From Lemma 3.13 and bounds (3.11), (3.12), (3.16), (3.45) we also get the
following pointwise estimate for the differentiated weak null terms.

######  Proposition 3.15.

There exists a constant $C>0$ such that, under the a-priori assumptions
(3.5)-(3.8), we have that

(3.48) $\displaystyle\big{|}Z^{\leq N-3}P_{\alpha\beta}(\partial h,\partial
h)(t)\big{|}_{\frac{1}{2}}$ $\displaystyle\lesssim
C_{0}^{2}\epsilon^{2}\big{(}t^{-1+2C\epsilon}+t^{-1+2\sigma}\sqrt{l(t)}\big{)}.$

###  3.6. Propagation of the energy estimates

We now proceed to the proof of proposition 3.1. We recall that for any multi-
index $K$, the differentiated coefficients $Z^{K}h^{1}_{\alpha\beta}$ solve
(3.9) with source term (3.10). We set $i=1$ if $Z^{K}=\partial Z^{K^{\prime}}$
and $|K^{\prime}|\leq N$, $i=0$ if simply $|K|\leq N$. Thanks to the smallness
of $H$ provided by (3.11) and (3.17), we apply (A.2) with
$\mathbf{W}=Z^{K}h^{1}_{\alpha\beta}$,
$\mathbf{F}=F^{K}_{\alpha\beta}+F^{0,K}_{\alpha\beta}$,
$w(q)=(2+r-t)^{1+2(i+\kappa)}$ and $\omega=x/|x|$. For every $t\in[2,T_{0})$
we get the following energy inequality

(3.49) $\displaystyle E^{e,i+\kappa}(t,Z^{K}h^{1}_{\alpha\beta})$
$\displaystyle+\int_{\tilde{\mathscr{H}}_{2,t}}(2+r-t)^{1+2(i+\kappa)}\Big{[}\Big{(}\frac{1}{2(1+r^{2})}+\chi\left(\frac{r}{t}\right)\chi(r)\frac{M}{2r}\Big{)}|\partial_{t}Z^{K}h^{1}_{\alpha\beta}|^{2}+|\underline{\nabla}Z^{K}h^{1}_{\alpha\beta}|^{2}\Big{]}dxdy$
$\displaystyle\lesssim
E^{e,i+\kappa}(2,Z^{K}h^{1}_{\alpha\beta})+\iint_{\mathscr{D}^{\text{e}}_{[2,t]}}(2+r-\tau)^{1+2(i+\kappa)}|(F^{K}_{\alpha\beta}+\partial^{\mu}{H_{\mu}}^{\sigma}\cdot\partial_{\sigma}Z^{K}h^{1}_{\alpha\beta})\partial_{t}Z^{K}h^{1}_{\alpha\beta}|d\tau
dxdy$
$\displaystyle+\iint_{\mathscr{D}^{\text{e}}_{[2,t]}}(2+r-\tau)^{1+2(i+\kappa)}\Big{[}|F^{0,K}_{\alpha\beta}\,\partial_{t}Z^{K}h^{1}_{\alpha\beta}|+|\partial_{t}{H_{\mu}}^{\sigma}\cdot\partial_{\sigma}Z^{K}h^{1}\cdot\
\partial^{\mu}Z^{K}h^{1}_{\alpha\beta}|\Big{]}\,d\tau dxdy$
$\displaystyle+\iint_{\mathscr{D}^{\text{e}}_{[2,t]}}(2+r-\tau)^{2(i+\kappa)}|H^{\rho\sigma}\cdot\partial_{\rho}Z^{K}h^{1}_{\alpha\beta}\cdot\partial_{\sigma}Z^{K}h^{1}_{\alpha\beta}|\,d\tau
dxdy$
$\displaystyle+\iint_{\mathscr{D}^{\text{e}}_{[2,t]}}(2+r-\tau)^{2(i+\kappa)}|(-H^{0\sigma}+\omega_{\bm{j}}H^{{\bm{j}}\sigma})\partial_{\sigma}Z^{K}h^{1}_{\alpha\beta}\cdot\partial_{t}Z^{K}h^{1}_{\alpha\beta}|d\tau
dxdy.$

The above inequality is satisfied for all $t\in[2,T_{0})$ and the implicit
constant is a universal constant.

###### Proof of proposition 3.1.

We recall the definition of the source term
$F^{K}_{\alpha\beta}=Z^{K}F_{\alpha\beta}$ where

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

The combination of estimates (3.20), (3.21) and (3.46) yields that for $i=0,1$

$\|(2+r-t)^{\frac{1}{2}+i+\kappa}F^{K}_{\alpha\beta}\|_{L^{2}(\Sigma^{\text{e}}_{t})}\lesssim
C_{0}^{2}\epsilon^{2}\big{(}t^{-1+C\epsilon}+t^{-1+2\sigma}\sqrt{l(t)}\big{)}$

which together with the Cauchy-Schwarz inequality and energy assumptions
(3.1), (3.2) gives

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

From estimate (3.35) we also get that

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

By injecting the above bounds, together with (3.41) and (3.42), into (3.49) we
deduce the existence of a constant $\tilde{C}>0$ such that for all
$t\in[2,T_{0})$

$\displaystyle E^{\text{e},i+\kappa}(t,Z^{K}h^{1}_{\alpha\beta})$
$\displaystyle+\int_{\tilde{\mathscr{H}}_{2,t}}(2+r-t)^{1+2(i+\kappa)}\Big{[}\Big{(}\frac{1}{2(1+r^{2})}+\chi\left(\frac{r}{t}\right)\chi(r)\frac{M}{2r}\Big{)}|\partial_{t}Z^{K}h^{1}_{\alpha\beta}|^{2}+|\underline{\nabla}Z^{K}h^{1}_{\alpha\beta}|^{2}\Big{]}dxdy$
$\displaystyle\leq\tilde{C}E^{\text{e},i+\kappa}(2,Z^{K}h^{1}_{\alpha\beta})+\tilde{C}C_{0}\epsilon^{2}+\tilde{C}C_{0}^{3}\epsilon^{3}t^{C\epsilon+\sigma}+\tilde{C}\epsilon\int_{2}^{t}\frac{E^{\text{e},i+\kappa}(\tau,Z^{K}h^{1}_{\alpha\beta})}{\tau}\,d\tau.$

By Grönwall’s inequality we then deduce that

$E^{\text{e},i+\kappa}(t,Z^{K}h^{1}_{\alpha\beta})\leq\tilde{C}\big{(}E^{\text{e},i+\kappa}(2,Z^{K}h^{1}_{\alpha\beta})+C_{0}\epsilon^{2}+C_{0}^{3}\epsilon^{3}t^{C\epsilon+\sigma}\big{)}t^{\tilde{C}\epsilon}.$

We denote the sum $C+\tilde{C}$ simply by $C$. Finally, we choose $C_{0}\gg 1$
sufficiently large so that
$3\tilde{C}E^{\text{e},i+\kappa}(2,Z^{K}h^{1}_{\alpha\beta})\leq(C_{0}\epsilon)^{2}$
and $3\tilde{C}<C_{0}$, then $\epsilon_{0}>0$ sufficiently small so that
$3\tilde{C}C_{0}^{3}\epsilon_{0}<1$ and $2C\epsilon_{0}<\sigma$ to infer that

$E^{\text{e},i+\kappa}(t,Z^{K}h^{1}_{\alpha\beta})\leq
C_{0}^{2}\epsilon^{2}t^{\sigma+C\epsilon}.$

As a byproduct, we also deduce that

(3.50)
$\int_{\tilde{\mathscr{H}}_{2,t}}(2+r-t)^{1+2(i+\kappa)}\Big{[}\Big{(}\frac{1}{2(1+r^{2})}+\chi\left(\frac{r}{t}\right)\chi(r)\frac{M}{2r}\Big{)}|\partial_{t}Z^{K}h^{1}_{\alpha\beta}|^{2}+|\underline{\nabla}Z^{K}h^{1}_{\alpha\beta}|^{2}\Big{]}dxdy\lesssim
C_{0}^{2}\epsilon^{2}t^{\sigma+C\epsilon}.$

∎

##  4\. The interior region

The goal of this section is to prove the existence in the interior region
$\mathscr{D}^{\text{i}}$ of the solution $h^{1}_{\alpha\beta}$ to (1.8) with
data satisfying the hypothesis of theorem 1.2. The proof is based on a
bootstrap argument in which the a-priori assumptions on the solutions are
bounds on the higher order energies on truncated hyperboloids
$\mathscr{H}_{s}$ as well as pointwise decay bounds on a certain number of $Z$
derivatives acting on it.

We define the interior energy functional as follows

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

where $\mathscr{S}=t\partial_{t}+x\cdot\nabla_{x}$ is the scaling vector field
and
$\Omega_{{\bm{i}}{\bm{j}}}=x_{\bm{i}}\partial_{\bm{j}}-x_{\bm{j}}\partial_{\bm{i}}$
are the Euclidean rotations in ${\mathbb{R}}^{3}$. Using Parseval’s identity,
we also define the energy functional associated to the zero-mode
$h^{1,\flat}_{\alpha\beta}$ of the solution as well as that of its zero-
average component $h^{1,\natural}_{\alpha\beta}$, so that

$E^{\text{i}}(s,h^{1}_{\alpha\beta})=E^{\text{i}}(s,h^{1,\flat}_{\alpha\beta})+E^{\text{i}}(s,h^{1,\natural}_{\alpha\beta}).$

We fix $N,N_{1}\in{\mathbb{N}}$ two integers sufficiently large with $N\geq
14$ and $N_{1}=N-5$ and assume the existence of two positive constants $1\ll
C_{1}\ll C_{2}$, as well as of a finite and increasing sequence of parameters
$0<\zeta_{k},\gamma_{k},\delta_{k}\ll 1$ with

(4.2) $\zeta_{i}\ll\gamma_{j}\ll\delta_{k},\quad\forall
i,j,k,\qquad\gamma_{i}+\delta_{j}\ll\delta_{k},\quad\forall i,j<k,$

such that:

$\bullet$ for $s_{0}$ close to 2 (e.g. $s_{0}=21/10$) and for any arbitrarily
fixed $S_{0}>s_{0}$, the solution $h^{1}_{\alpha\beta}$ exists in the
hyperbolic strip $\mathscr{H}_{[s_{0},S_{0})}$,

$\bullet$ for any $s\in[s_{0},S_{0})$, any multi-index $K=(I,J)$ of type
$(N,k)$111111We recall that a multi-index $K=(I,J)$ is said to be of type
$(N,k)$ if $|I|+|J|\leq N$ and $|J|\leq k$, it satisfies the following energy
bounds

(4.3) $\displaystyle E^{\text{i}}(s,\partial
Z^{K}h^{1}_{\alpha\beta})^{\frac{1}{2}}+E^{\text{i}}(s,Z^{K}h^{1}_{\alpha\beta})^{\frac{1}{2}}\leq
2C_{1}\epsilon s^{\frac{1}{2}+\zeta_{k}}$ (4.4) $\displaystyle
E^{\text{i}}(s,Z^{K}h^{1,\flat}_{\alpha\beta})^{\frac{1}{2}}\leq
2C_{1}\epsilon s^{\zeta_{k}}$

and for multi-indexes $K$ of type $(N,k)$ with $k\leq N_{1}$

(4.5) $E^{\text{i}}(s,Z^{K}h^{1}_{\alpha\beta})^{\frac{1}{2}}\leq
2C_{1}\epsilon s^{\delta_{k}}$

$\bullet$ for any $s\in[s_{0},S_{0})$, it satisfies the following pointwise
bounds

(4.6)
$\|t\,\Gamma^{J}h^{1,\flat}_{\alpha\beta}\|_{L^{\infty}_{x}(\mathscr{H}_{s})}\lesssim
2C_{2}\epsilon s^{\gamma_{k}}\text{ with }|J|=k\leq N_{1},$ (4.7)
$\begin{split}\|t^{\frac{1}{2}}s\,\partial_{tx}(\partial^{I}\Gamma^{J}h^{1,\natural}_{\alpha\beta})\|_{L^{\infty}_{x}L^{2}_{y}(\mathscr{H}_{s})}&+\|t^{\frac{3}{2}}\partial^{\leq
1}_{y}(\partial^{I}\Gamma^{J}h^{1,\natural}_{\alpha\beta})\|_{L^{\infty}_{x}L^{2}_{y}(\mathscr{H}_{s})}\\\
&\leq\begin{cases}2C_{2}\epsilon,\ &\text{if }|I|\leq N_{1},\ |J|=0\\\
2C_{2}\epsilon s^{\gamma_{k}},\ &\text{if }|I|+|J|\leq N_{1}+1,\ |J|=k\leq
N_{1}.\\\ \end{cases}\end{split}$

The result we aim to prove states the following

######  Proposition 4.1.

There exist two constants $1\ll C_{1}\ll C_{2}$ sufficiently large, a finite
and increasing sequence of parameters $0\leq\zeta_{k},\gamma_{k},\delta_{k}\ll
1$ satisfying (4.2) and $0<\epsilon_{0}\ll 1$ sufficiently small such that for
every $0<\epsilon<\epsilon_{0}$, if $h^{1}_{\alpha\beta}$ is solution to (1.8)
in the hyperbolic strip $\mathscr{H}_{[s_{0},S_{0})}$ that satisfies the
bounds (4.3)-(4.7) for all $s\in[s_{0},S_{0})$ and the energy bounds
(3.1)-(3.2) globally in the exterior region, then for every
$s\in[s_{0},S_{0})$ it actually satisfies the following:
for multi-indexes $K$ of type $(N,k)$

(4.8) $\displaystyle E^{\text{i}}(s,\partial
Z^{K}h^{1}_{\alpha\beta})^{\frac{1}{2}}+E^{\text{i}}(s,Z^{K}h^{1}_{\alpha\beta})^{\frac{1}{2}}\leq
C_{1}\epsilon s^{\frac{1}{2}+\zeta_{k}}$ (4.9) $\displaystyle
E^{\text{i}}(s,Z^{K}h^{1,\flat}_{\alpha\beta})^{\frac{1}{2}}\leq C_{1}\epsilon
s^{\zeta_{k}};$

for multi-indexes $K$ of type $(N,k)$ with $k\leq N_{1}$

(4.10) $E^{\text{i}}(s,Z^{K}h^{1}_{\alpha\beta})^{\frac{1}{2}}\leq
C_{1}\epsilon s^{\delta_{k}}$

and finally

(4.11)
$\|t\,\Gamma^{J}h^{1,\flat}_{\alpha\beta}\|_{L^{\infty}_{x}(\mathscr{H}_{s})}\leq
C_{2}\epsilon s^{\gamma_{k}}\quad\text{if }|J|=k\leq N_{1},$ (4.12)
$\displaystyle\|t^{\frac{1}{2}}s\,\partial_{tx}(\partial^{I}\Gamma^{J}h^{1,\natural}_{\alpha\beta})\|_{L^{\infty}_{x}L^{2}_{y}(\mathscr{H}_{s})}$
$\displaystyle+\|t^{\frac{3}{2}}\partial^{\leq
1}_{y}(\partial^{I}\Gamma^{J}h^{1,\natural}_{\alpha\beta})\|_{L^{\infty}_{x}L^{2}_{y}(\mathscr{H}_{s})}$
$\displaystyle\leq\begin{cases}C_{2}\epsilon,\ &\text{if }|I|\leq N_{1},\
|J|=0\\\ C_{2}\epsilon s^{\gamma_{k}},\ &\text{if }|I|+|J|\leq N_{1}+1,\
|J|=k\leq N_{1}\\\ \end{cases}$

######  Remark 4.2.

The a-priori assumptions (4.3)-(4.7) are satisfied when $s=s_{0}$ as a
consequence of the assumptions on the initial data and the local existence
result for the Einstein equations. The hyperbolic time $S_{0}$ in the above
proposition is arbitrary. This implies the existence of the solution in the
unbounded region $\mathscr{H}_{[s_{0},\infty)}$, hence in the full interior
region.

######  Remark 4.3.

The result stated above builds upon the energy and pointwise estimates the
solution has been proved to satisfy in the exterior region. This can be
already seen in the energy inequality (4.16) below, where the energy flux
through the separating hypersurface $\mathscr{H}_{[s_{0},s]}$, which is
controlled by the exterior energies, appears in the right hand side of the
inequality. Constants $C_{1},\gamma_{k},\delta_{k}$ in proposition 4.1 will in
particular be chosen relative to $C_{0},\sigma,\kappa$ so that $C_{1}\gg
C_{0}$, $\sigma\ll\gamma_{k}\ll\delta_{k}\ll\kappa$ and
$\delta_{k}\ll\kappa-\sigma$ for all $k=0,\dots,N$. For this reason and
throughout the rest of this section, we will often replace $C_{0}$ by $C_{1}$
in the inequalities obtained using bounds recovered in the exterior region.

In order to recover the enhanced energy bounds (4.8)-(4.10), we compare the
equation satisfied by $Z^{K}h^{1}_{\alpha\beta}$ and
$Z^{K}h^{1,\flat}_{\alpha\beta}$ respectively with (A.1) and apply the energy
inequality of proposition A.2. We recall that $Z^{K}h^{1}_{\alpha\beta}$
satisfies the following quasilinear wave equation

(4.13)
$\tilde{\Box}_{g}Z^{K}h^{1}_{\alpha\beta}=F^{K}_{\alpha\beta}+F^{0,K}_{\alpha\beta}$

with source terms

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

and that the equation of $Z^{K}h^{1,\flat}_{\alpha\beta}$ is obtained by
averaging (4.13) over ${\mathbb{S}}^{1}$

(4.15)
$\Box_{x}Z^{K}h^{1,\flat}_{\alpha\beta}+(H^{{\bm{\mu}}{\bm{\nu}}})^{\flat}\cdot\partial_{\bm{\mu}}\partial_{\bm{\nu}}Z^{K}h^{1,\flat}_{\alpha\beta}+\big{(}(H^{\mu\nu})^{\natural}\cdot\partial_{\mu}\partial_{\nu}Z^{K}h^{1,\natural}_{\alpha\beta}\big{)}^{\flat}=F^{K,\flat}_{\alpha\beta}+F^{0,K}_{\alpha\beta}$

where
$F^{K,\flat}_{\alpha\beta}=\fint_{{\mathbb{S}}^{1}}F^{K}_{\alpha\beta}dy$. If
the tensor $H$ satisfies suitable decay bounds in
$\mathscr{H}_{[s_{0},S_{0})}$, e.g. if for some $\delta>0$

$|H(t,x,y)|\lesssim\frac{\epsilon}{(1+t+r)^{\frac{3}{4}}},\quad|H^{1}_{LL}(t,x,y)|\lesssim\frac{\epsilon}{(1+t+r)^{1+\delta}}$

we derive the following two energy inequalities, which hold for any
$s\in[s_{0},S_{0})$

(4.16) $\displaystyle E^{\text{i}}(s,Z^{K}h^{1}_{\alpha\beta})\lesssim
E^{\text{i}}(s_{0},Z^{K}h^{1}_{\alpha\beta})$
$\displaystyle+\int_{\tilde{\mathscr{H}}_{s_{0}s}}\Big{(}\frac{1}{2(1+r^{2})}+\chi\left(\frac{r}{t}\right)\chi(r)\frac{M}{2r}\Big{)}|\partial_{t}Z^{K}h^{1}_{\alpha\beta}|^{2}+|\underline{\nabla}Z^{K}h^{1}_{\alpha\beta}|^{2}\,dxdy$
$\displaystyle+\iint_{\mathscr{H}_{[s_{0},s]}}|F^{K}_{\alpha\beta}+F^{0,K}_{\alpha\beta}+{\partial^{\mu}H_{\mu}}^{\sigma}\cdot\partial_{\sigma}Z^{K}h^{1}_{\alpha\beta}||\partial_{t}Z^{K}h^{1}_{\alpha\beta}|+\frac{1}{2}|{\partial_{t}H_{\mu}}^{\sigma}\cdot\partial_{\sigma}Z^{K}h^{1}_{\alpha\beta}\cdot\partial^{\mu}Z^{K}h^{1}_{\alpha\beta}|\,dtdxdy$

and

(4.17) $\displaystyle E^{\text{i}}(s,Z^{K}h^{1,\flat}_{\alpha\beta})\lesssim
E^{\text{i}}(s_{0},Z^{K}h^{1,\flat}_{\alpha\beta})$
$\displaystyle+\int_{\tilde{\mathscr{H}}_{s_{0}s}}\Big{(}\frac{1}{2(1+r^{2})}+\chi\left(\frac{r}{t}\right)\chi(r)\frac{M}{2r}\Big{)}|\partial_{t}Z^{K}h^{1,\flat}_{\alpha\beta}|^{2}+|\underline{\nabla}_{x}Z^{K}h^{1,\flat}_{\alpha\beta}|^{2}\,dx$
$\displaystyle+\iint_{\mathscr{H}_{[s_{0},s]}}|F^{K,\flat}_{\alpha\beta}+F^{0,K}_{\alpha\beta}+{\partial^{\mu}H_{\mu}}^{\sigma}\cdot\partial_{\sigma}Z^{K}h^{1,\flat}_{\alpha\beta}||\partial_{t}Z^{K}h^{1,\flat}_{\alpha\beta}|+\frac{1}{2}|{\partial_{t}H_{\mu}}^{\sigma}\cdot\partial_{\sigma}Z^{K}h^{1,\flat}_{\alpha\beta}\cdot\partial^{\mu}Z^{K}h^{1,\flat}_{\alpha\beta}|\,dtdx$
$\displaystyle+\iint_{\mathscr{H}_{[s_{0},s]}}|\big{(}(H^{\mu\nu})^{\natural}\cdot\partial_{\mu}\partial_{\nu}Z^{K}h^{1,\natural}_{\alpha\beta}\big{)}^{\flat}||\partial_{t}Z^{K}h^{1,\flat}_{\alpha\beta}|\,dtdx.$

The energy flux through the boundary $\tilde{\mathscr{H}}_{s_{0}s}$, which
appears in the right hand side of both of the above inequalities, is suitably
controlled using (3.50) with $t=t_{s}=s^{2}/2$

(4.18)
$\int_{\tilde{\mathscr{H}}_{s_{0}s}}\Big{(}\frac{1}{2(1+r^{2})}+\chi\left(\frac{r}{t}\right)\chi(r)\frac{M}{2r}\Big{)}|\partial_{t}Z^{K}h^{1}_{\alpha\beta}|^{2}+|\underline{\nabla}_{xy}Z^{K}h^{1}_{\alpha\beta}|^{2}\,dxdy\lesssim
C_{0}^{2}\epsilon^{2}s^{2\sigma+C\epsilon}.$

The current section is therefore mainly devoted to estimating the remaining
integrals in the right hand side of (4.16) and (4.17).

###  4.1. First sets of bounds

Below is a list of $L^{2}$ and $L^{\infty}$ bounds for
$h^{1}_{\alpha\beta},h^{1,\flat}_{\alpha\beta}$ and
$h^{1,\natural}_{\alpha\beta}$, which are a straightforward consequence of the
a-priori bounds. All bounds stated below hold true also for tensor
coefficients $H^{1,\mu\nu}$, after decomposition (1.12) and bound (3.11).

