$X=X^{n}\partial_{u}+X^{\ell}\ell^{m_{B}}$. There is $\epsilon_{0}^{*}$ so
that if the hypotheses of Theorem 6.1 hold with
$\epsilon_{0}<\epsilon_{0}^{*}$, we have the following bounds. First, for
$|I|\leq N_{C}$, the quantity $\gamma$ appearing in (6.18) satisfies the
following estimates.

$|\gamma|\lesssim\frac{\epsilon_{C}}{(1+v)(1+s)^{1/2}},\qquad|X^{\ell}||\gamma|\lesssim\epsilon_{C}|X^{n}|,$
(8.23)
$\int_{t_{0}}^{t_{1}}\|\nabla\gamma\|_{L^{\infty}(D^{C}_{t})}+\left\|\frac{1}{1+s}\gamma\right\|_{L^{\infty}(D_{t}^{C})}\,dt\\\
+\int_{t_{0}}^{t_{1}}\left\|(1+v)^{1/2}(1+s)^{1/2}\nabla_{\ell^{m_{B}}}\gamma\right\|_{L^{\infty}(D_{t}^{C})}+\left\|(1+v)^{1/2}(1+s)^{1/2}\not{\nabla}\gamma\right\|_{L^{\infty}(D_{t}^{C})}\,dt\lesssim\epsilon_{C}.$
(8.24)

The currents $P_{I,C},P_{I,null}$ from (6.18) satisfy the following estimates,

$\sum_{|I|\leq N_{C}}\int_{t_{0}}^{t_{1}}\||X^{n}_{m_{B}}|^{1/2}\nabla
P_{I,C}\|_{L^{2}(D_{t}^{C})}+\|(1+s)^{-1}|X^{n}_{m_{B}}|^{1/2}P_{I,C}\|_{L^{2}(D_{t}^{C})}\,dt\\\
+\sum_{|I|\leq
N_{C}}\int_{t_{0}}^{t_{1}}\|(1+v)^{-1/2}\nabla_{\ell^{m_{B}}}(P_{I,C}+P_{I,null})\|_{L^{2}(D_{t}^{C})}+\|(1+v)^{-1/2}\not{\nabla}(P_{I,C}+P_{I,null})\|_{L^{2}(D_{t}^{C})}\,dt\\\
\lesssim\epsilon_{C}^{2}+c_{0}(\epsilon_{0})\epsilon_{C},$ (8.25)

as well as the bounds

$\displaystyle\sup_{t_{0}\leq t\leq
t_{1}}\int_{D^{C}_{t}}v|P_{I,C}|^{2}+\int_{t_{0}}^{t_{1}}\int_{\Gamma^{L}_{t}}v|P_{I,C}|^{2}\,dSdt+\int_{t_{0}}^{t_{1}}\int_{\Gamma^{R}_{t}}v|P_{I,C}|^{2}\,dSdt$
$\displaystyle\lesssim\epsilon_{C}^{3},\qquad|I|\leq N_{C}.$ (8.26)

The quantities on the right-hand side of (6.18) satisfies the following
estimates. The remainders $F_{C,I}$ and $F_{\Sigma,I}$ satisfy

$\int_{t_{0}}^{t_{1}}\||X^{\ell}_{m_{B}}|^{1/2}F_{C,I}\|_{L^{2}(D^{C}_{t})}+\||X^{\ell}_{m_{B}}|^{1/2}F_{\Sigma,I}\|_{L^{2}(D^{C}_{t})}^{2}\,dt\lesssim
c_{0}(\epsilon_{0})+(c_{0}(\epsilon_{0})+\epsilon_{C})\epsilon_{C}^{2},$
(8.27)

while, for any $\delta>0$, $F_{m_{B},I}^{2}$ satisfies

$\int_{t_{0}}^{t_{1}}\int_{D^{C}_{t}}|F_{m_{B},I}^{2}||X\psi_{C}^{I}|\,dt\lesssim\delta\sum_{|J|\leq|I|}S_{J}^{C}(t_{1})+\left(\frac{1}{\delta}+1\right)c_{0}(\epsilon_{0})\epsilon_{C}^{2}.$
(8.28)

###### Proof.

Part 1: Estimates for $\gamma$

We start by noting that the first bound in (8.23) implies the second one,
because if the first bound holds,

$|X^{\ell}||\gamma|\lesssim\frac{\epsilon_{C}}{(1+s)^{1/2}}\lesssim\epsilon_{C}|X^{n}|,$
(8.29)

for both $X=X_{C},X_{T}$. To prove the first bound in (8.23), we use
(C.59)-(C.60) from Lemma C.4, which give

$|\gamma|\lesssim\frac{1}{1+v}|\partial\psi_{C}|+\frac{1}{(1+v)^{2}},$ (8.30)

so by (7.2), we have

$(1+v)(1+s)^{1/2}|\gamma|\lesssim(1+s)^{1/2}|\partial\psi_{C}|+\frac{(1+s)^{1/2}}{1+v}\lesssim\epsilon_{C}+c_{0}(\epsilon_{0}).$
(8.31)

where $c_{0}(0)=0$. Taking $\epsilon_{0}$ small enough that
$c_{0}(\epsilon_{0})\leq\epsilon_{C}$ gives the first bound in (8.23).

We now prove the time-integrated bounds. For this we use the bound in (C.60)
which gives

$(1+s)^{1/2}(1+v)\left((1+s)|\partial_{u}\gamma|+|\gamma|+(1+v)|\partial_{v}\gamma|+(1+v)|\not{\nabla}\gamma|\right)\\\
\lesssim\sum_{|I|\leq 1}(1+s)^{1/2}|\partial
Z_{m_{B}}^{I}\psi_{C}|+\frac{(1+s)^{1/2}}{(1+v)^{2}}\lesssim\epsilon_{C}+c_{0}(\epsilon_{0})$
(8.32)

where $c_{0}(0)=0$ and where we used the pointwise bound (7.2). In particular
this gives

$|\nabla\gamma|+\frac{1}{1+s}|\gamma|+(1+v)^{1/2}(1+s)^{1/2}|\nabla_{\ell^{m_{B}}}\gamma|+(1+v)^{1/2}(1+s)^{1/2}|\not{\nabla}\gamma|\\\
\lesssim\frac{1}{1+t}\frac{1}{(1+\log
t)^{3/2}}\left(\epsilon_{C}+c_{0}(\epsilon_{0})\right),$ (8.33)

which gives (8.24) after taking $\epsilon_{0}$ smaller if needed. Here we
bounded
$|\nabla_{\ell^{m_{B}}}\gamma|\lesssim|\partial_{v}\gamma|+\frac{1}{vs^{1/2}}|\partial_{u}\gamma|+\frac{1}{v}|\gamma|$.

Part 2: Estimates for the currents $P_{I,C},P_{I,null}$ We start with the
bound

$\int_{t_{0}}^{t_{1}}\||X^{n}_{m_{B}}|^{1/2}\nabla
P_{I,C}\|_{L^{2}(D^{C}_{t})}+\||X^{n}_{m_{B}}|^{1/2}(1+s)^{-1}P_{I,C}\|_{L^{2}(D^{C}_{t})}\,dt\\\
+\int_{t_{0}}^{t_{1}}\||X^{\ell}_{m_{B}}|^{1/2}\nabla_{\ell^{m_{B}}}P_{I,C}\|_{L^{2}(D^{C}_{t})}+\||X^{\ell}_{m_{B}}|^{1/2}\not{\nabla}P_{I,C}\|_{L^{2}(D^{C}_{t})}\,dt\\\
\lesssim\sum_{|J|\leq 1}\int_{t_{0}}^{t_{1}}\frac{1}{1+t}\frac{1}{(1+\log
t)^{3/2}}\|(1+v)(1+s)^{1/2}|X^{n}_{m_{B}}|^{1/2}Z_{m_{B}}P_{I,C}\|_{L^{2}(D^{C}_{t})}\,dt,$
(8.34)

where we used $(1+s)|\partial
q|+(1+v)|\partial_{v}q|+(1+v)|\not{\nabla}q|\lesssim|Z_{m_{B}}q|$ and that
$(1+v)^{-1/2}\lesssim(1+s)^{1/4}\lesssim|X^{n}_{m_{B}}|^{1/2}$ for both our
multipliers.

Using the estimate (C.64) to control $Z_{m_{B}}P_{I,C}$, we have

$(1+v)(1+s)^{1/2}|X^{n}_{m_{B}}|^{1/2}|Z_{m_{B}}P_{I,C}|\\\
\lesssim\sum_{\begin{subarray}{c}|I_{1}|+|I_{2}|\leq|I|+1,\\\
|I_{1}|,|I_{2}|\leq|I|\end{subarray}}(1+s)^{1/2}|\partial\psi^{I_{1}}_{C}|\left(|X^{n}_{m_{B}}|^{1/2}|\partial\psi^{I_{2}}_{C}|\right)+\sum_{|J|\leq|I|}\frac{1}{(1+s)^{1/2}}\left(|X^{n}_{m_{B}}|^{1/2}|\partial\psi^{J}_{C}|\right)\\\
\lesssim\left(\epsilon_{C}+c_{0}(\epsilon_{0})\right)\sum_{|J|\leq|I|}|X^{n}_{m_{B}}|^{1/2}|\partial\psi^{J}_{C}|,$
(8.35)

using the bound (8.1) and bounding $(1+s)^{-1/2}\leq c_{0}(\epsilon_{0})$.
Since
$\||X^{n}_{m_{B}}|^{1/2}\partial\psi^{J}_{C}\|_{L^{2}(D^{C}_{t})}\lesssim\epsilon_{C}(1+\log\log
t)$ if $|J|\leq N_{C}-1$ and $X=X_{C}$ or $|J|\leq N_{C}$ and $X=X_{T}$ (the
factor $\log\log t$ is only needed for the case $|J|=N_{C}-1,X=X_{C}$),
inserting (8.35) into (8.34) we find

$\int_{t_{0}}^{t_{1}}\frac{1}{1+t}\frac{1}{(1+\log
t)^{3/2}}\|(1+v)(1+s)^{1/2}|X^{n}_{m_{B}}|^{1/2}Z_{m_{B}}P_{I,C}\|_{L^{2}(D^{C}_{t})}\,dt\lesssim(\epsilon_{C}+c_{0}(\epsilon_{0}))\epsilon_{C},$
(8.36)

which is bounded by the right-hand side of (8.25). The bounds for the
contribution from $P_{I,null}$, which only involve the derivatives
$\ell^{m_{B}}$ and $\not{\nabla}$, into (8.25) follow easily from (C.68).

The bound (8.26) follows in a straightforward way from the pointwise bound
(C.62). We omit the proof.

Part 3: Bounds for $F_{C,I}$, $F_{\Sigma,I}$, $F_{m_{B},I}^{2}$

We now move onto controlling the remainder terms on the right-hand side of
(6.18). We recall from Lemma C.4 that these quantities satisfy the following
bounds. First, $F_{C,I}$ collects various nonlinear error terms and satisfies

$|F_{C,I}|\lesssim\frac{1}{(1+v)^{2}}\sum_{|I_{1}|+|I_{2}|\leq|I|}|\partial\psi_{C}^{I_{1}}||\partial\psi_{C}^{I_{2}}|+\frac{1}{(1+v)^{4}}\sum_{|I_{1}|+|I_{2}|\leq|I|}|\psi_{C}^{I_{1}}||\psi_{C}^{I_{2}}|\\\
+\frac{1}{(1+v)^{2}}\sum_{|J|\leq|I|}|\partial\psi_{C}^{J}|+\frac{1}{(1+v)^{2}(1+s)}\sum_{|J|\leq|I|}|\psi_{C}^{J}|.$
(8.37)

The remainder $F_{\Sigma,I}$ collects the error terms involving the model
profile $\Sigma$ alone and satisfies

$|F_{\Sigma,I}|\lesssim\frac{1}{(1+v)^{2}},$ (8.38)

Finally, $F_{m_{B},I}$ collects the error terms that we generated when we
commuted the angular Laplacian with our fields. With
$Z_{m_{B}}^{I}=X^{k}\Omega^{K}$, where $X^{k}$ denotes an arbitrary $k$-fold
product of the fields $X\in\\{X_{1},X_{2}\\}$, it satisfies

$|F_{m_{B},I}|\lesssim\frac{1}{1+v}\sum_{j\leq
k-1}\sum_{|J|\leq|K|+1}|\not{\nabla}X^{j}\Omega^{J}\psi_{C}|$ (8.39)

We start by proving the bound for $F_{C,I}$ in (8.27). The contribution from
the terms on the first line of (8.37) is straightforward to handle so we skip
it. For the contribution from the terms on the second line, we bound

$\sum_{|J|\leq|I|}\int_{t_{0}}^{t_{1}}\left(\int_{D^{C}_{t}}\frac{1}{(1+v)^{4}}|X^{\ell}||\partial\psi_{C}^{J}|^{2}\right)^{1/2}+\sum_{|J|\leq|I|}\int_{t_{0}}^{t_{1}}\left(\int_{D^{C}_{t}}\frac{1}{(1+v)^{4}(1+s)^{2}}|X^{\ell}||\psi_{C}^{J}|^{2}\right)^{1/2}\\\
\lesssim\sum_{|J|\leq|I|}\int_{t_{0}}^{t_{1}}\frac{1}{(1+t)^{3/2}}\left(\int_{D^{C}_{t}}|\partial\psi_{C}^{J}|^{2}+\frac{1}{(1+s)^{2}}|\psi_{C}^{J}|^{2}\right)^{1/2}\,dt,$
(8.40)

and this is easily bounded by the right-hand side of (8.37) after using the
bootstrap assumptions (6.78)-(6.80) for the energies (6.32)-(6.37) and
additionally using (7.14) to control $\|\psi^{J}_{C}\|_{L^{2}(D^{C}_{t})}$.

