^{u}-{\partial_{u}^{B}}X^{r}-\Big{(}\frac{X^{r}}{r}-X\ln\Omega\Big{)},$
(4.5b) $\displaystyle\mathcal{A}^{ur}$
$\displaystyle=\frac{1}{2}\partial_{r}^{B}X^{u}+\Big{(}\frac{X^{r}}{r}-X\ln\Omega\Big{)},$
(4.5c) $\displaystyle\mathcal{A}^{bc}$
$\displaystyle=\Big{(}\frac{X^{r}}{r}-\frac{1}{2}{\partial_{u}^{B}}X^{u}-\frac{1}{2}\partial_{r}^{B}X^{r}-\Big{(}\frac{X^{r}}{r}-X\ln\Omega\Big{)}\Big{)}\frac{\delta^{bc}}{r^{2}}.$
(4.5d)

######  Lemma 4.2 (Identities for the flux terms).

Let $\Omega\geq 0$ be a weight and $X$ be a smooth vector field $X$ in Bondi
coordinates. Then one has the following identity for boundary terms on the
equation (3.15) (where $\Omega>0$)

$\displaystyle{}^{(X)}\\!\widetilde{P}_{\alpha}N{{}^{\prime}}^{\alpha}[\widetilde{\phi}]$
$\displaystyle=X^{u}(1-{r\over
t})(\partial^{B}_{u}\widetilde{\phi})^{2}+\frac{1}{2}(X^{u}+X^{r}(1+{r\over
t}))(\partial^{B}_{r}\widetilde{\phi})^{2}-X^{u}(1-{r\over
t})\partial^{B}_{u}\widetilde{\phi}\partial_{r}^{B}\widetilde{\phi}$ (4.6)
$\displaystyle\quad+\frac{1}{2}(X^{u}+X^{r}(1-{r\over
t}))|{\,{/\\!\\!\\!\nabla}}\widetilde{\phi}|^{2}+\frac{\Lambda}{\Omega^{2}}\mathcal{B}\widetilde{\phi}^{2}.$

Assume that $\mathcal{B}\geq 0$ and $X$ is of the form

$X=X^{u}\partial^{B}_{u}+X^{r}\partial^{B}_{r}\quad{\text{with
}}X^{u}(u,r)\geq 0,\ X^{r}(u,r)\geq 0.$ (4.7)

Then, within the domain $\mathcal{K}$, the flux term
${}^{(X)}\\!\widetilde{P}_{\alpha}N{{}^{\prime}}^{\alpha}$ is equivalent to a
weighted sum of squares, namely

${}^{(X)}\\!\widetilde{P}_{\alpha}N{{}^{\prime}}^{\alpha}\approx
X^{u}(1-{r\over
t})(\partial^{B}_{u}\widetilde{\phi})^{2}+\big{(}X^{u}+X^{r}(1+{r\over
t})\big{)}(\partial^{B}_{r}\widetilde{\phi})^{2}+\big{(}X^{u}+X^{r}(1-{r\over
t})\big{)}|{\,{/\\!\\!\\!\nabla}}\widetilde{\phi}|^{2}+\frac{\Lambda}{\Omega^{2}}\mathcal{B}\widetilde{\phi}^{2}.$
(4.8)

We now specialize our identities to the weights (2.17).

######  Proposition 4.3 (Vanishing of the conformal potentials).

Recall that $V=\Omega^{3}\Box(\Omega^{-1})$. For the conformal weights
$\Omega_{r}=r$ and $\Omega_{s}=s^{2}$ and for all $r>0$ one has
$\Box(\Omega_{r}^{-1})=0$ and $\Box(\Omega_{s}^{-1})=0$. In particular, the
corresponding potentials, say $V_{r}$ and $V_{s}$, vanish identically in
(3.12)– (3.17), and on each hypersurface of constant $s$ the flux term in the
conformal multiplier identity (3.15) is positive definite.

As we will see later, for appropriate choices of vector field $X$, the
divergence term in the identity (3.15) also has a good sign.

###### Proof.

For the weight of $\Omega_{r}=r$ it suffices to check that $V_{r}$ vanishes
away from the singular set $r=0$. A simple calculation using Bondi coordinates
gives:
$\Box(\Omega_{r}^{-1})=((\partial^{B}_{r})^{2}+2r^{-1}\partial^{B}_{r})(r^{-1})=0$
valid on the set $\mathcal{K}\cap\\{r>0\\}$. For the second weight, since
$\mathcal{K}\cap\\{s^{2}=0\\}=\emptyset$, it suffices to check that $V_{s}=0$
throughout $\mathcal{K}$. A calculation gives, for all $(t,x)\in\mathcal{K}$:

$\displaystyle\Box(\Omega_{s}^{-1})$
$\displaystyle=(-2\partial^{B}_{u}\partial^{B}_{r}+(\partial^{B}_{r})^{2}-2r^{-1}\partial^{B}_{u}+2r^{-1}\partial^{B}_{r})((u^{2}+2ru)^{-1})$
$\displaystyle=(u^{2}+2ur)^{-3}(-16(u^{2}+ur)+4(u^{2}+2ur)+8u^{2})+4(u+2ur)^{-2}=0.\qed$

##  5 Conformal hyperboloidal energy estimates

###  5.1 Organization of this section

Our main estimates for the wave equation are established in the present
section. After preliminary material on functional or algebraic inequalities,
we consider our three choices of conformal weight in (5.6), (5.7), and (5.8),
below, and we establish multiplier estimates in Proposition 5.5. In turn, we
are in a position to give a proof of Theorem 2.2 about the fractional Morawetz
energy estimate for the wave equation, as stated in the introduction.

###  5.2 Preliminary estimates

We begin by restating a Hardy inequality established in [20, Lemma 2.4].

######  Lemma 5.1 (Hardy inequality).

For all sufficiently regular and compactly supported functions $\phi$ defined
in $\mathcal{K}_{[s_{0},s_{1}]}$ and for all $0<s_{0}\leq s$ one has

$\|\,r^{-1}\phi\,\|_{L^{2}(\mathcal{H}_{s})}\lesssim\sum_{i}\|\,\underline{\partial}_{i}\phi\,\|_{L^{2}(\mathcal{H}_{s})}.$
(5.1)

######  Lemma 5.2 (Estimates for the weights).

The following positivity condition holds in the domain $\mathcal{K}$

$q(\underline{u},u)_{a}:=\Big{(}\frac{\underline{u}^{2a}-u^{2a}}{r}-2a(\underline{u}^{2a-1}+u^{2a-1})\Big{)}\geq
0,\qquad a\in[1/2,1],$ (5.2)

in which the expression vanishes identically at the end points $a=1$ and
$a=1/2$. Additionally, one has

$(\underline{u}^{2a}-u^{2a})\approx a(rt^{2a-1}),\qquad a\in[1/2,1].$ (5.3)

###### Proof.

As in [30], we introduce $\underline{v}(\underline{u})=\underline{u}^{2a}$,
$v(u)=u^{2a}$. For each fixed $t$, we introduce the function
$f(r)=\underline{v}(t+r)-v(t-r)$. This function satisfies $f(0)=0$ and
$f^{\prime}(r)=\dot{\underline{v}}+\dot{v}$. The inequality (5.2) is then
equivalent to $f(0)-f(r)-f^{\prime}(r)(0-r)\leq 0$. An immediate computation
gives us

$\displaystyle f^{\prime}(r)\ $
$\displaystyle=2a(t+r)^{2a-1}+2a(t-r)^{2a-1},\qquad
f^{\prime\prime}(r)=(2a-1)2a(\underline{u}^{2a-2}-|u|^{2a-2}).$ (5.4)

By our conditions on $a$, we have $0\leq(2a-1)2a$ and $(2a-2)\leq 0$.
Therefore, $f^{\prime\prime}(r)\leq 0$ and the desired inequality is a
convexity inequality. Next, to derive the equation (5.3) we rewrite the
expression:
$\underline{u}^{2a}-u^{2a}=t^{2a}\big{(}(1+t^{-1}r)^{2a}-(1-t^{-1}r)^{2a}\big{)}$
and introduce the function $g(\omega):=(1+\omega)^{2a}-(1-\omega)^{2a}$ with
$\omega=t^{-1}r$. It suffices now to show that there exist constants
$C_{0},C_{1}>0$ such that $C_{0}a\omega\leq g(\omega)\leq C_{1}a\omega,$ with
$0\leq\omega<1$ inside the domain $\mathcal{K}$. Taking the derivative of this
function gives us

$g^{\prime}(\omega)=2a\big{(}(1+\omega)^{2a-1}+(1-\omega)^{2a-1}\big{)}.$

For all $a\in[1/2,1]$, $g^{\prime}(\omega)$ clearly satisfies the explicit
bound $2a\leq g^{\prime}(\omega)\leq 2a(2^{2a-1}+1)$. Integrating these
inequalities yields (5.2) with $C_{0}=2$ and $C_{1}=2(2^{2a-1}+1)$, which
completes the proof of the lemma. ∎

######  Remark 5.3.

In the range $a\in[0,1/2)$, we have $q(\underline{u},u)_{a}\leq 0$, and the
expression vanishes identically for $a=0$. Comparing this result with the
inequality (5.2) we see why we need distinct vector field multipliers in order
to cover both of the ranges $a\in[0,1/2)$ and $a\in[1/2,1]$.