####  4.1.1. $L^{2}_{xy}$ bounds on hyperboloids

From a-priori energy assumptions (4.3)-(4.4), the Parseval identity and
Poincaré inequality applied to the zero-average components
$h^{1,\natural}_{\alpha\beta}$, we derive the following $L^{2}$ bounds on
$\mathscr{H}_{s}$, for any multi-index of type $(N,k)$ and $i=0,1$,

(4.19)
$\displaystyle\left\|(s/t)\partial\partial^{i}Z^{K}h^{1}_{\alpha\beta}\right\|_{L^{2}(\mathscr{H}_{s})}+\left\|\underline{\partial}\partial^{i}Z^{K}h^{1}_{\alpha\beta}\right\|_{L^{2}(\mathscr{H}_{s})}\leq
2C_{1}\epsilon s^{\frac{1}{2}+\zeta_{k}}$ (4.20)
$\displaystyle\left\|(s/t)\partial
Z^{K}h^{1,\flat}_{\alpha\beta}\right\|_{L^{2}(\mathscr{H}_{s})}+\left\|\underline{\partial}Z^{K}h^{1,\flat}_{\alpha\beta}\right\|_{L^{2}(\mathscr{H}_{s})}\leq
2C_{1}\epsilon s^{\zeta_{k}}$ (4.21) $\displaystyle\left\|(s/t)\partial
Z^{K}h^{1,\natural}_{\alpha\beta}\right\|_{L^{2}(\mathscr{H}_{s})}+\left\|\underline{\partial}Z^{K}h^{1,\natural}_{\alpha\beta}\right\|_{L^{2}(\mathscr{H}_{s})}+\left\|Z^{K}h^{1,\natural}_{\alpha\beta}\right\|_{L^{2}(\mathscr{H}_{s})}\leq
2C_{1}\epsilon s^{\frac{1}{2}+\zeta_{k}}$

and

(4.22)
$\displaystyle\left\|t^{-1}\mathscr{S}Z^{K}h^{1,\flat}_{\alpha\beta}\right\|_{L^{2}(\mathscr{H}_{s})}+\left\|t^{-1}\Gamma
Z^{K}h^{1,\flat}_{\alpha\beta}\right\|_{L^{2}(\mathscr{H}_{s})}\leq
2C_{1}\epsilon s^{\zeta_{k}}$ (4.23)
$\displaystyle\left\|t^{-1}\mathscr{S}Z^{K}h^{1,\natural}_{\alpha\beta}\right\|_{L^{2}(\mathscr{H}_{s})}+\left\|t^{-1}\Gamma
Z^{K}h^{1,\natural}_{\alpha\beta}\right\|_{L^{2}(\mathscr{H}_{s})}\leq
2C_{1}\epsilon s^{\frac{1}{2}+\zeta_{k}}.$

For multi-indexes $K$ of type $(N,k)$ with $k\leq N_{1}$ we have

(4.24) $\displaystyle\left\|(s/t)\partial
Z^{K}h^{1}_{\alpha\beta}\right\|_{L^{2}(\mathscr{H}_{s})}+\left\|\underline{\partial}Z^{K}h^{1}_{\alpha\beta}\right\|_{L^{2}(\mathscr{H}_{s})}\leq
2C_{1}\epsilon s^{\delta_{k}}$ (4.25) $\displaystyle\left\|(s/t)\partial
Z^{K}h^{1,\natural}_{\alpha\beta}\right\|_{L^{2}(\mathscr{H}_{s})}+\left\|\underline{\partial}Z^{K}h^{1,\natural}_{\alpha\beta}\right\|_{L^{2}(\mathscr{H}_{s})}+\left\|Z^{K}h^{1,\natural}_{\alpha\beta}\right\|_{L^{2}(\mathscr{H}_{s})}\leq
2C_{1}\epsilon s^{\delta_{k}}$

and

(4.26)
$\displaystyle\left\|t^{-1}\mathscr{S}Z^{K}h^{1}_{\alpha\beta}\right\|_{L^{2}(\mathscr{H}_{s})}+\left\|t^{-1}\Gamma
Z^{K}h^{1}_{\alpha\beta}\right\|_{L^{2}(\mathscr{H}_{s})}\leq 2C_{1}\epsilon
s^{\delta_{k}}.$

Moreover, provided that $\sigma,\epsilon\ll 1$ are sufficiently small so that
$\sigma+C\epsilon\leq\zeta_{k}$ for all $1\leq k\leq N$, from the Hardy
inequality (B.7), energy assumption (4.19) and the exterior energy bound (3.3)
(recall that $t_{s}=s^{2}/2$) we also deduce the following bound when
$|J|=k\leq N$

(4.27)
$\displaystyle\left\|r^{-1}\Gamma^{J}h^{1}_{\alpha\beta}\right\|_{L^{2}_{xy}(\mathscr{H}_{s})}\lesssim
2C_{1}\epsilon s^{\frac{1}{2}+\zeta_{k}}+C_{0}\epsilon
s^{\sigma+C\epsilon}\lesssim 2C_{1}\epsilon s^{\frac{1}{2}+\zeta_{k}}$ (4.28)
$\displaystyle\left\|r^{-1}\Gamma^{J}h^{1,\flat}_{\alpha\beta}\right\|_{L^{2}_{x}(\mathscr{H}_{s})}\lesssim
2C_{1}\epsilon s^{\zeta_{k}}+C_{0}\epsilon s^{\sigma+C\epsilon}\lesssim
2C_{1}\epsilon s^{\zeta_{k}}.$

For $|J|=k\leq N_{1}$, we instead get from (4.24) that

(4.29)
$\left\|r^{-1}\Gamma^{J}h^{1}_{\alpha\beta}\right\|_{L^{2}_{xy}(\mathscr{H}_{s})}\lesssim
2C_{1}\epsilon s^{\delta_{k}}+C_{0}\epsilon s^{\sigma+C\epsilon}\lesssim
2C_{1}\epsilon s^{\delta_{k}}.$

####  4.1.2. $L^{\infty}_{x}L^{2}_{y}$ bounds on hyperboloids

These are obtained using the Poincaré inequality, lemma B.5, relation
$\underline{\partial}_{{\bm{i}}}=t^{-1}\Omega_{0{\bm{i}}}$ and energy
assumption (4.21).
For multi-indexes $K$ of type $(N-2,k)$

(4.30) $\displaystyle\left\|t^{\frac{3}{2}}\,\partial_{y}^{\leq
1}Z^{K}h^{1,\natural}_{\alpha\beta}\right\|_{L^{\infty}_{x}L^{2}_{y}(\mathscr{H}_{s})}+\left\|t^{\frac{1}{2}}s\,\partial_{tx}Z^{K}h^{1,\natural}_{\alpha\beta}\right\|_{L^{\infty}_{x}L^{2}_{y}(\mathscr{H}_{s})}\lesssim
C_{1}\epsilon s^{\frac{1}{2}+\zeta_{k+2}};$

for multi-indices $K$ of type $(N-3,k)$

(4.31) $\displaystyle\left\|t^{\frac{5}{2}}\,\partial_{y}^{\leq
1}\underline{\partial}_{x}Z^{K}h^{1,\natural}_{\alpha\beta}\right\|_{L^{\infty}_{x}L^{2}_{y}(\mathscr{H}_{s})}+\left\|t^{\frac{3}{2}}s\,\partial_{tx}\underline{\partial}_{x}Z^{K}h^{1,\natural}_{\alpha\beta}\right\|_{L^{\infty}_{x}L^{2}_{y}(\mathscr{H}_{s})}\lesssim
C_{1}\epsilon s^{\frac{1}{2}+\zeta_{k+3}};$

for multi-indexes $K$ of type $(N-4,k)$

(4.32) $\displaystyle\left\|t^{\frac{7}{2}}\,\partial_{y}^{\leq
1}\underline{\partial}^{2}_{x}Z^{K}h^{1,\natural}_{\alpha\beta}\right\|_{L^{\infty}_{x}L^{2}_{y}(\mathscr{H}_{s})}+\left\|t^{\frac{5}{2}}s\,\partial_{tx}\underline{\partial}^{2}_{x}Z^{K}h^{1,\natural}_{\alpha\beta}\right\|_{L^{\infty}_{x}L^{2}_{y}(\mathscr{H}_{s})}\lesssim
C_{1}\epsilon s^{\frac{1}{2}+\zeta_{k+4}}.$

Moreover, for multi-indexes $K$ of type $(N-2,k)$ with $k\leq N_{1}-2$

(4.33) $\displaystyle\left\|t^{\frac{3}{2}}\,\partial_{y}^{\leq
1}Z^{K}h^{1,\natural}_{\alpha\beta}\right\|_{L^{\infty}_{x}L^{2}_{y}(\mathscr{H}_{s})}+\left\|t^{\frac{1}{2}}s\,\partial_{tx}Z^{K}h^{1,\natural}_{\alpha\beta}\right\|_{L^{\infty}_{x}L^{2}_{y}(\mathscr{H}_{s})}\lesssim
C_{1}\epsilon s^{\delta_{k+2}};$

for multi-indices $K$ of type $(N-3,k)$ with $k\leq N_{1}-3$

(4.34) $\displaystyle\left\|t^{\frac{5}{2}}\,\partial_{y}^{\leq
1}\underline{\partial}_{x}Z^{K}h^{1,\natural}_{\alpha\beta}\right\|_{L^{\infty}_{x}L^{2}_{y}(\mathscr{H}_{s})}+\left\|t^{\frac{3}{2}}s\,\partial_{tx}\underline{\partial}_{x}Z^{K}h^{1,\natural}_{\alpha\beta}\right\|_{L^{\infty}_{x}L^{2}_{y}(\mathscr{H}_{s})}\lesssim
C_{1}\epsilon s^{\delta_{k+3}};$

for multi-indexes $K$ of type $(N-4,k)$ with $k\leq N_{1}-4$

(4.35) $\displaystyle\left\|t^{\frac{7}{2}}\,\partial_{y}^{\leq
1}\underline{\partial}^{2}_{x}Z^{K}h^{1,\natural}_{\alpha\beta}\right\|_{L^{\infty}_{x}L^{2}_{y}(\mathscr{H}_{s})}+\left\|t^{\frac{5}{2}}s\,\partial_{tx}\underline{\partial}^{2}_{x}Z^{K}h^{1,\natural}_{\alpha\beta}\right\|_{L^{\infty}_{x}L^{2}_{y}(\mathscr{H}_{s})}\lesssim
C_{1}\epsilon s^{\delta_{k+4}}.$

####  4.1.3. $L^{\infty}_{xy}$ bounds on hyperboloids

These are obtained from the energy assumptions using Poincaré inequality,
lemma B.5 and Sobolev embedding on ${\mathbb{S}}^{1}$.
For any multi-index $K$ of type $(N-3,k)$ and $i=0,1$

(4.36)
$\displaystyle\left\|t^{\frac{1}{2}}s^{\frac{1}{2}}\,\partial\partial^{i}Z^{K}h^{1}_{\alpha\beta}\right\|_{L^{\infty}_{xy}(\mathscr{H}_{s})}+\left\|t^{\frac{3}{2}}s^{-\frac{1}{2}}\,\underline{\partial}\partial^{i}Z^{K}h^{1}_{\alpha\beta}\right\|_{L^{\infty}_{xy}(\mathscr{H}_{s})}\lesssim
C_{1}\epsilon s^{\zeta_{k+3}}$ (4.37)
$\displaystyle\left\|t^{\frac{1}{2}}s\,\partial
Z^{K}h^{1,\natural}_{\alpha\beta}\right\|_{L^{\infty}_{xy}(\mathscr{H}_{s})}+\left\|t^{\frac{3}{2}}\,\underline{\partial}Z^{K}h^{1,\natural}_{\alpha\beta}\right\|_{L^{\infty}_{xy}(\mathscr{H}_{s})}+\left\|t^{\frac{3}{2}}Z^{K}h^{1,\natural}_{\alpha\beta}\right\|_{L^{\infty}_{xy}(\mathscr{H}_{s})}\lesssim
C_{1}\epsilon s^{\frac{1}{2}+\zeta_{k+3}}$ (4.38)
$\displaystyle\left\|t^{\frac{1}{2}}s\,\partial
Z^{K}h^{1,\flat}_{\alpha\beta}\right\|_{L^{\infty}_{x}(\mathscr{H}_{s})}+\left\|t^{\frac{3}{2}}\,\underline{\partial}Z^{K}h^{1,\flat}_{\alpha\beta}\right\|_{L^{\infty}_{x}(\mathscr{H}_{s})}\lesssim
C_{1}\epsilon s^{\zeta_{k+2}}$

and

(4.39)
$\displaystyle\|t^{\frac{1}{2}}\,\mathscr{S}\partial^{i}Z^{K}h^{1}_{\alpha\beta}\|_{L^{\infty}_{xy}(\mathscr{H}_{s})}+\|t^{\frac{1}{2}}\,\Gamma\partial^{i}Z^{K}h^{1}_{\alpha\beta}\|_{L^{\infty}_{xy}(\mathscr{H}_{s})}\lesssim
C_{1}\epsilon s^{\frac{1}{2}+\zeta_{k+3}}$ (4.40)
$\displaystyle\|t^{\frac{1}{2}}\,\mathscr{S}Z^{K}h^{1,\flat}_{\alpha\beta}\|_{L^{\infty}_{x}(\mathscr{H}_{s})}+\|t^{\frac{1}{2}}\,\Gamma
Z^{K}h^{1,\flat}_{\alpha\beta}\|_{L^{\infty}_{x}(\mathscr{H}_{s})}\lesssim
C_{1}\epsilon s^{\zeta_{k+2}}.$

From the pointwise bounds (4.7) and the Sobolev embedding on
${\mathbb{S}}^{1}$ we also have that

(4.41)
$\left\|t^{\frac{3}{2}}\,\partial^{I}\Gamma^{J}h^{1,\natural}_{\alpha\beta}\right\|_{L^{\infty}_{xy}(\mathscr{H}_{s})}\lesssim\begin{cases}C_{2}\epsilon\
&\text{ if }|I|\leq N_{1},\ |J|=0\\\ C_{2}\epsilon s^{\gamma_{k}}\ &\text{ if
}|I|+|J|\leq N_{1}+1,\ |J|\leq N_{1}\end{cases}$

which coupled to (4.38) gives that, for any $|I|+|J|\leq N_{1}\leq
N-3,|J|=k\geq 0$,

(4.42)
$\|t^{\frac{1}{2}}s\,\partial(\partial^{I}\Gamma^{J}h^{1}_{\alpha\beta})\|_{L^{\infty}_{xy}(\mathscr{H}_{s})}+\|t^{\frac{3}{2}}\underline{\partial}(\partial^{I}\Gamma^{J}h^{1}_{\alpha\beta})\|_{L^{\infty}_{xy}(\mathscr{H}_{s})}\lesssim
C_{2}\epsilon s^{\max(\zeta_{k+2},\gamma_{k})}.$

For $|J|=k\leq N-3$, we also have the following bound on coefficients without
derivatives

(4.43)
$\|t^{\frac{1}{2}}\,\Gamma^{J}h^{1}_{\alpha\beta}\|_{L^{\infty}_{xy}(\mathscr{H}_{s})}\lesssim
C_{1}\epsilon s^{\delta_{k+2}}.$

Such a bound is satisfied by $\Gamma^{J}h^{1,\natural}_{\alpha\beta}$ thanks
to (4.37), while for $\Gamma^{K}h^{1,\flat}_{\alpha\beta}$ it is obtained by
integration. More precisely, on the initial truncated hyperboloid
$\mathscr{H}_{s_{0}}$ such an estimate is obtained by integrating (4.38) along
the hyperboloid itself and up to the boundary
$\partial\mathscr{H}_{s_{0}}=S_{s_{0},r_{0}}$ where
$r_{0}:=\max\\{r>0:S_{s_{0},r}\subset\mathscr{H}_{s_{0}}\\}=\mathscr{O}(1)$.
In fact, for any $r\leq r_{0}$ and $\omega=x/|x|$

$\displaystyle\big{|}\Gamma^{J}h^{1,\flat}_{\alpha\beta}(\sqrt{s_{0}^{2}+|x|^{2}},r\omega)\big{|}$
$\displaystyle\leq\big{|}\Gamma^{J}h^{1,\flat}_{\alpha\beta}(\sqrt{s_{0}^{2}+r_{0}^{2}},r_{0}\omega)\big{|}+\int_{r}^{r_{0}}\big{|}\underline{\partial}\Gamma^{J}h^{1,\flat}_{\alpha\beta}(\sqrt{s_{0}^{2}+\rho^{2}},\rho\omega)\big{|}d\rho$
$\displaystyle\lesssim\big{|}\Gamma^{J}h^{1,\flat}_{\alpha\beta}(\sqrt{s_{0}^{2}+r_{0}^{2}},r_{0}\omega)\big{|}+\int_{r}^{\infty}C_{1}\epsilon(s_{0}^{2}+\rho^{2})^{-\frac{3}{4}+\frac{\zeta_{k+2}}{2}}d\rho$
$\displaystyle\lesssim
C_{1}\epsilon(s_{0}^{2}+r^{2})^{-\frac{1}{4}+\frac{\zeta_{k+2}}{2}}$

where we estimated the first term in the above right hand side using the
exterior bound (3.14). For all other points
$(t,x)\in\mathscr{H}_{(s_{0},S_{0})}$, the decay bound (4.43) is instead
obtained by integrating (4.38) along the rays with $t+r$ and $\omega$ fixed,
i.e. along

$\delta:\lambda\in[r,\lambda^{*}]\mapsto\delta(\lambda)=(t+r-\lambda,\lambda\omega)$

where $\lambda^{*}$ is the first time $\delta(\lambda)$ intersects the lateral
boundary $\tilde{\mathscr{H}}$ (in which case
$\lambda^{*}=\frac{(t+r)(t+r-2)}{2(t+r-1)}$) or the initial hyperboloid
$\mathscr{H}_{s_{0}}$ (in which case
$\lambda^{*}=\frac{(t+r)^{2}-s_{0}^{2}}{2(t+r)}$). In both cases
$\lambda^{*}=O(t+r)$, so from the estimates on $\tilde{\mathscr{H}}$ following
from (3.14) or on the initial truncated hyperboloid $\mathscr{H}_{s_{0}}$
derived above, we get

$\displaystyle|\Gamma^{J}h^{1,\flat}_{\alpha\beta}(t,x)|\lesssim|\Gamma^{J}h^{1,\flat}_{\alpha\beta}(t+r-\lambda^{*},\lambda^{*}\omega)|+\int_{r}^{\lambda^{*}}|(\partial\Gamma^{J}h^{1,\flat}_{\alpha\beta})(\delta(\lambda))|d\lambda$
$\displaystyle\lesssim(C_{0}+C_{1})\epsilon\,(1+t+r)^{-\frac{1}{2}}s^{\zeta_{k+2}}+C_{1}\epsilon(1+t+r)^{-1+\zeta_{k+2}}\int_{r}^{\lambda^{*}}(t+r-2\lambda)^{-\frac{1}{2}+\zeta_{k+2}}d\lambda$
$\displaystyle\lesssim(C_{0}+C_{1})\epsilon\,(1+t+r)^{-\frac{1}{2}}s^{\zeta_{k+2}}.$

####  4.1.4. $L^{\infty}_{xy}$ bounds for the good metric coefficients

These refer to the enhanced bounds satisfied by the metric coefficients
$H^{1}_{LT}$ as a consequence of the wave condition, more precisely of
inequality (2.10), and of the pointwise bounds obtained above. As remarked
above, these bounds are also satisfied by the $h^{1}_{LT}$ metric
coefficients.

######  Proposition 4.4.

Under the assumptions of proposition 4.1, we have that for any
$s\in[s_{0},S_{0})$, any multi-index $K$ of type $(N-3,k)$ and $i=0,1$

(4.44)
$\|t\,\partial\partial^{i}Z^{K}H^{1}_{LT}\|_{L^{\infty}(\mathscr{H}_{s})}\lesssim
C_{1}\epsilon s^{\zeta_{k+3}}$

and for any multi-index $K$ of type $(N_{1},k)$

(4.45) $\|t^{\frac{3}{2}}\partial
Z^{K}H^{1}_{LT}\|_{L^{\infty}(\mathscr{H}_{s})}\lesssim C_{1}\epsilon
s^{\delta_{k+2}}.$

Furthermore, for any multi-index $K$ of type $(N-2,k)$

(4.46) $\displaystyle\|t^{\frac{3}{2}}\partial
Z^{K}(H^{1}_{LT})^{\flat}\|_{L^{\infty}(\mathscr{H}_{s})}\lesssim
C_{1}\epsilon s^{\zeta_{k+2}}$ (4.47)
$\displaystyle\|t^{\frac{1}{2}}(t/s)^{2}Z^{K}(H^{1}_{LT})^{\flat}\|_{L^{\infty}(\mathscr{H}_{s})}\lesssim
C_{1}\epsilon s^{\zeta_{k+2}}.$

###### Proof.

The proof of the above estimates is based on inequality (2.10). Estimate
(4.44) (resp. (4.45)) is in fact obtained using (4.36) (resp. (4.42)) and
(4.43). Estimate (4.46) is deduced similarly, after taking the zero norm of
both left and right hand side of (2.10). We recall, in particular, that for
any two integrable functions $f$ and $g$ defined on ${\mathbb{S}}^{1}$, we
have

(4.48)
$(fg)^{\flat}=f^{\flat}g^{\flat}+\big{(}f^{\natural}g^{\natural}\big{)}^{\flat},\qquad(fg)^{\natural}=f^{\flat}g^{\natural}+f^{\natural}g^{\flat}+(f^{\natural}g^{\natural})^{\natural}.$

Therefore, (4.46) follows from (4.37), (4.38) and (4.43).

Finally, estimate (4.47) is satisfied in the interior of the cone $t=2r$ after
(4.43). In the portion of interior region where $t<2r$, it is instead obtained
from the integration of (4.46) along the rays with $t+r=const$ and
$\omega=const$ and up to the boundary of the interior region. From (3.17) we
derive that

$\displaystyle|Z^{K}(H^{1}_{LT})^{\flat}(t,x)|$
$\displaystyle\lesssim|Z^{K}(H^{1}_{LT})^{\flat}(\delta(\lambda^{*}))|+\int_{r}^{\lambda^{*}}|\partial
Z^{K}(H^{1}_{LT})^{\flat}(\zeta(\lambda))|d\lambda$
$\displaystyle\lesssim|Z^{K}(H^{1}_{LT})^{\flat}(\delta(\lambda^{*}))|+\int_{r}^{\lambda^{*}}C_{1}\epsilon(t+r)^{-\frac{3}{2}+\frac{\zeta_{k+2}}{2}}(t+r-2\lambda)^{\frac{\zeta_{k+2}}{2}}d\lambda$
$\displaystyle\lesssim
C_{0}\epsilon(1+t+r)^{-\frac{3}{2}+2\sigma}(t-r)^{\frac{1}{2}-\kappa}+C_{1}\epsilon(t+r)^{-\frac{3}{2}+\frac{\zeta_{k+2}}{2}}(t-r)^{1+\frac{\zeta_{k+2}}{2}}$
$\displaystyle\lesssim
C_{1}\epsilon\frac{(t^{2}-r^{2})^{1+\frac{\zeta_{k+2}}{2}}}{t^{2}}t^{-\frac{1}{2}}.$

∎

######  Remark 4.5.

By combining together the wave gauge estimate (2.10), with the energy bounds
(4.20) and (4.28) (respectively (4.19) and (4.27)) and the pointwise bounds
(4.38) and (4.43) (respectively (4.42) and (4.43)), we obtain the following
estimate (resp. the second)

(4.49) $\big{\|}\partial
Z^{K}H^{1,\flat}_{LT}\big{\|}_{L^{2}(\mathscr{H}_{s})}\lesssim
C_{1}\epsilon\begin{cases}s^{2\zeta_{k}}\quad&\text{if }K\text{ is of type
}(N,k),\\\ s^{\frac{1}{2}+2\zeta_{k}}\quad&\text{if }K\text{ is of type
}(N+1,k)\text{ with }k\leq N.\end{cases}$

###  4.2. The null and cubic terms

The $L^{2}$ and $L^{\infty}$ bounds deduced in subsection 4.1 from the
a-priori energy bounds, coupled with the a-priori pointwise bounds, allow us
to suitably estimate the null and cubic contributions appearing in the
equations for $Z^{K}h^{1}_{\alpha\beta}$ and $Z^{K}h^{1,\flat}_{\alpha\beta}$.
Quadratic and cubic interactions involving a $h^{0}$ factor are the simplest
ones to analyze. They satisfy the following estimates, which follow from a
straightforward application of the energy bounds (4.19), (4.27) and the
pointwise bounds (3.11), (4.42) and (4.43).

######  Lemma 4.6.

Let $Q=Q(\psi,\phi)$ and $C=C(\delta)(\phi,\psi)$ denote a quadratic and a
cubic form respectively. Under the a-priori assumptions (4.3)-(4.7), there
exists some small constant $0<\eta\ll 1$ depending linearly on
$\gamma_{k},\delta_{k},\zeta_{k}$, such that for $i=0,1$

(4.50) $\displaystyle\sum_{0\leq l+m\leq 1}\|\partial^{i}Z^{\leq N}Q(\partial
h^{l},\partial h^{m})\|_{L^{2}(\mathscr{H}_{s})}\lesssim
C_{1}\epsilon^{2}s^{-3/2+\eta}$ (4.51)
$\displaystyle\sum_{\begin{subarray}{c}0\leq l+m+n\leq 2\\\ l,m,n\leq
1\end{subarray}}\|\partial^{i}Z^{\leq N}C(h^{l})(\partial h^{m},\partial
h^{n})\|_{L^{2}(\mathscr{H}_{s})}\lesssim C_{1}^{2}\epsilon^{3}s^{-2+\eta}.$

######  Proposition 4.7.

Under the a-priori assumptions (4.3)-(4.7) there exists some small constant
$0<\eta\leq 3\delta_{N}\ll 1$ depending linearly on
$\zeta_{k},\gamma_{k},\delta_{k}$, such that for $i=0,1$

(4.52) $\displaystyle\|\partial^{i}Z^{\leq N}\mathbf{Q}_{\alpha\beta}(\partial
h,\partial
h)\|_{L^{2}(\mathscr{H}_{s})}\lesssim(C_{1}\epsilon)^{2}s^{-1+\eta}$ (4.53)
$\displaystyle\|\partial^{i}Z^{\leq N}G_{\alpha\beta}(h)(\partial h,\partial
h)\|_{L^{2}(\mathscr{H}_{s})}\lesssim(C_{1}\epsilon)^{3}s^{-3/2+\eta}.$

and multi-indexes $K$ of type $(N,k)$ with $k\leq N_{1}$

(4.54) $\displaystyle\|Z^{K}\mathbf{Q}_{\alpha\beta}(\partial h,\partial
h)\|_{L^{2}(\mathscr{H}_{s})}\lesssim(C_{1}\epsilon)^{2}s^{-3/2+\eta}$ (4.55)
$\displaystyle\|Z^{K}G_{\alpha\beta}(h)(\partial h,\partial
h)\|_{L^{2}(\mathscr{H}_{s})}\lesssim(C_{1}\epsilon)^{3}s^{-2+\eta}.$

Moreover, for multi-indexes $K$ of type $(N,k)$

(4.56) $\|Z^{K}\mathbf{Q}^{\flat}_{\alpha\beta}(\partial h,\partial
h)\|_{L^{2}(\mathscr{H}_{s})}\lesssim(C_{1}\epsilon)^{2}\left(\sum_{i=1}^{4}s^{-1+\gamma_{i}+\zeta_{k-i}}+s^{-1+\zeta_{k}}.\right)$

###### Proof.

Throughout the proof, $\eta$ will denote a small positive constant that
depends linearly on $\gamma_{k}$ and $\delta_{k}$. We do not need to keep
track of the explicit value of $\eta$, which may change from line to line.