For the remainder $F_{I,\Sigma}$ we just use that $Vol(D^{C}_{t})\lesssim
s^{1/2}\lesssim(1+v)^{1/5}$ (recall that we are using the measure $r^{-2}dx$),
and use (8.38) to bound

$\int_{t_{0}}^{t_{1}}\left(\int_{D^{C}_{t}}|X^{\ell}||F_{\Sigma,I}|^{2}\right)^{1/2}\,dt\lesssim\int_{t_{0}}^{t_{1}}\left(\int_{D^{C}_{t}}\frac{1}{(1+v)^{3}}\right)^{1/2}\,dt\lesssim\int_{t_{0}}^{t_{1}}\frac{1}{(1+t)^{5/4}}\,dt\lesssim
c_{0}(\epsilon_{0}),$ (8.41)

which completes the proof of (8.27).

We now control the contribution from $F_{I,m_{B}}^{2}$. By (C.72),

$|F_{m_{B},I}^{2}|\lesssim\frac{1+s}{(1+v)^{2}}\sum_{|J|\leq|I|}|\not{\nabla}\psi^{J}_{C}|+\frac{1}{(1+v)^{2}}\sum_{|J|\leq|I|-1}|\Omega\psi^{J}_{C}|.$
(8.42)

Bounding
$|X\psi^{I}_{C}|\lesssim(1+v)|\ell^{m_{B}}\psi^{I}_{C}|+(1+s)|\partial\psi^{I}_{C}|$
for either of our multipliers $X=X_{C},X_{T}$, for $|J|\leq|I|$ we have

$\int_{t_{0}}^{t_{1}}\int_{D^{C}_{t}}\frac{1+s}{(1+v)^{2}}|\not{\nabla}\psi^{J}_{C}||X\psi_{C}^{I}|\,dt\\\
\lesssim\int_{t_{0}}^{t_{1}}\frac{1+\log
t}{(1+t)^{3/2}}\left(\int_{D^{C}_{t}}|\not{\nabla}\psi_{C}^{J}||v^{1/2}\ell^{m_{B}}\psi_{C}^{I}|\right)\,dt+\int_{t_{0}}^{t_{1}}\frac{(1+\log
t)^{3/2}}{(1+t)^{2}}\left(\int_{D^{C}_{t}}|\not{\nabla}\psi_{C}^{J}||X^{n}|^{1/2}|\partial\psi_{C}^{I}|\right)\,dt\\\
\lesssim\delta\int_{t_{0}}^{t_{1}}\int_{D^{C}_{t}}|\not{\nabla}\psi_{C}^{J}|^{2}\,dt+\frac{1}{\delta}c_{0}(\epsilon_{0})\sup_{t_{0}\leq
t\leq t_{1}}\int_{D^{C}_{t}}|\partial\psi_{C}^{J}|^{2}_{X,m_{B}}\lesssim\delta
S_{J}^{C}(t_{1})+\frac{c_{0}(\epsilon_{0})}{\delta}\epsilon_{C}^{2}.$ (8.43)

As for the second term in (8.42), we use the Poincaré-type inequality (F.20)
combined with our bootstrap assumptions to bound

$\int_{D^{C}_{t}}|\Omega\psi_{C}^{J}|^{2}\\\ \lesssim(\log
t)^{1/2}\int_{D^{C}_{t_{0}}}|\Omega\psi_{C}^{J}|^{2}+(\log
t)^{3/2}\int_{t_{0}}^{t_{1}}\int_{\Gamma^{L}_{t^{\prime}}}v|\ell^{m_{B}}\Omega\psi_{C}^{J}|^{2}+\frac{1}{vs}|\partial_{u}\psi_{C}^{J}|^{2}\,dSdt+\log
t\int_{D^{C}_{t}}|\partial\psi_{C}^{J}|^{2}\\\ \lesssim(\log
t)^{3/2}\epsilon_{C}^{2},$ (8.44)

for $|J|\leq|I|-1\leq N_{C}$. It easily follows that

$\int_{t_{0}}^{t_{1}}\int_{D^{C}_{t}}\frac{1}{(1+v)^{2}}|\Omega\psi_{C}^{J}||X\psi_{C}^{I}|\,dt\lesssim\int_{t_{0}}^{t_{1}}\int_{D^{C}_{t}}\frac{1}{(1+v)^{3/2}}|\Omega\psi_{C}^{J}||\partial\psi_{C}^{I}|_{X,m_{B}}\,dt\lesssim\epsilon_{C}^{2}\int_{t_{0}}^{t_{1}}\frac{(1+\log
t)^{3/2}}{(1+t)^{3/2}}\,dt\\\ \lesssim c_{0}(\epsilon_{0})\epsilon_{C}^{2}.$
(8.45)

∎

We now handle the contribution from the component $P^{u}_{I,null}$, which is
responsible for the double-logarithmic growth in some of our estimates.

######  Lemma 8.3.

Let $X=X_{C}$ or $X=X_{T}$ with notation as in Section 2.1 and write
$X=X^{n}\partial_{u}+X^{\ell}\ell^{m_{B}}$. There is $\epsilon_{0}^{*}$ so
that if the hypotheses of Theorem 6.1 hold with
$\epsilon_{0}<\epsilon_{0}^{*}$, the $u$-component of $P_{I,null}$, defined in
Lemma C.4 satisfies the following bounds.

  * •

For $|I|\leq N_{C}$ and any $\delta>0$,

$\int_{t_{0}}^{t_{1}}\int_{D^{C}_{t}}|X^{n}_{T,m_{B}}|^{1/2}\left(|\nabla
P^{u}_{I,null}|+(1+s)^{-1}|P^{u}_{I,null}|\right)|\partial\psi^{I}_{C}|_{X_{T},m_{B}}\,dt\lesssim\delta
S_{I}^{C}(t_{1})+\frac{1}{\delta}c_{0}(\epsilon_{0})\epsilon_{C}^{2},$ (8.46)

where $S_{I}^{C}$ is the spacetime integral defined in (6.33).

  * •

For $|I|\leq N_{C}-1$ and any $\delta>0$,

$\int_{t_{0}}^{t_{1}}\int_{D^{C}_{t}}|X^{n}_{D,m_{B}}|^{1/2}\left(|\nabla
P^{u}_{I,null}|+(1+s)^{-1}|P^{u}_{I,null}|\right)|\partial\psi^{I}_{C}|_{X_{C},m_{B}}\,dt\\\
\lesssim\delta
E_{I,D}^{C}(t_{1})+\frac{1}{\delta}\sum_{|K|\leq|I|-1}E_{K,D}^{C}(t_{1})+\left(\delta+\frac{1}{\delta}\right)c_{0}(\epsilon_{0})\epsilon_{C}^{2}+\epsilon_{C}(1+\log\log
t_{1})\sum_{|J|\leq|I|-1}\sup_{t_{0}\leq t\leq
t_{1}}\left(E_{J,D}^{C}(t)\right)^{1/2},$ (8.47)

  * •

for $|I|\leq N_{C}-2$, and any $\delta>0$,

$\int_{t_{0}}^{t_{1}}\int_{D^{C}_{t}}|X^{n}_{D,m_{B}}|^{1/2}\left(|\nabla
P^{u}_{I,null}|+(1+s)^{-1}|P^{u}_{I,null}|\right)|\partial\psi^{I}_{C}|_{X_{C},m_{B}}\,dt\\\
\lesssim\delta
E_{I,D}^{C}(t_{1})+\frac{1}{\delta}\sum_{|K|\leq|I|-1}E_{K,D}^{C}(t_{1})+\left(\delta+\frac{1}{\delta}\right)c_{0}(\epsilon_{0})\epsilon_{C}^{2}$
(8.48)

We also have

$\sup_{t_{0}\leq t\leq
t_{1}}\int_{D^{C}_{t}}v|P_{I,null}|^{2}+\int_{t_{0}}^{t_{1}}\int_{\Gamma^{L}_{t}}v|P_{I,null}|^{2}\,dSdt+\int_{t_{0}}^{t_{1}}\int_{\Gamma^{R}_{t}}v|P_{I,null}|^{2}\,dSdt\\\
\lesssim
c_{0}(\epsilon_{0})\epsilon_{C}^{2}+\sum_{|J|\leq|I|-1}\sup_{t_{0}\leq t\leq
t_{1}}E_{X_{C},J}^{C}(t).$ (8.49)

###### Proof.

We recall that for our multipliers $X\in\\{X_{C},X_{T}\\}$, by definition

$|\partial q|_{X,m_{B}}=|X^{n}_{m_{B}}|^{1/2}|\partial
q|+v^{1/2}(|\ell^{m_{B}}q|+|\not{\nabla}q|).$ (8.50)

We also recall that from Lemma C.4, the components $P^{u}_{I,null}$ enjoy the
following estimates,

$\displaystyle|P_{I,null}^{u}|$
$\displaystyle\lesssim\frac{1}{1+v}\sum_{|J|\leq|I|-1}\left(\frac{1}{(1+s)^{1/2}}|\partial\psi_{C}^{J}|+|\overline{\partial}\psi_{C}^{J}|\right)+\frac{1}{1+v}\sum_{|J|\leq|I|-2}|\partial\psi_{C}^{J}|,$
(8.51) $(1+s)|\nabla P_{I,null}^{u}|+(1+v)|\partial_{v}P_{I,a}^{u}|+|\Omega
P_{I,null}^{u}|\\\
\lesssim\frac{1}{1+v}\sum_{|J|\leq|I|}\left(\frac{1}{(1+s)^{1/2}}|\partial\psi_{C}^{J}|+|\overline{\partial}\psi_{C}^{J}|\right)+\frac{1}{1+v}\sum_{|J|\leq|I|-1}|\partial\psi_{C}^{J}|,$
(8.52)

where $\overline{\partial}=(\not{\nabla},\ell^{m_{B}})$. Note the above imply

$\displaystyle|P_{I,null}^{u}|$
$\displaystyle\lesssim\frac{1}{(1+v)^{3/2}}\sum_{|J|\leq|I|-1}|\partial\psi_{C}^{J}|_{X,m_{B}}+\frac{1}{(1+v)(1+s)^{1/2}}\sum_{|J|\leq|I|-1}|\partial\psi_{C}^{J}|+\frac{1}{1+v}\sum_{|J|\leq|I|-2}|\partial\psi_{C}^{J}|,$
(8.53)

and

$\displaystyle(1+s)|\nabla
P_{I,null}^{u}|+(1+v)|\partial_{v}P_{I,a}^{u}|+|\Omega P_{I,null}^{u}|$
$\displaystyle\lesssim\frac{1}{(1+v)^{3/2}}\sum_{|J|\leq|I|}|\partial\psi_{C}^{J}|_{X,m_{B}}$
(8.54)
$\displaystyle\quad+\frac{1}{(1+v)(1+s)^{1/2}}\sum_{|J|\leq|I|-1}|\partial\psi_{C}^{J}|+\frac{1}{1+v}\sum_{|J|\leq|I|-1}|\partial\psi_{C}^{J}|.$
(8.55)

By these estimates, for either multiplier $X$ we have

$\displaystyle\int_{t_{0}}^{t_{1}}\int_{D^{C}_{t}}|X^{n}_{m_{B}}|^{1/2}$
$\displaystyle\left(|\nabla
P^{u}_{I,null}|+(1+s)^{-1}|P^{u}_{I,null}|\right)|\partial\psi_{C}^{I}|_{X,m_{B}}\,dt$
(8.56)
$\displaystyle\lesssim\sum_{|J|\leq|I|}\int_{t_{0}}^{t_{1}}\int_{D^{C}_{t}}\frac{|X^{n}_{m_{B}}|^{1/2}}{1+s}\frac{1}{(1+v)^{3/2}}|\partial\psi_{C}^{J}|_{X,m_{B}}^{2}\,dt$
(8.57)
$\displaystyle+\sum_{|J|\leq|I|}\int_{t_{0}}^{t_{1}}\int_{D^{C}_{t}}\frac{1}{(1+v)(1+s)^{3/2}}|X^{n}_{m_{B}}|^{1/2}|\partial\psi_{C}^{J}||\partial\psi_{C}^{I}|_{X,m_{B}}\,dt$
(8.58)
$\displaystyle+\sum_{|J|\leq|I|-1}\int_{t_{0}}^{t_{1}}\int_{D^{C}_{t}}\frac{1}{(1+v)(1+s)}|X^{n}_{m_{B}}|^{1/2}|\partial\psi_{C}^{J}||\partial\psi_{C}^{I}|_{X,m_{B}}\,dt.$
(8.59)

By definition
$|X^{n}_{m_{B}}|^{1/2}|\partial\psi_{C}^{J}|\lesssim|\partial\psi_{C}^{J}|_{X,m_{B}}$,
and since both our multipliers satisfy $|X^{n}_{m_{B}}|\lesssim 1+s$, for the
terms on the second and third lines we can just bound

$\sum_{|J|\leq|I|}\int_{t_{0}}^{t_{1}}\int_{D^{C}_{t}}\frac{|X^{n}_{m_{B}}|^{1/2}}{1+s}\frac{1}{(1+v)^{3/2}}|\partial\psi_{C}^{J}|_{X,m_{B}}^{2}+\frac{1}{(1+v)(1+s)^{3/2}}|X^{n}_{m_{B}}|^{1/2}|\partial\psi_{C}^{J}||\partial\psi_{C}^{I}|_{X,m_{B}}\,dt\\\
\lesssim\left(\int_{t_{0}}^{t_{1}}\frac{1}{(1+t)(1+\log
t)^{9/8}}\,dt\right)\frac{1}{1+\log\log t_{1}}\sup_{t_{0}\leq t\leq
t_{1}}\left(\sum_{|J|\leq|I|}\int_{D^{C}_{t}}|\partial\psi_{C}^{J}|^{2}_{X,m_{B}}\right)\lesssim
c_{0}(\epsilon_{0})\epsilon_{C}^{2},$ (8.60)

where the double-logarithmic factor is only needed in the case $X=X_{C}$ and
$|I|=N_{C}-1$, and where we bounded
$(1+s)^{-3/2}|X^{n}_{m_{B}}|^{1/2}(1+\log\log t)\lesssim(1+\log
t)^{-3/2}(1+\log t)^{1/4}(1+\log\log t)\lesssim(1+\log t)^{-9/8}$.