The following Sobolev estimate is a consequence of the standard Sobolev
estimate on hyperboloids. Observe that it allows any non-negative value of the
exponent $a$. It is derived from the standard Sobolev estimate on hyperboloids
applied to $(s^{a}/t)\phi$ and we omit the details.

######  Lemma 5.4 (Weighted Sobolev estimate on hyperboloids).

Let $s_{1}\geq s\geq s_{0}$ and $a\geq 0$. For any sufficiently regular
function $\phi$ supported in the interior of the future light cone $u=0$, one
has

$\|\,s^{a}\tau_{+}^{1/2}\phi(s)\,\|_{L^{\infty}(\mathcal{H}_{s})}\lesssim\sum_{|J|\leq
2}\|\,\frac{s^{a}}{t}L^{J}\phi(s)\,\|_{L^{2}(\mathcal{H}_{s})}.$ (5.5)

###  5.3 Conformal hyperboloidal energy multipliers

We make now three choices of weights and vector fields and apply the conformal
multiplier identity (3.15):

$\displaystyle K^{a}$
$\displaystyle=(1+u^{2a})\partial^{B}_{u}+\frac{1}{2}(\underline{u}^{2a}-u^{2a})\partial^{B}_{r}\,\qquad$
$\displaystyle\Omega=r,\qquad\Lambda=1,\qquad$ $\displaystyle 1/2\leq a\leq
1,$ (5.6) $\displaystyle Y^{a}$
$\displaystyle=(1+2a\tau_{+}^{2a}\tau_{0})\partial^{B}_{u}+r^{2a}\partial^{B}_{r},\qquad$
$\displaystyle\Omega=r,\qquad\Lambda=1,\qquad$ $\displaystyle 0\leq a\leq
1/2,$ (5.7) $\displaystyle K$
$\displaystyle=(1+u^{2})\partial^{B}_{u}+2(u+r)r\partial^{B}_{r},\qquad$
$\displaystyle\Omega=s^{2},\qquad\Lambda=s^{2a-2},\qquad$ $\displaystyle 0\leq
a\leq 1.$ (5.8)

This choice of vector fields can be motivated as follows: (5.6) smoothly
interpolates between the scaling vector field ($a=1/2$) and the classical
Morawetz vector field $(a=1)$. The second choice (5.7) is analogous to the
scaling vector field with a scaling that is chosen to produce a good sign in
the divergence terms (cf. Remark 5.3). Furthermore, this vector field smoothly
interpolates between the (Minkowski) Killling field $\partial_{t}$ and the
scaling vector field. Last, for the third choice (5.8) there is some
flexibility: as long the vector field $\nabla^{\alpha}\Lambda$ is timelike and
future directed we would have a good sign for the flux terms. However, the
decay statement needed in the present paper requires us to chose a weight
$\Lambda$ of the form above.

######  Proposition 5.5 (Main multiplier estimates).

Let $\phi$ be a sufficiently regular and compactly supported solution to the
equation (2.24). For any $s_{0}\leq s\leq s_{1}$, $0\leq a\leq 1$ the
following estimates hold:

$\displaystyle\sup_{s_{0}\leq s\leq s_{1}}\|\,\phi(s)\,\|_{E_{\text{W}}}$
$\displaystyle\lesssim\|\,\phi(s_{0})\,\|_{E_{\text{W}}}+\|\,F\,\|_{N_{W}^{0}[s_{0},s_{1}]},$
(5.9) $\displaystyle\sup_{s_{0}\leq s\leq s_{1}}\|\,r\phi(s)\,\|_{E_{r}^{a}}$
$\displaystyle\lesssim\|\,r\phi(s_{0})\,\|_{E_{r}^{a}}+\|\,F\,\|_{N_{W}^{a}[s_{0},s_{1}]},$
(5.10) $\displaystyle\sup_{s_{0}\leq s\leq
s_{1}}\|\,s^{2}\phi(s)\,\|_{E_{s}^{a}}$
$\displaystyle\lesssim\|\,s_{0}^{2}\phi(s_{0})\,\|_{E_{s}^{a}}+\|\,F\,\|_{N_{W}^{a}[s_{0},s_{1}]}.$
(5.11)

###### Proof of the estimate (5.9).

Substituting the vector field $X=\partial^{B}_{u}$ as well as $\Omega=1$ into
the equation (3.6) and using the fact that
$\mathcal{L}_{\partial^{B}_{u}}\eta=0$ give us

$\int_{s=s_{1}}\\!{}^{(X)}\\!P_{\alpha}N^{\alpha}\,dV=\int_{s=s_{0}}\\!{}^{(X)}\\!P_{\alpha}N^{\alpha}\,dV-\int_{s_{0}}^{s_{1}}\\!\\!\\!\int_{s=s^{\prime}}F\partial^{B}_{u}\phi\
dVds^{\prime}.$ (5.12)

1\. Boundary terms. In view of the equation (3.9) it suffices to compute the
integral of ${}^{(X)}\\!P_{\alpha}N{{}^{\prime}}^{\alpha}dx$. Using the
equation (4.6) gives us

$\displaystyle{}^{(X)}\\!P_{\alpha}N{{}^{\prime}}^{\alpha}$
$\displaystyle=(1-{r\over
t})(\partial^{B}_{u}\phi)^{2}+\frac{1}{2}((\partial^{B}_{r}\phi)^{2}+|{\,{/\\!\\!\\!\nabla}}\phi|^{2})-(1-{r\over
t})\partial^{B}_{u}\phi\partial_{r}^{B}\phi.$

Applying the equation (4.8) and integrating then yields (with
$\widetilde{\phi}=\phi$ in this case)

$\displaystyle\int_{s=s^{\prime}}{}^{(X)}\\!P_{\alpha}N^{\alpha}dV=\int_{s=s^{\prime}}{}^{(X)}\\!P_{\alpha}N{{}^{\prime}}^{\alpha}dx\approx\int_{s=s^{\prime}}(1-{r\over
t})(\partial^{B}_{u}\phi)^{2}+(\partial^{B}_{r}\phi)^{2}+|{\,{/\\!\\!\\!\nabla}}\phi|^{2}dx\approx\|\,\phi(s)\,\|_{E_{\text{W}}}^{2},$
(5.13)

where we used

$1-{r\over
t}=\Big{(}\frac{s}{t}\big{)}^{2}\frac{t}{t+r}\approx\Big{(}\frac{s}{t}\big{)}^{2}\quad\text{
in the domain $\mathcal{K}_{[s_{0},+\infty)}$.}$ (5.14)

2\. Source term. The term containing $F\partial^{B}_{u}\phi$ has no sign
therefore we take absolute value and apply Cauchy-Schwarz to get

$\displaystyle\big{|}\int_{s_{0}}^{s_{1}}\\!\\!\\!\int_{s=s^{\prime}}F\partial^{B}_{u}\phi\
dVds^{\prime}\big{|}$
$\displaystyle\leq\int_{s_{0}}^{s_{1}}\\!\\!\\!\int_{s=s^{\prime}}\big{|}F\frac{s}{t}\partial^{B}_{u}\phi\big{|}\
dxds^{\prime}\leq\|\,F\,\|_{N_{W}[s_{0},s_{1}]}\sup_{s_{0}\leq s\leq
s_{1}}\|\,\phi(s)\,\|_{E_{\text{W}}}.$ (5.15)

3\. Putting it all together. Substituting (5.13) and (5.15) into the equation
(5.12) yields us

$\|\,\phi(s_{1})\,\|_{E_{\text{W}}}^{2}\lesssim\|\,\phi(s_{0})\,\|_{E_{\text{W}}}^{2}+\|\,F\,\|_{N_{W}[s_{0},s_{1}]}\sup_{s_{0}\leq
s\leq s_{1}}\|\,\phi(s)\,\|_{E_{\text{W}}}.$