We start by decomposing each occurrence of $h$ in $\mathbf{Q}_{\alpha\beta}$
and $G_{\alpha\beta}$ into $h^{0}+h^{1}$. Owing to lemma 4.6, we only need to
prove that the above estimates are satisfied for null and cubic interactions
involving $h^{1}$ factors only.

We recall that the admissible vector fields $Z$ preserve the null structure
and that for any null form $Q$

(4.57)
$|Q(\partial\phi,\partial\psi)|\leq|\underline{\partial}\phi||\partial\psi|+|\partial\phi||\underline{\partial}\psi|+\frac{|t^{2}-r^{2}|}{t^{2}}|\partial\phi||\partial\psi|.$

For any $M\in\mathbb{N}$, we then have

$|\partial^{i}Z^{\leq M}\mathbf{Q}_{\alpha\beta}(\partial h^{1},\partial
h^{1})|\lesssim\sum_{\begin{subarray}{c}|K_{1}|+|K_{2}|\leq M\\\
|I_{1}|+|I_{2}|=i\end{subarray}}|\underline{\partial}\partial^{I_{1}}Z^{K_{1}}h^{1}||\partial\partial^{I_{2}}Z^{K_{2}}h^{1}|+\frac{|t^{2}-r^{2}|}{t^{2}}|\partial\partial^{I_{1}}Z^{K_{1}}h^{1}||\partial\partial^{I_{2}}Z^{K_{2}}h^{1}|.$

We observe that at least one of the two indexes $K_{j}$ in the above right
hand side has length smaller than $\lfloor M/2\rfloor$. When $M=N$ and since
$N$ and $N_{1}$ are such that $\lfloor N/2\rfloor+1\leq N_{1}$, we deduce from
energy bound (4.19) and pointwise estimate (4.42) that

$\displaystyle\sum_{\begin{subarray}{c}|K_{1}|+|K_{2}|\leq N\\\
|I_{1}|+|I_{2}|=i\end{subarray}}\left\|\underline{\partial}\partial^{I_{1}}Z^{K_{1}}h^{1}\cdot\partial\partial^{I_{2}}Z^{K_{2}}h^{1}\right\|_{L^{2}_{xy}(\mathscr{H}_{s})}+\left\|(s/t)^{2}\partial\partial^{I_{1}}Z^{K_{1}}h^{1}\cdot\partial\partial^{I_{2}}Z^{K_{2}}h^{1}\right\|_{L^{2}_{xy}(\mathscr{H}_{s})}$
$\displaystyle\lesssim\sum_{\begin{subarray}{c}|K_{1}|+|K_{2}|\leq N\\\
|K_{1}|\leq\lfloor N/2\rfloor\\\
|I_{1}|+|I_{2}|=i\end{subarray}}\Big{(}\left\|(t/s)\underline{\partial}\partial^{I_{1}}Z^{K_{1}}h^{1}_{\alpha\beta}\right\|_{L^{\infty}_{xy}(\mathscr{H}_{s})}+\left\|(s/t)\partial\partial^{I_{1}}Z^{K_{1}}h^{1}_{\alpha\beta}\right\|_{L^{\infty}_{xy}(\mathscr{H}_{s})}\Big{)}\left\|(s/t)\partial\partial^{I_{2}}Z^{K_{2}}h^{1}_{\alpha\beta}\right\|_{L^{2}_{xy}(\mathscr{H}_{s})}$
$\displaystyle+\sum_{\begin{subarray}{c}|K_{1}|+|K_{2}|\leq N\\\
|K_{2}|\leq\lfloor N/2\rfloor\\\
|I_{1}|+|I_{2}|=i\end{subarray}}\left\|\partial\partial^{I_{2}}Z^{K_{2}}h^{1}_{\alpha\beta}\right\|_{L^{\infty}_{xy}(\mathscr{H}_{s})}\left\|\underline{\partial}\partial^{I_{1}}Z^{K_{1}}h^{1}_{\alpha\beta}\right\|_{L^{2}_{xy}(\mathscr{H}_{s})}\lesssim(C_{1}\epsilon)^{2}s^{-1+\eta}.$

Similarly, when $K$ is of type $(N,k)$ with $k\leq N_{1}$, we get from energy
bound (4.24) and pointwise bound (4.42) that

$\displaystyle\sum_{\begin{subarray}{c}|K_{1}|+|K_{2}|\leq N\\\
|I_{1}|+|I_{2}|=i\end{subarray}}\left\|\underline{\partial}\partial^{I_{1}}Z^{K_{1}}h^{1}\cdot\partial\partial^{I_{2}}Z^{K_{2}}h^{1}\right\|_{L^{2}_{xy}(\mathscr{H}_{s})}+\left\|(s/t)^{2}\partial\partial^{I_{1}}Z^{K_{1}}h^{1}\cdot\partial\partial^{I_{2}}Z^{K_{2}}h^{1}\right\|_{L^{2}_{xy}(\mathscr{H}_{s})}$
$\displaystyle\lesssim(C_{1}\epsilon)^{2}s^{-\frac{3}{2}+\eta}.$

Estimate (4.52) can be slightly improved if we only consider the zero mode of
the quadratic null interactions. We recall that the zero-mode of a product
decomposes as in (4.48), hence

$\mathbf{Q}^{\flat}_{\alpha\beta}(\partial h^{1},\partial
h^{1})=\mathbf{Q}_{\alpha\beta}(\partial h^{1,\flat},\partial
h^{1,\flat})+\big{(}\mathbf{Q}_{\alpha\beta}(\partial h^{1,\natural},\partial
h^{1,\natural})\big{)}^{\flat}.$

The pure zero-mode interactions are treated using the null structure. From the
energy bound (4.20) and the pointwise bound (4.38) we derive that

$\displaystyle\left\|Z^{\leq N}\mathbf{Q}_{\alpha\beta}(\partial
h^{1,\flat},\partial
h^{1,\flat})\right\|_{L^{2}_{x}(\mathscr{H}_{s})}\lesssim\epsilon^{2}s^{-\frac{3}{2}+\eta}.$

The null structure is instead irrelevant when estimating the quadratic
interactions of pure non-zero modes. Using the Cauchy-Schwartz and Poincaré
inequalities and assuming $N,N_{1}$ are such that $\lfloor N/2\rfloor+1\leq
N_{1}$, we derive from the pointwise bound (4.7), the energy bounds (4.19),
(4.21) and (4.25) (recall that $N_{1}=N-5$) and relation (4.2) that

$\displaystyle\left\|\big{(}Z^{K}\mathbf{Q}_{\alpha\beta}(\partial
h^{1,\natural},\partial
h^{1,\natural})\big{)}^{\flat}\right\|_{L^{2}_{x}(\mathscr{H}_{s})}$
$\displaystyle\lesssim\sum_{|K_{1}|+|K_{2}|\leq|K|}\left\|\partial
Z^{K_{1}}h^{1,\natural}\cdot\partial
Z^{K_{2}}h^{1,\natural}\right\|_{L^{1}_{y}L^{2}_{x}(\mathscr{H}_{s})}$
$\displaystyle\lesssim\sum_{\begin{subarray}{c}|K_{1}|+|K_{2}|\leq|K|\\\
|K_{1}|\leq\lfloor|K|/2\rfloor\end{subarray}}\big{\|}\partial
Z^{K_{1}}h^{1,\natural}\big{\|}_{L^{\infty}_{x}L^{2}_{y}(\mathscr{H}_{s})}\big{\|}\partial
Z^{K_{2}}h^{1,\natural}\big{\|}_{L^{2}_{xy}(\mathscr{H}_{s})}$
$\displaystyle\lesssim
C_{1}C_{2}\epsilon^{2}(s^{-\frac{3}{2}+\eta}+s^{-1+\zeta_{k}}+\sum_{i=0}^{4}s^{-1+\gamma_{i}+\zeta_{k-i}}).$

As concerns the cubic terms, we have that

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

When $M=N$, we get from energy bound (4.19) and pointwise bounds (4.42),
(4.43) that

$\sum_{\begin{subarray}{c}|K_{1}|+|K_{2}|+|K_{3}|\leq N\\\
|K_{1}|+|K_{2}|\leq\lfloor N/2\rfloor\\\
|I_{1}|+|I_{2}|+|I_{3}|=i\end{subarray}}\left\|\partial^{I_{1}}Z^{K_{1}}h^{1}\cdot\partial\partial^{I_{2}}Z^{K_{2}}h^{1}\cdot\partial\partial^{I_{3}}Z^{K_{3}}h^{1}\right\|_{L^{2}_{xy}(\mathscr{H}_{s})}\\\
\lesssim\sum_{\begin{subarray}{c}|K_{1}|+|K_{2}|+|K_{3}|\leq N\\\
|K_{1}|+|K_{2}|\leq\lfloor N/2\rfloor\\\
|I_{1}|+|I_{2}|+|I_{3}|=i\end{subarray}}\left\|(t/s)\partial^{I_{1}}Z^{K_{1}}h^{1}\cdot\partial\partial^{I_{2}}Z^{K_{2}}h^{1}\right\|_{L^{\infty}_{xy}(\mathscr{H}_{s})}\left\|(s/t)\partial\partial^{I_{3}}Z^{K_{3}}h^{1}\right\|_{L^{2}_{xy}(\mathscr{H}_{s})}\\\
\lesssim(C_{1}\epsilon)^{3}s^{-\frac{3}{2}+\eta}$

and from (4.27), (4.36) and (4.42)

$\displaystyle\sum_{\begin{subarray}{c}|K_{1}|+|K_{2}|+|K_{3}|\leq N\\\
|K_{2}|+|K_{3}|\leq\lfloor N/2\rfloor\\\
|I_{2}|+|I_{3}|=i\end{subarray}}\left\|Z^{K_{1}}h^{1}\cdot\partial\partial^{I_{2}}Z^{K_{2}}h^{1}\cdot\partial\partial^{I_{3}}Z^{K_{3}}h^{1}\right\|_{L^{2}_{xy}(\mathscr{H}_{s})}$
$\displaystyle\lesssim\sum_{\begin{subarray}{c}|K_{1}|+|K_{2}|+|K_{3}|\leq
N\\\ |K_{2}|+|K_{3}|\leq\lfloor N/2\rfloor\\\
|I_{2}|+|I_{3}|=i\end{subarray}}\left\|r^{-1}Z^{K_{1}}h^{1}\right\|_{L^{2}_{xy}(\mathscr{H}_{s})}\left\|r\partial\partial^{I_{2}}Z^{K_{2}}h^{1}\cdot\partial\partial^{I_{3}}Z^{K_{3}}h^{1}\right\|_{L^{\infty}_{xy}(\mathscr{H}_{s})}\lesssim(C_{1}\epsilon)^{3}s^{-\frac{3}{2}+\eta}.$

Similarly, we deduce from (4.24), (4.42), (4.43) that when $K$ is of type
$(N,k)$ with $k\leq N_{1}$

$\sum_{\begin{subarray}{c}|K_{1}|+|K_{2}|+|K_{3}|\leq N\\\
|K_{1}|+|K_{2}|\leq\lfloor N/2\rfloor\\\
|I_{1}|+|I_{2}|+|I_{3}|=i\end{subarray}}\left\|\partial^{I_{1}}Z^{K_{1}}h^{1}\cdot\partial\partial^{I_{2}}Z^{K_{2}}h^{1}\cdot\partial\partial^{I_{3}}Z^{K_{3}}h^{1}\right\|_{L^{2}_{xy}(\mathscr{H}_{s})}\lesssim(C_{1}\epsilon)^{3}s^{-2+\eta}$

while from (4.29) and (4.42)

$\sum_{\begin{subarray}{c}|K_{1}|+|K_{2}|+|K_{3}|\leq N\\\
|K_{2}|+|K_{3}|\leq\lfloor N/2\rfloor\\\
|I_{2}|+|I_{3}|=i\end{subarray}}\left\|Z^{K_{1}}h^{1}\cdot\partial\partial^{I_{2}}Z^{K_{2}}h^{1}\cdot\partial\partial^{I_{3}}Z^{K_{3}}h^{1}\right\|_{L^{2}_{xy}(\mathscr{H}_{s})}\lesssim(C_{1}\epsilon)^{3}s^{-2+\eta}.$

∎

######  Lemma 4.8.

There exists some small constant $0<\eta\leq 3\delta_{N}\ll 1$ depending
linearly on $\zeta_{k},\gamma_{k},\delta_{k}$ such that for multi-indexes $K$
of type $(N_{1},k)$ we have

(4.58) $\displaystyle\|t^{2}Z^{K}\mathbf{Q}_{\alpha\beta}(\partial h,\partial
h)\|_{L^{\infty}_{xy}(\mathscr{H}_{s})}$ $\displaystyle\lesssim
C_{2}^{2}\epsilon^{2}s^{-1+\eta}$ (4.59)
$\displaystyle\|t^{\frac{3}{2}}Z^{K}G_{\alpha\beta}(h)(\partial h,\partial
h)\|_{L^{\infty}_{xy}(\mathscr{H}_{s})}$ $\displaystyle\lesssim
C_{1}C_{2}^{2}\epsilon^{3}s^{-2+\eta}.$

while for any quadratic form $N$ we have

(4.60) $\left\|t\,\partial Z^{K}N(\partial h,\partial
h)\right\|_{L^{\infty}_{xy}(\mathscr{H}_{s})}\lesssim
C_{1}^{2}\epsilon^{2}s^{-2+\eta}.$

###### Proof.

This is a direct consequence of bounds (4.42) and (4.43). ∎

###  4.3. Second order derivatives of the zero modes

As expected for solutions to wave equations on ${\mathbb{R}}^{1+3}$, the
second order derivatives of the differentiated coefficients
$Z^{K}h^{1,\flat}_{\alpha\beta}$ enjoy better decay estimates compared to
(4.20) and (4.42) respectively.

######  Lemma 4.9.

There exists some small constant $0<\eta\leq 2\delta_{N}\ll 1$ depending
linearly on $\zeta_{k},\gamma_{k},\delta_{k}$ such that for any multi-index
$K$ of type $(N_{1},k)$

(4.61)
$\left\|t^{\frac{3}{2}}(s/t)^{2}\partial_{t}^{2}Z^{K}h^{1,\flat}_{\alpha\beta}\right\|_{L^{\infty}(\mathscr{H}_{s})}\lesssim
C_{2}\epsilon s^{-1+\eta}.$

###### Proof.

The flat wave operator can be expressed in terms of the hyperbolic derivatives
as

(4.62)
$-\Box_{tx}=(s/t)^{2}\partial^{2}_{t}+2(x^{\bm{a}}/t)\underline{\partial}_{\bm{a}}\partial_{t}-\underline{\partial}^{\bm{a}}\underline{\partial}_{\bm{a}}+\frac{r^{2}}{t^{3}}\partial_{t}+\frac{3}{t}\partial_{t}$

and from (1.15) the curved part can be written as

$(H^{{\bm{\mu}}{\bm{\nu}}})^{\flat}\partial_{\bm{\mu}}\partial_{\bm{\nu}}=(H^{UV})^{\flat}c^{{\bm{\alpha}}{\bm{\beta}}}_{UV}\underline{\partial}_{\bm{\alpha}}\underline{\partial}_{\bm{\beta}}+(H^{UV})^{\flat}d^{\bm{\mu}}_{UV}\underline{\partial}_{\bm{\mu}}.$

Equation (4.15) for $Z^{K}h^{1,\flat}_{\alpha\beta}$ becomes

(4.63)
$\displaystyle\left((s/t)^{2}+(H^{UV})^{\flat}c_{UV}^{00}\right)\partial_{t}^{2}Z^{K}h^{1,\flat}_{\alpha\beta}$
$\displaystyle=S_{1}(Z^{K}h^{1,\flat}_{\alpha\beta})+S_{2}(Z^{K}h^{1,\flat}_{\alpha\beta})$
$\displaystyle+F^{K,\flat}_{\alpha\beta}+(F^{0,K}_{\alpha\beta})^{\flat}-\big{(}(H^{\mu\nu})^{\natural}\cdot\partial_{\mu}\partial_{\nu}Z^{K}h^{1,\natural}_{\alpha\beta}\big{)}^{\flat}$

where $F^{K,\flat}_{\alpha\beta}$ is the average over ${\mathbb{S}}^{1}$ of
the source term in (4.14), and $S_{1}(p),S_{2}(p)$ are defined as follows for
an arbitrary two tensor $p$

(4.64)
$\begin{gathered}S_{1}(p):=-\big{(}2(x^{\bm{a}}/t)\underline{\partial}_{\bm{a}}\partial_{t}-\underline{\partial}^{\bm{a}}\underline{\partial}_{\bm{a}}+\frac{r^{2}}{t^{3}}\partial_{t}+\frac{3}{t}\partial_{t}\big{)}p\\\
S_{2}(p):=-\big{(}(H^{UV})^{\flat}c_{UV}^{{\bm{a}}{\bm{\beta}}}\underline{\partial}_{\bm{a}}\underline{\partial}_{\bm{\beta}}+(H^{UV})^{\flat}c_{UV}^{{\bm{\alpha}}{\bm{b}}}\underline{\partial}_{\bm{\alpha}}\underline{\partial}_{\bm{b}}+(H^{UV})^{\flat}d_{UV}^{{\bm{\mu}}}\underline{\partial}_{\bm{\mu}}\big{)}p.\end{gathered}$

We note that, if $\epsilon$ is sufficiently small, relation (1.18) with
$\pi=(Z^{K}H^{UV})^{\flat}$, bounds (3.11), (4.43) and (4.47) yield

(4.65) $\big{|}(Z^{K}H^{UV})^{\flat}c_{UV}^{00}\big{|}\lesssim C_{1}\epsilon
t^{-1/2}(s/t)^{2}s^{\delta_{k+2}}\lesssim(1/2)(s/t)^{2},$

hence it is enough to prove that the right hand side in (4.63) is bounded by
$C_{1}\epsilon t^{-3/2}s^{-1+2\zeta_{k+3}}$.

This is the case for the $S_{1},S_{2}$ terms with
$p=Z^{K}h^{1,\flat}_{\alpha\beta}$, as follows by using the pointwise bounds
(4.38), (4.43) and (1.17) together with the fact that
$\underline{\partial}_{\bm{a}}=\Omega_{0{\bm{a}}}/t$.

All quadratic and cubic terms in $F^{K,\flat}_{\alpha\beta}$, except for the
commutator terms, are estimated using (4.60) and (4.59) respectively, while
from (3.36), (3.11) and (4.38) we have

$\|t^{3}(F^{0,K}_{\alpha\beta})^{\flat}\|_{L^{\infty}(\mathscr{H}_{s})}\lesssim\epsilon.$

From the pointwise bounds (4.7), (4.37) and relation (4.2), we have that for
multi-indexes $K$ of type $(N_{1},k)$

$\displaystyle\big{|}Z^{K}\big{(}(H^{\mu\nu})^{\natural}\cdot\partial_{\mu}\partial_{\nu}h^{1,\natural}_{\alpha\beta}\big{)}^{\flat}\big{|}$
$\displaystyle\lesssim\sum_{\begin{subarray}{c}|K^{\prime}|\leq|K|\end{subarray}}\big{\|}H^{1,\natural}\big{\|}_{L^{\infty}_{x}L^{2}_{y}}\
\big{\|}\partial_{\mu}\partial_{\nu}Z^{K^{\prime}}h^{1,\natural}_{\alpha\beta}\big{\|}_{L^{\infty}_{x}L^{2}_{y}}+\sum_{\begin{subarray}{c}|K_{1}|+|K_{2}|\leq|K|\\\
|K_{2}|<|K|\end{subarray}}\big{\|}(Z^{K_{1}}H^{1,\mu\nu})^{\natural}\big{\|}_{L^{\infty}_{x}L^{2}_{y}}\big{\|}\partial_{\mu}\partial_{\nu}Z^{K_{2}}h^{1,\natural}_{\alpha\beta}\big{\|}_{L^{\infty}_{x}L^{2}_{y}}$
$\displaystyle\lesssim
C_{1}C_{2}\epsilon^{2}t^{-2}s^{-\frac{1}{2}+\zeta_{k+3}}+C_{2}^{2}\epsilon^{2}t^{-2}s^{-1+\gamma_{k}}.$

We observe that the above bound can be improved if $|K|\leq N_{1}-1$ to
$C_{2}^{2}\epsilon^{2}t^{-2}s^{-1+\gamma_{k}}$.

Finally, for the purpose of this proof it is enough to write the commutator
terms which only involve zero-modes as

$\sum_{\begin{subarray}{c}|K_{1}|+|K_{2}|\leq|K|\\\
|K_{2}|<|K|\end{subarray}}Z^{K_{1}}H^{1,\flat}\cdot\partial^{2}Z^{K_{2}}h^{1,\flat}_{\alpha\beta}$

which is estimated by $C_{1}C_{2}\epsilon^{2}t^{-3/2}s^{-1+2\zeta_{k+2}}$
using (4.38) whenever $Z^{K_{1}}$ contains at least a usual derivative, and
(4.6) together with (4.38) otherwise.

The conclusion of the proof follows by assuming $\epsilon$ sufficiently small
so that $C_{2}\epsilon<1$. ∎

We highlight that the proof of the above Lemma also yields the following

######  Corollary 4.10.

For every multi-index $K$ of type $(N_{1}-1,k)$ we have

(4.66)
$\big{\|}t^{2}\big{(}(H^{1,\mu\nu})^{\natural}\cdot\partial_{\mu}\partial_{\nu}Z^{K}h^{1,\natural}_{\alpha\beta}\big{)}^{\flat}\big{\|}_{L^{\infty}(\mathscr{H}_{s})}+\big{\|}t^{2}\big{(}[Z^{K},H^{1,\mu\nu}\partial_{\mu}\partial_{\nu}]h^{1}_{\alpha\beta}\big{)}^{\flat}\big{\|}_{L^{\infty}(\mathscr{H}_{s})}\lesssim
C_{1}^{2}\epsilon^{2}s^{-1+\eta}.$

######  Lemma 4.11.

There exists some small constant $0<\eta\leq 2\delta_{N}\ll 1$ depending
linearly on $\zeta_{k},\gamma_{k},\delta_{k}$ such that for any multi-index
$K$ of the type $(N-1,k)$ we have

(4.67)
$\displaystyle\left\|(s/t)^{2}\partial^{2}_{t}Z^{K}h^{1,\flat}_{\alpha\beta}\right\|_{L^{2}(\mathscr{H}_{s})}\lesssim
C_{1}\epsilon s^{-1+\eta}.$ (4.68)
$\displaystyle\left\|(s/t)^{2}\partial^{2}_{t}\partial
Z^{K}h^{1,\flat}_{\alpha\beta}\right\|_{L^{2}(\mathscr{H}_{s})}\lesssim
C_{1}\epsilon s^{-\frac{1}{2}+\eta}.$

###### Proof.

This is based on (4.63) and (4.64). We only detail the proof of estimate
(4.67), since (4.68) is obtained in a similar way by replacing energy bound
(4.20) with (4.19) whenever it occurs.