It remains to control the terms on the last line of (8.56). This is
straightforward when $X=X_{T}$, since then we have
$|X^{n}_{T,m_{B}}|^{1/2}\lesssim(1+s)^{-1/4}$ and in that case

$\sum_{|J|\leq|I|-1}\int_{t_{0}}^{t_{1}}\int_{D^{C}_{t}}\frac{1}{(1+v)(1+s)}|X^{n}_{T,m_{B}}|^{1/2}|\partial\psi_{C}^{J}||\partial\psi_{C}^{I}|_{X_{T},m_{B}}\,dt\\\
\lesssim\int_{t_{0}}^{t_{1}}\int_{D^{C}_{t}}\frac{1}{(1+v)(1+s)^{3/2}}\left(\sum_{|J|\leq|I|-1}|\partial\psi_{C}^{J}|_{X_{C},m_{B}}\right)|\partial\psi_{C}^{I}|_{X_{T},m_{B}}^{2}\,dt\lesssim
c_{0}(\epsilon_{0})\epsilon_{C}^{2},$ (8.61)

after, similarly to the above, bounding $\int_{t_{0}}^{t_{1}}\frac{1+\log\log
t}{(1+t)(1+\log t)^{3/2}}\lesssim c_{0}(\epsilon_{0})$.

The argument is more complicated when $X=X_{C}$, because we cannot afford to
directly use the bootstrap assumptions to handle this term, as that would lead
to a bound of size $\epsilon_{C}^{2}(1+\log\log t)^{2}$, which is too large
for our purposes. We are going to instead prove the improved estimate: for any
$\delta>0$ and $|J|\leq|I|\leq N_{C}-1$,

$\sum_{|J|\leq|I|-1}\int_{t_{0}}^{t_{1}}\int_{D^{C}_{t}}\frac{1}{(1+v)(1+s)}|X^{n}_{C,m_{B}}|^{1/2}|\partial\psi^{J}_{C}||\partial\psi^{I}_{C}|_{X_{C},m_{B}}\,dt\\\
\lesssim\delta
E_{I,D}^{C}(t_{1})+\frac{1}{\delta}\sum_{|K|\leq|I|-1}E_{K,D}^{C}(t_{1})+\left(\delta+\frac{1}{\delta}\right)c_{0}(\epsilon_{0})\epsilon_{C}^{2}+\epsilon_{C}(1+\log\log
t_{1})^{a_{I}}\sum_{|K|\leq|I|-1}\sup_{t_{0}\leq t\leq
t_{1}}(E_{K,D}^{C}(t))^{1/2},$ (8.62)

with $a_{I}=1$ when $|I|=N_{C}-1$ and $a_{I}=0$ otherwise.

The idea is to exploit the fact that since we only need to consider
$|J|\leq|I|-1\leq N_{C}-2$, we can afford to integrate to the shock using
Lemma F.5. Since the domain has width $\sim s^{1/2}$ and since we control
$s\partial_{u}$ applied to the solution, the interior term we generate is
easily handled. It turns out that the boundary term this generates is exactly
of the form controlled by our energy, which allows us to close the estimate.
When $|I|=N_{C}-1$ there is the added complication that we cannot afford to
integrate in both factors because the bounds we have for the top-order
energies $E^{C}_{T}$ are not strong enough to control the resulting quantities
(recall that $E^{C}_{T,I}\gtrsim(1+\log
t)^{-1/2}\|\partial\psi_{C}^{I}\|_{L^{2}(D^{C}_{t})}^{2}$).

By the bound (F.19) from Lemma F.5 and the fact that $(1+\log
t)^{1/2}|\partial q|\lesssim(1+\log
t)^{-1/2}\sum_{Z_{m_{B}}\in\mathcal{Z}_{m_{B}}}|Z_{m_{B}}q|$, we have the
bound

$\int_{D^{C}_{t}}|q|^{2}\lesssim\int_{\Gamma^{L}_{t}}(1+\log
t)^{1/2}|q|^{2}\,dS+\frac{1}{1+\log
t}\sum_{Z_{m_{B}}\in\mathcal{Z}_{m_{B}}}\int_{D^{C}_{t}}|Z_{m_{B}}q|^{2},$
(8.63)

and in particular, since $E_{K,D}(t)\gtrsim(1+\log
t)\int_{D^{C}_{t}}|\partial\psi_{C}^{K}|^{2}$,

$\int_{D_{t}^{C}}|\partial\psi_{C}^{J}|^{2}\lesssim\int_{\Gamma^{L}_{t}}s^{1/2}|\partial\psi_{C}^{J}|^{2}\,dS+\frac{1}{(1+\log
t)^{2}}\sum_{|K|\leq|J|+1}E^{C}_{K,D}(t),$ (8.64)

as well as the similar estimate

$\int_{D^{C}_{t}}|\partial\psi_{C}^{J}|_{X,m_{B}}^{2}\lesssim\int_{\Gamma^{L}_{t}}s^{1/2}|\partial\psi_{C}^{J}|_{X,m_{B}}^{2}\
dS+\frac{1}{(1+\log t)^{2}}\sum_{|K|\leq|J|+1}E^{C}_{K,D}(t),$ (8.65)

We can now prove (8.62). We will need to handle the two cases $|I|=N_{C}-1$
and $|I|\leq N_{C}-2$ separately, with the first of these being slightly more
involved and responsible for the (slow) growth of our energies.

_The proof of ( 8.62) when $|I|=N_{C}-1$_

For $|J|\leq|I|-1$, since $|X^{n}_{C,m_{B}}|\lesssim 1+\log t$, we have

$\displaystyle\int_{t_{0}}^{t_{1}}\int_{D^{C}_{t}}$
$\displaystyle\frac{1}{(1+v)(1+s)}|X^{n}_{C,m_{B}}|^{1/2}|\partial\psi_{C}^{J}||\partial\psi_{C}^{I}|_{X,m_{B}}\,dt$
(8.66) $\displaystyle\lesssim\int_{t_{0}}^{t_{1}}\frac{1}{1+t}\frac{1}{(1+\log
t)^{1/2}}\left(\int_{D^{C}_{t}}|\partial\psi_{C}^{J}|^{2}\right)^{1/2}\left(E^{C}_{I,D}(t)\right)^{1/2}\,dt$
(8.67) $\displaystyle\lesssim\int_{t_{0}}^{t_{1}}\frac{1}{1+t}\frac{1}{(1+\log
t)^{1/2}}\left(\int_{\Gamma^{L}_{t}}s^{1/2}|\partial\psi_{C}^{J}|^{2}\,dS+\frac{1}{(1+\log
t)^{2}}\sum_{|K|\leq|J|+1}E_{K,D}^{C}(t)\right)^{1/2}(E^{C}_{I,D}(t))^{1/2}\,dt,$
(8.68)

by (8.64). For the second term here, we just note that by (6.79) we have

$\int_{t_{0}}^{t_{1}}\frac{1}{1+t}\frac{1}{(1+\log
t)^{3/2}}\sum_{|K|\leq|J|+1}E_{K,D}^{C}(t)^{1/2}E_{I,D}^{C}(t)^{1/2}\,dt\lesssim\epsilon_{C}\int_{t_{0}}^{t_{1}}\frac{1}{1+t}\frac{1+\log\log
t}{(1+\log t)^{3/2}}\,dt\lesssim c_{0}(\epsilon_{0})\epsilon_{C}.$ (8.69)

For the first term in (8.68), we bound

$\int_{t_{0}}^{t_{1}}\frac{1}{1+t}\frac{1}{(1+\log
t)^{1/2}}\left(\int_{\Gamma^{L}_{t}}s^{1/2}|\partial\psi^{J}_{C}|^{2}\,dS\right)^{1/2}(E^{C}_{I,D}(t))^{1/2}\,dt\\\
\lesssim\epsilon_{C}(1+\log\log
t_{1})^{1/2}\left(\int_{t_{0}}^{t_{1}}\frac{1}{1+t}\frac{1}{1+\log
t}\right)^{1/2}\left(\int_{t_{0}}^{t_{1}}\int_{\Gamma^{L}_{t}}\frac{s^{1/2}}{v}|\partial\psi_{C}^{J}|^{2}\,dS\,dt\right)^{1/2}\\\
\lesssim\epsilon_{C}(\log\log t_{1})\sum_{|K|\leq|I|-1}\sup_{t_{0}\leq t\leq
t_{1}}E_{K,D}^{C}(t)^{1/2},$ (8.70)

and combining this with the previous inequality we get (8.62). We remark that
it is to handle this term that we needed to allow the norms $E_{I,D}^{C}(t)$
to grow slowly when $|I|=N_{C}-1$.

The proof of (8.62) when $|I|\leq N_{C}-2$

The argument in this case is similar but a bit simpler, because we can afford
to integrate as in (8.64) in both factors. For $|I|\leq N_{C}-2$ and
$|J|\leq|I|-1$, we first bound
$|X^{n}_{C,m_{B}}|^{1/2}|\partial\psi_{C}^{J}||\partial\psi_{C}^{I}|_{X,m_{B}}\leq|\partial\psi_{C}^{J}||\partial\psi_{C}^{I}|_{X,m_{B}}$,
so for any $\delta>0$ we have

$\int_{t_{0}}^{t_{1}}\int_{D^{C}_{t}}\frac{1}{(1+v)(1+s)}|X^{n}_{C,m_{B}}|^{1/2}|\partial\psi_{C}^{J}||\partial\psi_{C}^{I}|_{X,m_{B}}\,dt\leq\int_{t_{0}}^{t_{1}}\int_{D^{C}_{t}}\frac{1}{(1+v)(1+s)}|\partial\psi_{C}^{J}|_{X,m_{B}}|\partial\psi_{C}^{I}|_{X,m_{B}}\,dt\\\
\lesssim\delta\int_{t_{0}}^{t_{1}}\int_{D^{C}_{t}}\frac{1}{(1+v)(1+s)}|\partial\psi_{C}^{I}|_{X,m_{B}}^{2}\,dt+\frac{1}{\delta}\int_{t_{0}}^{t_{1}}\int_{D^{C}_{t}}\frac{1}{(1+v)(1+s)}|\partial\psi_{C}^{J}|_{X,m_{B}}^{2}\,dt.$
(8.71)

Using (8.65), we find

$\int_{t_{0}}^{t_{1}}\int_{D^{C}_{t}}\frac{1}{(1+v)(1+s)}|\partial\psi_{C}^{K}|_{X,m_{B}}^{2}\,dt\,dt\\\
\lesssim\int_{t_{0}}^{t_{1}}\int_{\Gamma^{L}_{t}}\frac{1}{(1+v)(1+s)^{1/2}}|\partial\psi_{C}^{K}|_{X,m_{B}}^{2}\,dSdt+\sum_{|K^{\prime}|\leq|K|+1}\int_{t_{0}}^{t_{1}}\frac{1}{(1+t)(1+\log
t)^{3}}E_{K^{\prime},D}^{C}(t)\,dt\\\ \lesssim
E_{K,D}^{C}(t_{1})+c_{0}(\epsilon_{0})\epsilon_{C}^{2},$ (8.72)

for $|K|\leq N_{C}-2$. From the above bounds we find

$\int_{t_{0}}^{t_{1}}\int_{D^{C}_{t}}\frac{1}{(1+v)(1+s)}|X^{n}_{C,m_{B}}|^{1/2}|\partial\psi_{C}^{J}||\partial\psi_{C}^{I}|_{X,m_{B}}\,dt\\\
\lesssim\delta
E_{I,D}^{C}(t_{1})+\frac{1}{\delta}\sum_{|J|\leq|I|-1}E_{J,D}(t_{1})+\left(\delta+\frac{1}{\delta}\right)c_{0}(\epsilon_{0})\epsilon_{C}^{2},$
(8.73)

which concludes the proof of (8.62).