Taking the supremum and dividing by $\sup\limits_{s_{0}\leq s\leq
s_{1}}\|\,\phi(s)\,\|_{E_{\text{W}}}$ complete the proof of the estimate
(5.9). ∎

###### Proof of the estimate (5.10) in the range $\frac{1}{2}\leq a\leq 1$.

We use the multiplier identity (3.15) with the choice (5.6), that is,

$X=K^{a}=(1+u^{2a})\partial^{B}_{u}+\frac{1}{2}(\underline{u}^{2a}-u^{2a})\partial^{B}_{r}\,\qquad\Omega=r,\qquad\Lambda=1.$

Setting $\widetilde{\phi}=r\phi$ and using the formulas in line (3.17) we find

$\displaystyle\int_{s=s_{1}}\\!\\!{}^{(X)}\widetilde{P}_{\alpha}[\widetilde{\phi}]N^{\alpha}r^{-2}\
dV+\int_{s_{0}}^{s_{1}}\\!\\!\\!\int_{s=s^{\prime}}r^{-2}\mathcal{A}^{\alpha\beta}\partial_{\alpha}\widetilde{\phi}\partial_{\beta}\widetilde{\phi}\
dVds^{\prime}$ (5.16)
$\displaystyle=\int_{s=s_{0}}\\!\\!{}^{(X)}\widetilde{P}_{\alpha}[\widetilde{\phi}]N^{\alpha}r^{-2}\
dV-\int_{s_{0}}^{s_{1}}\\!\\!\\!\int_{s=s^{\prime}}Fr^{-1}X\widetilde{\phi}\
dVds^{\prime},$

where we used Lemma 4.3 to see that $\mathcal{B}(X,r)\equiv 0$.

1\. Terms $\mathcal{A}^{\alpha\beta}$. A simple computation gives us

$\frac{(K^{a})^{r}}{r}-K^{a}(\ln
r)=\frac{(K^{a})^{r}}{r}(1-\partial^{B}_{r}r)=0,$ (5.17)

and substituting this (4.5) yields us

$\displaystyle\mathcal{A}^{uu}$
$\displaystyle=-\partial_{r}^{B}(X^{u})=-\partial_{r}^{B}(u^{2a})=0,\qquad\qquad\qquad\mathcal{A}^{ur}=\frac{1}{2}\partial_{r}^{B}X^{u}=\frac{1}{2}\partial_{r}^{B}(u^{2a})=0,\,$
$\displaystyle\mathcal{A}^{rr}$
$\displaystyle=\frac{1}{2}\partial_{r}^{B}X^{r}-\frac{1}{2}{\partial_{u}^{B}}X^{u}-{\partial_{u}^{B}}X^{r}=\frac{1}{4}\partial^{B}_{r}(\underline{u}^{2a}-u^{2a})-\frac{1}{2}\partial^{a}_{u}(u^{2a})-\frac{1}{2}\partial^{B}_{u}(\underline{u}^{2a}-u^{2a})$
$\displaystyle=\frac{1}{4}(4a\underline{u}^{2a-1})-\frac{1}{2}(2au^{2a-1})-\frac{1}{2}(2a\underline{u}^{2a-1}-2au^{2a-1})=0,$
$\displaystyle\mathcal{A}^{bc}$
$\displaystyle=\big{(}\frac{X^{r}}{r}-\frac{1}{2}{\partial_{u}^{B}}X^{u}-\frac{1}{2}\partial_{r}^{B}X^{r}\big{)}\frac{\delta^{bc}}{r^{2}}=\big{(}\frac{1}{2r}(\underline{u}^{2a}-u^{2a})-\frac{1}{2}(2a\underline{u}^{2a-1})-\frac{1}{4}(4a\underline{u}^{2a-1})\big{)}\frac{\delta^{bc}}{r^{2}}$
$\displaystyle=\big{(}\frac{1}{2r}(\underline{u}^{2a}-u^{2a})-a(\underline{u}^{2a-1}+u^{2a-1})\big{)}\frac{\delta^{bc}}{r^{2}}=q(\underline{u},u)\frac{\delta^{bc}}{2r^{2}}.$

We then apply Lemma 5.2 to the weight in the last line above and conclude
that, in the range $\frac{1}{2}\leq a\leq 1$, $\mathcal{A}^{bc}\geq 0$ and
therefore

$A^{\alpha\beta}\partial_{\alpha}\widetilde{\phi}\partial_{\beta}\widetilde{\phi}\geq
0,$ (5.19)

where equality holds in the cases $a=1/2$ and $a=1$.

2\. Boundary terms. We compute
${}^{(X)}\widetilde{P}_{\alpha}N{{}^{\prime}}^{\alpha}$ using the equation
(3.7) in Bondi coordinates:

$\displaystyle{}^{(X)}\\!\widetilde{P}_{\alpha}N{{}^{\prime}}^{\alpha}$
$\displaystyle=(1+u^{2a})(1-{r\over
t})(\partial^{B}_{u}\widetilde{\phi})^{2}+\frac{1}{2}((1+u^{2a})+\frac{1}{2}(\underline{u}^{2a}-u^{2a})(1+{r\over
t}))(\partial^{B}_{r}\widetilde{\phi})^{2}$
$\displaystyle\quad+\frac{1}{2}((1+u^{2a})+\frac{1}{2}(\underline{u}^{2a}-u^{2a})(1-{r\over
t}))|{\,{/\\!\\!\\!\nabla}}\widetilde{\phi}|^{2}-(1+u^{2a})(1-{r\over
t})\partial^{B}_{u}\widetilde{\phi}\partial_{r}^{B}\widetilde{\phi}.$

Multiplying by $r^{-2}$, integrating, and applying the equation (4.8), we
obtain

$\displaystyle\int_{s=s^{\prime}}{}^{(X)}\\!P_{\alpha}N^{\alpha}\,r^{-2}dV=\int_{s=s^{\prime}}{}^{(X)}\\!P_{\alpha}N{{}^{\prime}}^{\alpha}\,r^{-2}dx$
(5.20) $\displaystyle\approx\int_{s=s^{\prime}}\Big{(}(1+u^{2a})(1-{r\over
t})(\partial^{B}_{u}\widetilde{\phi})^{2}+((1+u^{2a})+(\underline{u}^{2a}-u^{2a})(1+{r\over
t}))(\partial^{B}_{r}\widetilde{\phi})^{2}$
$\displaystyle\qquad\qquad\quad+((1+u^{2a})+(\underline{u}^{2a}-u^{2a})(1-{r\over
t}))|{\,{/\\!\\!\\!\nabla}}\widetilde{\phi}|^{2}\Big{)}\,r^{-2}dx$
$\displaystyle\approx\|\,r\phi(s)\,\|_{E_{r}^{a}}^{2}.$

Here, we used the equations (5.3) and (5.14) in order to control the weights
on the last line.

3\. Source term. We take absolute value and apply Cauchy-Schwarz to get

$\displaystyle\big{|}\int_{s_{0}}^{s_{1}}\\!\\!\\!\int_{s=s^{\prime}}F\frac{K^{a}\widetilde{\phi}}{r}\
dVds^{\prime}\big{|}$
$\displaystyle\lesssim\int_{s_{0}}^{s_{1}}\\!\\!\\!\int_{s=s^{\prime}}\big{|}F(1+u^{2a})\frac{s}{t}\frac{\partial^{B}_{u}\widetilde{\phi}}{r}\big{|}+\big{|}F\tau_{+}^{2a}\frac{s}{t}\frac{\partial^{B}_{r}\widetilde{\phi}}{r}\big{|}\
dxds^{\prime}$ (5.21)
$\displaystyle\lesssim\|\,F\,\|_{N_{W}^{a}[s_{0},s_{1}]}\sup_{s_{0}\leq s\leq
s_{1}}\|\,r\phi(s)\,\|_{E_{r}^{a}}.$

4\. Putting it all together. Substituting the results of Eqs. (5.19), (5.20),
and (5.21) into the equation (5.16) yields:

$\|\,r\phi(s_{1})\,\|_{E_{r}^{a}}^{2}\lesssim\|\,r\phi(s_{0})\,\|_{E_{r}^{a}}^{2}+\|\,F\,\|_{N_{W}^{a}[s_{0},s_{1}]}\sup_{s_{0}\leq
s\leq s_{1}}\|\,r\phi(s)\,\|_{E_{r}^{a}}.$