We make use of (4.63), (4.64) and (4.65) to estimate the $S_{1},S_{2}$ terms.
From $\underline{\partial}_{\bm{a}}=(1/t)\Omega_{0{\bm{a}}}$, the energy bound
(4.20) and the pointwise bounds (3.11), (4.43) we derive that

$\sum_{i=0}^{1}\left\|sS_{i}(Z^{K}h^{1,\flat}_{\alpha\beta})\right\|_{L^{2}(\mathscr{H}_{s})}\lesssim\|(s/t)\partial\Gamma^{\leq
1}Z^{K}h^{1,\flat}_{\alpha\beta}\|_{L^{2}(\mathscr{H}_{s})}\lesssim
C_{1}\epsilon s^{\zeta_{k+1}}.$

We recall that the quadratic null terms satisfy (4.56), while the cubic terms
verify (4.53). In general, the zero-mode of a quadratic interaction can be
estimated as follows: using (4.20) and (4.38) we find

$\sum_{|K_{1}|+|K_{2}|\leq|K|}\|\partial Z^{K_{1}}h^{1,\flat}\cdot\partial
Z^{K_{2}}h^{1,\flat}\|_{L^{2}_{x}(\mathscr{H}_{s})}\lesssim\sum_{\begin{subarray}{c}|K^{\prime}|\leq|K|\end{subarray}}C_{1}\epsilon
s^{-1+\delta_{N}}\|(s/t)\partial
Z^{K^{\prime}}h^{1,\flat}\|_{L^{2}(\mathscr{H}_{s})}\lesssim
C_{1}^{2}\epsilon^{2}s^{-1+2\delta_{N}}$

while, since $\lfloor N/2\rfloor+1\leq N_{1}$, from (4.7) and (4.21) we get

$\displaystyle\sum_{|K_{1}|+|K_{2}|\leq|K|}\|\big{(}\partial
Z^{K_{1}}h^{1,\natural}\cdot\partial
Z^{K_{2}}h^{1,\natural}\big{)}^{\flat}\|_{L^{2}_{x}(\mathscr{H}_{s})}\lesssim\sum_{|K^{\prime}|\leq|K|}C_{2}\epsilon
s^{-\frac{3}{2}+\gamma_{N_{1}}}\|(s/t)\partial
Z^{K^{\prime}}h^{1,\natural}\|_{L^{2}_{xy}(\mathscr{H}_{s})}$
$\displaystyle\lesssim C_{1}C_{2}\epsilon^{2}s^{-1+2\delta_{N}}.$

From (4.7) and (4.21) we also have

$\left\|\big{(}(H^{\mu\nu})^{\natural}\cdot\partial_{\mu}\partial_{\nu}Z^{K}h^{1,\natural}_{\alpha\beta}\big{)}^{\flat}\right\|_{L^{2}(\mathscr{H}_{s})}\lesssim
C_{1}C_{2}\epsilon^{2}s^{-1+\zeta_{k}}.$

Finally, from (3.36), estimate (3.11), the Hardy inequality (B.7) and energy
estimates (3.1) and (4.20) we get

(4.69)
$\|Z^{K}\tilde{\Box}_{g}h^{0}\|_{L^{2}(\mathscr{H}_{s})}\lesssim\epsilon
s^{-\frac{3}{2}}.$

∎

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

In this subsection we show that for any $T\in\mathscr{T}$ and
$U\in\mathscr{U}$, the differentiated metric coefficients $\partial Z^{\leq
N_{1}-1}h^{1}_{TU}$ satisfy better lower order decay bounds than the ones in
(4.42) obtained via Klainerman-Sobolev inequalities. More precisely, we want
to prove the following:

######  Proposition 4.12.

There exists a constant $C>0$ such that, for any multi-index $K$ of type
$(N_{1}-1,k)$ and any multi-indexes $I$ with $|I|\leq N_{1}-1$, we have that

(4.70)
$\displaystyle\sup_{{\mathbb{S}}^{1}}|\partial_{\alpha}Z^{K}h^{1}_{TU}|\lesssim
C_{1}\epsilon t^{-1+C\epsilon}$ (4.71)
$\displaystyle\sup_{{\mathbb{S}}^{1}}|\partial_{\alpha}\partial^{I}h^{1}_{TU}|\lesssim
C_{1}\epsilon t^{-1}.$

We observe that the above bounds are satisfied by any $\partial
Z^{K}h^{1,\natural}_{TU}$ with $|K|\leq N_{1}$ as a consequence of (4.41). In
the interior of the cone $t=2r$ they follow immediately from (4.38), where one
has $t^{2}-r^{2}\geq 3t^{2}/4$ and consequently

$|\partial Z^{\leq N-3}h^{1,\flat}_{a\beta}(t,x,y)|\lesssim C_{1}\epsilon
t^{-3/2+\zeta_{N}}.$

For all other points in the interior region, (4.70)-(4.71) are instead
obtained by integration along characteristics, see lemma 4.13. The difference
between $h^{1}_{TU}$ and any general coefficient $h^{1}_{\alpha\beta}$ relies
on the fact that weak null terms do not appear in the equation (3.31)
satisfied by the former. We only sketch the proof of the following lemma, see
[40] for additional details.

######  Lemma 4.13.

Let $s>s_{0}$ and $\mathscr{D}_{s}$ be the set of points $(t,x)$ in the cone
$t/2<|x|<2t-3$ with $t\geq 2$ such that

$|x|\leq\sqrt{t^{2}-s^{2}}\text{ if }(t,x)\in\mathscr{D}^{\text{i}}\quad\text{
or }\quad t\leq s^{2}/2\text{ otherwise.}$

We denote by $\partial_{B}\mathscr{D}_{s}$ the lateral boundary of
$\mathscr{D}_{s}$, i.e.

$\partial_{B}\mathscr{D}_{s}:=\\{(t,x):|x|=t/2\text{ and }t\leq 8/3\text{ or
}|x|=2t-3\text{ and }t\leq s^{2}/2\\}.$

Let $u$ be a solution to the wave equation on the curved 4-dimensional
spacetime $\tilde{\Box}_{g}u=F$, where $g=(g_{{\bm{\mu}}{\bm{\nu}}})$ is a
Lorenztian metric, $g^{-1}=(g^{{\bm{\mu}}{\bm{\nu}}})$ is its inverse and $F$
is some smooth source term. Let $m=(m_{{\bm{\mu}}{\bm{\nu}}})$ denote the
Minkowski metric, $m^{-1}=(m^{{\bm{\mu}}{\bm{\nu}}})$ its inverse and
$\pi^{{\bm{\mu}}{\bm{\nu}}}:=g^{{\bm{\mu}}{\bm{\nu}}}-m^{{\bm{\mu}}{\bm{\nu}}}$.
For any spacetime point $(t,x)\in\mathscr{D}_{s}$, let
$(\tau,\varphi(\tau;t,x))$ be the integral curve of the vector field

$\partial_{t}+\frac{1-\pi_{LL}/4}{1+\pi_{LL}/4}\partial_{r}$

passing through $(t,x)$, i.e. $\varphi(t;t,x)=x$. Assume that

$|\pi^{L\underline{L}}(t,x)|<1/4\quad\text{and}\quad|\pi_{LL}(t,x)|\leq\epsilon\frac{|t-r|}{t+r},\qquad\forall(t,x)\in\mathscr{D}_{s}.$

Then for any $(t,x)\in\mathscr{D}_{s}$ one has that

(4.72) $\displaystyle t|(\partial_{t}-\partial_{r})u(t,x)|$
$\displaystyle\lesssim\sup_{\partial_{B}\mathscr{D}_{s}}|(\partial_{t}-\partial_{r})(ru)|+\int_{2}^{t}|M[u,\pi](\tau)|d\tau+\int_{2}^{t}\tau|F|_{(\tau,\varphi(\tau;t,x))}d\tau$

where

(4.73)
$M[u,\pi](\tau)=\Big{(}r|\Delta_{{\mathbb{S}}^{1}}u|+|\pi|_{\mathscr{L}\mathscr{T}}\big{(}r|\overline{\partial}\partial
u|+|\partial
u|\big{)}+|\pi|r|\overline{\partial}^{2}u|\Big{)}|_{(\tau,\varphi(\tau;t,x))}.$

###### Proof.

From the hypothesis on $\pi$,
$-2g^{L\underline{L}}=1-2\pi^{L\underline{L}}>1/2$ and
$\tilde{g}^{\alpha\beta}:=\frac{g^{\alpha\beta}}{-2g^{L\underline{L}}}$ is
well-defined. From $\tilde{\Box}_{g}u=F$ one has that

$\Box_{x}u+\theta^{\alpha\beta}\partial_{\alpha}\partial_{\beta}u=\frac{F}{-2g^{L\underline{L}}}$

where $\theta^{\alpha\beta}:=\tilde{g}^{\alpha\beta}-m^{\alpha\beta}$
satisfies the following

$\begin{gathered}\theta_{L\underline{L}}=0,\quad\theta_{LT}=(-2g^{L\underline{L}})^{-1}\pi_{LT},\quad\overline{\text{tr}}\theta=(-2g^{L\underline{L}})^{-1}(\overline{\text{tr}}\theta+\pi_{L\underline{L}}),\quad|\theta|\lesssim|\pi|\\\
\Big{|}\theta^{\alpha\beta}\partial_{\alpha}\partial_{\beta}u-\frac{1}{r}\theta^{\underline{L}\underline{L}}\underline{L}^{2}(ru)\Big{|}\lesssim|\pi|_{\mathscr{L}\mathscr{T}}|\overline{\partial}\partial
u|+|\pi||\overline{\partial}^{2}u|+|\pi|r^{-1}|\partial u|.\end{gathered}$

We recall that the flat wave operator can be written as follows

$\Box_{x}u=\frac{1}{r}(\partial_{t}+\partial_{r})(\partial_{t}-\partial_{r})(ru)+\Delta_{{\mathbb{S}}^{1}}u.$

Therefore

$\displaystyle\Big{|}\Big{(}\partial_{t}+\frac{1-\pi_{LL}/4}{1+\pi_{LL}/4}\partial_{r}\Big{)}(\partial_{t}-\partial_{r})(ru)\Big{|}\lesssim
r|\Delta_{{\mathbb{S}}^{1}}u|+|\pi|_{\mathscr{L}\mathscr{T}}\big{(}r|\overline{\partial}\partial
u|+|\partial u|\big{)}+|\pi|r|\overline{\partial}^{2}u|+|rF|.$

Due to the smallness assumption on $\pi_{LL}$, any integral curve
$(\tau,\varphi(\tau;t,x))$ passing through a point $(t,x)\in\mathscr{D}_{s}$
must intersect the boundary $\partial_{B}\mathscr{D}_{s}$. The result of the
lemma finally follows from integration along the characteristic curve, from
which we get

$|(\partial_{t}-\partial_{r})(ru)(t,x)|\lesssim|(\partial_{t}-\partial_{r})(ru)(t_{0},x_{0})|+\int_{t_{0}}^{t}|M[u,\pi](\tau)|+\tau|F|_{(\tau,\varphi(\tau;t,x))}d\tau$

where $(t_{0},x_{0})$ is the first point at which the intersection with the
lateral boundary occurs. ∎

###### Proof of Proposition 4.12.

Throughout this proof we will denote by $\eta$ any small positive constant
that linearly depends on $\zeta_{k},\gamma_{k},\delta_{k}$. After the above
observations, we only need to prove that (4.70) and (4.71) are satisfied in
the exterior of the cone $t=2r$. Such estimates are satisfied by the
$(s/t)^{2}\partial_{t}$ and $\underline{\partial}_{\bm{a}}$ derivatives as a
consequence of (4.38). Moreover, since

$\displaystyle\partial_{t}$
$\displaystyle=\frac{t-r}{t}\partial_{t}+\frac{x^{\bm{a}}}{t+r}\underline{\partial}_{\bm{a}}+\frac{r}{t+r}(\partial_{t}-\partial_{r}),\qquad\partial_{\bm{a}}$
$\displaystyle=\underline{\partial}_{\bm{a}}-\frac{x_{\bm{b}}}{t}\partial_{t}$

we can reduce to proving them for the $\partial_{t}-\partial_{r}$ derivative,
which we do by applying Lemma 4.13. We remark that for a point
$(t,x)\in\mathscr{D}_{s}\cap\mathscr{D}^{\text{i}}$, the integral curve
$(\tau,\varphi(\tau;t,x))$ may have a non-empty intersection with the exterior
region, which explains why in the following we invoke some pointwise estimates
obtained in section 3.

The integration of (3.31) along ${\mathbb{S}}^{1}$ shows that
$Z^{K}h^{1,\flat}_{\alpha\beta}$ is solution to the following equation

(4.74)
$\Box_{x}Z^{K}h^{1,\flat}_{TU}+(H^{{\bm{\mu}}{\bm{\nu}}})^{\flat}\partial_{\bm{\mu}}\partial_{\bm{\nu}}Z^{K}h^{1,\flat}_{TU}=F^{K,\flat}_{TU}+F^{0,K}_{TU}-\big{(}(H^{\mu\nu})^{\natural}\cdot\partial_{\mu}\partial_{\nu}Z^{K}h^{1,\natural}_{TU}\big{)}^{\flat}$

where $F^{K,\flat}_{TU}=(F^{K}_{TU})^{\flat}$ and $F^{K}_{TU}$ is given by
(3.32). We recall that tensor $H^{\mu\nu}$ decomposes as in (1.12). The
hypothesis of Lemma 4.13 are met thanks to the pointwise bounds (3.11), (3.17)
and (4.47), therefore for all $s>s_{0}$ and all $(t,x)\in\mathscr{D}_{s}$

(4.75)
$\displaystyle|(\partial_{t}-\partial_{r})Z^{K}h^{1,\flat}_{TU}(t,x)|\leq
t^{-1}\sup_{\partial_{B}\mathscr{D}_{s}}|(\partial_{t}-\partial_{r})(rZ^{K}h^{1,\flat}_{TU})|+t^{-1}\int_{2}^{t}|M[Z^{K}h^{1,\flat}_{TU},H^{\flat}](\tau)|d\tau$
$\displaystyle+t^{-1}\int_{2}^{t}\tau\,(\text{RHS of }\eqref{eq_ZKhTU})d\tau$

where $M[\cdot,\cdot]$ is given by the formula in (4.73).

From the interior pointwise bounds (4.38), (4.43) and the exterior pointwise
bounds (3.12), (3.14) we see that

$\sup_{\partial_{B}\mathscr{D}_{s}}|(\partial_{t}-\partial_{r})(rZ^{K}h^{1,\flat}_{TU})|\lesssim
C_{1}\epsilon t^{-\frac{1}{2}+\eta}.$

As concerns the contribution coming from $M[Z^{K}h^{1,\flat}_{TU},H^{\flat}]$,
we see from formula (4.73) together with the fact that
$\not{\partial}_{\bm{j}}=\frac{x^{\bm{i}}}{r^{2}}\Omega_{{\bm{i}}{\bm{j}}}$
and the exterior pointwise bounds (3.12)-(3.14) that for points
$(t,x)\in\mathscr{D}_{s}\cap\mathscr{D}^{\text{e}}$

$|M[Z^{K}h^{1,\flat}_{TU},H^{\flat}](t,x)|\lesssim C_{0}\epsilon
t^{-1+2\sigma}\sqrt{l(t)}+C_{0}^{2}\epsilon^{2}t^{-2+2\sigma}.$

In $\mathscr{D}_{s}\cap\mathscr{D}^{\text{i}}$, we rewrite (4.73) using
inequality (1.20)

$\displaystyle|M[Z^{K}h^{1,\flat}_{TU},H^{\flat}](t,x)|\lesssim|\underline{\partial}Z^{\leq
1}Z^{K}h^{1,\flat}_{TU}|+|H^{\flat}|_{\mathscr{L}\mathscr{T}}\Big{(}r\Big{(}\frac{s}{t}\Big{)}^{2}|\partial^{2}Z^{K}h^{1,\flat}_{TU}|+|\partial
Z^{\leq 1}Z^{K}h^{1,\flat}_{TU}|\Big{)}$
$\displaystyle+|H^{\flat}|\Big{(}r\Big{(}\frac{s}{t}\Big{)}^{4}|\partial^{2}Z^{K}h^{1,\flat}_{TU}|+\Big{(}\frac{s}{t}\Big{)}^{2}|\partial
Z^{\leq 1}Z^{K}h^{1,\flat}_{TU}|+|\underline{\partial}Z^{\leq
1}Z^{K}h^{1,\flat}_{TU}|\Big{)}$

and hence deduce from pointwise bounds (3.11), (4.38), (4.38), (4.43), (4.47)
that

$|M[Z^{K}h^{1,\flat}_{TU},H^{\flat}](t,x)|\lesssim C_{1}\epsilon
t^{-\frac{3}{2}+\eta}.$

Overall, $M[Z^{K}h^{1,\flat}_{TU},H^{\flat}]|_{(\tau,\varphi(\tau;t,x))}$ is
an integrable function of $\tau$.

We next show that the right hand side of (4.74) multiplied by $t$ is
integrable in $\tau$ along the characteristic curve. Concerning the
contributions to $F^{K,\flat}_{TU}$, see formula (3.32): the weak null terms
do not appear in $F^{K,\flat}_{TU}$, hence from (3.22), (3.23) and (4.58),
(4.59) we have that

$|Z^{K}F^{\flat}_{TU}(t,x)|\lesssim
C_{1}^{2}\epsilon^{2}t^{-\frac{5}{2}+\eta}+C_{0}^{2}\epsilon^{2}t^{-2+2\sigma}\sqrt{l(t)}.$

The terms arising from the commutation of the null frame with the wave
operator are estimated, on the one hand, using (3.11), pointwise interior
bounds (4.38), (4.43) and the exterior bounds (3.13), (3.14)

$\sum_{|K^{\prime}|\leq|K|}C^{{\bm{i}}\alpha\beta}_{TU,K^{\prime}}\big{|}\not{\partial}_{\bm{i}}Z^{K^{\prime}}h^{1,\flat}_{\alpha\beta}\big{|}+\big{|}D^{\alpha\beta}_{TU,K^{\prime}}Z^{K^{\prime}}h^{1,\flat}_{\alpha\beta}\big{|}\lesssim
C_{1}\epsilon t^{-\frac{5}{2}+\eta}+C_{0}\epsilon t^{-2+\sigma}\sqrt{l(t)}.$

On the other hand, using additionally the a-priori bound (4.6) we see that

$\displaystyle\sum_{|K_{1}|+|K_{2}|\leq|K|}\big{|}E^{{\bm{i}}\alpha\beta}_{TU\mu\nu,K_{1}K_{2}}\big{(}Z^{K_{1}}H^{\mu\nu}\cdot\not{\partial}_{\bm{i}}Z^{K_{2}}h^{1}_{\alpha\beta}\big{)}^{\flat}\big{|}+\big{|}F^{\alpha\beta}_{TU\mu\nu,K_{1}K_{2}}\big{(}Z^{K_{1}}H^{\mu\nu}\cdot
Z^{K_{2}}h^{1}_{\alpha\beta}\big{)}^{\flat}\big{|}$ $\displaystyle\lesssim
C_{1}C_{2}\epsilon^{2}t^{-\frac{5}{2}+\eta}+C_{0}^{2}\epsilon^{2}t^{-2+2\sigma}\sqrt{l(t)}.$

From (4.66), together with (3.12) and (3.14), we have that

$\big{|}\big{(}(H^{1,\mu\nu})^{\natural}\cdot\partial_{\mu}\partial_{\nu}Z^{K}h^{1,\natural}_{TU}\big{)}^{\flat}(t,x)\big{|}+\big{|}\big{(}[Z^{K},H^{1,\mu\nu}\partial_{\mu}\partial_{\nu}]h^{1,\natural}_{TU}\big{)}^{\flat}(t,x)\big{|}\lesssim
C_{1}^{2}\epsilon^{2}t^{-\frac{5}{2}+\eta}.$

Thus all together

$t|F^{K,\flat}_{TU}-[Z^{K},H^{0,{\bm{\mu}}{\bm{\nu}}}\partial_{\bm{\mu}}\partial_{\bm{\nu}}]h^{1,\flat}_{TU}|+t\big{|}\big{(}(H^{1,\mu\nu})^{\natural}\cdot\partial_{\mu}\partial_{\nu}Z^{K}h^{1,\natural}_{TU}\big{)}^{\flat}(t,x)\big{|}\\\
\lesssim C_{1}\epsilon t^{-\frac{3}{2}+\eta}+C_{0}\epsilon
t^{-1+2\sigma}\sqrt{l(t)}.$

Using the structure highlighted in Lemma 3.9, one can easily show that

$|F^{0,K}_{TU}(t,x)|\lesssim\epsilon t^{-3}.$

Finally, if $K$ is such that $Z^{K}=\partial^{I}$ is a product of derivatives
only, with $|I|\leq N_{1}$, we derive from (3.11), (3.12) and (4.38) that

$t\big{|}[\partial^{I},(H^{0,{\bm{\mu}}{\bm{\nu}}})\partial_{\bm{\mu}}\partial_{\bm{\nu}}]h^{1,\flat}_{TU}\big{|}\lesssim\sum_{\begin{subarray}{c}|I_{1}|+|I_{2}|=|I|\\\
|I_{1}|\geq
1\end{subarray}}t|\partial^{I_{1}}H^{0}|\,|\partial^{2}\partial^{I_{2}}h^{1,\flat}_{TU}|\lesssim
C_{1}\epsilon^{2}t^{-2+\delta_{k+2}}.$

This is an integrable quantity, as all others above, therefore we obtain
(4.71).

Bound (4.70) follows then by induction on the number $k$ of vector fields in
$Z^{K}$, since

$\displaystyle|[Z^{K},(H^{0,{\bm{\mu}}{\bm{\nu}}})\partial_{\bm{\mu}}\partial_{\bm{\nu}}]h^{1,\flat}_{TU}|$
$\displaystyle\leq\sum_{\begin{subarray}{c}|I_{1}|+|K^{\prime}_{2}|\leq|K|\\\
|I_{1}|\geq
1,\,|K^{\prime}_{2}|<|K|\end{subarray}}|\partial^{I_{1}}H^{0}||\partial^{2}Z^{K^{\prime}_{2}}h^{1,\flat}_{TU}|+\sum_{\begin{subarray}{c}|K_{1}|+|K^{\prime\prime}_{2}|\leq|K|\\\
|K^{\prime\prime}_{2}|<|K|\end{subarray}}|Z^{K_{1}}H^{0}||\partial^{2}Z^{K^{\prime\prime}_{2}}h^{1,\flat}_{TU}|$
$\displaystyle\leq\sum_{\begin{subarray}{c}|K^{\prime}|<|K|\end{subarray}}\frac{1}{(1+t+r)^{2}}|\partial^{2}Z^{K^{\prime}}h^{1,\flat}_{TU}|+\sum_{\begin{subarray}{c}|K^{\prime\prime}|<|K|\end{subarray}}\frac{1}{(1+t+r)}|\partial^{2}Z^{K^{\prime\prime}}h^{1,\flat}_{TU}|$

where $K^{\prime}$ is a multi-index of type $(|K|-1,k)$ and $K^{\prime\prime}$
is of type $(|K|-1,k-1)$.

∎

###  4.5. The weak null terms

In this section we prove estimates on the weak null terms.

######  Lemma 4.14.

For any multi-index $K$ of type $(N,k)$ and $i=0,1$, we have that

(4.76) $\left\|\partial^{i}Z^{K}P_{\alpha\beta}(\partial h,\partial
h)\right\|_{L^{2}_{xy}(\mathscr{H}_{s})}\lesssim
C_{1}^{2}\epsilon^{2}s^{-\frac{1}{2}+\zeta_{k}},$

while for any multi-index $K$ of type $(N,k)$ with $k\leq N_{1}$

(4.77) $\displaystyle\left\|Z^{K}P_{\alpha\beta}(\partial h,\partial
h)\right\|_{L^{2}_{xy}(\mathscr{H}_{s})}\lesssim
C_{1}^{2}\epsilon^{2}s^{-1+\delta_{k}}.$

Moreover, for $K$ a multi-index of type $(N,k)$ we have

(4.78) $\left\|Z^{K}P^{\flat}_{\alpha\beta}(\partial h,\partial
h)\right\|_{L^{2}_{xy}(\mathscr{H}_{s})}\\\ \lesssim C_{1}\epsilon
s^{-1}\sum_{K^{\prime}}E^{\text{i}}(s,Z^{K^{\prime}}h^{1,\flat})^{1/2}+C_{1}\epsilon
s^{-1+C\epsilon}\sum_{K^{\prime\prime}}E^{\text{i}}(s,Z^{K^{\prime\prime}}h^{1,\flat})^{1/2}+C_{1}^{2}\epsilon^{2}s^{-\frac{3}{2}+2\delta_{N}}\\\
+C_{1}^{2}\epsilon^{2}\delta_{k>N_{1}}\left(\sum_{i=1}^{4}s^{-1+\gamma_{i}+\zeta_{k-i}}+s^{-1+\zeta_{k}}\right)$

where $K^{\prime}$ is of type $(|K|,k)$, $K^{\prime\prime}$ of type
$(|K|-1,k-1)$, and where $\delta_{k>N_{1}}=1$ when $k>N_{1}$, 0 otherwise.