It remains only to prove (8.49). This follows after using the simple estimate

$|P_{I,null}|\lesssim\frac{1}{1+v}\sum_{|J|\leq|I|-1}|\partial\psi^{J}_{C}|,$
(8.74)

which follows directly from the estimate (C.68) from Lemma C.4, and then
bounding

$\int_{D^{C}_{t}}(1+v)|P_{I,null}|^{2}\lesssim\frac{1}{1+t}\sum_{|J|\leq|I|}\int_{D^{C}_{t}}|\partial\psi^{J}_{C}|^{2}\,dt\lesssim
c_{0}(\epsilon_{0})\epsilon_{C}^{2},$ (8.75)

and, for $A=L,R$,

$\int_{t_{0}}^{t_{1}}\int_{\Gamma^{A}_{t}}(1+v)|P_{I,null}|^{2}\,dSdt\lesssim\sum_{|J|\leq|I|-1}\int_{t_{0}}^{t_{1}}\int_{\Gamma^{A}_{t}}\frac{1}{1+v}|\partial\psi_{C}^{J}|^{2}\,dSdt\\\
\lesssim\frac{1}{(1+\log
t_{0})^{1/2}}\int_{t_{0}}^{t_{1}}\int_{\Gamma^{A}_{t}}\frac{(1+s)^{1/2}}{1+v}|\partial\psi_{C}^{J}|^{2}\,dSdt\lesssim
c_{0}(\epsilon_{0})\epsilon_{C}^{2}.$ (8.76)

Here, we used (7.51) to control the angular derivatives along the shock.

∎

We now control the term $F_{m_{B},I}^{1}$.

######  Lemma 8.4.

Under the hypotheses of Theorem 6.1, for either $X=X_{T}$ or $X=X_{C}$ and
$|I|\leq N_{C}$, for any $\delta>0$, we have

$-\int_{t_{0}}^{t_{1}}\int_{D^{C}_{t}}F_{I,m_{B}}^{1}\,dt\\\ \lesssim
c_{0}(\epsilon_{0})\epsilon_{C}^{2}+\delta\sum_{|J|\leq|I|}\left(E_{J,X_{T}}(t_{1})+S_{I}^{C}(t_{1})\right)+\frac{1}{\delta}\sum_{|J|\leq|I|-1}E_{J,X_{T}}(t_{1})+\left(1+\frac{1}{\delta}\right)\sum_{|J|\leq|I|-1}S_{J}^{C}(t_{1})+\epsilon_{C}^{3}.$
(8.77)

###### Proof.

By the definition (C.71) of $F_{m_{B},I}^{1}$,

$-\int_{t_{0}}^{t_{1}}\int_{D^{C}_{t}}F_{m_{B},I}^{1}X\psi^{I}\,dt=\sum_{|J_{1}|+|J_{2}|\leq|I|-1}\int_{t_{0}}^{t_{1}}\int_{D^{C}_{t}}a^{I}_{J_{1}J_{2}}\not{\Delta}Z_{m_{B}}^{J}\psi_{C}(v\partial_{v}Z_{m_{B}}^{J_{1}}(v\partial_{v})Z_{m_{B}}^{J_{2}}\psi_{C})\,dt\\\
+\sum_{|J_{1}|+|J_{2}|\leq|I|-1}\int_{t_{0}}^{t_{1}}\int_{D^{C}_{t}}a^{I}_{J_{1}J_{2}}\not{\Delta}Z_{m_{B}}^{J}\psi_{C}(X^{u}\partial_{u}Z_{m_{B}}^{J_{1}}(v\partial_{v})Z_{m_{B}}^{J_{2}}\psi_{C})\,dt,$
(8.78)

where $a^{I}_{J_{1}J_{2}}=1$ if
$Z_{m_{B}}^{I}=Z_{m_{B}}^{J_{1}}(v\partial_{v})Z_{m_{B}}^{J_{2}}$ and
$a^{I}_{J_{1}J_{2}}=0$ otherwise, and where
$Z_{m_{B}}^{J}=Z_{m_{B}}^{J_{1}}Z_{m_{B}}^{J_{2}}$.

We start by dealing with the first term in (8.78). For this, write

$\not{\Delta}Z_{m_{B}}^{J}\psi_{C}(v\partial_{v}Z_{m_{B}}^{J_{1}}(v\partial_{v})Z_{m_{B}}^{J_{2}}\psi_{C})=a^{I}_{J_{1}J_{2}}\not{\Delta}Z_{m_{B}}^{J}\psi_{C}(v\partial_{v})^{2}Z_{m_{B}}^{J}\psi_{C}\\\
+a^{I}_{J_{1}J_{2}}\not{\Delta}Z_{m_{B}}^{J}\psi_{C}v\partial_{v}\left([Z_{m_{B}}^{J_{1}},v\partial_{v}]Z_{m_{B}}^{J_{2}}\psi_{C}\right).$
(8.79)

The first term here needs to be handled carefully. We write

$\displaystyle\not{\Delta}Z^{J}_{m_{B}}\psi_{C}(v\partial_{v})^{2}Z_{m_{B}}^{J}\psi_{C}$
$\displaystyle=\not{\nabla}\cdot\left(\not{\nabla}Z^{J}_{m_{B}}\psi_{C}(v\partial_{v})^{2}Z_{m_{B}}^{J}\psi_{C}\right)-\not{\nabla}Z^{J}_{m_{B}}\psi_{C}\cdot
v\partial_{v}\not{\nabla}(v\partial_{v}Z_{m_{B}}^{J}\psi_{C})$ (8.80)
$\displaystyle\qquad+\not{\nabla}Z_{m_{B}}^{J}\psi_{C}\cdot[v\partial_{v},\not{\nabla}]v\partial_{v}Z_{m_{B}}^{J}\psi_{C}$
(8.81)
$\displaystyle=\not{\nabla}\cdot\left(v\not{\nabla}Z^{J}_{m_{B}}\psi_{C}\partial_{v}(v\partial_{v}Z_{m_{B}}^{J}\psi_{C})\right)-\partial_{v}\left(v\not{\nabla}Z^{J}_{m_{B}}\psi_{C}\cdot\not{\nabla}(v\partial_{v}Z_{m_{B}}^{J}\psi_{C})\right)$
(8.82)
$\displaystyle\qquad+\not{\nabla}Z_{m_{B}}^{J}\psi_{C}\cdot\not{\nabla}(v\partial_{v}Z_{m_{B}}^{J}\psi_{C})+|\not{\nabla}(v\partial_{v}Z_{m_{B}}^{J}\psi_{C})|^{2}$
(8.83)
$\displaystyle\qquad+\not{\nabla}Z_{m_{B}}^{J}\psi_{C}\cdot[v\partial_{v},\not{\nabla}]v\partial_{v}Z_{m_{B}}^{J}\psi_{C}+[v\partial_{v},\not{\nabla}]Z^{J}_{m_{B}}\psi_{C}\cdot\not{\nabla}(v\partial_{v}Z_{m_{B}}^{J}\psi_{C})$
(8.84)

We start by handling the spacetime integrals of the terms on the first line.
By Stokes’ theorem,

$\int_{t_{0}}^{t_{1}}\int_{D^{C}_{t}}\not{\nabla}\cdot\left(\not{\nabla}Z^{J}_{m_{B}}\psi_{C}(v\partial_{v})^{2}Z_{m_{B}}^{J}\psi_{C}\right)\,dt\\\
=\int_{t_{0}}^{t_{1}}\int_{\Gamma^{L}_{t}}v\not{\nabla}B^{L}\cdot\not{\nabla}Z^{J}_{m_{B}}\psi_{C}\partial_{v}(v\partial_{v}Z_{m_{B}}^{J}\psi_{C})\,dSdt-\int_{t_{0}}^{t_{1}}\int_{\Gamma^{R}_{t}}v\not{\nabla}B^{R}\cdot\not{\nabla}Z^{J}_{m_{B}}\psi_{C}\partial_{v}(v\partial_{v}Z_{m_{B}}^{J}\psi_{C})\,dSdt.$
(8.85)

Since $|v\not{\nabla}B^{A}|\lesssim s^{1/2}$, we have

$\int_{t_{0}}^{t_{1}}\int_{\Gamma^{A}_{t}}|v\not{\nabla}B^{A}\cdot\not{\nabla}Z^{J}_{m_{B}}\psi_{C}\partial_{v}(v\partial_{v}Z_{m_{B}}^{J}\psi_{C})|\,dSdt\\\
\lesssim\int_{t_{0}}^{t_{1}}\int_{\Gamma^{A}_{t}}\frac{(1+s)^{1/2}}{(1+v)^{1/2}}|\not{\nabla}Z^{J}_{m_{B}}\psi_{C}||v^{1/2}\partial_{v}(v\partial_{v}Z_{m_{B}}^{J}\psi_{C})|\,dSdt\lesssim
c_{0}(\epsilon_{0})\epsilon_{C}^{2},$ (8.86)

using the bounds (7.47)-(7.50) for the boundary terms in the energies.

Also by Stokes’ theorem,

$\displaystyle\int_{t_{0}}^{t_{1}}$
$\displaystyle\int_{D_{t}^{C}}\partial_{v}\big{(}v\not{\nabla}Z^{J}_{m_{B}}\psi_{C}\cdot\not{\nabla}(v\partial_{v})Z_{m_{B}}^{J}\psi_{C}\big{)}\,dt$
(8.87)
$\displaystyle=\int_{t_{0}}^{t_{1}}\int_{\Gamma^{L}_{t}}v\partial_{v}B^{L}\not{\nabla}Z^{J}_{m_{B}}\psi_{C}\cdot\not{\nabla}(v\partial_{v})Z_{m_{B}}^{J}\psi_{C}\,dSdt-\int_{t_{0}}^{t_{1}}\int_{\Gamma^{R}_{t}}v\partial_{v}B^{R}\not{\nabla}Z^{J}_{m_{B}}\psi_{C}\cdot\not{\nabla}(v\partial_{v})Z_{m_{B}}^{J}\psi_{C}\,dSdt$
(8.88)
$\displaystyle+\int_{D^{C}_{t_{1}}}v\not{\nabla}Z^{J}_{m_{B}}\psi_{C}\cdot\not{\nabla}(v\partial_{v}Z_{m_{B}}^{J}\psi_{C})-\int_{D^{C}_{t_{0}}}v\not{\nabla}Z^{J}_{m_{B}}\psi_{C}\cdot\not{\nabla}(v\partial_{v}Z_{m_{B}}^{J}\psi_{C}).$
(8.89)

For the terms along the shocks, we use that $|v\partial_{v}B^{A}|\lesssim
s^{-1/2}$ and write $v\partial_{v}Z_{m_{B}}^{J}=Z_{m_{B}}^{J^{\prime}}$, which
gives

$\displaystyle\left|\int_{t_{0}}^{t_{1}}\int_{\Gamma^{A}_{t}}v\partial_{v}B^{A}\not{\nabla}Z^{J}_{m_{B}}\psi_{C}\cdot\not{\nabla}(v\partial_{v}Z_{m_{B}}^{J}\psi_{C})\,dSdt\right|$
$\displaystyle\lesssim\int_{t_{0}}^{t_{1}}\int_{\Gamma^{A}_{t}}\frac{1}{(1+s)^{1/2}}|\not{\nabla}Z^{J}_{m_{B}}\psi_{C}||\not{\nabla}Z_{m_{B}}^{J^{\prime}}\psi_{C}|\,dSdt$
(8.90)
$\displaystyle\lesssim(E_{J,T}^{C}(t))^{1/2}(E_{J^{\prime},T}^{C}(t))^{1/2}\lesssim\frac{1}{\delta}E_{J,T}^{C}(t_{1})+\delta
E_{J^{\prime},T}^{C}(t_{1})$ (8.91)

for arbitrary $\delta>0$ (recall here $|J|\leq|I|-1$ and $|J^{\prime}|=|I|$).
For the terms in (8.87) along the time slices, we just bound

$\int_{D^{C}_{t}}v|\not{\nabla}Z^{J}_{m_{B}}\psi_{C}||\not{\nabla}(v\partial_{v}Z_{m_{B}}^{J}\psi_{C})|\lesssim(E_{J,T}^{C}(t))^{1/2}(E_{J^{\prime},T}^{C}(t))^{1/2}\lesssim\frac{1}{\delta}E_{J,T}^{C}(t)+\delta
E_{J^{\prime},T}^{C}(t).$ (8.92)

For the first two terms on the last line of (8.80) we just note that, again
writing $v\partial_{v}Z_{m_{B}}^{J}=Z_{m_{B}}^{J^{\prime}}$,

$\displaystyle\int_{t_{0}}^{t_{1}}\int_{D^{C}_{t}}\not{\nabla}Z_{m_{B}}^{J}\psi_{C}\cdot\not{\nabla}$
$\displaystyle(v\partial_{v}Z_{m_{B}}^{J}\psi_{C})+|\not{\nabla}(v\partial_{v}Z_{m_{B}}^{J}\psi_{C})|^{2}\,dt$
(8.93) $\displaystyle\geq
S_{X_{T}}[Z^{J^{\prime}}_{m_{B}}\psi_{C}]-S_{X_{T}}[Z^{J^{\prime}}_{m_{B}}\psi]^{1/2}S_{X_{T}}[Z^{J}_{m_{B}}\psi_{C}]^{1/2},$
(8.94)
$\displaystyle\geq\frac{1}{2}S_{X_{T}}[Z^{J^{\prime}}_{m_{B}}\psi_{C}]-\frac{1}{2}S_{X_{T}}[Z^{J}_{m_{B}}\psi_{C}]$
(8.95) $\displaystyle\geq-\frac{1}{2}S_{X_{T}}[Z^{J}_{m_{B}}\psi_{C}],$ (8.96)

which is the crucial step. This last term is of the correct form since
$|J|\leq|I|-1$.