Taking the supremum and dividing by $\sup\limits_{s_{0}\leq s\leq
s_{1}}\|\,r\phi(s)\,\|_{E_{r}^{a}}$ complete the proof of (5.10). ∎

###### Proof of the estimate (5.10) for the range $0<a\leq 1/2$.

We now use the multiplier identity the equation (3.15) with the choice (5.7),
that is,

$X=Y^{a}=(1+2a\tau_{+}^{2a}\tau_{0})\partial^{B}_{u}+r^{2a}\partial^{B}_{r},\qquad\Omega=r,\qquad\Lambda=1.$

Setting $\widetilde{\phi}=r\phi$ and using the formulas in line (3.17), once
again we get

$\displaystyle\int_{s=s_{1}}\\!\\!{}^{(X)}\widetilde{P}_{\alpha}[\widetilde{\phi}]N^{\alpha}r^{-2}\
dV+\int_{s_{0}}^{s_{1}}\\!\\!\\!\int_{s=s^{\prime}}r^{-2}A^{\alpha\beta}\partial_{\alpha}\widetilde{\phi}\partial_{\beta}\widetilde{\phi}\
dVds^{\prime}$
$\displaystyle=\int_{s=s_{0}}\\!\\!{}^{(X)}\widetilde{P}_{\alpha}[\widetilde{\phi}]N^{\alpha}r^{-2}\
dV-\int_{s_{0}}^{s_{1}}\\!\\!\\!\int_{s=s^{\prime}}Fr^{-1}X\widetilde{\phi}\
dVds^{\prime},$

where we used Lemma 4.3 in order to conclude that $\mathcal{B}(X,r)\equiv 0$.

1\. Terms $\mathcal{A}^{\alpha\beta}$. Since $\Omega_{r}=r$ we apply the
equation (5.17), use (4.5), and find

$\displaystyle\mathcal{A}^{uu}$
$\displaystyle=-\partial_{r}^{B}(X^{u})=-\partial_{r}^{B}(2a\tau_{+}^{2a}\tau_{0})=2a(1-2a)\tau^{2a-1}_{+}\tau_{0},$
$\displaystyle\mathcal{A}^{ur}$
$\displaystyle=\frac{1}{2}\partial_{r}^{B}X^{u}=\frac{1}{2}\partial_{r}^{B}(2a\tau_{+}^{2a}\tau_{0})=-a(1-2a)\tau^{2a-1}_{+}\tau_{0},$
and $\displaystyle\mathcal{A}^{rr}$
$\displaystyle=\frac{1}{2}\partial_{r}^{B}X^{r}-\frac{1}{2}{\partial_{u}^{B}}X^{u}-{\partial_{u}^{B}}X^{r}=ar^{2a-1}-a\tau_{+}^{2a-1}\frac{u}{\tau_{-}}-a(2a-1)\tau_{+}^{2a-1}\tau_{0}$
$\displaystyle=a(1-2a)\tau_{+}^{2a-2}\underline{u}\tau_{0}+a(r^{2a-1}-\tau_{+}^{2a-1}\frac{u}{\tau_{-}})\geq
a(1-2a)\tau_{+}^{2a-1}\tau_{0},$ $\displaystyle\mathcal{A}^{bc}$
$\displaystyle=\big{(}\frac{X^{r}}{r}-\frac{1}{2}{\partial_{u}^{B}}X^{u}-\frac{1}{2}\partial_{r}^{B}X^{r}\big{)}\frac{\delta^{bc}}{r^{2}}=\big{(}r^{2a-1}-a\tau_{+}^{2a-1}\frac{u}{\tau_{-}}-a(2a-1)\tau_{+}^{2a-1}\tau_{0}-ar^{2a-1}\big{)}\frac{\delta^{bc}}{r^{2}}$
$\displaystyle=\big{(}(1-a)r^{2a-1}-a\tau_{+}^{2a-1}\frac{u}{\tau_{-}}+a(1-2a)\tau_{+}^{2a-1}\tau_{0}\big{)}\frac{\delta^{bc}}{r^{2}}\geq
0,$

where we used that $r^{2a-1}\geq\tau_{+}^{2a-1}$ in the range
$0<a\leq\frac{1}{2}$. Combining these results gives us

$\mathcal{A}^{uu}\geq 0,\qquad\mathcal{A}^{rr}\geq
0,\qquad|\mathcal{A}^{ur}|\leq\sqrt{\mathcal{A}^{uu}\mathcal{A}^{rr}},\qquad\mathcal{A}^{bc}\geq
0.$ (5.23)

An application of the classical inequality of arithmetic and geometric means
then yields the favorable sign

$A^{\alpha\beta}\partial_{\alpha}\widetilde{\phi}\partial_{\beta}\widetilde{\phi}\geq
0.$ (5.24)

2\. Boundary terms. We compute ${}^{(X)}\\!P_{\alpha}N^{\alpha}$ using the
equation (3.7) in Bondi coordinates

${}^{(X)}\\!\widetilde{P}_{\alpha}N{{}^{\prime}}^{\alpha}=(1+2a\tau^{2a}_{+}\tau_{0})(1-{r\over
t})(\partial^{B}_{u}\widetilde{\phi})^{2}+\frac{1}{2}(1+2a\tau^{2a}_{+}\tau_{0}+r^{2a}(1+{r\over
t}))(\partial^{B}_{r}\widetilde{\phi})^{2}\\\
+\frac{1}{2}(1+2a\tau^{2a}_{+}\tau_{0}+r^{2a}(1-{r\over
t}))|{\,{/\\!\\!\\!\nabla}}\widetilde{\phi}|^{2}-(1+2a\tau^{2a}_{+}\tau_{0})(1-{r\over
t})\partial^{B}_{u}\widetilde{\phi}\partial_{r}^{B}\widetilde{\phi}.$ (5.25)

Multiplying by $r^{-2}$, integrating, and applying the equation (4.8) yields

$\displaystyle\int_{s=s^{\prime}}{}^{(X)}\\!P_{\alpha}N^{\alpha}\,r^{-2}dV=\int_{s=s^{\prime}}{}^{(X)}\\!P_{\alpha}N{{}^{\prime}}^{\alpha}\,r^{-2}dx$
(5.26)
$\displaystyle\approx\int_{s=s^{\prime}}\Big{(}1+2a\tau^{2a}_{+}\tau_{0}(1-{r\over
t})(\partial^{B}_{u}\widetilde{\phi})^{2}+(1+2a\tau^{2a}_{+}\tau_{0}+r^{2a}(1+{r\over
t}))(\partial^{B}_{r}\widetilde{\phi})^{2}$ $\displaystyle\hskip
72.26999pt+(1+2a\tau^{2a}_{+}\tau_{0}+r^{2a}(1-{r\over
t}))|{\,{/\\!\\!\\!\nabla}}\widetilde{\phi}|^{2}\Big{)}\,r^{-2}dx\hskip
56.9055pt\approx\|\,r\phi(s)\,\|_{E_{r}^{a}}^{2},$

where we have used (5.14) in order to bound the weights on the last line.

3\. Source term. Similarly to what was done before, we have now

$\displaystyle\big{|}\int_{s_{0}}^{s_{1}}\\!\\!\\!\int_{s=s^{\prime}}F\frac{X\widetilde{\phi}}{r}\
dVds^{\prime}\big{|}$
$\displaystyle\lesssim\int_{s_{0}}^{s_{1}}\\!\\!\\!\int_{s=s^{\prime}}\big{|}F\tau^{2a}_{+}\tau_{0}\frac{s}{t}\frac{\partial^{B}_{u}\widetilde{\phi}}{r}\big{|}+\big{|}F\tau_{+}^{2a}\frac{s}{t}\frac{\partial^{B}_{r}\widetilde{\phi}}{r}\big{|}\
dxds^{\prime}$ (5.27)
$\displaystyle\lesssim\|\,F\,\|_{N_{W}^{a}[s_{0},s_{1}]}\sup_{s_{0}\leq s\leq
s_{1}}\|\,r\phi(s)\,\|_{E_{r}^{a}}^{2}.$

4\. Putting it all together. The conclusion follows from substituting the
estimates (5.24), (5.26), and (5.27) into (5.3) and simplifying the result. ∎

###### Proof of the estimate (5.11) for $0<a\leq 1$.

We now use the multiplier identity (3.15) with the choice (5.8), that is,

$X=K=(1+u^{2})\partial^{B}_{u}+2(u+r)r\partial^{B}_{r},\qquad\Omega=s^{2},\qquad\Lambda=s^{2a-2}.$