###### Proof.

We start by decomposing each occurrence of $h$ into the sum $h^{0}+h^{1}$ and
observe that the quadratic interactions involving at least one factor $h^{0}$
verify (4.50). We hence focus on estimating the weak null terms only involving
factors $h^{1}$ and distinguish between the region inside the cone $t=2r$ and
its complement in $\mathscr{D}^{\text{i}}$.

In the interior of the cone $t=2r$ (where $s\approx t$), the bounds of the
statement do not depend on the weak null structure: consequently they are the
same as the bounds for the null terms (4.52), (4.54) and (4.56).

The estimates in the region $\mathscr{D}^{\text{i}}\cap\\{t<2r\\}$ follow from
the particular structure of the weak null terms and the wave condition. We
have already seen that these two yield inequality (3.47) which, after (1.20),
can be also written in the following form

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

We observe that the quadratic terms on the second line of (4.79) are null and
hence satisfy (4.52), (4.54) and (4.56); the ones on the third line are cubic
(or cubic-like) and satisfy (4.53), (4.55); moreover

$\left\|\frac{M\chi_{0}(t/2\leq r\leq
3t/4)}{(1+t+r)^{2}}\,\partial\partial^{i}Z^{\leq
K}h^{1}\right\|_{L^{2}}\lesssim\epsilon
s^{-2}\left\|(s/t)\partial\partial^{i}Z^{\leq K}h^{1}\right\|_{L^{2}}\lesssim
C_{1}\epsilon^{2}s^{-\frac{3}{2}+\delta_{k}}.$

Finally, from the enhanced bounds (4.70), (4.71) satisfied by the $h^{1}_{TU}$
coefficients we derive

$\displaystyle\sum_{\begin{subarray}{c}|K_{1}|+|K_{2}|\leq|K|\\\
|I_{1}|+|I_{2}|=i\end{subarray}}\left\|\partial\partial^{I_{1}}Z^{K_{1}}h^{1}_{TU}\cdot\partial\partial^{I_{2}}Z^{K_{2}}h^{1}_{TU}\right\|_{L^{2}}\lesssim\sum_{\begin{subarray}{c}|I_{1}|+|K_{2}|\leq|K|+i\\\
|I_{1}|\leq\lfloor(|K|+i)/2\rfloor\end{subarray}}\|(t/s)\partial\partial^{I_{1}}h^{1}_{TU}\|_{L^{\infty}}\|(s/t)\partial
Z^{K_{2}}h^{1}_{TU}\|_{L^{2}}$
$\displaystyle+\sum_{\begin{subarray}{c}|I_{1}|+|J_{1}|+|K_{2}|\leq|K|+i\\\
|I_{1}|+|J_{1}|\leq\lfloor(|K|+i)/2\rfloor\\\ |J_{1}|\geq
1\end{subarray}}\|(t/s)\partial\partial^{I_{1}}\Gamma^{J_{1}}h^{1}_{TU}\|_{L^{\infty}}\|(s/t)\partial
Z^{K_{2}}h^{1}_{TU}\|_{L^{2}}$ $\displaystyle\lesssim C_{1}\epsilon
s^{-1}\sum_{j=0}^{i}\|(s/t)\partial\partial^{j}Z^{\leq
K}h^{1}_{TU}\|_{L^{2}}+C_{1}\epsilon
s^{-1+2C\epsilon}\sum_{j=0}^{i}\|(s/t)\partial\partial^{j}Z^{\leq
K-1}h^{1}_{TU}\|_{L^{2}}.$

∎

###  4.6. The commutator terms

The goal of this section is to get suitable estimates for the trilinear terms
involving commutators in the right hand side of energy inequalities (4.16) and
(4.17). More precisely, we will prove the following.

######  Proposition 4.15.

There exist some small positive parameters $\delta_{k}$ with
$\sigma\ll\delta_{k}\ll\delta_{k+1}\ll\kappa$ and $\delta_{k}\ll\kappa-\sigma$
for $k=0,\dots,N$ such that, under the assumptions of Proposition 4.1 we have
the following inequalities for all $s\in[s_{0},S_{0})$:

for multi-indexes $K$ of type $(N+1,k)$ with $k\leq N$

(4.80)
$\iint_{\mathscr{H}_{[s_{0},s]}}\big{|}[Z^{K},H^{1,\mu\nu}\partial_{\mu}\partial_{\nu}]h^{1}_{\alpha\beta}\big{|}|\partial_{t}Z^{K}h^{1}_{\alpha\beta}|dxdydt\lesssim(C_{0}^{2}+C_{1}^{2})C_{2}^{2}\epsilon^{3}{\Big{(}\sum_{i=1}^{4}s^{1+\gamma_{i}+\zeta_{k-i}+\zeta_{k}}+s^{1+2\zeta_{k}}\Big{)}},$

for multi-indexes $K$ of type $(N,k)$

(4.81)
$\iint_{\mathscr{H}_{[s_{0},s]}}\big{|}\big{(}[Z^{K},H^{1,\mu\nu}\partial_{\mu}\partial_{\nu}]h^{1}_{\alpha\beta}\big{)}^{\flat}\big{|}|\partial_{t}Z^{K}h^{1,\flat}_{\alpha\beta}|dxdt\\\
+\iint_{\mathscr{H}_{[s_{0},s]}}\big{|}\big{(}(H^{1,\mu\nu})^{\natural}\cdot\partial_{\mu}\partial_{\nu}Z^{K}h^{1,\natural}_{\alpha\beta}\big{)}^{\flat}\big{|}|\partial_{t}Z^{K}h^{1,\flat}_{\alpha\beta}|dxdt\lesssim(C_{0}^{2}+C_{1}^{2})C_{2}^{2}\epsilon^{3}{\Big{(}\sum_{i=1}^{4}s^{\gamma_{i}+\zeta_{k-i}+\zeta_{k}}+s^{2\zeta_{k}}\Big{)}}$

for multi-indexes $K$ of type $(N,k)$ with $k\leq N_{1}$

(4.82)
$\iint_{\mathscr{H}_{[s_{0},s]}}\big{|}[Z^{K},H^{1,\mu\nu}\partial_{\mu}\partial_{\nu}]h^{1}_{\alpha\beta}\big{|}|\partial_{t}Z^{K}h^{1}_{\alpha\beta}|dxdydt\lesssim(C_{0}^{2}+C_{1}^{2})C_{2}^{2}\epsilon^{3}s^{2\delta_{k}}$

and for multi-indexes $K$ of type $(N-1,k)$ with $k\leq N_{1}$

(4.83)
$\iint_{\mathscr{H}_{[s_{0},s]}}\big{|}\big{(}[Z^{K},H^{1,\mu\nu}\partial_{\mu}\partial_{\nu}]h^{1}_{\alpha\beta}\big{)}^{\flat}\big{|}|\partial_{t}Z^{K}h^{1,\flat}_{\alpha\beta}|dxdt\\\
+\iint_{\mathscr{H}_{[s_{0},s]}}\big{|}\big{(}(H^{1,\mu\nu})^{\natural}\cdot\partial_{\mu}\partial_{\nu}Z^{K}h^{1,\natural}_{\alpha\beta}\big{)}^{\flat}\big{|}|\partial_{t}Z^{K}h^{1,\flat}_{\alpha\beta}|dxdt\lesssim(C_{0}^{2}+C_{1}^{2})C_{2}^{2}\epsilon^{3}.$

We postpone the proof of the above proposition and first observe that, because
of (4.48), we will need to estimate quadratic terms (in fact, commutators)
that are either pure products of zero modes, or pure products of non-zero
modes, or mixed products. We proceed to the analysis of those separately, in
the lemmas that follow.

######  Lemma 4.16.

There exists $0<\eta\leq 2\delta_{N}\ll 1$, linearly depending on
$\zeta_{k},\gamma_{k},\delta_{k}$, such that for any multi-index $K$ of type
$(N,k)$ we have

(4.84)
$\displaystyle\big{\|}[\partial^{i}Z^{K},(H^{1,{\bm{\mu}}{\bm{\nu}}})^{\flat}\partial_{\bm{\mu}}\partial_{\bm{\nu}}]h^{1,\flat}_{\alpha\beta}-\sum_{\begin{subarray}{c}|K_{1}|+|K_{2}|\leq|K|\\\
|K_{2}|\leq\lfloor|K|/2\rfloor\end{subarray}}Z^{K_{1}}H^{1,\flat}_{LL}\cdot\partial^{2}_{t}\partial^{i}Z^{K_{2}}h^{1,\flat}_{\alpha\beta}\big{\|}_{L^{2}_{x}(\mathscr{H}_{s})}$
$\displaystyle\lesssim C_{1}C_{2}\epsilon^{2}\begin{cases}s^{-1+\eta}&\text{if
}i=1\text{ and }|K|=N\\\ s^{-\frac{3}{2}+\eta}&\text{if }i=0\end{cases}$

###### Proof.

The proof is based on inequality (3.25) applied to $\pi=H^{1,\flat}$ and
$\phi=h^{1,\flat}_{\alpha\beta}$. We will focus on discussing only the
following terms, the remaining ones being simpler:

$\sum_{\begin{subarray}{c}|K_{1}|+|K_{2}|\leq|K|\\\
|K_{1}|\leq\lfloor|K|/2\rfloor,|K_{2}|<|K|\end{subarray}}\big{\|}Z^{K_{1}}H^{1,\flat}_{LL}\cdot\partial^{2}_{t}\partial^{i}Z^{K_{2}}h^{1,\flat}_{\alpha\beta}\big{\|}_{L^{2}(\mathscr{H}_{s})}$

and

$\sum_{\begin{subarray}{c}|K_{1}|+|K_{2}|\leq|K|\end{subarray}}\big{\|}\partial
Z^{K_{1}}H^{1,\flat}_{LL}\cdot\partial^{2}_{t}Z^{K_{2}}h^{1,\flat}_{\alpha\beta}\big{\|}_{L^{2}(\mathscr{H}_{s})}.$

We remark that the latter sum only appears in the case $i=1$.

Concerning the first sum with $i=0$, pointwise bound (4.47) and $L^{2}$ bound
(4.67) yield

$\displaystyle\sum_{\begin{subarray}{c}|K_{1}|+|K_{2}|\leq|K|\\\
|K_{1}|\leq\lfloor|K|/2\rfloor,\,|K_{2}|<|K|\end{subarray}}\big{\|}Z^{K_{1}}H^{1,\flat}_{LL}\cdot\partial^{2}_{t}Z^{K_{2}}h^{1,\flat}_{\alpha\beta}\big{\|}_{L^{2}(\mathscr{H}_{s})}\lesssim
C_{1}\epsilon
s^{-\frac{1}{2}+\eta}\sum_{|K^{\prime}|<|K|}\|(s/t)^{2}\partial^{2}_{t}Z^{K^{\prime}}h^{1,\flat}_{\alpha\beta}\big{\|}_{L^{2}}$
$\displaystyle\lesssim C_{1}^{2}\epsilon^{2}s^{-\frac{3}{2}+\eta}.$

When $i=1$, the $L^{2}$ bound (4.68) gives

$\sum_{\begin{subarray}{c}|K_{1}|+|K_{2}|\leq|K|\\\
|K_{1}|\leq\lfloor|K|/2\rfloor,\,|K_{2}|<|K|\end{subarray}}\big{\|}Z^{K_{1}}H^{1,\flat}_{LL}\cdot\partial^{2}_{t}\partial
Z^{K_{2}}h^{1,\flat}_{\alpha\beta}\big{\|}_{L^{2}(\mathscr{H}_{s})}\lesssim
C_{1}C_{2}\epsilon^{2}s^{-1+\eta}.$

The second sum is estimated using (4.19) and (4.46) whenever $|K_{1}|\leq
N-2$, and using (4.38) and (4.49) otherwise

$\sum_{\begin{subarray}{c}|K_{1}|+|K_{2}|\leq|K|\end{subarray}}\big{\|}\partial
Z^{K_{1}}H^{1,\flat}_{LL}\cdot\partial^{2}_{t}Z^{K_{2}}h^{1,\flat}_{\alpha\beta}\big{\|}_{L^{2}(\mathscr{H}_{s})}\lesssim
C_{1}^{2}\epsilon s^{-1+\eta}.$

∎

######  Lemma 4.17.

For any fixed fixed multi-index $K$ and any smooth function $\phi$, we have

(4.85) $\displaystyle\sum_{\begin{subarray}{c}|K_{1}|+|K_{2}|\leq|K|\\\
|K_{2}|\leq\lfloor|K|/2\rfloor\end{subarray}}\iint_{\mathscr{H}_{[s_{0},s]}}|Z^{K_{1}}H^{1,\flat}_{LL}||\partial^{2}_{t}Z^{K_{2}}h^{1,\flat}_{\alpha\beta}||\partial_{t}\phi|dxdt\lesssim(C_{0}^{2}+C_{1}^{2})C_{2}^{2}\epsilon^{3}s^{\kappa_{\phi}}$

with

(4.86) $\kappa_{\phi}=\begin{cases}1,\quad&\text{if
}\phi=Z^{K}h^{1}_{\alpha\beta}\text{ and K is of type }(N+1,k),\ k\leq
N\\\\[2.0pt] 0,\quad&\text{if }\phi=Z^{K}h^{1,\flat}_{\alpha\beta}\text{ and K
is of type }(N,k)\\\\[2.0pt] 0,\quad&\text{if
}\phi=Z^{K}h^{1}_{\alpha\beta}\text{ and K is of type }(N,k),\ k\leq
N_{1}\end{cases}$

###### Proof.

We restrict our attention to the case where $Z^{K_{1}}=\Gamma^{K_{1}}$, as the
ones where $Z^{K_{1}}=\partial^{I_{1}}\Gamma^{J_{1}}$ with $|I_{1}|\geq 1$ can
be estimated as in the proof of lemma 4.16.

For any $K_{1}$, $K_{2}$ in the selected range of indexes and any fixed $\nu$
such that $2\delta_{k}<\nu\ll 1$, we write the following

$\displaystyle\iint_{\mathscr{H}_{[s_{0},s]}}|Z^{K_{1}}H^{1,\flat}_{LL}||\partial^{2}_{t}Z^{K_{2}}h^{1,\flat}_{\alpha\beta}||\partial_{t}\phi|dxdt\leq\int_{s_{0}}^{s}E^{\text{i}}(\tau,\phi)^{\frac{1}{2}}\big{\|}Z^{K_{1}}H^{1,\flat}_{LL}\cdot\partial^{2}_{t}Z^{K_{2}}h^{1,\flat}_{\alpha\beta}\big{\|}_{L^{2}(\mathscr{H}_{\tau})}d\tau$
$\displaystyle\lesssim\int_{s_{0}}^{s}\epsilon\tau^{-1-\nu}E^{\text{i}}(\tau,\phi)d\tau+\frac{1}{\epsilon}\int_{s_{0}}^{s}\int_{\mathscr{H}_{\tau}}\tau^{1+\nu}|Z^{K_{1}}H^{1,\flat}_{LL}|^{2}|\partial^{2}_{t}Z^{K_{2}}h^{1,\flat}_{\alpha\beta}|^{2}dxd\tau$
$\displaystyle\lesssim
C_{1}^{2}\epsilon^{3}s^{\kappa_{\phi}}+\frac{1}{\epsilon}\int_{s_{0}}^{t_{s}}\int_{\mathscr{C}_{t}}|Z^{K_{1}}H^{1,\flat}_{LL}|^{2}|\partial^{2}_{t}Z^{K_{2}}h^{1,\flat}_{\alpha\beta}|^{2}t^{1+\nu}dxdt$

where
$\mathscr{C}_{t}=\\{x\in{\mathbb{R}}^{3}:\sqrt{(t^{2}-s^{2})^{+}}\leq|x|\leq\sqrt{(t-1)^{2}-1}\\}$
and $t_{s}=s^{2}/2$. The latter inequality is obtained by injecting the energy
assumptions (4.3), (4.4), (4.5) in the first integral on the second line and
by performing a change of coordinates in the second one.

We use (4.61) and the Hardy inequality of corollary B.8 with $\mu=1-\eta$ and
$\alpha=1-\eta-\nu$ to estimate the above integral. For any fixed
$\mu^{\prime}>0$, we get that

$\displaystyle\frac{1}{\epsilon}\int_{s_{0}}^{t_{s}}\int_{\mathscr{C}_{t}}|Z^{K_{1}}H^{1,\flat}_{LL}|^{2}|\partial^{2}_{t}Z^{K_{2}}h^{1,\flat}_{\alpha\beta}|^{2}t^{1+\nu}dxdt$
$\displaystyle\lesssim
C_{2}^{2}\epsilon\int_{s_{0}}^{t_{s}}\int_{\mathscr{C}_{t}}\frac{|Z^{K_{1}}H^{1,\flat}_{LL}|^{2}}{(1+t-r)^{2+(1-\eta)}}\frac{dxdt}{(1+t+r)^{1-\eta-\nu}}$
$\displaystyle\lesssim
C_{2}^{2}\epsilon\int_{s_{0}}^{t_{s}}\int_{\mathscr{H}_{\tau}}\frac{|\partial_{r}Z^{K_{1}}H^{1,\flat}_{LL}|^{2}}{\tau^{2(1-\eta-\nu)}}dxd\tau+C_{2}^{2}\epsilon\int_{s_{0}}^{t_{s}}\int_{\Sigma^{\text{e}}_{t}}|\partial_{r}Z^{K_{1}}H^{1,\flat}_{LL}|^{2}\frac{(1+|r-t|)^{1+\mu^{\prime}}}{(1+t+r)^{1-\eta-\nu}}dxdt.$

We estimate the first integral using (4.49):

$C_{2}^{2}\epsilon\int_{s_{0}}^{t_{s}}\int_{\mathscr{H}_{\tau}}\frac{|\partial_{r}Z^{K_{1}}H^{1,\flat}_{LL}|^{2}}{\tau^{2(1-\eta-\nu)}}dxd\tau\lesssim\int_{s_{0}}^{t_{s}}C_{1}^{2}C_{2}^{2}\epsilon^{3}\tau^{-2(1-\eta-\nu)+4\delta_{k_{1}}}d\tau\lesssim
C_{1}^{2}C_{2}^{2}\epsilon^{3}.$

For the latter one, we pick $2\delta_{k}\leq\nu\ll\kappa$ and
$\mu^{\prime}:=2\kappa-\eta-2\nu$ so that $\mu^{\prime}>0$. From inequality
(2.9) we get

$\displaystyle
C_{2}^{2}\epsilon\int_{s_{0}}^{t_{s}}\int_{\Sigma^{\text{e}}_{t}}|\partial_{r}Z^{K_{1}}H^{1,\flat}_{LL}|^{2}\frac{(1+|r-t|)^{1+\mu^{\prime}}}{(1+t+r)^{1-\eta-\nu}}dxdt$
$\displaystyle\lesssim
C_{2}^{2}\epsilon\sum_{|K^{\prime}|\leq|K_{1}|}\int_{s_{0}}^{t_{s}}\int_{\Sigma^{\text{e}}_{t}}t^{-\nu}(2+r-t)^{2\kappa}|\overline{\partial}Z^{K^{\prime}}H^{1,\flat}|^{2}dxdt+\int_{s_{0}}^{t_{s}}\int_{\Sigma^{\text{e}}_{t}}t^{-\nu}r^{-1}(2+r-t)^{2\kappa-1}|Z^{K^{\prime}}H^{1,\flat}|^{2}dxdt$
$\displaystyle+\sum_{|K^{\prime}_{1}|+|K^{\prime}_{2}|\leq|K_{1}|}C_{2}^{2}\epsilon\int_{s_{0}}^{t_{s}}\int_{\Sigma^{\text{e}}_{t}}t^{-\nu}(2+r-t)^{2\kappa}|\big{(}Z^{K^{\prime}_{1}}H^{1}\cdot\partial
Z^{K^{\prime}_{2}}H^{1}\big{)}^{\flat}|^{2}dxdt$
$\displaystyle+C_{2}^{2}\epsilon\int_{s_{0}}^{t_{s}}\int_{\Sigma^{\text{e}}_{t}}(2+r-t)^{2\kappa}M^{2}\chi_{0}^{2}(1/2\leq
r/t\leq 3/4)r^{-4-\nu}dxdt.$

The above right hand side is bounded by $C_{2}^{2}C_{0}^{2}\epsilon^{3}$. For
the first integral, this follows using (3.6) and the fact that
$\nu>2\delta_{k}>\sigma$; for the second one, it follows from the weighted
Hardy inequality (B.4) and (3.5); for the third one, from the pointwise bounds
(3.12), (3.14), the weighed Hardy inequality (B.4) and the energy bounds
(3.5), (3.6); for the last one, from the fact that the domain of integration
of the latter one is uniformly bounded in $(t,x)$. ∎

We now estimate the different contributions to the commutator
$[Z^{K},(H^{1,\mu\nu})^{\flat}\partial_{\mu}\partial_{\nu}]h^{1,\natural}_{\alpha\beta}$,
which appear in the equation satisfied by $Z^{K}h^{1}_{\alpha\beta}$.

######  Lemma 4.18.

For any fixed multi-index $K$ of type $(N+1,k)$ with $k\leq N$, one has

(4.87)
$\|[Z^{K},(H^{1,\mu\nu})^{\flat}\partial_{\mu}\partial_{\nu}]h^{1,\natural}_{\alpha\beta}\|_{L^{2}(\mathscr{H}_{s})}\lesssim
C_{1}C_{2}\epsilon^{2}\left(\sum_{i=1}^{4}s^{-\frac{1}{2}+\gamma_{i}+\zeta_{k-i}}+s^{-\frac{1}{2}+\zeta_{k}}\right)$

and for $K$ of type $(N,k)$ with $k\leq N_{1}$

(4.88)
$\|[Z^{K},(H^{1,\mu\nu})^{\flat}\partial_{\mu}\partial_{\nu}]h^{1,\natural}_{\alpha\beta}\|_{L^{2}(\mathscr{H}_{s})}\lesssim
C_{1}C_{2}\epsilon^{2}s^{-1+\delta_{k}}.$

###### Proof.

The terms in the commutator are of the form

$Z^{K_{1}}{H}^{1,\flat}\,\partial^{2}Z^{K_{2}}h^{1,\natural}_{\alpha\beta}$

with each $K_{j}$ of type $(|K_{j}|,k_{j})$ and such that
$|K_{1}|+|K_{2}|\leq|K|$, $|K_{2}|<|K|$. We focus on the case where
$|K_{1}|=k_{1}$, that is where $Z^{K_{1}}=\Gamma^{K_{1}}$, the remaining ones
being simpler, and also recall that $N=N_{1}+5$.

In the case where $|K_{1}|\leq N_{1}$, we estimate
$\Gamma^{K_{1}}{H}^{1,\flat}$ in $L^{\infty}$ with (4.6). Together with the
energy bounds (4.21) and (4.24), and the relation (4.2), this yields

$\displaystyle\|\Gamma^{K_{1}}{H}^{1,\flat}\,\partial^{2}Z^{K_{2}}h^{1,\natural}_{\alpha\beta}\|_{L^{2}(\mathscr{H}_{s})}\lesssim
C_{1}C_{2}\epsilon^{2}\begin{cases}s^{-1+\delta_{k}},\ &\text{if }|k_{2}|\leq
N_{1}\\\ \sum_{i=0}^{4}s^{-\frac{1}{2}+\gamma_{i}+\zeta_{k-i}},\ &\text{if
}N_{1}<|k_{2}|\leq N\end{cases}$

In the case where $N_{1}<|K_{1}|\leq N$, we estimate
$\partial^{2}Z^{K_{2}}h^{1,\natural}_{\alpha\beta}$ in $L^{\infty}$ with (4.7)
and $\Gamma^{K_{1}}H^{1,\flat}$ in $L^{2}$ using (4.28)

$\displaystyle\|\Gamma^{K_{1}}{H}^{1,\flat}\,\partial^{2}Z^{K_{2}}h^{1,\natural}_{\alpha\beta}\|_{L^{2}(\mathscr{H}_{s})}$
$\displaystyle\lesssim\epsilon^{2}C_{1}C_{2}\left(s^{-\frac{1}{2}+\zeta_{k}}+\sum_{i=1}^{4}s^{-\frac{1}{2}+\gamma_{i}+\zeta_{k-i}}\right)$

The conclusion of the proof follows from relation (4.2) and the observation
that, if $K$ is a multi-index of type $(N,k)$ with $k\leq N_{1}$ and
$|K_{1}|>N_{1}$, then $\Gamma^{K_{1}}$ contains at least one usual derivative.
∎

We conclude this subsection with some estimates on commutator terms involving
$H^{1,\natural}$.