To deal with the terms from (8.80) involving $[v\partial_{v},\not{\nabla}]$,
we use (A.19) to write

$[v\partial_{v},\not{\nabla}_{i}]=[v\partial_{v},\frac{\omega^{j}}{r}\Omega_{ij}]=-\frac{1}{2}\frac{v}{r^{2}}\omega^{j}\Omega_{ij}=-\frac{1}{2}\frac{v}{r}\not{\nabla}_{i},$
(8.97)

and since $|J|\leq|I|-1$, we have

$\int_{t_{0}}^{t_{1}}\int_{D^{C}_{t}}|\not{\nabla}Z_{m_{B}}^{J}\psi_{C}\cdot[v\partial_{v},\not{\nabla}]v\partial_{v}Z_{m_{B}}^{J}\psi_{C}|\,dt\lesssim\int_{t_{0}}^{t_{1}}\int_{D^{C}_{t}}|\not{\nabla}Z_{m_{B}}^{J}\psi_{C}||\not{\nabla}(v\partial_{v}Z_{m_{B}}^{J}\psi_{C})|\,dt\\\
\lesssim\frac{1}{\delta}\sum_{|K|\leq|I|-1}S_{K}^{C}(t_{1})+\delta\sum_{|K|\leq|I|}S_{K}^{C}(t_{1}),$
(8.98)

and similarly

$\int_{t_{0}}^{t_{1}}\int_{D^{C}_{t}}|[v\partial_{v},\not{\nabla}]Z_{m_{B}}^{J}\psi_{C}\cdot\not{\nabla}(v\partial_{v}Z_{m_{B}}^{J}\psi_{C})|\,dt\lesssim\frac{1}{\delta}\sum_{|K|\leq|I|-1}S_{K}^{C}(t_{1})+\delta\sum_{|K|\leq|I|}S_{K}^{C}(t_{1}),$
(8.99)

It remains only to deal with the second term in (8.78) and the second term in
(8.79). For the former, we just bound, for any $\delta>0$,

$\displaystyle\int_{t_{0}}^{t_{1}}\int_{D^{C}_{t}}|\not{\nabla}^{2}Z_{m_{B}}^{J}\psi_{C}|$
$\displaystyle|X^{u}||\partial Z_{m_{B}}^{I}\psi_{C}|\,dt$ (8.100)
$\displaystyle\lesssim\delta\int_{t_{0}}^{t_{1}}\int_{D^{C}_{t}}|\not{\nabla}\Omega
Z_{m_{B}}^{J}\psi_{C}|^{2}\,dt+\frac{1}{\delta}\int_{t_{0}}^{t_{1}}\frac{1}{(1+t)^{2}}\left(\int_{D^{C}_{t}}|X^{u}||\partial
Z_{m_{B}}^{I}\psi_{C}|^{2}\right)\,dt$ (8.101) $\displaystyle\lesssim\delta
S_{X_{T}}[Z_{m_{B}}^{J}\psi_{C}]+\frac{1}{\delta}c_{0}(\epsilon_{0})\epsilon_{C}^{2}.$
(8.102)

As for the latter term, using that
$[v\partial_{v},s\partial_{u}]=\partial_{u}$ and that otherwise
$[v\partial_{v},Z_{m_{B}}]=0$, we have
$|v\partial_{v}[Z^{J_{1}}_{m_{B}},v\partial_{v}]Z_{m_{B}}^{J_{2}}\psi_{C}|\lesssim\sum_{|K|\leq|J_{1}|+|J_{2}|}|v\partial_{v}\partial
Z_{m_{B}}^{K}\psi_{C}|\lesssim\sum_{|K|\leq|J_{1}|+|J_{2}|+1}|\partial
Z_{m_{B}}^{K}\psi_{C}|$, so that, since $|J_{1}|+|J_{2}|\leq|I|-1$,

$\displaystyle\int_{t_{0}}^{t_{1}}\int_{D^{C}_{t}}|\not{\nabla}^{2}Z_{m_{B}}^{J}\psi_{C}|$
$\displaystyle|v\partial_{v}Z_{m_{B}}^{J_{1}}[Z_{m_{B}}^{J_{2}},v\partial_{v}]\psi_{C}|\,dt$
(8.103)
$\displaystyle\lesssim\sum_{|K|\leq|I|}\int_{t_{0}}^{t_{1}}\int_{D^{C}_{t}}\frac{1}{1+v}|\not{\nabla}\Omega
Z_{m_{B}}^{J}\psi_{C}||\partial Z_{m_{B}}^{K}\psi_{C}|\,dt$ (8.104)
$\displaystyle\lesssim\delta\int_{t_{0}}^{t_{1}}\int_{D^{C}_{t}}|\not{\nabla}\Omega
Z_{m_{B}}^{K}\psi_{C}|^{2}\,dt+\frac{1}{\delta}\int_{t_{0}}^{t_{1}}\frac{1}{(1+t)^{2}}\left(\int_{D^{C}_{t}}|\partial
Z_{m_{B}}^{K}\psi_{C}|\right)\,dt$ (8.105) $\displaystyle\lesssim\delta
S_{X_{T}}[Z_{m_{B}}^{J}\psi_{C}]+\frac{1}{\delta}c_{0}(\epsilon_{0})\epsilon_{C}^{2},$
(8.106)

as needed.

∎

We now prove the corresponding bounds in the leftmost region.

######  Lemma 8.5.

Define $X=X_{L}$ or $X=X_{M}$ as in Section 2.1 and write
$X=X^{n}\partial_{u}+X^{\ell}\partial_{v}$. If the hypotheses of Proposition
6.1 hold, the quantities $\gamma,P_{I},F_{I}$ appearing in (C.11) satisfy the
following estimates.

$|\gamma|\lesssim\frac{\epsilon_{L}}{(1+v)(1+s)^{1/2}}\frac{(\log\log
s)^{\alpha}}{(\log
s)^{\alpha-1}},\qquad|X_{m}^{\ell}||\gamma|\lesssim\epsilon_{L}|X^{n}_{m}|,$
(8.107) $\int_{t_{0}}^{T}\|\nabla\gamma\|_{L^{\infty}(D^{L}_{t}\cap\\{|u|\leq
v/8\\})}+\left\|\frac{1}{1+|u|}\gamma\right\|_{L^{\infty}(D_{t}^{L}\cap\\{|u|\leq
v/8\\})}\,dt\\\
+\int_{t_{0}}^{t_{1}}\left\|\left(1+\frac{|X_{m}^{\ell}|^{1/2}}{|X^{n}|^{1/2}}\right)\nabla_{v}\gamma\right\|_{L^{\infty}(D_{t}^{L}\cap\\{|u|\leq
v/8\\})}+\left\|\left(1+\frac{|X_{m}^{\ell}|^{1/2}}{|X_{m}^{n}|^{1/2}}\right)\not{\nabla}\gamma\right\|_{L^{\infty}(D_{t}^{L}\cap\\{|u|\leq
v/8\\})}\,dt\lesssim\epsilon_{L},$ (8.108)

and

$\int_{t_{0}}^{t_{1}}\||X^{n}_{m}|^{1/2}\nabla
P_{I}\|_{L^{2}(D^{L}_{t}\cap\\{|u|\leq
v/8\\})}+\|(1+|u|)^{-1}|X^{n}_{m}|^{1/2}P_{I}\|_{L^{2}(D^{L}_{t}\cap\\{|u|\leq
v/8\\})}\,dt\\\
+\int_{t_{0}}^{t_{1}}\||X^{\ell}_{m}|^{1/2}\nabla_{v}P_{I}\|_{L^{2}(D^{L}_{t}\cap\\{|u|\leq
v/8\\})}+\||X^{\ell}_{m}|^{1/2}\not{\nabla}P_{I}\|_{L^{2}(D^{L}_{t}\cap\\{|u|\leq
v/8\\})}+\||X^{\ell}_{m}|^{1/2}F_{I}\|_{L^{2}(D^{L}_{t})}\,dt\lesssim\epsilon_{L}^{2},$
(8.109)

as well as the following bounds in the region $|u|\geq v/8$,

$\displaystyle\int_{t_{0}}^{t_{1}}\|\mathcal{L}_{X}\gamma\|_{L^{\infty}(D^{L}_{t}\cap\\{|u|\geq
v/8\\})}\,dt$ $\displaystyle\lesssim\epsilon_{L},$ (8.110)
$\displaystyle\int_{t_{0}}^{t_{1}}\|\mathcal{L}_{X}P_{I}\|_{L^{2}(D^{L}_{t}\cap\\{|u|\geq
v/8\\})}\,dt$ $\displaystyle\lesssim\epsilon_{L}^{2}.$ (8.111)

We also have

$\sup_{t_{0}\leq t\leq
t_{1}}\int_{D^{L}_{t}}|X^{\ell}_{m}||P_{I}|^{2}+\int_{t_{0}}^{t_{1}}\left(|X^{\ell}_{m}|+(1+v)(1+s)^{1/2}|X^{n}_{m}|\right)|P_{I}|^{2}\,dSdt\lesssim\epsilon_{L}^{3}.$
(8.112)

###### Proof.

We will need to argue slightly differently in the three regions

$D_{t,1}^{L}=D_{t}^{L}\cap\\{|u|\leq s^{3}\\},\qquad
D_{t,2}^{L}=D_{t}^{L}\cap\\{s^{3}\leq|u|\leq v/8\\},\qquad
D_{t,3}^{L}=D_{t}^{L}\cap\\{|u|\geq v/8\\}.$ (8.113)

We first consider the bounds in $D_{t,1}^{L}$. As in (8.6), we have the bounds

$\displaystyle|\gamma|$
$\displaystyle\lesssim\frac{1}{1+v}|\partial\psi_{L}|+\frac{1}{(1+v)^{2}}|\psi_{L}|,$
(8.114) $\displaystyle|\nabla_{Y}\gamma|$
$\displaystyle\lesssim\frac{1}{1+v}|\nabla_{Y}\partial\psi_{L}|+\frac{1}{(1+v)^{2}}|Y||\partial\psi_{L}|+\frac{1}{(1+v)^{3}}|Y||\psi_{L}|,$
(8.115)

where we used that $r\geq\tfrac{1}{4}v$, say, in this region.

The bounds (8.107) in the region $D_{t,1}^{L}$ then follow from (8.114) and
(7.4), with the crucial observation being that

$\frac{|X^{\ell}_{m}|}{1+v}|\partial\psi_{L}|\lesssim
f(v)|\partial\psi_{L}|\lesssim\epsilon_{L}\frac{f(v)}{|u|f(u)^{1/2}}\lesssim\epsilon_{L}\frac{s(\log
s)^{\alpha}}{s^{1/2}(\log s^{1/2}(\log\log
s)^{\alpha/2}}\lesssim\epsilon_{L}s^{1/2}\frac{(\log
s)^{\alpha-1/2}}{(\log\log s)^{\alpha/2}}\\\ \lesssim\epsilon_{L}s^{1/2}\log
s(\log\log s)^{\alpha}\lesssim\epsilon_{L}|u|f(u),$ (8.116)

where in the second-last step we used that $\alpha<3/2$ and in the last step
we used the lower bound for $|u|$ in $D^{L}_{t}$. To control the last term in
(8.114), we used the bound (7.8).

As in Lemmas 8.8-(8.9), the time-integrated bounds for $\gamma$ and its
derivatives in $D^{L}_{t,1}$ follow from (8.114)- (8.115) and the time-
integrated bounds in Lemma 7.5, but using (7.8) and the fact that we have a
bound for $B_{I}(t)$, in place of the bound (7.6) that we used in the right
region. The quantities $P_{I}$ and $F_{I}$ in this region can be handled using
similar arguments and so we skip them.

It remains only to prove the needed bounds in the region $D^{L}_{t,2}$ and
$D^{L}_{t,3}$. Here the estimates are less delicate because of the lower bound
for $|u|$. In this region, we work in terms of $\phi$ and recall from (C.12),

$|\gamma|\lesssim|\partial\phi_{L}|,\qquad|\nabla_{X}\gamma|\lesssim|\nabla_{X}\partial\phi_{L}|,\qquad|\mathcal{L}_{X}\gamma|\lesssim|\mathcal{L}_{X}\partial\phi_{L}|.$
(8.117)

By the pointwise bound (7.5), the bounds in (8.107) clearly hold in
$D^{L}_{t,2}$ and $D^{L}_{t,3}$. To get the time-integrated bound for
derivatives of $\gamma$ in $D^{L}_{t,2}$, we just bound

$|\nabla\partial\phi|\lesssim|\partial^{2}\phi|+\frac{1}{r}|\partial\phi|\lesssim|\partial^{2}\phi|+\frac{1}{1+v}|\partial\phi|,\qquad|u|\leq
v/8$ (8.118)

using that the Christoffel symbols in our coordinate system satisfy
$|\Gamma|\lesssim\frac{1}{r}$. By (7.5) this gives

$|\nabla\partial\phi|\lesssim\frac{\epsilon_{L}}{(1+v)(1+s)^{6}},\qquad|u|\geq
s^{3},$ (8.119)

which is more than enough to get the first bound in (8.108). The bound for
$(1+|u|)^{-1}\gamma$ is identical, and the bound for the terms on the second
line of (8.108) is easier since we can just bound
$|X^{\ell}|^{1/2}(|\partial_{v}q|+|\not{\nabla}q|)\lesssim(1+v)^{-1/4}\sum_{Z\in\mathcal{Z}}|Zq|$.