Setting$\widetilde{\phi}=s^{2}\phi$ and using the formulas in line (3.17), we
obtain

$\displaystyle\int_{s=s_{1}}\\!\\!{}^{(X)}\widetilde{P}_{\alpha}[\widetilde{\phi}]N^{\alpha}s^{-4}\
dV+\int_{s_{0}}^{s_{1}}\\!\\!\\!\int_{s=s^{\prime}}\big{(}s^{2a-2}\mathcal{A}^{\alpha\beta}\partial_{\alpha}\widetilde{\phi}\partial_{\beta}\widetilde{\phi}+s^{2-2a}\mathcal{C}^{\alpha}{}^{(X)}\widetilde{P}_{\alpha}\big{)}s^{-4}\
dVds^{\prime}$ (5.28)
$\displaystyle=\int_{s=s_{0}}\\!\\!{}^{(X)}\widetilde{P}_{\alpha}[\widetilde{\phi}]N^{\alpha}s^{-4}\
dV-\int_{s_{0}}^{s_{1}}\\!\\!\\!\int_{s=s^{\prime}}s^{2a-2}Fs^{-2}X\widetilde{\phi}\
dVds^{\prime},$

where we used Lemma 4.3 to see that $\mathcal{B}(X,s^{2})\equiv 0$.

1\. Terms $\mathcal{A}^{\alpha\beta}$. Since $\Omega=s^{2}$ we have

$\displaystyle\frac{K^{r}}{r}-K(\ln s^{2})$
$\displaystyle=2(u+r)-\frac{1}{u\underline{u}}(2(u+r)u^{2}+4(u+r)ur)=0.$
(5.29)

Combining this with the list of identities underneath the equation (5.17) with
$a=1$ we get

$\mathcal{A}^{uu}=0\qquad\mathcal{A}^{rr}=0,\qquad\mathcal{A}^{ur}=0,\qquad\mathcal{A}^{bc}=\big{(}\frac{1}{2r}(\underline{u}^{2}-u^{2})-(\underline{u}+u)\big{)}\frac{\delta^{bc}}{r^{2}}=0,$
(5.30)

therefore

$A^{\alpha\beta}\partial_{\alpha}\widetilde{\phi}\partial_{\beta}\widetilde{\phi}=0.$
(5.31)

2\. Term $\mathcal{C}^{\alpha}$. Expanding
$\mathcal{C}^{\alpha}=\nabla^{\alpha}s^{2a-2}=(2a-2)s^{2a-3}\nabla^{\alpha}s,$
and since the case $a=1$ is trivial we may assume $0<a<1$. We observe that the
vector field $-\nabla
s=-\eta^{\alpha\beta}\partial_{\alpha}s\partial_{\beta}=\frac{t}{s}\partial_{t}+\frac{r}{s}\partial_{r}$
is future directed and causal. In addition, the Morawetz vector field $K$ is
future directed and causal throughout the domain
$\mathcal{K}_{[s_{0},+\infty)}$. Since the classical energy momentum tensor
$Q_{\alpha\beta}$ satisfies the dominant energy condition, the sign condition

$\mathcal{C}^{\alpha}{}^{(X)}\widetilde{P}_{\alpha}s^{2-2a}=\mathcal{C}^{\alpha}{}^{(X)}{P}_{\alpha}=(2-2a)s^{2a-3}Q(-\nabla
s,X)\geq 0$ (5.32)

is valid throughout the domain $\mathcal{K}$.

3\. Boundary terms. We compute
${}^{(X)}\widetilde{P}_{\alpha}N{{}^{\prime}}^{\alpha}$ using the equation
(3.7). Multiplying by $s^{-4}$, integrating, and applying the equation (4.8),
we find

$\displaystyle\int_{s=s^{\prime}}{}^{(X)}\\!\widetilde{P}_{\alpha}N^{\alpha}\,s^{-4}dV=\int_{s=s^{\prime}}{}^{(X)}\\!\widetilde{P}_{\alpha}N{{}^{\prime}}^{\alpha}\,s^{-4}dx$
$\displaystyle\approx\int_{s=s^{\prime}}s^{2a-2}_{+}\Big{(}(1+u^{2})(1-{r\over
t})(\partial^{B}_{u}\widetilde{\phi})^{2}+((1+u^{2})+(\underline{u}^{2}-u^{2})(1+{r\over
t}))(\partial^{B}_{r}\widetilde{\phi})^{2}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad+((1+u^{2})+(\underline{u}^{2}-u^{2})(1-{r\over
t}))|{\,{/\\!\\!\\!\nabla}}\widetilde{\phi}|^{2}\Big{)}\,s^{-4}dx\approx\|\,s^{2}\phi(s)\,\|_{E_{s}^{a}}^{2},$

where we have used equation (5.3) and equation (5.14) to bound the weights on
the last line.

4\. Source term. Taking the absolute value and applying the Cauchy-Schwarz
inequality, we get

$\displaystyle\big{|}\int_{s_{0}}^{s_{1}}\\!\\!\\!\int_{s=s^{\prime}}Fs^{2a-2}_{+}\frac{X\widetilde{\phi}}{s^{2}}\
dVds^{\prime}\big{|}\lesssim\int_{s_{0}}^{s_{1}}\\!\\!\\!\int_{s=s^{\prime}}\big{|}Fs^{2a-2}\tau^{2}_{-}\frac{s}{t}\frac{\partial^{B}_{u}\widetilde{\phi}}{s^{2}}\big{|}+\big{|}Fs^{2a-2}\tau^{2}_{+}\frac{s}{t}\frac{\partial^{B}_{r}\widetilde{\phi}}{s^{2}}\big{|}\
dxds^{\prime}$ (5.33)
$\displaystyle\lesssim\|\,F\,\|_{N^{a}[s_{0},s_{1}]}\sup_{s_{0}\leq s\leq
s_{1}}\|\,s^{2}\phi(s)\,\|_{E_{s}^{a}},$

where, for each of the weights, we used

$s^{2a-2}\tau^{2}_{-}\lesssim
s^{a}(s^{a-1}\tau_{-})\lesssim\tau_{+}^{a}(s^{a-1}\tau_{-}),\qquad
s^{2a-2}\tau^{2}_{+}\frac{s}{t}\lesssim
s^{a}(s^{a-1}\tau_{+})\lesssim\tau_{+}^{a}(s^{a-1}\tau_{+}).$

5\. Putting it all together. The desired concusion now follows by substituting
the estimates (5.31), (5.32), (5.3), and (5.33) into (5.28) and simplifying
the result. ∎

######  Lemma 5.6 (Combining the energies).

For all $s_{0}\leq s\leq s_{1}$, $0<a\leq 1$ with
$\mathcal{K}_{[s_{0},s_{1}]}\subset\mathcal{K}$. the lower bound holds:

$\displaystyle\|\,\phi(s)\,\|_{E_{W}^{a}}\lesssim\|\,r\phi(s)\,\|_{E_{r}^{a}}+\|\,s^{2}\phi(s)\,\|_{E_{s}^{a}}+\|\,\phi(s)\,\|_{E_{W}}.$
(5.34)

###### Proof.

By recalling that $\partial_{t}=\partial^{B}_{u}$ and
$\partial^{B}_{r}=\partial_{t}+\partial_{r}$, a simple computation yields the
identities

$\displaystyle\frac{1}{r}\partial^{B}_{r}(r\phi)=\partial^{B}_{r}\phi+\frac{\phi}{r},$
$\displaystyle\frac{1}{r}\partial_{t}(r\phi)=\partial_{t}\phi,$
$\displaystyle\frac{1}{s^{2}}\partial^{B}_{r}(s^{2}\phi)=\partial^{B}_{r}\phi+\frac{2}{\underline{u}}\phi,$
$\displaystyle\frac{1}{s^{2}}\partial_{t}(s^{2}\phi)=\partial_{t}\phi+\frac{2t}{\underline{u}u}\phi.$

We multiply the two identities in the right-hand side above by the weight
$(s^{a}u/t)$ and subtract, namely

$\displaystyle
s^{a-1}\frac{s}{t}u\Big{(}\frac{1}{s^{2}}\partial_{t}(s^{2}\phi)-\frac{1}{r}\partial_{t}(r\phi)\Big{)}=\frac{s^{a}}{t}\,\frac{2t}{\underline{u}}\phi.$