######  Lemma 4.19.

We have that

(4.89)
$\big{\|}Z^{K}\big{(}(H^{1,\mu\nu})^{\natural}\cdot\partial_{\mu}\partial_{\nu}h^{1,\natural}_{\alpha\beta}\big{)}^{\flat}\big{\|}_{L^{2}(\mathscr{H}_{s})}\\\
\lesssim
C_{1}C_{2}\epsilon^{2}\begin{cases}\sum_{i=1}^{4}s^{-1+\gamma_{i}+\zeta_{k-i}}+s^{-1+\zeta_{k}}\
&\text{if }K\text{ of type }(N,k)\\\ s^{-\frac{3}{2}+\delta_{k}}\ &\text{if
}K\text{ of type }(N-1,k)\text{ with }k\leq N_{1}\end{cases}$

and

(4.90)
$\big{\|}\big{[}Z^{K},(H^{1,\mu\nu})^{\natural}\partial_{\mu}\partial_{\nu}\big{]}h^{1,\natural}_{\alpha\beta}\big{\|}_{L^{2}_{xy}(\mathscr{H}_{s})}\\\
\qquad\\\ \lesssim
C_{1}C_{2}\epsilon^{2}\begin{cases}\sum_{i=1}^{4}s^{-1+\gamma_{i}+\zeta_{k-i}}+s^{-1+\zeta_{k}}\
&\text{if }K\text{ of type }(N+1,k)\text{ with }k\leq N\\\
s^{-\frac{3}{2}+\delta_{k}}\ &\text{if }K\text{ of type }(N,k)\text{ with
}k\leq N_{1}\end{cases}$

###### Proof.

We observe that if $K$ is a multi-index of type $(N-1,k)$ then $\partial
Z^{K}=Z^{K^{\prime}}$ with $K^{\prime}$ of type $(N,k)$. The first bound
(resp. the second) in (4.89) simply follows from (4.2), (4.7), (4.21) (resp.
(4.25)) and the fact that $\lfloor N/2\rfloor+2\leq N_{1}$. Similarly, the
first bound (resp. the second) in (4.90) follows from (4.19) (resp. (4.25))
and (4.41). ∎

######  Lemma 4.20.

We have that

(4.91)
$\big{\|}\big{[}Z^{K},(H^{1,{\bm{\mu}}{\bm{\nu}}})^{\natural}\partial_{\bm{\mu}}\partial_{\bm{\nu}}\big{]}h^{1,\flat}_{\alpha\beta}\big{\|}_{L^{2}_{xy}(\mathscr{H}_{s})}\lesssim
C_{1}^{2}\epsilon^{2}\begin{cases}s^{-1+2\delta_{N}}\quad&\text{if }K\text{ of
type }(N+1,k)\text{ with }k\leq N\\\
s^{-\frac{3}{2}+2\delta_{N}}\quad&\text{if }K\text{ of type }(N,k)\text{ with
}k\leq N_{1}.\end{cases}$

###### Proof.

We begin by writing that

$\big{|}\big{[}Z^{K},(H^{1,{\bm{\mu}}{\bm{\nu}}})^{\natural}\partial_{\bm{\mu}}\partial_{\bm{\nu}}\big{]}h^{1,\flat}_{\alpha\beta}\big{|}\lesssim\sum_{\begin{subarray}{c}|K_{1}|+|K_{2}|\leq|K|\\\
|K_{2}|<|K|\end{subarray}}|Z^{K_{1}}H^{1,\natural}||\partial^{2}Z^{K_{2}}h^{1,\flat}_{\alpha\beta}|.$

For any multi-index $K$ of type $(N+1,k)$ with $k\leq N$, energy bound (4.20)
and pointwise bound (4.30) yield

$\sum_{\begin{subarray}{c}|K_{1}|+|K_{2}|\leq|K|\\\
|K_{1}|\leq\lfloor|K|/2\rfloor,|K_{2}|<|K|\end{subarray}}\big{\|}Z^{K_{1}}H^{1,\natural}\cdot\partial^{2}Z^{K_{2}}h^{1,\flat}_{\alpha\beta}\big{\|}_{L^{2}_{xy}}\\\
\lesssim\sum_{\begin{subarray}{c}|K_{1}|+|K_{2}|\leq|K|\\\
|K_{1}|\leq\lfloor|K|/2\rfloor,|K_{2}|<|K|\end{subarray}}\big{\|}(t/s)Z^{K_{1}}H^{1,\natural}\big{\|}_{L^{\infty}_{x}L^{2}_{y}}\big{\|}(s/t)\partial^{2}Z^{K_{2}}h^{1,\flat}_{\alpha\beta}\big{\|}_{L^{2}_{x}}\lesssim
C_{1}^{2}\epsilon^{2}s^{-\frac{3}{2}+2\delta_{N}}$

while energy bound (4.19) and pointwise bound (4.42) yield

$\sum_{\begin{subarray}{c}|K_{1}|+|K_{2}|\leq|K|\\\
|K_{2}|\leq\lfloor|K|/2\rfloor\end{subarray}}\big{\|}Z^{K_{1}}H^{1,\natural}\cdot\partial^{2}Z^{K_{2}}h^{1,\flat}_{\alpha\beta}\big{\|}_{L^{2}_{xy}}\\\
\lesssim\sum_{\begin{subarray}{c}|K_{1}|+|K_{2}|\leq|K|\\\
|K_{2}|\leq\lfloor|K|/2\rfloor\end{subarray}}\big{\|}\partial^{2}Z^{K_{2}}h^{1,\flat}_{\alpha\beta}\big{\|}_{L^{\infty}_{x}}\big{\|}Z^{K_{1}}H^{1,\natural}\big{\|}_{L^{2}_{xy}}\lesssim
C_{1}^{2}\epsilon^{2}s^{-1+2\delta_{N}}.$

If $K$ is instead a multi-index of type $(N,k)$ with $k\leq N_{1}$, the above
estimate can be improved to
$C_{1}^{2}\epsilon^{2}s^{-\frac{3}{2}+2\delta_{N}}$ using energy bound (4.25).

∎

###### Proof of Proposition 4.15.

We decompose $H^{1,\mu\nu}$ and $h^{1}_{\alpha\beta}$ appearing in the
commutator terms into their zero mode and their zero-average component and
express all commutators involving $(H^{1,\mu\nu})^{\flat}$ with respect to the
null framework.

From (4.84), (4.85) and the energy bounds (4.19), (4.20) and (4.24) we derive
that

$\iint_{\mathscr{H}_{[s_{0},s]}}|[Z^{K},(H^{1,{\bm{\mu}}{\bm{\nu}}})^{\flat}\partial_{\bm{\mu}}\partial_{\bm{\nu}}]h^{1,\flat}_{\alpha\beta}||\partial_{t}\phi|dxdydt\lesssim(C_{1}^{3}+C_{0}^{2})\epsilon^{3}s^{\kappa_{\phi}}$

with $\kappa_{\phi}$ given by (4.86), while from (4.89) and (4.20)

$\iint_{\mathscr{H}_{[s_{0},s]}}|Z^{K}\big{(}(H^{1,\mu\nu})^{\natural}\cdot\partial_{\mu}\partial_{\nu}h^{1,\natural}_{\alpha\beta}\big{)}^{\flat}||\partial_{t}Z^{K}h^{1,\flat}_{\alpha\beta}|dxdt\lesssim
C_{1}^{3}\epsilon^{2}\Big{(}\sum_{i=0}^{4}s^{\mu(\gamma_{i}+\zeta_{k-i}+\zeta_{k})}+s^{2\mu\zeta_{k}}\Big{)}$

where $\mu=0$ if $K$ is of type $(N-1,k)$ with $k\leq N_{1}$, $\mu=1$
otherwise. These two estimates together imply (4.81) and (4.83).

Additionally, for $K$ of type $(N+1,k)$ with $k\leq N$ we get from (4.87) and
(4.19) that

$\iint_{\mathscr{H}_{[s_{0},s]}}|[Z^{K},(H^{1,\mu\nu})^{\flat}\partial_{\mu}\partial_{\nu}]h^{1,\natural}_{\alpha\beta}||\partial_{t}Z^{K}h^{1}_{\alpha\beta}|dxdydt\lesssim(C_{1}^{3}+C_{0}^{2})\epsilon^{3}s\Big{(}\sum_{i=0}^{4}s^{(\gamma_{i}+\zeta_{k-i}+\zeta_{k})}+s^{2\zeta_{k}}\Big{)}$

while for multi-indexes of type $(N,k)$ with $k\leq N_{1}$, estimates (4.88)
and (4.24) yield

$\iint_{\mathscr{H}_{[s_{0},s]}}|[Z^{K},(H^{1,\mu\nu})^{\flat}\partial_{\mu}\partial_{\nu}]h^{1,\natural}_{\alpha\beta}||\partial_{t}Z^{K}h^{1}_{\alpha\beta}|dxdydt\lesssim(C_{1}^{3}+C_{0}^{2})\epsilon^{3}s^{2\delta_{k}}.$

Finally, from (4.91) and the energy bounds (4.19), (4.24) we get

$\iint_{[s_{0},s]}|[Z^{K},(H^{1,{\bm{\mu}}{\bm{\nu}}})^{\natural}\partial_{\bm{\mu}}\partial_{\bm{\nu}}]h^{1,\flat}_{\alpha\beta}||\partial_{t}Z^{K}h^{1}_{\alpha\beta}|dxdydt\lesssim
C_{1}^{3}\epsilon^{3}s^{\mu}$

with $\mu=1$ if $K$ is of type $(N+1,k)$ with $k\leq N$, $\mu=1$ if $K$ is of
type $(N,k)$ with $k\leq N_{1}$. This concludes the proof of (4.80) and
(4.82). ∎

###  4.7. Propagation of the pointwise bound (4.7)

This section is devoted to the propagation of the pointwise estimates (4.7) on
the zero-average component $h^{1,\natural}_{\alpha\beta}$ of the solution. The
equation satisfied by $Z^{K}h^{1,\natural}_{\alpha\beta}$ is obtained from the
subtraction of (4.15) from (4.13):

(4.92)
$\displaystyle\Box_{xy}Z^{K}h^{1,\natural}_{\alpha\beta}+(H^{\mu\nu})^{\flat}\partial_{\mu}\partial_{\nu}Z^{K}h^{1,\natural}_{\alpha\beta}=$
$\displaystyle
F^{K,\natural}_{\alpha\beta}-(H^{1,\mu\nu})^{\natural}\partial_{\mu}\partial_{\nu}Z^{K}h^{1,\flat}_{\alpha\beta}-\big{(}(H^{\mu\nu})^{\natural}\partial_{\mu}\partial_{\nu}Z^{K}h^{1,\natural}_{\alpha\beta}\big{)}^{\natural}$

where
$F^{K,\natural}_{\alpha\beta}=F^{K}_{\alpha\beta}-F^{K,\flat}_{\alpha\beta}$
and $F^{K}_{\alpha\beta}$ is defined in (4.14). We observe that, after (4.48),
pure zero-mode interactions do not appear in the above right hand side.

The proof relies on the following result, which is motivated by the early work
of Klainerman [29] and can be found in slightly different forms in the works
of LeFloch-Ma [37], Dong-Wyatt [14] and Huneau-Stingo [21].

######  Proposition 4.21.

Suppose that $\phi$ is a solution of the equation

(4.93)
$\Box_{xy}\phi+(H^{\mu\nu})^{\flat}\partial_{\mu}\partial_{\nu}\phi=F,\quad(t,x,y)\in\mathbb{R}^{1+3}\times\mathbb{S}^{1}$

such that $\int_{\mathbb{S}^{1}}\phi\,dy=0$. For each $(t,x)$ in the cone
$\\{t>r\\}$, let $s=\sqrt{t^{2}-r^{2}}$ and $Y_{tx},A_{tx},B_{tx}$ be
functions defined as follows

$\displaystyle Y^{2}_{tx}(\lambda)$
$\displaystyle:=\int_{\mathbb{S}^{1}}\lambda\left|\frac{3}{2}\phi_{\lambda}+(\mathscr{S}\phi)_{\lambda}\right|^{2}+\lambda^{3}|\partial_{y}\phi_{\lambda}|^{2}dy$
$\displaystyle A_{tx}(\lambda)$
$\displaystyle:=\sup_{\mathbb{S}^{1}}\left|\lambda^{-1}\left(\mathscr{S}\left((t/s)^{2}(H^{1,UV})^{\flat}c^{00}_{UV}\right)\right)_{\lambda}\right|+\sup_{\mathbb{S}^{1}}\left|\lambda^{-1}(\mathscr{S}(H^{1,44})^{\flat})_{\lambda}\right|$
$\displaystyle\quad+\sup_{\mathbb{S}^{1}}\left|\lambda^{-1}(\chi(t/r)\chi^{\prime}(r)(t/s)^{2}M)_{\lambda}\right|$
$\displaystyle B^{2}_{tx}(\lambda)$
$\displaystyle:=\int_{\mathbb{S}^{1}}\lambda^{-1}|(R[\phi])_{\lambda}|^{2}dy,$

where $f_{\lambda}(t,x,y)=f(\tfrac{\lambda t}{s},\tfrac{\lambda x}{s},y)$ and
$R[\phi]:=R_{0}[\phi]+\sum_{i=1}^{2}R^{0}_{i}[\phi]+\sum_{i=1}^{3}R^{1}_{i}[\phi]-s^{2}F$
with

$\displaystyle R_{0}[\phi]$
$\displaystyle:=s^{2}\underline{\partial}^{\bm{a}}\underline{\partial}_{\bm{a}}\phi+x^{\bm{a}}x^{\bm{b}}\underline{\partial}_{\bm{a}}\underline{\partial}_{\bm{b}}\phi+\tfrac{3}{4}\phi+3x^{\bm{a}}\underline{\partial}_{\bm{a}}\phi$
$\displaystyle R^{0}_{1}[\phi]$
$\displaystyle:=s^{2}\chi(t/r)\chi(r)\frac{M}{r}\left((x^{\bm{a}}/t)\partial_{t}\underline{\partial}_{\bm{a}}\phi+(x^{\bm{a}}/t)\underline{\partial}_{\bm{a}}\partial_{t}\phi-\underline{\partial}^{\bm{a}}\underline{\partial}_{\bm{a}}\phi+(3/t)\partial_{t}\phi\right)$
$\displaystyle R^{0}_{2}[\phi]$
$\displaystyle:=\chi(t/r)\chi(r)\frac{M}{r}\frac{t^{2}+r^{2}}{s^{2}}\cdot
Q[\phi]$ $\displaystyle R^{1}_{1}[\phi]$
$\displaystyle:=s^{2}(H^{1,UV})^{\flat}\left(c_{UV}^{{\bm{a}}\beta}\underline{\partial}_{\bm{a}}\underline{\partial}_{\beta}\phi+c_{UV}^{\alpha{\bm{b}}}\underline{\partial}_{\alpha}\underline{\partial}_{\bm{b}}\phi+c_{UV}^{4{\bm{a}}}\partial_{y}\underline{\partial}_{\bm{a}}\phi+d_{UV}^{\mu}\underline{\partial}_{\mu}\phi\right)$
$\displaystyle R^{1}_{2}[\phi]$
$\displaystyle:=-(t/s)(H^{1,UV})^{\flat}c^{40}_{UV}\cdot
s\left(\tfrac{3}{2}\partial_{y}\phi+x^{\bm{a}}\partial_{y}\underline{\partial}_{\bm{a}}\phi\right)$
$\displaystyle R^{1}_{3}[\phi]$
$\displaystyle:=-(t/s)^{2}(H^{1,UV})^{\flat}c^{00}_{UV}\cdot Q[\phi]$

and

$Q[\phi]=\left(\tfrac{3}{4}+s^{2}\left(2(x^{\bm{a}}/t)\underline{\partial}_{\bm{a}}\partial_{t}+s^{-2}x^{\bm{a}}x^{\bm{b}}\underline{\partial}_{\bm{a}}\underline{\partial}_{\bm{b}}+(r^{2}/t^{3})\partial_{t}+(3/t)\partial_{t}+3s^{-2}x^{\bm{a}}\underline{\partial}_{\bm{a}}\right)\right)\phi.$

Then, in the hyperbolic region $\mathscr{H}_{[s_{0},S_{0})}$, the following
inequality holds

$s^{\frac{3}{2}}\left(\|\phi\|_{L^{2}(\mathbb{S}^{1})}+\|\partial_{y}\phi\|_{L^{2}(\mathbb{S}^{1})}\right)+s^{\frac{1}{2}}\|\mathscr{S}\phi\|_{L^{2}(\mathbb{S}^{1})}\lesssim\left(Y_{tx}(s_{0})+\int_{s_{0}}^{s}B_{tx}(\lambda)d\lambda\right)\exp^{\int_{s_{0}}^{s}A_{tx}(\lambda)d\lambda}.$

###### Proof.

The wave operator in question writes in terms of the $\partial_{t}$ and
$\underline{\partial}_{\bm{a}}$ derivatives as follows

$-\Box_{xy}=(s/t)^{2}\partial^{2}_{t}+2(x^{\bm{a}}/t)\underline{\partial}_{\bm{a}}\partial_{t}-\underline{\partial}^{\bm{a}}\underline{\partial}_{\bm{a}}+(r^{2}/t^{3})\partial_{t}+(3/t)\partial_{t}-\Delta_{y}.$

For some $\lambda>0$ and fixed $(t,x,y)$, we define
$\omega_{txy}(\lambda):=\lambda^{3/2}\phi(\tfrac{\lambda t}{s},\tfrac{\lambda
x}{s},y)$ to be the evaluation of $\phi$ on the hyperboloid
$\mathscr{H}_{\lambda}$ dilated by $\lambda^{3/2}$. We compute

$\displaystyle\dot{\omega}_{txy}$
$\displaystyle=\lambda^{1/2}\left(\tfrac{3}{2}\phi+(\mathscr{S}\phi)\right)_{\lambda}$
$\displaystyle\ddot{\omega}_{txy}$
$\displaystyle=\lambda^{-1/2}(P\phi)_{\lambda}:=\lambda^{-1/2}\left(\tfrac{3}{4}\phi+3(\mathscr{S}\phi)+(t^{2}\partial_{t}^{2}+2tx^{\bm{a}}\partial_{\bm{a}}\partial_{t}+x^{\bm{a}}x^{\bm{b}}\partial_{\bm{a}}\partial_{\bm{b}})\phi\right)_{\lambda}.$

A calculation shows that

(4.94) $\displaystyle P\phi$
$\displaystyle=s^{2}\left(-\Box_{txy}+\underline{\partial}^{\bm{a}}\underline{\partial}_{\bm{a}}+\Delta_{y}\right)\phi+x^{\bm{a}}x^{\bm{b}}\underline{\partial}_{\bm{a}}\underline{\partial}_{\bm{b}}\phi+3x^{\bm{a}}\underline{\partial}_{\bm{a}}\phi+\tfrac{3}{4}\phi.$

Using equation (4.93) we derive that $\omega_{txy}(\lambda)$ satisfies

$\ddot{\omega}_{txy}-\Delta_{y}\omega_{txy}=\lambda^{-1/2}(s^{2}(H^{\mu\nu})^{\flat}\partial_{\mu}\partial_{\nu}\phi)_{\lambda}+\lambda^{-1/2}R_{0}[\phi]_{\lambda}-\lambda^{3/2}F_{\lambda}.$

For the curved part in the above expression, we expand
$(H^{\mu\nu})^{\flat}=(H^{1,\mu\nu})^{\flat}+H^{0,\mu\nu}$ where $H^{0}$ is
defined in (1.12). Starting with the $H^{0}$ piece, we compute

$H^{0,\mu\nu}\partial_{\mu}\partial_{\nu}\phi=H^{0,{\bm{\mu}}{\bm{\nu}}}\partial_{\bm{\mu}}\partial_{\bm{\nu}}\phi=-a(t,x)\left(1+(r/t)^{2}\right)\partial_{t}^{2}\phi+s^{-2}R^{0}_{1}[\phi].$

where for simplicity we put $a(t,x):=\chi(r/t)\chi(r)\tfrac{M}{r}$. Using the
calculation for $P\phi$ given in (4.94), we find

$\displaystyle-\lambda^{-1/2}(s^{2}a(t,x)\left(1+(r/t)^{2}\right)\partial_{t}^{2}\phi)_{\lambda}$
$\displaystyle=-\lambda^{-1/2}\left(a(t,x)(t/s)^{2}(1+r^{2}/t^{2})\cdot
s^{2}(s/t)^{2}\partial_{t}^{2}\phi\right)_{\lambda}$
$\displaystyle=-\left(a(t,x)(t/s)^{2}(1+r^{2}/t^{2})\right)_{\lambda}\ddot{\omega}_{txy}+\lambda^{-1/2}R^{0}_{2}[\phi]_{\lambda}.$

For the $H^{1,\flat}$ part, we use (1.16) and (1.19). We find

$(H^{1,{\mu\nu}})^{\flat}\partial_{\mu}\partial_{\nu}\phi=\big{[}(H^{1,UV})^{\flat}c_{UV}^{00}\partial_{t}^{2}+(H^{1,UV})^{\flat}c_{UV}^{04}\partial_{t}\partial_{y}+(H^{1,44})^{\flat}\partial_{y}^{2}\big{]}\phi+s^{-2}R^{1}_{1}[\phi].$

Since we can write
$\dot{\omega}_{txy}=\lambda^{1/2}\left(\tfrac{3}{2}\phi+(s^{2}/t)\partial_{t}\phi+x^{\bm{a}}\underline{\partial}_{\bm{a}}\phi\right)_{\lambda}$,
we find

$\displaystyle\lambda^{-1/2}(s^{2}(H^{1,UV})^{\flat}c^{04}_{UV}\partial_{t}\partial_{y}\phi)_{\lambda}$
$\displaystyle=\left((t/s)(H^{1,UV})^{\flat}c^{04}_{UV}\partial_{y}(\lambda^{1/2}\tfrac{s^{2}}{t}\partial_{t}\phi)\right)_{\lambda}$
$\displaystyle=\left((t/s)(H^{1,UV})^{\flat}c^{04}_{UV}\right)_{\lambda}\partial_{y}\dot{\omega}_{txy}+\lambda^{-1/2}R^{1}_{2}[\phi]_{\lambda}.$

In a similar way, using also the calculation for $P\phi$ given in (4.94), we
find

$\displaystyle\lambda^{-1/2}(s^{2}(H^{1,UV})^{\flat}c^{00}_{UV}\partial_{t}^{2}\phi)_{\lambda}$
$\displaystyle=\left((t/s)^{2}(H^{1,UV})^{\flat}c^{00}_{UV}\right)_{\lambda}\ddot{\omega}_{txy}+\lambda^{-1/2}R^{1}_{3}[\phi]_{\lambda}.$

For simplicity, we henceforth write $\omega_{txy}=\omega(\lambda)$ and
suppress also the $|_{\lambda}$ notation. Putting the above computations
together, we have

(4.95) $\displaystyle b(t,x)\ddot{\omega}$
$\displaystyle-(1+(H^{1,44})^{\flat})\Delta_{y}\omega-(t/s)(H^{1,UV})^{\flat}c^{04}_{UV}\partial_{y}\dot{\omega}=\lambda^{-1/2}R[\phi]$

where

(4.96)
$b(t,x):=1-(t/s)^{2}(H^{1,UV})^{\flat}c^{00}_{UV}+\chi(t/r)\chi(r)\frac{M}{r}(t/s)^{2}(1+r^{2}/t^{2}).$

We multiply (4.95) by $\partial_{\lambda}\omega$, integrate over
$\mathbb{S}^{1}$ and integrate by parts to get:

$\displaystyle\int_{\mathbb{S}^{1}}\partial_{\lambda}\omega\Big{(}b\partial_{\lambda}^{2}\omega-(1+(H^{1,44})^{\flat})\Delta_{y}\omega-(t/s)(H^{1,UV})^{\flat}c^{04}_{UV}\partial_{y}\partial_{\lambda}\omega\Big{)}dy$
$\displaystyle=\frac{d}{d\lambda}\Big{(}\frac{1}{2}\int_{\mathbb{S}^{1}}b|\partial_{\lambda}\omega|^{2}dy+(1+(H^{1,44})^{\flat})|\partial_{y}\omega|^{2}\Big{)}-\frac{1}{2}\int_{{\mathbb{S}}^{1}}(\partial_{\lambda}b)|\partial_{\lambda}\omega|^{2}dy.$

Recalling the definition of $b$ in (4.96), we obtain

$\displaystyle\frac{d}{d\lambda}\Big{(}\int_{\mathbb{S}^{1}}b|\partial_{\lambda}\omega|^{2}+\left(1+(H^{44})^{\flat}\right)|\partial_{y}\omega|^{2}dy\Big{)}$
$\displaystyle=-\int_{\mathbb{S}^{1}}\partial_{\lambda}\left((t/s)^{2}(H^{1,UV})^{\flat}c^{00}_{UV}\right)|\partial_{\lambda}\omega|^{2}dy+\int_{\mathbb{S}^{1}}\partial_{\lambda}\left(\chi(t/r)\chi(r)(t/s)^{2}(1+r^{2}/t^{2})\right)\tfrac{M}{r}|\partial_{\lambda}\omega|^{2}dy$
$\displaystyle+\int_{\mathbb{S}^{1}}\chi(t/r)\chi(r)(t/s)^{2}(1+r^{2}/t^{2})\partial_{\lambda}\left(\tfrac{M}{r}\right)|\partial_{\lambda}\omega|^{2}dy+2\int_{\mathbb{S}^{1}}\lambda^{-1/2}R[\phi]\partial_{\lambda}\omega
dy.$

We crucially can drop the third term on the RHS above using the fact that
$\chi\geq 0$, $M>0$ and the identity
$\partial_{\lambda}(\tfrac{M}{r})_{\lambda}=-(\tfrac{M}{sr})_{\lambda}$. Note
also that the cut-off function $\chi$ is supported for $2r>t>2$ and so in the
region $\\{t\geq r+1\\}$ we have
$|\chi(t/r)\chi(r)\tfrac{M}{r}(t^{2}/s^{2})|\lesssim\epsilon$.