To get the time-integrated bound for $\mathcal{L}_{X}\gamma$, which is needed
in $D^{L}_{t,3}$, we bound

$|\mathcal{L}_{X}\partial\phi|\lesssim\sum_{\nu\in\\{u,v,\theta^{1},\theta^{2}\\}}|X^{\mu}\partial_{\mu}\partial_{\nu}\phi|+|\partial_{\nu}X^{\mu}\partial_{\mu}\phi|.$
(8.120)

For the first term, we bound

$|X^{\mu}\partial_{\mu}\partial_{\nu}\phi|\lesssim|X^{v}||\partial_{v}\partial\phi|+|X^{u}||\partial_{u}\partial\phi|\lesssim
f(v)\sum_{Z\in\mathcal{Z}}|\partial Z\phi|,$ (8.121)

where we used that for either multiplier we have
$|X^{v}|/v+|X^{u}|/|u|\lesssim f(v)$. By the pointwise bound (7.5) and the
definition of $f$ from (2.12) in the region $D^{L}_{t,1}$ we have

$f(v)|\partial Z\phi|\lesssim\frac{s(\log
s)^{\alpha}}{(1+v)(1+s)^{3}}\epsilon_{L}\lesssim\frac{1}{1+t}\frac{1}{(1+\log
t)^{2}}\epsilon_{L},$ (8.122)

which is time-integrable. For the second term in (8.120), we just use that
$v|f^{\prime}(v)|+f(u)+|uf^{\prime}(u)|\lesssim f(v)$ and then

$\sum_{\nu\in\\{u,v,\theta^{1},\theta^{2}\\}}|\partial_{\nu}X^{\mu}\partial_{\mu}\phi|\lesssim
f(v)|\partial\phi|,$ (8.123)

which can be bounded just as in (8.122). The bound for $|\nabla_{X}\gamma|$ in
the region $D^{L}_{t,2}$ can be proven in a nearly identical way.

The needed bounds for $P_{I}$ and $F_{I}$ can be proven using the same
arguments we have now used many times, after using the bounds from Lemma C.1.
∎

###  8.2  Control of the scalar currents

We now use the results of the previous section to control the scalar currents
$\widetilde{K}_{X,\gamma,P}$ which appear in the energy estimates (6.103),
(6.120), (6.121) and (6.138).

By Proposition 3.1, in the regions $D^{L}$ and $D^{R}$, when $|u|\leq v/8$ for
our multipliers $X=X_{L},X_{M},X_{R}$, this quantity satisfies

$|\widetilde{K}_{X,\gamma,P}[\psi]|\lesssim\left(|\nabla\gamma|+\frac{1}{1+|u|}|\gamma|+\frac{|X^{\ell}_{m}|^{1/2}}{|X^{n}_{m}|^{1/2}}|\nabla_{{\ell^{m}}}\gamma|+\frac{|X^{\ell}_{m}|^{1/2}}{|X^{n}_{m}|^{1/2}}|\not{\nabla}\gamma|\right)|\partial\psi|_{X,m}^{2}+|X^{n}_{m}||F||\partial\psi|_{X,m}\\\
+\left(|\nabla
P|+\frac{1}{1+|u|}|P|+\frac{|X^{\ell}_{m}|^{1/2}}{|X^{n}_{m}|^{1/2}}|\nabla_{{\ell^{m}}}P|+|X^{\ell}_{m}||\not{\nabla}P|\right)|X^{n}_{m}|^{1/2}|\partial\psi|_{X,m}\\\
+|P||\partial_{u}X^{v}||{\ell^{m}}\psi|+|P||X|\left(|F|+\frac{1}{1+v}|P|\right)$
(8.124)

and in the region $|u|\geq v/8$, we instead have the bound

$|\widetilde{K}_{X,\gamma,P}[\psi]|\lesssim|\mathcal{L}_{X}\gamma||\partial\psi|^{2}+|\gamma||\partial
X||\partial\psi|^{2}+|\mathcal{L}_{X}P||\partial\psi|+\frac{1}{1+v}|X|\left(|\gamma||\partial\psi|^{2}+|P||\partial\psi|\right).$
(8.125)

In the central region $D^{C}_{t}$ by Propositition 3.2, we have

$\displaystyle|\widetilde{K}_{X,\gamma,P}[\psi]|$
$\displaystyle\lesssim\left(|\nabla\gamma|+\frac{|\gamma|}{1+s}+\frac{|X^{\ell}_{m_{B}}|^{1/2}}{|X^{n}_{m_{B}}|^{1/2}}\left(|\nabla_{\ell^{m_{B}}}\gamma|+|\not{\nabla}\gamma|\right)\right)|\partial\psi|_{X,m_{B}}^{2}+\frac{1}{(1+v)^{1/4}}|F||\partial\psi|_{X,m_{B}}$
(8.126) $\displaystyle+\left(|\nabla
P^{u}|+\frac{|P^{u}|}{1+s}+|X^{\ell}_{m_{B}}|^{1/2}\left(|\nabla_{\ell^{m_{B}}}P|+|\not{\nabla}P|\right)\right)|X^{n}_{m_{B}}|^{1/2}|\partial\psi|_{X,m_{B}}$
(8.127)
$\displaystyle+\epsilon_{C}\left(\frac{1}{(1+v)^{3/2}}|\partial\psi|^{2}+\frac{1}{(1+v)^{1/2}}(|\ell^{m_{B}}\psi|^{2}+|\not{\nabla}\psi|^{2})\right)$
(8.128) $\displaystyle+\frac{1}{(1+s)^{1/2}}\left(|\nabla
P^{u}|+\frac{|P^{u}|}{1+v}\right)|\partial\psi|+v|P|\left(|\nabla
P|+\frac{|P|}{1+v}+|F|\right).$ (8.129)

We also note that by (H.185), we have the following bound for the scalar
current $K_{\gamma_{a},X}$ generated by the linear term (6.6) satisfying the
null condition,

$|K_{\gamma_{a},X}[\psi]|\lesssim\frac{1}{(1+v)^{3/2}}|\partial\psi|^{2}+\frac{1}{(1+v)^{1/2}}|\partial_{v}\psi|^{2}+\frac{1}{(1+v)^{1/2}}|\not{\nabla}\psi|^{2}.$
(8.130)

We now record the needed $L^{1}_{t}L^{1}_{x}$ bounds for these quantities in
each region. The main ingredients needed for these bounds are the time-
integrated bounds in Section 7.2.

As a result of Lemma 8.1, we have the following bound in the right-most
region.

######  Lemma 8.6 (Estimates for the scalar currents in the rightmost region).

Under the hypotheses of Proposition 6.1, with $X_{R}$ defined as in Section
(2.1), we have the following bound,

$\sum_{|I|\leq
N_{R}}\int_{t_{0}}^{t_{1}}\int_{D^{R}_{t}}|\widetilde{K}_{X_{R},\gamma,P_{I}}[Z^{I}\psi_{R}]|+|F_{I}||X_{R}\psi^{I}_{R}|\,dt\lesssim\epsilon_{R}^{3}.$
(8.131)

###### Proof.

We just prove the bound in the region $|u|\leq v/8$, as the bound in the
region $|u|\geq v/8$ is simpler in light of the strong decay estimates (7.1)
for $\psi$ in that region. For the multiplier $X=X_{R}$ we have the bound
$\frac{|X^{\ell}_{m}|^{1/2}}{|X^{n}_{m}|^{1/2}}\lesssim
1+\frac{r^{1/2}}{(1+|u|)^{\mu/2}}$, and so by Lemma 8.1, we have

$\displaystyle\int_{t_{0}}^{t_{1}}\int_{D^{R}_{t}\cap\\{|u|\leq v/8\\}}$
$\displaystyle\left(|\nabla\gamma|+\frac{1}{1+|u|}|\gamma|+\frac{|X^{\ell}_{m}|^{1/2}}{|X^{n}_{m}|^{1/2}}|\nabla_{{\ell^{m}}}\gamma|+\frac{|X^{\ell}_{m}|^{1/2}}{|X^{n}_{m}|^{1/2}}|\not{\nabla}\gamma|\right)|\partial\psi|_{X,m}^{2}\,dt$
(8.132)
$\displaystyle\lesssim\int_{t_{0}}^{t_{1}}\left(\|\nabla\gamma\|_{L^{\infty}(D^{R}_{t})}+\|(1+|u|)^{-1}\gamma\|_{L^{\infty}(D^{R}_{t})}\right)E^{R}_{I}(t)\,dt$
(8.133)
$\displaystyle+\int_{t_{0}}^{t_{1}}\left(\left\|\left(1+\frac{r^{1/2}}{(1+|u|)^{\mu/2}}\right)\partial_{v}\gamma\right\|_{L^{\infty}(D^{R}_{t})}+\left\|\left(1+\frac{r^{1/2}}{(1+|u|)^{\mu/2}}\right)\not{\nabla}\gamma\right\|_{L^{\infty}(D^{R}_{t})}\right)E^{R}_{I}(t)\,dt\lesssim\epsilon_{R}^{3},$
(8.134)

using Lemma 8.1. By the same result, we have

$\int_{t_{0}}^{t_{1}}\int_{D^{R}_{t}\cap\\{|u|\leq v/8\\}}\left(|\nabla
P_{I}|+(1+|u|)^{-1}|P_{I}|+\frac{|X^{\ell}_{m}|^{1/2}}{|X^{n}_{m}|^{1/2}}|\nabla_{{\ell^{m}}}P|+|X^{\ell}_{m}||\not{\nabla}P|\right)|X^{n}_{m}|^{1/2}|\partial\psi|_{X,m}\,dt\\\
\lesssim\int_{t_{0}}^{t_{1}}\left(\||X^{n}_{m}|^{1/2}\nabla
P_{I}\|_{L^{2}(D^{R}_{t})}+\|(1+|u|)^{-1}|X^{n}_{m}|^{1/2}P_{I}\|_{L^{2}(D^{R}_{t})}\right)E_{I}^{R}(t)\,dt\\\
+\int_{t_{0}}^{t_{1}}\left(\||X^{\ell}_{m}|^{1/2}\nabla_{{\ell^{m}}}P_{I}\|_{L^{2}(D^{R}_{t})}+\||X^{\ell}_{m}|^{1/2}\not{\nabla}P_{I}\|_{L^{2}(D^{R}_{t})}\right)E_{I}^{R}(t)\,dt\lesssim\epsilon_{R}^{3}.$
(8.135)

By our choice of $\mu,\nu$ in (6.20), in $D^{R}_{t}$ we have the bound
$|\partial_{u}X^{v}_{m}|\lesssim 1+|u|^{\mu-1}+(\log r)^{\nu-1}\lesssim
1+|u|^{\mu-1}$ in $D^{R}_{t}$ and so we also have

$\int_{t_{0}}^{t_{1}}\int_{D^{R}_{t}}|P_{I}||\partial_{u}X^{v}||{\ell^{m}}\psi_{R}^{I}|\,dt\lesssim\int_{t_{0}}^{t_{1}}\int_{D^{R}_{t}}\frac{1+|u|^{\mu-1}}{r^{1/2}}|P_{I}||\partial\psi_{R}^{I}|_{X_{R},m}\,dt\lesssim\epsilon_{R}^{3},$
(8.136)

using Lemma 8.1 to bound the contribution from $P_{I}$. To complete the bounds
in the region $|u|\leq v/8$, it remains only to bound the last term in (8.124)
and (8.131), and the bounds for these quantities follow easily from the
estimates (8.4) (8.11).

∎

In the central region, the analogous result is the following.

######  Lemma 8.7 (Estimates for the scalar currents in the central region).

Under the hypotheses of Proposition 6.1, there is a continuous function
$c_{0}$ with $c_{0}(0)=0$ so that for any $\delta>0$ we have the following
estimates.

If $|I|\leq N_{C}$,

$\int_{t_{0}}^{t_{1}}\int_{D^{C}_{t}}|\widetilde{K}_{X_{T},\gamma,P_{I}+P_{I,null}}[Z_{m_{B}}^{I}\psi_{C}]|+|K_{X_{T},\gamma_{a}}[Z_{m_{B}}^{I}\psi_{C}]|+\left(|F_{C,I}|+|F_{\Sigma,I}|+|{F_{m_{B},I}^{2}}|\right)|X_{T}\psi^{I}_{C}|\,dt\\\
\lesssim\epsilon_{C}^{3}+c_{0}(\epsilon_{0})\left(1+\frac{1}{\delta}\right)\epsilon_{C}^{2}+c_{0}(\epsilon_{0})+\delta\sum_{|J|\leq|I|}\left(\sup_{t_{0}\leq
t\leq t_{1}}E_{J,T}^{C}(t)+S_{J}^{C}(t_{1})\right)$ (8.137)

If $|I|\leq N_{C}-1$,

$\int_{t_{0}}^{t_{1}}\int_{D^{C}_{t}}|\widetilde{K}_{X_{C},\gamma,P_{I}+P_{I,null}}[Z_{m_{B}}^{I}\psi_{C}]|+|K_{X_{C},\gamma_{a}}[Z_{m_{B}}^{I}\psi_{C}]|+\left(|F_{C,I}|+|F_{\Sigma,I}|+|{F_{m_{B},I}^{2}}|\right)|X_{C}\psi^{I}_{C}|\,dt\\\
\lesssim\epsilon_{C}^{3}(1+\log\log
t_{1})+c_{0}(\epsilon_{0})\left(1+\frac{1}{\delta}\right)\epsilon_{C}^{2}+c_{0}(\epsilon_{0})+\delta
S_{I}^{C}(t_{1})\\\ +\epsilon_{C}(1+\log\log
t_{1})\sum_{|J|\leq|I|-1}\sup_{t_{0}\leq t\leq t_{1}}(E_{J,D}^{C}(t))^{1/2},$
(8.138)

and if $|I|\leq N_{C}-2$,

$\int_{t_{0}}^{t_{1}}\int_{D^{C}_{t}}|\widetilde{K}_{X_{C},\gamma,P_{I}+P_{I,null}}[Z_{m_{B}}^{I}\psi_{C}]|+|K_{X_{C},\gamma_{a}}[Z_{m_{B}}^{I}\psi_{C}]|+\left(|F_{C,I}|+|F_{\Sigma,I}|+|{F_{m_{B},I}^{2}}|\right)|X_{C}\psi^{I}_{C}|\,dt\\\
\lesssim\epsilon_{C}^{3}+c_{0}(\epsilon_{0})\left(1+\frac{1}{\delta}\right)\epsilon_{C}^{2}+c_{0}(\epsilon_{0})+\delta\sum_{|J|\leq|I|}S_{J}^{C}(t_{1}).$
(8.139)