Squaring both sides, expanding, using Young inequality and the fact that
$1\leq 2t/\underline{u}\leq 2$ gives us

$\big{|}\frac{s^{a}\phi}{t}\big{|}^{2}\lesssim
s^{2a-2}(1+u^{2})\big{|}\frac{s}{t}\cdot\frac{1}{s^{2}}\partial_{t}(s^{2}\phi)\big{|}^{2}+\tau^{2a}_{+}\tau^{\max\\{2a,1\\}}_{0}\big{|}\frac{s}{t}\,\frac{1}{r}\partial_{t}(r\phi)\big{|}^{2},$
(5.35)

where on the weight for the last term in the right-hand side we used the
following inequalities valid in the domain $\mathcal{K}_{[s_{0},+\infty)}$:

$\displaystyle\frac{s^{a}u}{t}=s^{a-1}u\frac{s}{t}=u^{a}(\frac{u}{s})^{1-a}\frac{s}{t}\leq
u^{a}\frac{s}{t},$ $\displaystyle\frac{1}{2}\leq a\leq 1,$
$\displaystyle\frac{s^{a}u}{t}\leq\tau_{+}^{a}\frac{u}{s}\frac{s}{t}=\tau_{+}^{a}\big{(}\frac{u}{\underline{u}}\big{)}^{1/2}\frac{s}{t}\lesssim\tau_{+}^{a}\tau_{0}^{1/2}\frac{s}{t},$
$\displaystyle 0<a<\frac{1}{2}.$

Thus we have the lower bound

$\displaystyle\tau^{2a}_{+}\tau^{\max\\{2a,1\\}}_{0}\big{|}\frac{s}{t}\partial_{t}\phi\big{|}^{2}+\big{|}\frac{s^{a}\phi}{t}\big{|}^{2}\lesssim
s^{2a-2}\tau_{-}^{2}\big{|}\frac{s}{t}\frac{1}{s^{2}}\partial_{t}(s^{2}\phi)\big{|}^{2}+\tau^{2a}_{+}\tau^{\max\\{2a,1\\}}_{0}\big{|}\frac{s}{t}\frac{1}{r}\partial_{t}(r\phi)\big{|}^{2}.$
(5.36)

Next we derive a bound for tangential derivatives to the hyperboloids
$\underline{\nabla}_{x}$. Observe that

$\partial^{B}_{r}=\partial_{t}+\partial_{r}=\big{(}1-{r\over
t}\big{)}\partial_{t}+\big{(}{r\over
t}\partial_{t}+\partial_{r}\big{)}=\big{(}\frac{u}{t}\big{)}\partial_{t}+\underline{\partial}_{r}.$

Multiplying by $s^{a-1}\underline{u}(s/t)$, conjugating by the weight
$\Omega_{s}=s^{2}$ and rearranging yields

$s^{a}\frac{\underline{u}}{t}\frac{\underline{\partial}_{r}(s^{2}\phi)}{s^{2}}=s^{a-1}\underline{u}\frac{s}{t}\big{(}\frac{\partial^{B}_{r}(s^{2}\phi)}{s^{2}}\big{)}-\frac{s^{a-1}u\underline{u}}{t}\big{(}\frac{s}{t}\frac{1}{s^{2}}\partial_{t}(s^{2}\phi)\big{)}.$
(5.37)

Squaring both sides, using Young inequality and the fact that
$\underline{\partial}_{r}(s)=0$ together with the bound
$1\leq\underline{u}/t\leq 2$ gives the estimate

$\displaystyle s^{2a}\big{|}\underline{\partial}_{r}\phi\big{|}^{2}\lesssim
s^{2a-2}\tau_{-}^{2}\big{|}\frac{s}{t}\cdot\frac{1}{s^{2}}\partial_{t}(s^{2}\phi)\big{|}^{2}+s^{2a-2}\tau_{+}^{2}\big{|}\frac{\partial^{B}_{r}(s^{2}\phi)}{s^{2}}\big{|}^{2}.$
(5.38)

To add the angular derivatives we observe that, for any function on a fixed
hyperboloid $s=const$, we have
$\partial_{i}\phi(s)=\underline{\partial}_{i}\phi(s)$. Therefore, after
integrating on hyperboloids we can identify
$\underline{{\,{/\\!\\!\\!\nabla}}}\phi(s)={\,{/\\!\\!\\!\nabla}}\phi(s)$.
Consequently, adding (5.36) and (5.38), integrating, and using this
observation completes the proof of inequality (5.34). ∎

######  Remark 5.7.

There is a small penalty for removing the conjugation and going back to the
semi-hyperboloidal frame. This is to be expected since we require a weight of
$(s/t)$ to use the $\partial_{t}$ derivative in $L^{2}(\mathcal{H}_{s})$.

##  6 Boundeness property for the wave-Klein-Gordon model

###  6.1 Main linear estimates

We are now in a position to combine our results so far and establish (2.25),
(2.26), and (2.27). For each of these inequalities, we first prove the case
$k=0$ then add regularity by commuting the equation with the vector fields in
the Lie algebra $\mathbb{L}$. Using the commutation relations

$\displaystyle\left[\Box,\partial^{I_{0}}_{0}\partial^{I_{1}}_{1}\partial^{I_{2}}_{2}\partial^{I_{3}}_{3}\right]=0,$
$\displaystyle I_{0}+I_{1}+I_{2}+I_{3}=|I|,$ (6.1)
$\displaystyle\big{[}\Box,L^{J_{1}}_{1}L^{J_{2}}_{2}L^{J_{3}}_{3}\big{]}=0,$
$\displaystyle J_{1}+J_{2}+J_{3}=|J|,$ (6.2)

where $|I|+|J|=k$ then yields each corresponding result for $k=0,1,\ldots$

###### Proof of Theorem 2.2.

The estimate (2.25) follows by adding estimates (5.9)+(5.10)+(5.11). The
estimate (2.25) for each $k=0,1,\ldots$ follows, via induction, by commuting
the equation with products of vector fields in $\mathbb{L}$ and applying the
relations (6.1) and (6.2). ∎

###### Proof of Corollary 2.3.

Estimate (2.26) in the case $k=0$ follows by applying estimate (5.34) to the
left-hand side of (2.25) with $k=0$. The estimate (2.26) for each
$k=0,1,\ldots$ then follows, via induction, by commuting the equation with
products of vector fields in $\mathbb{L}$ and using the relations (6.1) and
(6.2).

On the other hand, the estimate (2.27) in the case $k=0$ follows by using
estimate (5.5) followed by an application of the estimate (2.26) with $k=2$.
The estimate (2.27) for each $k=0,1,\ldots$ then follows, via induction, by
commuting the equation with products of vector fields in $\mathbb{L}$ and
using the relations (6.1) and (6.2). ∎

###  6.2 Closing the bootstrap argument

We consider the local-in-time solution $(u,v)$ to the Cauchy problem
associated with the system (1.2) and, as usual, we proceed by bootstrap and
assume that this solution is defined up to some hyperboloidal time
$s_{1}>s_{0}$ which is chosen to be maximal for the bounds under
consideration. That is, the following energy bounds333In the high-order energy
bound for the wave equation, the exponent $\max(1,k))\delta$ provides a small
growth of order $\delta$ for all $k$, and we refer the reader to Appendix A in
[23]. are assumed on the interval $[s_{0},s_{1}]$ (for some constants
$C_{1},\epsilon,\delta>0$, yet to be determined):

$\displaystyle E_{\text{W}}^{a}(s,\partial^{I}L^{J}u)^{1/2}\leq C_{1}\epsilon
s^{(\max(a-1/2,0)+\max(1,k))\delta},$ $\displaystyle|J|=k,\quad$
$\displaystyle|I|+|J|\leq N,$ (6.3) $\displaystyle
E_{\text{W}}^{a}(s,\partial^{I}L^{J}u)^{1/2}\leq C_{1}\epsilon
s^{\max(a-1/2,0)\delta},$ $\displaystyle|I|+|J|\leq N-4,$ $\displaystyle
E_{\text{KG}}(s,\partial^{I}L^{J}v)^{1/2}\leq C_{1}\epsilon
s^{(\max(-a+1/2,0)+k)\delta},$ $\displaystyle|J|=k,\quad$
$\displaystyle|I|+|J|\leq N,$ $\displaystyle
E_{\text{KG}}(s,\partial^{I}L^{J}v)^{1/2}\leq C_{1}\epsilon s^{k\delta},$
$\displaystyle|J|=k,\quad$ $\displaystyle|I|+|J|\leq N-4.$

We will establish improved bounds, with $C_{1}$ replaced by $C_{1}/2$, on the
same interval of time (provided $\epsilon$ is sufficiently small and the
constants are suitably chosen).