By relation (1.18) with $\pi=H^{1,\flat}$, the estimates (1.17), (4.43), and
(4.47), and the fact that $t/s^{2}\leq 1$ in the interior of the light cone,
we find

$\sup_{\mathbb{S}^{1}}|(t/s)^{2}(H^{1,UV})^{\flat}c^{00}_{UV}|+|(H^{1,44})^{\flat}|\lesssim\varepsilon.$

All together, we obtain

$\frac{d}{d\lambda}Y^{2}_{tx}(\lambda)\lesssim
A_{tx}(\lambda)Y^{2}_{tx}(\lambda)+B_{tx}(\lambda)Y_{tx}(\lambda)$

with $A_{tx},B_{tx},Y_{tx}$ as in the statement. From Grönwall’s lemma, we
have

(4.97)
$Y_{tx}(s)\lesssim\left(Y_{tx}(s_{0})+\int_{s_{0}}^{s}B_{tx}(\lambda)d\lambda\right)\exp\left(\int_{s_{0}}^{s}A_{tx}(\lambda)d\lambda\right),$

and the conclusion follows from the Poincaré inequality. ∎

######  Proposition 4.22.

There exists a constant $C_{2}$ sufficiently large, a finite and increasing
sequence of parameters $0<\gamma_{k},\delta_{k},\zeta_{k}\ll 1$ satisfying
(4.2), and $0<\epsilon_{0}\ll 1$ sufficiently small such that, under the
assumptions of Proposition 4.1, we have (4.12).

###### Proof.

Throughout the proof, $0<\eta\leq 2\delta_{N}\ll 1$ will denote a constant
that linearly depends on $\zeta_{k},\gamma_{k},\delta_{k}$. We apply
Proposition 4.21 to the variable
$\mathbf{W}=\partial^{I}\Gamma^{J}h^{1,\natural}_{\alpha\beta}$ with
$|I|+|J|\leq N_{1}+1=N-4$ and $|J|\leq N_{1}$, governed by the PDE (4.92).

1\. The $A_{tx}(\lambda)$ term: this is the same for all values of
$\mathbf{W}$. Bound (4.40) gives that

$\sup_{\mathbb{S}^{1}}\lambda^{-1}|(\mathscr{S}(H^{1,44})^{\flat})_{\lambda}|\lesssim\epsilon\lambda^{-\frac{3}{2}+\delta_{2}}.$

For the other piece of $A_{tx}(\lambda)$ we use (1.18) with $\pi=H^{1,\flat}$,
the identities $\mathscr{S}(s)=s$ and $\mathscr{S}(t+r)=t+r$, the pointwise
bounds (4.40), (4.45), (4.47) and the fact that $t/s^{2}\leq 1$ in the
interior of the lightcone, to derive

$\displaystyle|\mathscr{S}\left((t/s)^{2}(H^{1,UV})^{\flat}c^{00}_{UV}\right)|$
$\displaystyle\lesssim|(s/t)^{2}\mathscr{S}H^{1}_{\underline{L}\underline{L}}|+|(t/s)^{2}\mathscr{S}(H^{1}_{LL})^{\flat}|+|\mathscr{S}H^{1}_{L\underline{L}}|\lesssim\epsilon
s^{-\frac{1}{2}+\gamma_{1}}.$

Note that the estimate of the second term in the above right hand side is
obtained using also the following decomposition of the scaling vector field

(4.98)
$\mathscr{S}=(t-r)\partial_{t}+(r-t)\partial_{r}+(x^{\bm{a}}/r)\Omega_{0{\bm{a}}}.$

Consequently

$\displaystyle\lambda^{-1}|\left(\mathscr{S}\left((t/s)^{2}(H^{1,UV})^{\flat}c^{00}_{UV}\right)\right)_{\lambda}|$
$\displaystyle\lesssim\epsilon\lambda^{-\frac{3}{2}+\gamma_{1}}.$

Finally, we use that $\chi(r)=O(1)$ on its support, so that we can bound
$|\chi^{\prime}(r)|\lesssim r^{-2}$ and thus get

$\lambda^{-1}\left|(\chi(t/r)\chi^{\prime}(r)(t/s)^{2}M)_{\lambda}\right|\lesssim\epsilon\lambda^{-2}.$

Bringing all this together gives

(4.99) $\int_{s_{0}}^{s}A_{tx}(\lambda)d\lambda\lesssim\epsilon.$

2\. The $Y_{tx}(s_{0})$ term: We evaluate all the expressions here on the
hyperboloid $\mathscr{H}_{s_{0}}$ for $s_{0}$ close to 2. We observe that
$1+t+r=O(1)$ on $\mathscr{H}_{s_{0}}$, so by (4.98) and lemma B.5

(4.100)
$|Y_{tx}(s_{0})|\lesssim\|\mathbf{W}_{s_{0}}\|_{L^{2}_{y}}+\|(\mathscr{S}\mathbf{W})_{s_{0}}\|_{L^{2}_{y}}+\|(\partial_{y}\mathbf{W})_{s_{0}}\|_{L^{2}_{y}}\lesssim\Big{(}\frac{s}{t}\Big{)}^{\frac{3}{2}}E^{\text{i}}(s_{0},Z^{\leq
2}\mathbf{W})^{\frac{1}{2}}.$

3.a. The $R_{0}$ term in $B_{tx}(\lambda)$: By (4.30), (4.31), and (4.32) we
obtain

(4.101) $\|\lambda^{-1/2}(R_{0}[\mathbf{W}])_{\lambda}\|_{L^{2}_{y}}\lesssim
C_{1}\epsilon\lambda^{-\frac{3}{2}+\zeta_{k+4}}\Big{(}\frac{s}{t}\Big{)}^{\frac{3}{2}}.$

3.b. The $R^{1}_{1}$ term in $B_{tx}(\lambda)$: First note that in the region
$\\{r<t\\}$ we have $|c^{{\bm{a}}\beta}_{UV}|\lesssim 1$ using (1.17) and
straightforward computations. Thus, by (4.6), (4.30) and (4.31),

(4.102)
$\displaystyle\|\lambda^{-1/2}(R^{1}_{1}[\mathbf{W}])_{\lambda}\|_{L^{2}_{y}}$
$\displaystyle\lesssim\lambda^{\frac{3}{2}}\left|H^{1,\flat}\right|\Big{(}\left\|\partial\underline{\partial}\mathbf{W}\right\|_{L^{2}_{y}}+t^{-1}\|\mathbf{W}\|_{L^{2}_{y}}\Big{)}$
$\displaystyle\lesssim
C_{1}^{2}\epsilon^{2}\lambda^{-\frac{3}{2}+\gamma_{0}+\zeta_{k+3}}\Big{(}\frac{s}{t}\Big{)}^{\frac{5}{2}}.$

3.c The $R^{1}_{2}$ term in $B_{tx}(\lambda)$: using (4.6), (4.30), (4.31) we
get

$\displaystyle\|\lambda^{-1/2}(R^{1}_{2}[\mathbf{W}])_{\lambda}\|_{L^{2}_{y}}$
$\displaystyle\lesssim\lambda^{\frac{1}{2}}\big{|}\big{(}(t/s)((H^{1,UV})^{\flat}c^{40}_{UV}\big{)}_{\lambda}\big{|}\big{(}\left\|\partial_{y}\mathbf{W}\right\|_{L^{2}_{y}}+t\left\|\partial_{y}\underline{\partial}\mathbf{W}\right\|_{L^{2}_{y}}\Big{)}$
$\displaystyle\lesssim
C_{1}^{2}\epsilon^{2}\lambda^{-\frac{3}{2}+\gamma_{0}+\zeta_{k+3}}\Big{(}\frac{s}{t}\Big{)}^{\frac{3}{2}}.$

3.d.i The $R^{1}_{3}$ term in $B_{tx}(\lambda)$: we observe that from (4.30),
(4.31), (4.32)

$\displaystyle\|Q[\mathbf{W}]\|_{L^{2}_{y}}$
$\displaystyle\lesssim\big{(}\|\mathbf{W}\|_{L^{2}_{y}}+\left\|x^{2}\underline{\partial}^{2}\mathbf{W}\right\|_{L^{2}_{y}}+\left\|x\underline{\partial}\mathbf{W}\right\|_{L^{2}_{y}}\big{)}+\big{(}\left\|s^{2}\underline{\partial}\partial\mathbf{W}\right\|_{L^{2}_{y}}+\left\|(s^{2}/t)\partial\mathbf{W}\right\|_{L^{2}_{y}}\big{)}$
$\displaystyle\lesssim C_{1}\epsilon
t^{-\frac{3}{2}}s^{\frac{1}{2}+\zeta_{k+4}}+C_{1}\epsilon
t^{-\frac{5}{2}}s^{\frac{5}{2}+\zeta_{k+3}}$

which, coupled to (4.6) and the fact that $s^{-2}\lesssim t^{-1}$ yields again

$\displaystyle\|\lambda^{-1/2}(R^{1}_{3}[\mathbf{W}])_{\lambda}\|_{L^{2}_{y}}$
$\displaystyle\lesssim\big{|}\big{(}s^{-\frac{1}{2}}(t/s)^{2}(H^{1,UV})^{\flat}c^{00}_{UV}\big{)}_{\lambda}\big{|}\|Q[\mathbf{W}]_{\lambda}\|_{L^{2}_{y}}\lesssim
C_{1}^{2}\epsilon^{2}\lambda^{-\frac{3}{2}+\gamma_{0}+\zeta_{k+3}}\Big{(}\frac{s}{t}\Big{)}^{\frac{3}{2}}.$

3.e The $R^{0}_{1}$ and $R^{0}_{2}$ terms in $B_{tx}(\lambda)$: satisfy the
same bounds as $R^{1}_{1}$ and $R^{1}_{3}$ respectively.

In summary, for
$\mathbf{W}=\partial^{I}\Gamma^{J}h^{1,\natural}_{\alpha\beta}$ with
$|I|+|J|\leq N-4=N_{1}+1$

(4.103)
$\|\lambda^{-1/2}(R[\mathbf{W}]+s^{2}F)_{\lambda}\|_{L^{2}({\mathbb{S}}^{1})}\lesssim
C_{1}\epsilon\lambda^{-\frac{3}{2}+\eta}\Big{(}\frac{s}{t}\Big{)}^{\frac{3}{2}}.$

4.a The source term $F$: we simply distribute derivatives and vector fields
across the nonlinearities given in (4.92). We begin by analyzing the quadratic
interactions of zero-average component with itself, which are of the form

$\sum_{\begin{subarray}{c}|I_{1}|+|I_{2}|=|I|\\\
|J_{1}|+|J_{2}|\leq|J|\end{subarray}}\Big{(}\partial(\partial^{I_{1}}\Gamma^{J_{1}}h^{1,\natural})\cdot\partial(\partial^{I_{2}}\Gamma^{J_{2}}h^{1,\natural})\Big{)}^{\natural}+\Big{(}(\partial^{I_{1}}\Gamma^{J_{1}}h^{1,\natural})\cdot\partial^{2}(\partial^{I_{2}}\Gamma^{J_{2}}h^{1,\natural})\Big{)}^{\natural}$

and recall that there must exist an index $l=1,2$ such that
$|I_{l}|+|J_{l}|\leq\lfloor N_{1}+1/2\rfloor$.

To estimate the first sum, we use (4.7) and (4.41) to obtain that

$\displaystyle\sum_{\begin{subarray}{c}|I_{1}|+|I_{2}|=|I|\\\
|J_{1}|+|J_{2}|\leq|J|\end{subarray}}\lambda^{\frac{3}{2}}\left\|\Big{(}\partial(\partial^{I_{1}}\Gamma^{J_{1}}h^{1,\natural})\cdot\partial(\partial^{I_{2}}\Gamma^{J_{2}}h^{1,\natural})\Big{)}^{\natural}\right\|_{L^{2}_{y}}$
$\displaystyle\lesssim\lambda^{\frac{3}{2}}\left\|\partial Z^{\leq
N_{1}-1}h^{1,\natural}\right\|_{L^{\infty}_{y}}\left\|\partial Z^{\leq
N_{1}}h^{1,\natural}\right\|_{L^{2}_{y}}$ $\displaystyle\lesssim
C_{1}^{2}\epsilon^{2}\lambda^{-\frac{3}{2}+\eta}\Big{(}\frac{s}{t}\Big{)}^{2}.$

All terms in the second sum are estimated in the same way, besides the one
corresponding to $|I_{2}|+|J_{2}|=|I|+|J|=N_{1}+1$. For this one we use (4.30)
and (4.41), together with the assumption (4.2) on the parameters (i.e.
$\zeta_{i}\ll\gamma_{j}$ for any $i,j$), and derive that

$\lambda^{\frac{3}{2}}\left\|\Big{(}h^{1,\natural}\cdot\partial^{2}(\partial^{I}\Gamma^{J}h^{1,\natural})\Big{)}^{\natural}\right\|_{L^{2}_{y}}\lesssim\lambda^{\frac{3}{2}}\|h^{1,\natural}\|_{L^{\infty}_{y}}\left\|\partial^{2}(\partial^{I}\Gamma^{J}h^{1,\natural})\right\|_{L^{2}_{y}}\lesssim
C_{1}C_{2}\epsilon^{2}\lambda^{-1+\gamma_{k}}\Big{(}\frac{s}{t}\Big{)}^{2}.$

Let us note that this term is highly specific to the Kaluza–Klein problem and
is absent in the Einstein-Klein Gordon equations. We also observe that, in the
case where $|I|+|J|=N_{1}+1$ but $|J|\leq N_{1}-2$ we can use (4.33) instead
of (4.41) to obtain

$\lambda^{\frac{3}{2}}\left\|\Big{(}h^{1,\natural}\cdot\partial^{2}(\partial^{I}\Gamma^{J}h^{1,\natural})\Big{)}^{\natural}\right\|_{L^{2}_{y}}\lesssim
C_{1}C_{2}\epsilon^{2}\lambda^{-\frac{3}{2}+\eta}\Big{(}\frac{s}{t}\Big{)}^{2}.$

Next we turn to the mixed interactions between the zero mode and the zero-
average component. We begin with the commutator terms
$[\partial^{I}\Gamma^{J},(H^{1,\mu\nu})^{\flat}\partial_{\mu}\partial_{\nu}]h^{1,\natural}_{\alpha\beta}$,
which we rewrite using (3.25) with $\pi=H^{1,\flat}$. We focus on treating the
following products

(4.104)
$\displaystyle\partial^{I_{1}}\Gamma^{J_{1}}H^{1,\flat}_{LL}\cdot\partial^{2}_{t}(\partial^{I_{2}}\Gamma^{J_{2}}h^{1,\natural}_{\alpha\beta}),\quad\partial^{I_{1}}\Gamma^{J_{1}}H^{1,\flat}_{4L}\cdot\partial_{t}\partial_{y}(\partial^{I_{2}}\Gamma^{J_{2}}h^{1,\natural}_{\alpha\beta}),\quad\partial^{I_{1}}\Gamma^{J_{1}}H^{1,\flat}_{44}\cdot\partial^{2}_{y}(\partial^{I_{2}}\Gamma^{J_{2}}h^{1,\natural}_{\alpha\beta})$
$\displaystyle\text{for
}|I_{1}|+|I_{2}|=|I|,\quad|J_{1}|+|J_{2}|\leq|J|,\quad|I_{2}|+|J_{2}|<|I|+|J|$

the remaining ones being simpler. The latter term can be rewritten using the
equation, i.e.

$\displaystyle\partial_{y}^{2}(\partial^{I_{2}}\Gamma^{J_{2}}h^{1,\natural}_{\alpha\beta})=\Big{(}(s/t)^{2}\partial^{2}_{t}+2(x^{\bm{a}}/t)\underline{\partial}_{\bm{a}}\partial_{t}-\underline{\partial}^{\bm{a}}\underline{\partial}_{\bm{a}}+(r^{2}/t^{3})\partial_{t}+(3/t)\partial_{t}\Big{)}(\partial^{I_{2}}\Gamma^{J_{2}}h^{1,\natural}_{\alpha\beta})$
$\displaystyle+\Box_{xy}(\partial^{I_{2}}\Gamma^{J_{2}}h^{1,\natural}_{\alpha\beta}).$

If $|I_{1}|>0$, we estimate the first two terms in (4.104) using (4.7) and
(4.46) and the latter one using (4.38) and (4.7), hence getting that they are
all bounded by $C_{1}C_{2}\epsilon^{2}\lambda^{-\frac{3}{2}+\eta}(s/t)^{2}$.
If $|I_{1}|=0$ and $|J_{1}|>0$, we use (4.6) and (4.7) for all terms, together
with the algebraic relation (4.2), obtaining

$\lambda^{\frac{3}{2}}\left|\Gamma^{J_{1}}h^{1,\flat}\right|\left\|\partial^{2}(\partial^{I}\Gamma^{J_{2}}h^{1,\natural}_{\alpha\beta})\right\|_{L^{2}_{y}}\lesssim
C_{2}^{2}\epsilon^{2}\Big{(}\frac{s}{t}\Big{)}^{\frac{3}{2}}\lambda^{-1+\gamma_{k}}.$

Turning now to the semilinear interactions between the zero-mode and the zero-
average component, we immediately obtain from (4.7) and (4.38) that

$\lambda^{\frac{3}{2}}|\partial(\partial^{I_{1}}\Gamma^{J_{1}}h^{1,\flat})|\left\|\partial(\partial^{I_{2}}\Gamma^{J_{2}}h^{1,\natural})\right\|_{L^{2}_{y}}\lesssim
C_{1}C_{2}\epsilon^{2}\lambda^{-\frac{3}{2}+\eta}\Big{(}\frac{s}{t}\Big{)}^{2},\quad|I_{2}|+|J_{2}|\leq
N_{1}.$

When $|I_{2}|+|J_{2}|=|I|+|J|=N_{1}+1$, (4.7) does not provide us with the
right power of $(s/t)$, which we instead get using the structure of the
semilinear terms. On the one hand, using (4.57) together with (4.7), relation
$\underline{\partial}_{\bm{a}}=(1/t)\Omega_{0{\bm{a}}}$, (4.37) and (4.38), we
easily derive that

$\lambda^{\frac{3}{2}}\left\|\mathbf{Q}(\partial
h^{1,\flat},\partial(\partial^{I}\Gamma^{J}h^{1,\natural}))\right\|_{L^{2}_{y}}\lesssim
C_{1}C_{2}\epsilon^{2}\lambda^{-\frac{3}{2}+\eta}\Big{(}\frac{s}{t}\Big{)}^{2}.$

The cubic terms also satisfy the same estimate as above, we leave the details
to the reader.

On the other hand, using lemma 3.13 we see that

$\left\|P_{\alpha\beta}(\partial
h^{1,\flat},\partial(\partial^{I}\Gamma^{J}h^{1,\natural}))\right\|_{L^{2}_{y}}\\\
\lesssim|\partial
h^{1,\flat}_{TU}|\|\partial(\partial^{I}\Gamma^{J}h^{1,\natural})_{TU}\|_{L^{2}_{y}}+|\partial
h^{1,\flat}_{LL}|\|\partial(\partial^{I}\Gamma^{J}h^{1,\natural})_{\underline{L}\underline{L}}\|_{L^{2}_{y}}+|\partial
h^{1,\flat}_{\underline{L}\underline{L}}|\|\partial(\partial^{I}\Gamma^{J}h^{1,\natural})_{LL}\|_{L^{2}_{y}}.$

From (4.7) and (4.71)

$\lambda^{\frac{3}{2}}|\partial
h^{1,\flat}_{TU}|\|\partial(\partial^{I}\Gamma^{J}h^{1,\natural})_{TU}\|_{L^{2}_{y}}\lesssim
C_{1}C_{2}\lambda^{-1+\gamma_{k}}\Big{(}\frac{s}{t}\Big{)}^{\frac{3}{2}};$

from (4.7) and (4.46)

$\lambda^{\frac{3}{2}}|\partial
h^{1,\flat}_{LL}|\|\partial(\partial^{I}\Gamma^{J}h^{1,\natural})_{\underline{L}\underline{L}}\|_{L^{2}_{y}}\lesssim
C_{1}C_{2}\epsilon^{2}\lambda^{-\frac{3}{2}+\eta}\Big{(}\frac{s}{t}\Big{)}^{2};$

finally, since

$|\partial(\partial^{I}\Gamma^{J}h^{1,\natural})_{LL}|\leq|\partial(\partial^{I}\Gamma^{J}h^{1})_{LL}|+|\partial(\partial^{I}\Gamma^{J}h^{1,\flat})_{LL}|$

from lemma 2.2 and pointwise estimates (4.6), (4.7), (4.38), (4.46) we deduce
that

$\|\partial(\partial^{I}\Gamma^{J}h^{1,\natural})_{LL}\|_{L^{2}_{y}}\lesssim
C_{2}\epsilon t^{-\frac{3}{2}}s^{\gamma_{k}}$

and consequently that

$|\partial
h^{1,\flat}_{\underline{L}\underline{L}}|\|\partial(\partial^{I}\Gamma^{J}h^{1,\natural})_{LL}\|_{L^{2}_{y}}\lesssim
C_{1}C_{2}\epsilon^{2}\lambda^{-\frac{3}{2}+\eta}\Big{(}\frac{s}{t}\Big{)}^{2}.$

In summary,

$\lambda^{\frac{3}{2}}\|F_{\lambda}\|_{L^{2}_{y}}\lesssim
C_{2}^{2}\epsilon^{2}\Big{(}\frac{s}{t}\Big{)}^{\frac{3}{2}}\begin{cases}\lambda^{-\frac{3}{2}+\eta},\
&\text{if }|I|\leq N_{1},\ |J|=0\\\ \lambda^{-1+\gamma_{k}},\ &\text{if
}|I|+|J|\leq N_{1}+1,\ |J|=k\leq N_{1}\\\ \end{cases}$

and

$\int_{s_{0}}^{s}B_{tx}(\lambda)d\lambda\lesssim(C_{1}\epsilon+C_{2}^{2}\epsilon^{2})\Big{(}\frac{s}{t}\Big{)}^{\frac{3}{2}}\begin{cases}1\
&\text{if }|I|\leq N_{1},\ |J|=0\\\ s^{\gamma_{k}},\ &\text{if }|I|+|J|\leq
N_{1}+1,|J|=k\leq N_{1}\\\ \end{cases}$

Finally, from Proposition 4.21 we obtain that there exists a constant $C>0$
such that

$\displaystyle s^{\frac{3}{2}}\|\partial_{y}^{\leq
1}(\partial^{I}\Gamma^{J}h^{1,\natural})\|_{L^{2}}+s^{\frac{1}{2}}\|\mathscr{S}(\partial^{I}\Gamma^{J}h^{1,\natural})\|_{L^{2}}\leq
C\Big{(}\frac{s}{t}\Big{)}^{\frac{3}{2}}E^{\text{i}}(s_{0},Z^{\leq
2}(\partial^{I}\Gamma^{J}h^{1,\natural}))^{\frac{1}{2}}$
$\displaystyle+C(C_{1}\epsilon+C_{2}^{2}\epsilon^{2})\Big{(}\frac{s}{t}\Big{)}^{\frac{3}{2}}\begin{cases}1\
&\text{if }|I|\leq N_{1},\ |J|=0\\\ s^{\gamma_{k}},\ &\text{if }|I|+|J|\leq
N_{1}+1,|J|=k\leq N_{1}\\\ \end{cases}$

The conclusion of the proof then follows from the following relation

$\partial_{t}=\frac{t}{s^{2}}\mathscr{S}-\frac{tx^{\bm{j}}}{s^{2}}\underline{\partial}_{\bm{j}},\quad\partial_{\bm{j}}=\underline{\partial}_{\bm{j}}-\frac{x_{\bm{j}}}{s^{2}}\mathscr{S}+\frac{x_{\bm{j}}x^{\bm{k}}}{s^{2}}\underline{\partial}_{\bm{k}}$

and by choosing $C_{2}$ sufficiently large so that $CC_{1}\ll C_{2}$ and
$CE^{\text{i}}(s_{0},Z^{\leq
2}(\partial^{I}\Gamma^{J}h^{1,\natural}))^{\frac{1}{2}}\ll(C_{2}\epsilon)$,
together with $\epsilon_{0}$ sufficiently small so that $CC_{2}\epsilon\ll 1$.
∎

######  Remark 4.23.

It will be useful in view of Proposition 4.32 to observe that the loss
$s^{\gamma_{k}}$ in the estimate of
$\partial^{I}\Gamma^{J}h^{1,\natural}_{\alpha\beta}$ when $|I|+|J|\leq N_{1}$
and $|J|=k$ is only due to the following contributions, which arise from the
commutator term between zero modes and zero-average components

$\Gamma^{J_{1}}h^{1,\flat}\cdot\partial^{2}(\partial^{I}\Gamma^{J_{2}}h^{1,\natural}_{\alpha\beta}),\quad|J_{1}|>0.$

###  4.8. Enhanced energy bounds for the zero modes

The goal of this subsection is to show that the lower order energies of the
zero-modes enjoy enhanced energy estimates compared to (4.4).