###### Proof.

We use (8.126) and Lemma 8.2, 8.3. First, by Lemma (8.2), regardless of the
multiplier $X$ we bound

$\displaystyle\int_{t_{0}}^{t_{1}}\int_{D^{C}_{t}}$
$\displaystyle\left(|\nabla\gamma|+\frac{1}{1+s}|\gamma|+(1+v)^{1/2}(1+s)^{1/2}|\nabla_{\ell^{m_{B}}}\gamma|+(1+v)^{1/2}(1+s)^{1/2}|\not{\nabla}\gamma|\right)|\partial
Z_{m_{B}}^{I}\psi|_{X,m_{B}}^{2}\,dt$
$\displaystyle\lesssim\int_{t_{0}}^{t_{1}}\|\nabla\gamma\|_{L^{\infty}(D^{C}_{t})}+\left\|\frac{1}{1+s}\gamma\right\|_{L^{\infty}(D_{t}^{C})}E_{X,I}(t)\,dt$
(8.140)
$\displaystyle+\int_{t_{0}}^{t_{1}}\left\|(1+v)^{1/2}(1+s)^{1/2}\nabla_{\ell^{m_{B}}}\gamma\right\|_{L^{\infty}(D_{t}^{C})}+\left\|(1+v)^{1/2}(1+s)^{1/2}\not{\nabla}\gamma\right\|_{L^{\infty}(D_{t}^{C})}E_{X,I}(t)\,dt,$
(8.141) $\displaystyle\lesssim\epsilon_{C}\sup_{t_{0}\leq t\leq
t_{1}}E_{X,I}(t),$ (8.142)

with the notation $E_{X_{T},I}=E_{T,I}^{C}$ and $E_{X_{C},I}=E_{D,I}^{C}$. By
the bootstrap assumptions (6.78)-(6.79) for the energies, this is bounded by
(8.137) when $X=X_{T}$ and $|I|\leq N_{C}$, by (8.138) when $X=X_{C}$ and
$|I|\leq N_{C}-1$, and by (8.139) when $X=X_{C}$ and $|I|\leq N_{C}-2$.

We now control the terms on the second line of (8.126). Regardless of which
multiplier we use, by (8.25) we have

$\displaystyle\int_{t_{0}}^{t_{1}}\int_{D^{C}_{t}}$
$\displaystyle|X^{n}_{m_{B}}|^{1/2}\left(|\nabla
P_{I}|+\frac{1}{1+s}|P_{I}|\right)|\partial
Z_{m_{B}}^{I}\psi_{C}|_{X,m_{B}}\,dt$ (8.143)
$\displaystyle+\int_{t_{0}}^{t_{1}}\int_{D^{C}_{t}}|X^{\ell}_{m_{B}}|^{1/2}\left(|\nabla_{\ell^{m_{B}}}P_{I}|+|\not{\nabla}P_{I}|\right)|\partial
Z_{m_{B}}^{I}\psi_{C}|_{X,m_{B}}\,dt$ (8.144)
$\displaystyle\lesssim\int_{t_{0}}^{t_{1}}\||X^{n}_{m_{B}}|^{1/2}\nabla
P_{I}\|_{L^{2}(D^{C}_{t})}+\||X^{n}_{m_{B}}|^{1/2}(1+s)^{-1}P_{I}\|_{L^{2}(D^{C}_{t})}E_{X,I}(t)^{1/2}\,dt$
(8.145)
$\displaystyle+\int_{t_{0}}^{t_{1}}\left(\|(1+v)^{1/2}\nabla_{\ell^{m_{B}}}P_{I}\|_{L^{2}(D^{C}_{t})}+\|(1+v)^{1/2}\not{\nabla}P_{I}\|_{L^{2}(D^{C}_{t})}\right)E_{X,I}(t)^{1/2}\,dt$
(8.146)
$\displaystyle\lesssim(\epsilon_{C}^{2}+c_{0}(\epsilon_{0})\epsilon_{C})\sup_{t_{0}\leq
t\leq t_{1}}E_{X,I}(t)^{1/2},$ (8.147)

which is bounded by the right-hand side of (8.137)-(8.139). To control the
$L^{1}_{t}L^{1}_{x}$ norm of the terms on the second line of (8.126), it
remains to control the contribution from $P_{I,null}^{u}$, and the needed
bound follows directly from Lemma 8.3.

For the terms on the third line of (8.126) we just bound

$\int_{t_{0}}^{t_{1}}\int_{D^{C}_{t}}\left(\frac{1}{(1+v)^{3/2}}|\partial
Z_{m_{B}}^{I}\psi_{C}|^{2}+\frac{1}{(1+v)^{1/2}}(|\ell^{m_{B}}Z_{m_{B}}^{I}\psi_{C}|^{2}+|\not{\nabla}Z_{m_{B}}^{I}\psi_{C}|^{2})\right)\,dt\\\
\lesssim\int_{t_{0}}^{t_{1}}\frac{1}{(1+t)^{5/4}}\int_{D^{C}_{t}}|X^{n}_{m_{B}}||\partial
Z_{m_{B}}^{I}\psi_{C}|^{2}+|X^{\ell}_{m_{B}}|(|\ell^{m_{B}}Z_{m_{B}}^{I}\psi_{C}|^{2}+|\not{\nabla}Z_{m_{B}}^{I}\psi_{C}|^{2})\,dt\lesssim
c_{0}(\epsilon_{0})\epsilon_{C}^{2}.$ (8.148)

To finish the bounds for the scalar current $\widetilde{K}$, it remains to
control the terms on the last line of (8.126). These terms are easier to
handle than the above after using the pointwise estimates from Lemma C.4,
noting in particular that $(1+s)^{1/2}\leq|X^{n}_{m_{B}}|^{1/2}$ for both
estimates, and we skip them. The bounds for the linear scalar currents
$K_{X,\gamma_{a}}$ follow easily from the bound (8.130) and the definitions of
our energies,

$\int_{t_{0}}^{t_{1}}\int_{D^{C}_{t}}\frac{1}{(1+v)^{3/2}}|\partial_{u}\psi_{C}^{I}|^{2}+\frac{1}{(1+v)^{1/2}}\left(|\partial_{v}\psi_{C}^{I}|^{2}+|\not{\nabla}\psi_{C}^{I}|^{2}\right)\,dt\\\
\lesssim\int_{t_{0}}^{t_{1}}\frac{1}{(1+t)^{5/4}}\int_{D^{C}_{t}}X^{n}_{m_{B}}|\partial_{u}\psi_{C}^{I}|^{2}+X^{\ell}_{m_{B}}\left(|\ell^{m_{B}}\psi_{C}^{I}|^{2}+|\not{\nabla}\psi_{C}^{I}|^{2}\right)\,dt\lesssim
c_{0}(\epsilon_{0})\sup_{t_{0}\leq t\leq t_{1}}E_{X,I}(t),$ (8.149)

where we used that $|X^{n}_{m_{B}}|\gtrsim(1+s)^{-1/2}$ and
$X^{\ell}_{m_{B}}=v$ for both multipliers.

The needed bounds for the remainder terms
$F_{C,I},F_{\Sigma,I},{F_{m_{B},I}^{2}}$ follow directly from the bounds
(8.27)-(8.28).

∎

Finally, in the leftmost region we need the following result.

######  Lemma 8.8 (Estimates for the scalar currents in the leftmost region).

Under the hypotheses of Proposition 6.1, with $X_{L}$ and $X_{M}$ defined as
in Section 2.1, we have

$\sum_{|I|\leq
N_{L}}\int_{t_{0}}^{t_{1}}\int_{D^{L}_{t}}|\widetilde{K}_{X_{L},\gamma,P_{I}}[\psi^{I}_{L}]|+|\widetilde{K}_{X_{M},\gamma,P_{I}}[\psi^{I}_{L}]|\,dt+\int_{t_{0}}^{t_{1}}\int_{D^{L}_{t}}|F_{I}||X_{L}\psi^{I}_{L}|+|F_{I}||X_{M}\psi^{I}_{L}|\,dt\lesssim\epsilon_{L}^{3}.$
(8.150)

###### Proof.

The proof follows in the same way as the above results, but using the
pointwise estimates (8.124)-(8.125) for the scalar current and Lemma 8.5 for
the needed time-integrated bounds. ∎

##  9 The higher-order boundary conditions

The goal of this section is to prove that under the hypotheses of Proposition
6.1, the bounds in (6.116) and (6.134) for the derivatives $\ell\psi$ along
the timelike sides of the shocks hold. Specifically we will be proving bounds
for the quantities

$B^{L}_{I}(t_{1})=\int_{t_{0}}^{t_{1}}\int_{\Gamma^{L}_{t}}vf(v)|{\ell^{m}}\psi_{L}^{I}|^{2}\,dSdt,\qquad
B^{C}_{I}(t_{1})=\int_{t_{0}}^{t_{1}}\int_{\Gamma^{R}_{t}}(1+v)|\ell^{m_{B}}\psi^{I}_{C}|^{2}\,dSdt.$
(9.1)

We remind the reader at this point that by the definitions of the energies in
(6.26) and (6.38), our bootstrap assumptions (6.78)-(6.81) imply the bounds

$\sum_{|I|\leq
N_{L}}B^{L}_{I}(t_{1})\lesssim\epsilon_{L}^{2},\qquad\sum_{|I|\leq
N_{C}}B^{C}_{I}(t_{1})\lesssim\epsilon_{C}^{2}.$ (9.2)

At the left shock, the result is the following.

######  Proposition 9.1.

There is $\epsilon_{0}^{*}>0$ with the following property. If the hypotheses
of Proposition 6.1 hold with $\epsilon_{0}<\epsilon_{0}^{*}$, then

$\sum_{|I|\leq N_{L}}B^{L}_{I}(t_{1})\lesssim\epsilon_{L}^{3}.$ (9.3)

The analogous result at the right shock is the following.

######  Proposition 9.2.

There is $\epsilon_{0}^{*}>0$ with the following property. If the hypotheses
of Proposition 6.1 hold with $\epsilon_{0}<\epsilon_{0}^{*}$, then writing
$X=X^{\ell}\ell^{m_{B}}+X^{n}n$, with $X=X^{\ell}_{C}$ or $X=X^{\ell}_{T}$, we
have

$\sum_{|I|\leq N_{C}}B^{C}_{I}(t_{1})\lesssim\epsilon_{C}^{3}.$ (9.4)

The idea behind these estimates is the following. Let
$(\ell_{-},\ell_{+})=({\ell^{m}},\ell^{m_{B}})$ at the left shock and
$(\ell^{m_{B}},{\ell^{m}})$ at the right shock. For both estimates, the basic
ingredient needed is a bound for a quantity of the form $\ell_{-}\psi^{I}_{-}$
where $\psi^{I}_{-}$ denotes some collection of vector fields $Z^{I}$ applied
to $\psi_{-}$, the potential along the timelike side of the shock. When no
vector fields are present, the jump conditions at each shock take the form

$\ell_{-}\psi_{-}+\frac{1}{1+v}Q(\partial\psi_{-},\partial\psi_{-})=\ell_{+}\psi_{+}+\frac{1}{1+v}Q(\partial\psi_{+},\partial\psi_{+}),$
(9.5)

where $Q$ is a quadratic nonlinearity and where we are omitting lower-order
terms (see (6.9) and (6.12)). This expresses $\ell_{-}\psi_{-}$ in terms of
the “boundary data” $\ell_{+}\psi_{+}$ and nonlinear terms. The weights
$X^{\ell}$ on the timelike sides of the shock are such that the contribution
from the nonlinear terms can be handled, and using our energy estimates on the
spacelike side of the shock the contribution from the boundary data
$\ell_{+}\psi_{+}$ can also be handled.

At higher order, the calculation is more involved because we do not directly
have an equation for $\ell_{-}\psi^{I}_{-}$ in terms of the higher-order
boundary data $\ell_{+}\psi^{I}_{+}$ since the vector fields we consider are
transverse to the shock. We therefore need to replace the vector fields
$Z^{I}$ with a product of vector fields $Z_{T}^{I}$ which are tangent to the
shock. For this we first need to commute the fields $Z^{I}$ with the
derivatives $\ell_{-}$ which generates lower-order terms. We then replace the
fields $Z^{I}$ with the $Z_{T}^{I}$, which generates error terms which involve
high-order derivatives of the boundary-definining functions $B^{L},B^{R}$,
which is where we need the bounds (6.82)-(6.83) for the geometry of the
shocks. We can then bound $Z_{T}^{I}\ell_{-}\psi_{-}$ by applying tangential
fields to (9.5) and replacing the fields $Z_{T}^{I}$ with the usual fields
$Z^{I}$ we can bound the quantities on the right-hand side of (9.5) by our
energies. This is slightly cumbersome because we are using different families
of vector fields in each region, and this is ultimately why we need to take
the parameter $\mu$ from (9.5) large.