The existence theory for the model (1.2) was already established in [23] and
what we need to check here is that the additional decay in space (assumed on
the initial data) is propagated in time. The proof in [23] requires some
adaptation which is now discussed. It is unnecessary to rewrite the proof
completely, and we now explain the additional arguments (for any given $0\leq
a\leq 1$).

  * •

Commutator in the Klein-Gordon equation. Let us consider the terms
$u\partial_{\alpha}\partial_{\beta}v$ arising in the Klein-Gordon equation. We
now control the wave component in a suitably weighted $L^{2}$ norm, namely the
norm $\|\,\frac{s^{a}}{\tau_{+}}u\,\|_{L^{2}(\mathcal{H}_{s})}$. Hence, the
higher-order commutator terms for the Klein-Gordon equation can now be
estimated as follows, in the case $N_{1}\geq N_{2}$ (even when $N_{1}=N$ the
maximum number of vector fields) and

For any combination of boosts and translations $Z^{N}=\partial^{I}L^{J}$ with
$|I+|J|\leq N$, we consider the quadratic expression
$Z^{N_{1}}u\partial^{2}Z^{N_{2}}v$

$\displaystyle\|\,Z^{N_{1}}u\partial^{2}Z^{N_{2}}v\,\|_{L^{2}(\mathcal{H}_{s})}$
$\displaystyle\lesssim\|\,\frac{s^{a}}{\tau_{+}}Z^{N_{1}}u\,\|_{L^{2}(\mathcal{H}_{s})}\,\|\,\frac{\tau_{+}}{s^{a}}\partial^{2}Z^{N_{2}}v\,\|_{L^{\infty}(\mathcal{H}_{s})}$
(6.4)
$\displaystyle\lesssim\|\,\frac{s^{a}}{\tau_{+}}Z^{N_{1}}u\,\|_{L^{2}(\mathcal{H}_{s})}\,\frac{\tau_{+}}{s^{a+1/2}t}=\|\,\frac{s^{a}}{\tau_{+}}Z^{N_{1}}u\,\|_{L^{2}(\mathcal{H}_{s})}\,\frac{1}{s^{a+1/2}},$

where, in the second inequality, we have used the $L^{\infty}$ decay enjoyed
by the Klein-Gordon component.

  * •

First-order quadratic nonlinearities in the wave equation. Let us consider
next the terms $\partial_{\alpha}v\partial_{\beta}v$ arising in the wave
equation, and recall that our norm involves the weight $s^{a-1}\tau_{+}$.
Independently from the number of vector fields applied to each term, one of
the two copies of the Klein-Gordon component is always estimated in
$L^{\infty}$, hence we write for general exponents $N_{1},N_{2}$

$\displaystyle\|\,s^{a-1}\tau_{+}Z^{N_{1}}\partial vZ^{N_{2}}\partial
v\,\|_{L^{2}(\mathcal{H}_{s})}\lesssim\|\,\partial
Z^{N_{1}}v\,\|_{L^{2}(\mathcal{H}_{s})}\|\,s^{a-1}\tau_{+}\partial
Z^{N_{2}}v\,\|_{L^{\infty}(\mathcal{H}_{s})}$
$\displaystyle\lesssim\|\,\partial
Z^{N_{1}}v\,\|_{L^{2}(\mathcal{H}_{s})}\frac{s^{a-1}\tau_{+}}{s^{1/2}t}=\|\,\partial
Z^{N_{1}}v\,\|_{L^{2}(\mathcal{H}_{s})}\frac{1}{s^{3/2-a}}.$

  * •

Quadratic nonlinearity in the wave equation. Similarly, for the wave equation
again we write

$\displaystyle\|\,s^{a-1}\tau_{+}Z^{N_{1}}vZ^{N}v\,\|_{L^{2}(\mathcal{H}_{s})}\lesssim\|\,Z^{N_{1}}v\,\|_{L^{2}(\mathcal{H}_{s})}\|\,s^{a-1}\tau_{+}Z^{N}v\,\|_{L^{\infty}(\mathcal{H}_{s})}$
(6.5)
$\displaystyle\lesssim\|\,Z^{M}v\,\|_{L^{2}(\mathcal{H}_{s})}\frac{s^{a-1}\tau_{+}}{s^{1/2}t}=\|\,Z^{M}v\,\|_{L^{2}(\mathcal{H}_{s})}\frac{1}{s^{3/2-a}}.$

Hence, for all nonlinearities of interest we can control the source-terms of
both the wave and the Klein-Gordon equations and the remaining arguments in
[23] applies.

#### Acknowledgments.

The first author (PLF) was partially supported by the ERC Innovative Training
Network grant 642768. Part of this work was done when the first author was
visiting the School of Mathematical Sciences at Fudan University, Shanghai,
and the Courant Institute of Mathematical Sciences, New York University. The
first author (PLF) is very grateful to the third author (YT) for the
opportunity of visiting Kyoto University (where this paper was completed) as
an international research fellow of the Japan Society for the Promotion of
Science (JSPS). The second author (JO) gratefully acknowledges support from
Institut Henri Poincaré, Paris.

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##  Appendix A Proof of technical results

###### Proof of Lemma 4.1.

To establish the equation (4.4) we expand
$X^{i}=\overline{X}^{i}+\omega^{i}\omega_{j}X^{j}$

$\partial^{B}_{\gamma}X^{\gamma}=\partial^{B}_{u}X^{u}+\partial^{B}_{i}(\omega^{i})\omega_{j}X^{j}+\omega^{i}\partial^{B}_{i}(\omega^{i}X^{j})+\partial^{B}_{i}\overline{X}^{i}=\partial^{B}_{u}X^{u}+\frac{2X^{r}}{r}+\partial^{B}_{r}X^{r}+\partial^{B}_{i}\overline{X}^{i}.$

Using this together with the basic identity
${}^{(X)}\widehat{\pi}^{\alpha\beta}=-\mathcal{L}_{X}\eta^{-1}-\eta^{\alpha\beta}\partial_{\gamma}X^{\gamma}$,
in (3.10) gives the result. To calculate all the components of $\mathcal{A}$
we use Eq. (4.4) together with the equation (3.10) to get

$\mathcal{A}^{\alpha\beta}=\frac{1}{2}\Big{(}\partial_{\gamma}^{B}(X^{\alpha})\eta^{\gamma\beta}+\partial_{\gamma}^{B}(X^{\beta})\eta^{\alpha\gamma}-X(\eta^{\alpha\beta})-({\partial_{u}^{B}}X^{u}+\partial_{r}^{B}X^{r})\eta^{\alpha\beta}\Big{)}-\Big{(}\frac{X^{r}}{r}-X\ln\Omega)\Big{)}\eta^{\alpha\beta}.$

We now compute all the components of this tensor. For $\mathcal{A}^{uu}$ we
use the fact that $\eta^{u\alpha}=-1$ if $\alpha=r$, and $\eta^{u\alpha}=0$
otherwise. This gives

$\displaystyle\mathcal{A}^{uu}$
$\displaystyle=\frac{1}{2}\Big{(}\partial_{\gamma}^{B}(X^{u})\eta^{\gamma
u}+\partial_{\gamma}^{B}(X^{u})\eta^{\gamma
u}-X(\eta^{uu})-({\partial_{u}^{B}}X^{u}+\partial_{r}^{B}X^{r})\eta^{uu}\Big{)}-\Big{(}\frac{X^{r}}{r}-X\ln\Omega\Big{)})\eta^{uu}=-\partial_{r}^{B}X^{u}.$

For $\mathcal{A}^{rr}$ we use the fact that $\eta^{ur}=-1$, $\eta^{rr}=1$, and
$\eta^{r\alpha}=0$ otherwise. This yields

$\displaystyle\mathcal{A}^{rr}$
$\displaystyle=\frac{1}{2}\big{(}\partial_{\gamma}^{B}(X^{r})\eta^{\gamma
r}+\partial_{\gamma}^{B}(X^{r})\eta^{\gamma
r}-X(\eta^{rr})-({\partial_{u}^{B}}X^{u}+\partial_{r}^{B}X^{r})\eta^{rr}\big{)}-\Big{(}\frac{X^{r}}{r}-X\ln\Omega\Big{)})\eta^{rr}$
$\displaystyle=\frac{1}{2}(2\partial_{r}^{B}X^{r}-2\partial_{u}^{B}X^{r})-({\partial_{u}^{B}}X^{u}+\partial_{r}^{B}X^{r})\big{)}-\Big{(}\frac{X^{r}}{r}-X\ln\Omega\Big{)})$
$\displaystyle=\frac{1}{2}\partial_{r}^{B}X^{r}-\frac{1}{2}{\partial_{u}^{B}}X^{u}-{\partial_{u}^{B}}X^{r}-\Big{(}\frac{X^{r}}{r}-X\ln\Omega\Big{)}).$