######  Proposition 4.24.

Under the assumptions of Proposition 4.1, we have that for any fixed
$s\in[s_{0},S_{0})$ and any multi-index $K$ of type $(N-1,N_{1})$

(4.105) $E^{\text{i}}(s,Z^{K}h^{1,\flat})^{1/2}\lesssim C_{1}\epsilon
s^{3\sigma}$

where $0<\sigma\ll\gamma_{0}$ is the rate of growth of the exterior energies.

The proof of the above statement is based on energy inequality (4.17). We
recall the estimates already obtained in the previous subsections on the
quadratic null terms (4.54), on quadratic weak null terms (4.78) and on the
cubic terms (4.55) appearing in $F^{K,\flat}_{\alpha\beta}$, as well as on the
commutator terms (4.83) and on the contributions coming from
$F^{0,K}_{\alpha\beta}$ (4.69). We complete the picture with the estimates of
the remaining trilinear terms appearing in the right hand side of (4.17).

######  Lemma 4.25.

For any multi-index $K$ of type $(N,k)$ we have that

(4.106)
$\displaystyle\iint_{\mathscr{H}_{[s_{0},s]}}|{\partial^{\mu}H_{\mu}}^{\sigma}\cdot\partial_{\sigma}Z^{K}h^{1}_{\alpha\beta}||\partial_{t}Z^{K}h^{1}_{\alpha\beta}|+\frac{1}{2}|{\partial_{t}H_{\mu}}^{\sigma}\cdot\partial_{\sigma}Z^{K}h^{1}_{\alpha\beta}\cdot\partial^{\mu}Z^{K}h^{1}_{\alpha\beta}|\,dtdx\lesssim
C_{1}^{3}\epsilon^{3}s^{\frac{1}{2}+3\delta_{N}}$ (4.107)
$\displaystyle\iint_{\mathscr{H}_{[s_{0},s]}}|{\partial^{\mu}H_{\mu}}^{\sigma}\cdot\partial_{\sigma}Z^{K}h^{1,\flat}_{\alpha\beta}||\partial_{t}Z^{K}h^{1,\flat}_{\alpha\beta}|+\frac{1}{2}|{\partial_{t}H_{\mu}}^{\sigma}\cdot\partial_{\sigma}Z^{K}h^{1,\flat}_{\alpha\beta}\cdot\partial^{\mu}Z^{K}h^{1,\flat}_{\alpha\beta}|\,dtdx\lesssim
C_{1}^{3}\epsilon^{3},$

and for multi-indexes $K$ of type $(N_{1},k)$

(4.108)
$\iint_{\mathscr{H}_{[s_{0},s]}}|{\partial^{\mu}H_{\mu}}^{\sigma}\cdot\partial_{\sigma}Z^{K}h^{1}_{\alpha\beta}||\partial_{t}Z^{K}h^{1}_{\alpha\beta}|+\frac{1}{2}|{\partial_{t}H_{\mu}}^{\sigma}\cdot\partial_{\sigma}Z^{K}h^{1}_{\alpha\beta}\cdot\partial^{\mu}Z^{K}h^{1}_{\alpha\beta}|\,dtdx\lesssim
C_{1}^{3}\epsilon^{3}.$

###### Proof.

It is a straightforward consequence of (3.37), (3.38) applied to
$\phi=Z^{K}h^{1}_{\alpha\beta}$ and $\phi=Z^{K}h^{1,\flat}_{\alpha\beta}$
respectively, coupled with the pointwise bounds (3.11), (4.38), (4.46) and
with the energy bounds (4.19) to get (4.106), with (4.20) to get (4.107) and
with (4.24) to get (4.108).

∎

###### Proof of Proposition 4.24.

This follows by plugging the estimates obtained so far in the energy
inequality (4.17). In fact, from Lemma 4.6, estimates (4.54), (4.55), (4.78),
(4.83) and the a-priori energy bound (4.20) there exists a constant $C>0$ such
that for any fixed multi-index $K$ of type $(N_{1}-1,k)$ we have

$\displaystyle\iint_{\mathscr{H}_{[s_{0},s]}}|F^{K,\flat}_{\alpha\beta}(h)(\partial
h,\partial
h)||\partial_{t}Z^{K}h^{1,\flat}_{\alpha\beta}|dxdt+\iint_{\mathscr{H}_{[s_{0},s]}}\big{|}(H^{1,\mu\nu})^{\natural}\cdot\partial_{\mu}\partial_{\nu}Z^{K}h^{1,\natural}_{\alpha\beta}\big{)}^{\flat}\big{|}|\partial_{t}Z^{K}h^{1,\flat}_{\alpha\beta}|dxdt$
$\displaystyle\leq\int_{s_{0}}^{s}CC_{1}\epsilon\tau^{-1}\sum_{K^{\prime}}E^{\text{i}}(\tau,Z^{K^{\prime}}h^{1,\flat})^{1/2}E^{\text{i}}(\tau,Z^{K}h^{1,\flat}_{\alpha\beta})^{1/2}d\tau$
$\displaystyle+\int_{s_{0}}^{s}CC_{1}\epsilon\tau^{-1+C\epsilon}\sum_{K^{\prime\prime}}E^{\text{i}}(\tau,Z^{K^{\prime\prime}}h^{1,\flat})^{1/2}E^{\text{i}}(\tau,Z^{K}h^{1,\flat}_{\alpha\beta})^{1/2}d\tau+(C_{0}^{2}+C_{1}^{2})C_{2}^{2}\epsilon^{3}$

where $K^{\prime}$ denote multi-indexes of type $(|K|,k)$ and
$K^{\prime\prime}$ multi-indexes of type $(|K|-1,k-1)$. Furthermore, from
(3.11)

$\iint_{\mathscr{H}_{[s_{0}s]}}|[Z^{K},H^{0,{\bm{\mu}}{\bm{\nu}}}\partial_{\bm{\mu}}\partial_{\bm{\nu}}]h^{1,\flat}_{\alpha\beta}||\partial_{t}Z^{K}h^{1,\flat}_{\alpha\beta}|dxdt\\\
\leq\int_{s_{0}}^{s}C\epsilon\tau^{-1}\sum_{K^{\prime}}E^{\text{i}}(\tau,Z^{K^{\prime}}h^{1,\flat})^{1/2}E^{\text{i}}(\tau,Z^{K}h^{1,\flat}_{\alpha\beta})^{1/2}d\tau.$

Summing the above estimates up together with (4.18), (4.69) and (4.107) we get
that there exists some positive constant $C>0$ such that

$E^{\text{i}}(s,Z^{\leq K}h^{1,\flat}_{\alpha\beta})\leq
CE^{\text{i}}(s_{0},Z^{\leq
K}h^{1,\flat}_{\alpha\beta})+CC_{0}^{2}\epsilon^{2}s^{2\sigma+C\epsilon}+C(C_{0}^{2}+C_{1}^{2})C_{2}^{2}\epsilon^{3}\\\
+\int_{s_{0}}^{s}CC_{1}\epsilon\tau^{-1}\sum_{K^{\prime}}E^{\text{i}}(\tau,Z^{K^{\prime}}h^{1,\flat})^{1/2}E^{\text{i}}(\tau,Z^{\leq
K}h^{1,\flat})^{1/2}d\tau\\\
+\int_{s_{0}}^{s}CC_{1}\epsilon\tau^{-1+C\epsilon}\sum_{K^{\prime\prime}}E^{\text{i}}(\tau,Z^{K^{\prime\prime}}h^{1,\flat})^{1/2}E^{\text{i}}(\tau,Z^{\leq
K}h^{1,\flat})^{1/2}d\tau.$

Performing an induction argument on $k$, it then follows that there exist some
positive constants $c_{1}<c_{2}<\dots<c_{N}$ such that

$E^{\text{i}}(s,Z^{\leq K}h^{1,\flat}_{\alpha\beta})\leq
C\big{(}E^{\text{i}}(s_{0},Z^{\leq
K}h^{1,\flat}_{\alpha\beta})+CC_{0}^{2}\epsilon^{2}+(C_{0}^{2}+C_{1}^{2})C_{2}^{2}\epsilon^{3}\big{)}s^{2\sigma+c_{k}\epsilon}$

so the end of the proof follows from the smallness assumptions on the data and
after choosing $0<\epsilon_{0}\ll 1$ sufficiently small so that
$c_{N}\epsilon_{0}\leq\sigma$. ∎

The improved lower-order energy estimate (4.105) leads to the following
improved sup-norm estimates. These are obtained following the proofs of their
analogues with $s^{\delta_{k}}$ losses and using the energy bound (4.105) in
place of (4.20). For any multi-index $K$ of type $(N-3,N_{1})$, bounds (4.38)
and (4.43) are enhanced to the following ones

(4.109) $\displaystyle\big{\|}st^{\frac{1}{2}}\partial
Z^{K}h^{1,\flat}_{\alpha\beta}\big{\|}_{L^{\infty}(\mathscr{H}_{s})}+\big{\|}t^{\frac{3}{2}}\underline{\partial}Z^{K}h^{1,\flat}_{\alpha\beta}\big{\|}_{L^{\infty}(\mathscr{H}_{s})}+\big{\|}t^{\frac{1}{2}}Z^{K}h^{1,\flat}_{\alpha\beta}\big{\|}_{L^{\infty}(\mathscr{H}_{s})}$
$\displaystyle\lesssim C_{1}\epsilon s^{3\sigma}$

and bounds (4.46), (4.47) are improved to

(4.110) $\displaystyle\|t^{\frac{3}{2}}\partial
Z^{K}(H^{1}_{LT})^{\flat}\|_{L^{\infty}(\mathscr{H}_{s})}+\|t^{\frac{1}{2}}(t/s)^{2}Z^{K}(H^{1}_{LT})^{\flat}\|_{L^{\infty}(\mathscr{H}_{s})}\lesssim
C_{1}\epsilon s^{3\sigma}.$

Furthermore, for multi-indexes $K$ of type $(N-4,k)$ we can also improve
(4.61) to the following

(4.111)
$\|t^{\frac{3}{2}}(s/t)^{2}\partial^{2}_{t}Z^{K}h^{1,\flat}_{\alpha\beta}\|_{L^{\infty}(\mathscr{H}_{s})}\lesssim
C_{1}\epsilon s^{-1+6\sigma}.$

###  4.9. Propagation of pointwise bound (4.6)

This section is dedicated to the proof of the enhanced pointwise bound (4.6),
see proposition 4.31. We will make use of the following lemmas, which are due
respectively to Alinhac [1], Asakura [3] and Katayama-Yokoyama [28].

######  Lemma 4.26.

Let $u=u(t,x)$ be the solution to the inhomogeneous wave equation
$\Box_{tx}u=F$ on the flat space ${\mathbb{R}}^{1+3}$ with zero initial data.
Suppose that $F$ is spatially compacted supported and that there exist some
constants $C_{0}>0$, $\mu,\nu\geq 0$ such that the following pointwise bound
is satisfied

$|F(t,x)|\leq C_{0}t^{-\nu}(t-r)^{-\mu}.$

Defining $\Phi_{\mu}(s)=1,\log s,s^{1-\mu}/(1-\mu)$ according to $\mu>1,=1,<1$
respectively, we then have

  * (i)

If $\nu>2$,

$|u(t,x)|\leq CC_{0}\Phi_{\mu}\big{(}\langle t-r\rangle\big{)}\frac{\langle
t-r\rangle^{\nu-2}}{\nu-2}(1+t)^{-1}$

  * (ii)

If $\nu=2$,

$|u(t,x)|\leq CC_{0}\Phi_{\mu}\big{(}\langle
t-r\rangle\big{)}(1+t)^{-1}\log(1+t)$

  * (iii)

If $\nu<2$,

$|u(t,x)|\leq CC_{0}\Phi_{\mu}\big{(}\langle
t-r\rangle\big{)}\frac{(1+t)^{1-\nu}}{2-\nu}.$

######  Lemma 4.27.

Let $\phi,\psi$ be smooth functions on ${\mathbb{R}}^{3}$ such that

$|\phi|\leq C(1+|x|)^{-1-\kappa},\qquad|\nabla\phi|+|\psi|\leq
C(1+|x|)^{-2-\kappa}$

for some constant $C>0$ and some fixed $0<\kappa<1$. Let $u$ be the solution
to the homogeneous wave equation $\Box u=0$ with initial
$(u,\partial_{t}u)|_{t=0}=(\phi,\psi)$. There exists a constant $\tilde{C}>0$
such that $u$ satisfies the following inequality

$|u(t,x)|\leq\frac{C\tilde{C}}{(1+t+|x|)(1+|t-r|)^{\kappa}}.$

######  Lemma 4.28.

Let $u$ be the solution to the inhomogeneous wave equation $\Box u=F$ with
zero data and $F$ be a smooth function on ${\mathbb{R}}^{1+3}$. Let
$\mu,\nu>1$ be fixed constants. Provided the following right hand side is
finite, $u$ satisfies the following inequality

$\langle t+|x|\rangle\langle
t-|x|\rangle^{\mu-1}|u(t,x)|\lesssim\sup_{\tau\in[0,t]}\sup_{|x-y|\leq|t-\tau|}|y|\langle\tau+|y|\rangle^{\mu}\langle\tau-|y|\rangle^{\nu}|F(\tau,y)|.$

The idea of the proof of Proposition 4.31 is to look at
$u=\Gamma^{J}h^{1,\flat}_{\alpha\beta}$, for any fixed $J$ with $|J|=j\leq
N_{1}$, as the solution to a Cauchy problem of the following form

$\begin{cases}\Box_{tx}u=F\\\ (u,\partial_{t}u)|_{t=2}=(\phi,\psi)\end{cases}$

for some given smooth initial data $\phi,\psi$ and source term $F$, and to
successively decompose $u$ as the sum of three waves $v_{1},v_{2},v_{3}$ such
that

$\begin{cases}\Box_{tx}v_{1}=\chi((r+1/2)/t)F\\\
(v_{1},\partial_{t}v_{1})|_{t=0}=(0,0)\end{cases}\quad\text{and}\qquad\begin{cases}\Box_{tx}v_{2}=(1-\chi((r+1/2)/t))F\\\
(v_{2},\partial_{t}v_{2})|_{t=0}=(0,0)\end{cases}$

and $v_{3}$ is the solution to the homogeneous wave equation with data
$(\phi,\psi)$.

In the above systems, $\chi$ is a cut-off function equal to 1 on the ball
$B_{1/2}(0)$ and supported in $\overline{B_{1}(0)}$, so that the source term
in the equation of $v_{1}$ is supported in the interior of the cone $t=r+1/2$,
while the source term in the equation of $v_{2}$ is supported in the portion
of exterior region such that $t\leq r+1/2$.

The solutions $v_{1},v_{2},v_{3}$ are estimated using lemma 4.26, 4.27 and
4.28 respectively. The combination of such estimates will provide us with the
desired estimate on $u=Z^{K}h^{1,\flat}_{\alpha\beta}$.

Let us denote by $D^{J}_{\alpha\beta}$ the nonlinearity in equation (4.15)
satisfied by $\Gamma^{J}h^{1,\flat}_{\alpha\beta}$, i.e.

$D^{J,\flat}_{\alpha\beta}:=-(H^{{\bm{\mu}}{\bm{\nu}}})^{\flat}\cdot\partial_{\bm{\mu}}\partial_{\bm{\nu}}\Gamma^{J}h^{1,\flat}_{\alpha\beta}+F^{J,\flat}_{\alpha\beta}+F^{0,J}_{\alpha\beta}-\big{(}(H^{\mu\nu})^{\natural}\cdot\partial_{\mu}\partial_{\nu}\Gamma^{J}h^{1,\natural}_{\alpha\beta}\big{)}^{\flat}.$

We start by estimating the source term $D^{J,\flat}_{\alpha\beta}$ in the
interior of the cone $t=r+1/2$. Since the intersection of this cone with the
exterior region $\mathscr{D}^{\text{e}}$ is non-empty, we will make use of
some estimates obtained in Section 3.

######  Lemma 4.29.

For any multi-index $J$ with $|J|=k\leq N_{1}$, any $s\in[s_{0},S_{0})$ and
any $(t,x)$ with $t^{2}-r^{2}=s^{2}$ and $t>r+1/2$ we have that

$\displaystyle|D^{J,\flat}_{\alpha\beta}(t,x)|$
$\displaystyle\lesssim(C_{2}\epsilon)^{2}t^{-1}s^{-2+\gamma_{k}}$

###### Proof.

From the pointwise bounds (3.11), (3.12) and (3.14) we get that for any
$(t,x,y)\in\mathscr{D}^{\text{e}}$ such that $t>r+1/2$ and $t^{2}-r^{2}=s^{2}$

$\begin{gathered}\sum_{|J_{1}|+|J_{2}|\leq
k}|\partial\Gamma^{J_{1}}h\cdot\partial\Gamma^{J_{2}}h|\lesssim
C_{0}^{2}\epsilon^{2}t^{-2+2\sigma}(2+r-t)^{-2-2\kappa}\lesssim
C_{0}^{2}\epsilon^{2}t^{-1}s^{-2+4\sigma}\\\ \sum_{|J_{1}|+|J_{2}|\leq
k}|\Gamma^{J_{1}}H\cdot\partial^{2}\Gamma^{J_{2}}h|\lesssim
C_{0}^{2}\epsilon^{2}t^{-2+2\sigma}(2+r-t)^{-\frac{3}{2}-2\kappa}\lesssim
C_{0}^{2}\epsilon^{2}t^{-1}s^{-2+4\sigma}.\end{gathered}$

In the interior region, we recall the pointwise decay estimates already
obtained in lemma 4.6 for the quadratic and cubic terms involving at least one
$h^{0}$.

Turning next to the pure quadratic zero-mode interactions, we derive from
(4.109) that

$\sum_{|J_{1}|+|J_{2}|\leq
k}\big{|}\partial\Gamma^{J_{1}}h^{1,\flat}\cdot\partial\Gamma^{J_{2}}h^{1,\flat}\big{|}\lesssim
C_{1}^{2}\epsilon^{2}t^{-1}s^{-2+6\sigma}$

and from (3.25) and bounds (4.109), (4.111) that

$\displaystyle\big{|}\Gamma^{J}\big{(}(H^{1,{\bm{\mu}}{\bm{\nu}}})^{\flat}\cdot\partial_{\bm{\mu}}\partial_{\bm{\nu}}h^{1,\flat}_{\alpha\beta}\big{)}\big{|}\lesssim\sum_{\begin{subarray}{c}|J_{1}|+|J_{2}|\leq
k\end{subarray}}|\Gamma^{J_{1}}H^{1,\flat}_{LL}|\,|\partial^{2}_{t}\Gamma^{J_{2}}h^{1,\flat}_{\alpha\beta}|+\Big{(}\frac{s}{t}\Big{)}^{2}|\Gamma^{J_{1}}H^{1,\flat}|\,|\partial^{2}_{t}\Gamma^{J_{2}}h^{1,\flat}_{\alpha\beta}|$
$\displaystyle+\sum_{\begin{subarray}{c}|J_{1}|+|J_{2}|\leq
j\end{subarray}}(1+t+r)^{-1}|\Gamma^{J_{1}}H^{1,\flat}|\,|\partial\Gamma^{J_{2}}h^{1,\flat}_{\alpha\beta}|\lesssim
C_{1}^{2}\epsilon^{2}t^{-1}s^{-2+9\sigma}.$

As concerns instead the pure interaction of the zero-average components, we
derive from (4.7) and the algebraic relation (4.2) that

$\sum_{|J_{1}|+|J_{2}|=k}|(\partial\Gamma^{J_{1}}h^{1,\natural}\cdot\partial\Gamma^{J_{2}}h^{1,\natural})^{\flat}|+\big{|}\big{(}(H^{\mu\nu})^{\natural}\cdot\partial_{\mu}\partial_{\nu}\Gamma^{J}h^{1,\natural}_{\alpha\beta}\big{)}^{\flat}\big{|}\lesssim
C_{2}^{2}\epsilon^{2}t^{-3}s^{\gamma_{k}}.$

The conclusion of the proof follows from the fact that $9\sigma\ll\gamma_{0}$.
∎

######  Remark 4.30.

It is important to observe, in view of Proposition 4.32, that the loss
$s^{\gamma_{k}}$ in the above estimate for $D^{J}_{\alpha\beta}$ is only
caused by the pure interactions of the zero-average components of the metric
perturbations. All other interactions cause a smaller loss $s^{9\sigma}$.

We are now able to propagate the a-priori pointwise bound (4.6).

######  Proposition 4.31.

There exist two constants $1\ll C_{1}\ll C_{2}$ sufficiently large, a finite
and increasing sequence of parameters $0<\zeta_{k},\gamma_{k},\delta_{k}\ll 1$
satisfying (4.2) and $0<\epsilon_{0}\ll 1$ sufficiently small such that, under
the assumptions of Proposition 4.1, the enhanced estimate (4.11) is satisfied.

###### Proof.

We split $\Gamma^{J}h^{1,\flat}_{\alpha\beta}$ into the sum of three waves
$v^{J}_{1,\alpha\beta}$, $v^{J}_{2,\alpha\beta}$ and $v^{J}_{3,\alpha\beta}$,
where

$\begin{cases}\Box_{tx}v^{J}_{1,\alpha\beta}=\chi((r+1/2)/t)D^{J,\flat}_{\alpha\beta}\\\
(v^{J}_{1,\alpha\beta},\partial_{t}v^{J}_{1,\alpha\beta})|_{t=2}=(0,0)\end{cases}\begin{cases}\Box_{tx}v^{J}_{2,\alpha\beta}=\big{(}1-\chi((r+1/2)/t)\big{)}D^{J,\flat}_{\alpha\beta}\\\
(v^{J}_{2,\alpha\beta},\partial_{t}v^{J}_{2,\alpha\beta})|_{t=2}=(0,0)|_{t=2}\end{cases}$

and

$\begin{cases}\Box_{tx}v^{J}_{3,\alpha\beta}=0\\\
(v^{J}_{3,\alpha\beta},\partial_{t}v^{J}_{3,\alpha\beta})|_{t=2}=(\Gamma^{J}h^{1,\flat}_{\alpha\beta},\partial_{t}\Gamma^{J}h^{1,\flat}_{\alpha\beta})|_{t=2}.\end{cases}$

From Lemma 4.29, we get that in the interior of the cone $t=r+1/2$

(4.112) $|D^{J,\flat}_{\alpha\beta}(t,x)|\lesssim
C_{2}^{2}\epsilon^{2}t^{-1}s^{-2+\gamma_{k}}\lesssim
C_{2}^{2}\epsilon^{2}t^{-2+\frac{\gamma_{k}}{2}}\langle
t-r\rangle^{-1+\frac{\gamma_{k}}{2}}.$

Then, Lemma 4.26 with $\nu=2-\frac{\gamma_{k}}{2}$ and
$\mu=1-\frac{\gamma_{k}}{2}$ yields

(4.113) $|v^{J}_{\alpha y}(t,x)|\lesssim
C_{2}^{2}\epsilon^{2}(1+t)^{-1+\frac{\gamma_{k}}{2}}\langle
t-r\rangle^{\frac{{\gamma_{k}}}{2}}.$

As concerns the region exterior to the cone $t=r+1/2$, we recall the estimates
(3.22) for the quadratic null terms, (3.23) for the cubic interactions, (3.29)
for the commutator terms and (3.48) for the weak null interactions. For any
$(t,x)$ with $t<r+1/2$, we at least have