Similarly, for $\mathcal{A}^{ur}$ we get

$\displaystyle\mathcal{A}^{ur}$
$\displaystyle=\frac{1}{2}\big{(}\partial_{\gamma}^{B}(X^{u})\eta^{\gamma
r}+\partial_{\gamma}^{B}(X^{r})\eta^{\gamma
u}-X(\eta^{ur})-({\partial_{u}^{B}}X^{u}+\partial_{r}^{B}X^{r})\eta^{ur}\big{)}-\Big{(}\frac{X^{r}}{r}-X\ln\Omega\Big{)})\eta^{ur}$
$\displaystyle=\frac{1}{2}\Big{(}-\partial_{u}^{B}X^{u}+\partial_{r}^{B}X^{u}-\partial_{r}^{B}X^{r}+{\partial_{u}^{B}}X^{u}+\partial_{r}^{B}X^{r}\Big{)}+\Big{(}\frac{X^{r}}{r}-X\ln\Omega\Big{)}=\frac{1}{2}\partial_{r}^{B}X^{u}+\Big{(}\frac{X^{r}}{r}-X\ln\Omega\Big{)}.$

For $\mathcal{A}^{bc}$ we use the hypothesis that our vector field is radial
and get

$\displaystyle\mathcal{A}^{bc}$
$\displaystyle=\frac{1}{2}\Big{(}\partial_{\gamma}^{B}(X^{b})\eta^{\gamma
c}+\partial_{\gamma}^{B}(X^{c})\eta^{\gamma
b}-X(\eta^{bc})-({\partial_{u}^{B}}X^{u}+\partial_{r}^{B}X^{r}\Big{)}\eta^{bc}\big{)}-\Big{(}\frac{X^{r}}{r}-X\ln\Omega\Big{)})\eta^{bc}$
$\displaystyle=\Big{(}\frac{X^{r}}{r}-\frac{1}{2}{\partial_{u}^{B}}X^{u}-\frac{1}{2}\partial_{r}^{B}X^{r}-\Big{(}\frac{X^{r}}{r}-X\ln\Omega\Big{)}\Big{)}\frac{\delta^{bc}}{r^{2}}.\qed$

###### Proof of Lemma 4.2.

By Lemma 4.3, the conformal potential $V$ identically vanishes for the choices
of weight ${}^{I}\Omega$ and ${}^{I\\!I}\Omega$. Therefore we have
$Q_{\alpha\beta}[\widetilde{\phi}]=\widetilde{Q}_{\alpha\beta}[\widetilde{\phi}],$
where $Q$ is energy-momentum tensor in Eq. (3.1). Expanding this tensor in
polar Bondi coordinates and using the equation (4.1) for the metric
coefficients gives:

$\displaystyle
Q_{uu}[\widetilde{\phi}]=(\partial^{B}_{u}\widetilde{\phi})^{2}+\frac{1}{2}(\partial^{B}_{r}\widetilde{\phi})^{2}+\frac{1}{2}|{\,{/\\!\\!\\!\nabla}}\widetilde{\phi}|^{2}-\partial^{B}_{u}\widetilde{\phi}\partial^{B}_{r}\widetilde{\phi},$
$\displaystyle
Q_{ur}[\widetilde{\phi}]=\frac{1}{2}(\partial^{B}_{r}\widetilde{\phi})^{2}+\frac{1}{2}|{\,{/\\!\\!\\!\nabla}}\widetilde{\phi}|^{2},\qquad
Q_{rr}[\widetilde{\phi}]=(\partial^{B}_{r}\widetilde{\phi})^{2}.$

Since the rescaled normal is $N{{}^{\prime}}=(1-{r\over
t})\partial^{B}_{u}+{r\over t}\partial^{B}_{r}$ we use these identities to
get:

$\displaystyle{}^{(X)}\\!\widetilde{P}_{\alpha}N{{}^{\prime}}^{\alpha}$
$\displaystyle=X^{u}(1-{r\over t})Q_{uu}[\widetilde{\phi}]+X^{u}{r\over
t}Q_{ur}[\widetilde{\phi}]+X^{r}(1-{r\over
t})Q_{ru}[\widetilde{\phi}]+X^{r}{r\over
t}Q_{rr}[\widetilde{\phi}]+\frac{\Lambda}{\Omega^{2}}\mathcal{B}\widetilde{\phi}^{2}$
$\displaystyle=X^{u}\big{[}(1-{r\over
t})\big{(}(\partial^{B}_{u}\widetilde{\phi})^{2}+\frac{1}{2}(\partial^{B}_{r}\widetilde{\phi})^{2}+\frac{1}{2}|{\,{/\\!\\!\\!\nabla}}\widetilde{\phi}|^{2}-\partial^{B}_{u}\widetilde{\phi}\partial_{r}^{B}\widetilde{\phi}\big{)}+{r\over
t}\big{(}\frac{1}{2}(\partial^{B}_{r}\widetilde{\phi})^{2}+\frac{1}{2}|{\,{/\\!\\!\\!\nabla}}\widetilde{\phi}|^{2}\big{)}\big{]}$
$\displaystyle\qquad\qquad+X^{r}\big{[}(1-{r\over
t})(\frac{1}{2}(\partial^{B}_{r}\widetilde{\phi})^{2}+\frac{1}{2}|{\,{/\\!\\!\\!\nabla}}\widetilde{\phi}|^{2})+{r\over
t}(\partial^{B}_{r}\widetilde{\phi})^{2}\big{]}+\frac{\Lambda}{\Omega^{2}}\mathcal{B}\widetilde{\phi}^{2}.$

Combining terms yields the equation (4.6) immediately. To prove a coercive
lower bound we apply Young inequality with $\epsilon$ to the unsigned term
$-\partial^{B}_{u}\widetilde{\phi}\partial_{r}^{B}\widetilde{\phi}$ in the
second equation. This gives us:

$\displaystyle{}^{(X)}\\!\widetilde{P}_{\alpha}N{{}^{\prime}}^{\alpha}$
$\displaystyle\geq X^{u}(1-{r\over
t})\big{[}(1-\frac{1}{2\epsilon^{2}})(\partial^{B}_{u}\widetilde{\phi})^{2}+(\frac{1}{2}-\frac{1}{2}\epsilon^{2})(\partial^{B}_{r}\widetilde{\phi})^{2}\big{]}+\frac{1}{2}X^{u}({r\over
t}(\partial^{B}_{r}\widetilde{\phi})^{2}+|{\,{/\\!\\!\\!\nabla}}\widetilde{\phi}|^{2})$
(A.1) $\displaystyle\qquad\qquad+\frac{1}{2}X^{r}\big{[}(1+{r\over
t})(\partial^{B}_{r}\widetilde{\phi})^{2}+(1-{r\over
t})|{\,{/\\!\\!\\!\nabla}}\widetilde{\phi}|^{2})\big{]}+\frac{\Lambda}{\Omega^{2}}\mathcal{B}\widetilde{\phi}^{2}.$

Choosing $\epsilon$ satisfying the property $\frac{1}{2}<\epsilon^{2}<1$ and
using the positivity of $X^{u}$ yields the coercive bound

$\displaystyle{}^{(X)}\\!\widetilde{P}_{\alpha}N{{}^{\prime}}^{\alpha}\gtrsim_{\epsilon}X^{u}(1-{r\over
t})(\partial^{B}_{u}\widetilde{\phi})^{2}+\big{(}X^{u}+X^{r}(1+{r\over
t})\big{)}(\partial^{B}_{r}\widetilde{\phi})^{2}+\big{(}X^{u}+X^{r}(1-{r\over
t})\big{)}|{\,{/\\!\\!\\!\nabla}}\widetilde{\phi}|^{2}+\frac{\Lambda}{\Omega^{2}}\mathcal{B}\widetilde{\phi}^{2},$

where the implicit constant depends only on $\epsilon$. The upper bound
follows trivially from applying Young inequality to the right-hand side of the
equation (4.6). This completes the derivation of (4.8). ∎