There is an additional difficulty at the left shock, which is that the weight
$X^{\ell}$ we use on the timelike side is large relative to the weights we use
on the spacelike side, and so to deal with the contribution from the error
term $X^{\ell}|\ell_{+}\psi_{+}^{I}|^{2}$ we need to integrate to the right
shock. This generates a bulk term and a boundary term. The bulk term can be
handled since the main term we generate in this way is of the form
$n\ell_{+}\psi_{+}$ and we have an equation for this quantity. The boundary
term can be handled by using the above strategy to control the resulting term
on the timelike side of the right shock in terms of the data on the spacelike
side and nonlinear terms.

The estimates for replacing the $Z^{I}$ with the $Z^{I}_{T}$ are the content
of Section 9.1. In sections 9.2 and 9.3 we reduce the proofs of Propositions
9.1-9.2 to a sequence of lemmas which handle the nonlinear terms, the various
error terms we generate when commuting the fields $Z^{I}$ with the derivatives
$\ell_{-}$, and which give the needed bounds for higher-order derivatives of
the boundary data.

###  9.1 Estimates for derivatives in terms of tangential derivatives

Let $\mathcal{Z}$ denote a collection of vector fields. In what follows we
will take either $\mathcal{Z}=\mathcal{Z}_{m}$ or $\mathcal{Z}_{m_{B}}$, with
notation as in Section 2.1. It will be helpful to enlarge the collection
$\mathcal{Z}$ and to write
$\widehat{Z}\in\widehat{\mathcal{Z}}=\mathcal{Z}\cup\\{n\\}$. We will write
$\widehat{Z}^{K}$ for a $|K|$-fold product of the fields in $\widehat{Z}$.
Specifically, if $\mathcal{Z}=\\{Z_{1}\cdots Z_{m}\\}$ we let
$K=(K_{1},...,K_{r})$ where each $K_{j}\in\\{e_{1},\dots e_{m+1}\\}$ where
$e_{j}$ denotes the standard basis of $\mathbb{R}^{m+1}$. If $K_{j}=e_{p}$ we
then define $\widehat{Z}^{K_{j}}=\widehat{Z}_{p}$ with $\widehat{Z}_{p}=Z_{p}$
when $p\leq m$ and $\widehat{Z}_{p}=n$ when $p=m+1$. Finally we set
$\widehat{Z}^{K}=\widehat{Z}^{K_{1}}\cdots\widehat{Z}^{K_{r}}$.

Fix a function $\xi:\mathbb{R}^{4}\to\mathbb{R}$ with $d\xi\not=0$ and which
satisfies $n\xi=1$. Given a vector field $Z$, we define $Z_{T}=Z-Z\xi n$, so
that $Z_{T}\xi=d\xi(Z_{T})=0$. In particular $Z_{T}$ is tangent to the set
$\\{\xi=0\\}$. Let $Z_{T}^{I}$ denote a product of the fields $Z_{T}$ for
$Z\in\mathcal{Z}$. The following basic result then relates the tangential
fields $Z_{T}$ to the fields $Z$.

######  Lemma 9.1.

Let $\mathcal{Z}$ denote any collection of vector fields and define
$Z^{I},\widehat{Z}^{I}$ as in the above paragraph. Then we have

$|Z^{I}q-Z_{T}^{I}q-(Z^{I}\xi)nq|\lesssim\sum_{r\geq
1}\sum_{\begin{subarray}{c}|I_{1}|+\cdots|I_{r}|+|I_{r+1}|\leq|I|+1,\\\
|I_{k}|\geq 1,|I_{r+1}|\geq
2\end{subarray}}|\widehat{Z}^{I_{1}}\xi|\cdots|\widehat{Z}^{I_{r}}\xi||\widehat{Z}^{I_{r+1}}q|,$
(9.6)

where there are $r$ factors of $n$ present in the collection
$\widehat{Z}^{I_{1}},...,\widehat{Z}^{I_{r+1}}$ and at least one factor of $n$
in $\widehat{Z}^{I_{r+1}}$.

######  Remark 4.

Note that on the right-hand side of (9.6), there are no more than $|I|-1$ of
the $\widehat{Z}$ derivatives landing on $\xi$ by the last condition in the
sum and no more than $|I|$ of the $\widehat{Z}$ derivatives of $q$ since each
$|I_{k}|\geq 1$. We also note that the reason there are $r$ factors of $n$
present in $\widehat{Z}^{I_{r+1}}$ is that this bound follows from repeatedly
applying the definition $Z_{T}=Z-(Z\xi)n$, and every time we use this formula
on derivatives of $q$, the number of $n$ derivatives present and number of
factors of $Z\xi$ both increase by one. This counting is important because in
our applications we expect $Z\xi\sim|u|$ and so we need to gain a power of
$|u|^{-1}$ for each factor of $Z\xi$ (and thus factor of $n$) we encounter.
The vector fields we consider are such that schematically,
$n\sim\frac{1}{|u|}Z$ or better (see e.g (9.17)), and so we gain (at least)
one power of $u$ for each factor.

###### Proof.

When $|I|=1$ this is just the definition $Z_{T}q=Zq-(Z\xi)nq$. If the result
holds for all $I$ with $|I|\leq m$ for some $m\geq 1$, we fix a multi-index
$I$ with $|I|=m+1$ and write $Z^{I}=Z^{J}Z$ and then, writing
$Zq=Z_{T}q+(Z\xi)nq$, we have

$\displaystyle Z^{J}Zq$ $\displaystyle=Z_{T}^{J}Zq+(Z^{J}-Z_{T}^{J})Zq$ (9.7)
$\displaystyle=Z_{T}^{I}q+Z^{J}_{T}(Z\xi nq)+(Z^{J}-Z_{T}^{J})Zq$ (9.8)
$\displaystyle=Z_{T}^{I}q+(Z^{I}\xi)nq$ (9.9)
$\displaystyle\quad+\sum_{\begin{subarray}{c}|J_{1}|+|J_{2}|\leq|J|,\\\
|J_{1}|\leq|J|-1\end{subarray}}c^{J}_{J_{1}J_{2}}(Z_{T}^{J_{1}}Z\xi)(Z^{J_{2}}_{T}nq)+((Z_{T}^{J}-Z^{J})Z\xi)nq+(Z^{J}-Z_{T}^{J})Zq,$
(9.10)

for constants $c^{J}_{J_{1}J_{2}}$. It remains to show that the terms on the
last line here are of the appropriate form.

By the inductive assumption, whenever $|L|\leq|I|-1$ for any $q^{\prime}$ we
have the bound

$|(Z^{L}-Z_{T}^{L})q^{\prime}|\lesssim\sum_{s\geq
1}\sum_{\begin{subarray}{c}|L_{1}|+\cdots+|L_{s+1}|\leq|L|+1,\\\ |L_{i}|\geq
1\end{subarray}}|\widehat{Z}^{L_{1}}\xi|\cdots|\widehat{Z}^{L_{s}}\xi||\widehat{Z}^{L_{s+1}}q^{\prime}|,$
(9.11)

where there are $s$ factors of $n$ present in the collection
$\widehat{Z}^{L_{1}},\cdots\widehat{Z}^{L_{s+1}}$ and at least one factor in
$\widehat{Z}^{L_{s+1}}$. Applying this with $q^{\prime}$ replaced by $Zq$ we
find that

$|(Z^{J}-Z_{T}^{J})Zq|\lesssim\sum_{s\geq
1}\sum_{\begin{subarray}{c}|L_{1}|+\cdots+|L_{s+1}|\leq|L|+1,\\\ |L_{i}|\geq
1\end{subarray}}|\widehat{Z}^{L_{1}}\xi|\cdots|\widehat{Z}^{L_{s}}\xi||\widehat{Z}^{L_{s+1}}Zq|,$
(9.12)

which is the correct form. Applying this with $q^{\prime}$ replaced by $Z\xi$,
we also have that

$|((Z_{T}^{J}-Z^{J})Z\xi)nq|\lesssim\sum_{s\geq
1}\sum_{\begin{subarray}{c}|L_{1}|+\cdots+|L_{s+1}|\leq|L|+1,\\\ |L_{i}|\geq
1\end{subarray}}|\widehat{Z}^{L_{1}}\xi|\cdots|\widehat{Z}^{L_{s}}\xi||\widehat{Z}^{L_{s+1}}Z\xi||nq|,$
(9.13)

where there are now $s+1$ factors of $\xi$ present in the sums and $s$ factors
of $n$ present in the $\widehat{Z}^{L_{k}}$ along with one additional one in
the last factor, so this is also of the correct form. In the same way, to
handle the terms in the sum on the last line of (9.10) we apply (9.13) to
$q^{\prime}=Z\xi$ and $q^{\prime}=nq$ and note that there is already one
factor of $n$ present in each product there, and the result follows.

∎

From now on, we take $\xi=u-B^{A}$. In the next section we collect some
estimates for the quantities appearing in (9.6) when $Z^{I}$ denotes a product
of Minkowski fields and when $Z^{I}$ denotes a product of the fields from
$\mathcal{Z}_{m_{B}}$. At the left shock, the main result we need is Lemma 9.3
and at the right shock, the main result is Lemma 9.5.

####  9.1.1 Estimates for $Z^{I}-Z_{T}^{I}$ in the Minkowskian case

We start with the following simple result.

######  Lemma 9.2.

Let $\mathcal{Z}=\mathcal{Z}_{m}$ denote the Minkowskian fields. Fix a multi-
index $J$ and suppose that

$\sum_{|L|\leq|J|/2+1}\frac{|Z_{T}^{L}B|}{1+|u|}\leq M.$ (9.14)

Let $\widehat{Z}^{J}$ be as in the paragraph before Lemma 9.1. If there are
$j$ factors of $n$ present in $\widehat{Z}^{J}$ then with $\xi=u-B$,

$(1+|u|)^{j}|\widehat{Z}^{J}\xi|\leq
C(M)(1+|u|)\left(1+\sum_{|K|\leq|J|}\frac{|Z_{T}^{K}B|}{1+|u|}\right),\qquad\text{
if }t/2\leq r\leq 3t/2,t\geq 1.$ (9.15)

###### Proof.

By definition, $\widehat{Z}^{J}$ is a product of the form
$n^{j_{1}}Z^{J_{1}}\cdots n^{j_{k}}Z^{J_{k}}$ where $\sum j_{i}+|J_{i}|=|J|$
and where we recall that $n=\partial_{u}$. The idea in what follows is to
first use basic properties of the Minkowski fields $\mathcal{Z}_{m}$ to re-
write the vector fields $n$ in terms of powers of $(1+|u|)^{-1}$ and the
Minkowski fields, and then to use (9.6) and the fact that $nB=0$ to re-write
quantities of the form $Z^{K}B$ in terms of tangential derivatives.

For this, it will be helpful to recall some simple and well-known properties
of the vector fields $Z\in\mathcal{Z}_{m}$, which follow immediately from the
formulas (A.11)-(A.12). First, there are functions
$a^{Z},a^{Z}_{u},a^{Z}_{v},\not{a}^{Z}$ satisfying the (Minkowskian) symbol
condition

$|Z^{J}a|\leq C_{J}$ (9.16)

for constants $C_{J}$ so that we can write

$n=\frac{1}{1+|u|}\sum_{Z\in\mathcal{Z}_{m}}a^{Z}Z,\qquad
Z=a^{Z}_{u}(1+|u|)\partial_{u}+a^{Z}_{v}(1+v)\partial_{v}+\not{a}^{Z}(1+r)\cdot\not{\nabla},\quad\text{
if }t/2\leq r\leq 3t/2,t\geq 1.$ (9.17)

We will also use that there are constants $c_{ZZ^{\prime}}^{Z^{\prime\prime}}$
so that

$[Z,Z^{\prime}]=\sum_{Z^{\prime\prime}\in\mathcal{Z}_{m}}c_{ZZ^{\prime}}^{Z^{\prime\prime}}Z^{\prime\prime}.$
(9.18)

We now prove the bound (9.15). To start, we claim that if there are $j$
factors of $n$ present in $\widehat{Z}^{J}$, then

$(1+|u|)^{j}|\widehat{Z}^{J}\xi|\lesssim
1+|u|+\sum_{|J^{\prime}|\leq|J|-\widetilde{j}}|Z^{J^{\prime}}B|,\qquad\widetilde{j}=\begin{cases}1,\quad
j\geq 1.\\\ 0,\quad j=0\end{cases},$ (9.19)

Recalling $\xi=u-B$ and using (9.17) it is enough to prove this bound with
$\xi$ replaced by $B$. When $j=0$ there is nothing to prove since then
$\widehat{Z}^{J}=Z^{J}$. If $j\geq 1$, we write
$\widehat{Z}^{J}=n^{j_{1}}Z^{J_{1}}\cdots n^{j_{r}}Z^{J_{r}}$ where without
loss of generality $j_{r}\geq 1$. Using the first identity in (9.17) to
convert $n$ derivatives into $Z$ derivatives, $\widehat{Z}^{J}B$ can be
written as a sum of terms of the form

$\frac{a}{(1+|u|)^{j_{1}+\cdots
j_{r-1}+j_{r}-1}}Z^{J^{\prime}}nZ^{J^{\prime\prime}}B=\frac{a}{(1+|u|)^{j_{1}+\cdots
j_{r-1}+j_{r}-1}}Z^{J^{\prime}}[n,Z^{J^{\prime\prime}}]B$ (9.20)

where $a$ satisfies the symbol condition (9.16) and where
$|J^{\prime}|+|J^{\prime\prime}|\leq|J|-1$. Here we used that $nB\equiv 0$. To
handle the commutator, we just use (9.17) to express $n$ in terms of the
fields $Z$ and then use the algebra property (9.18). This gives
$|Z^{J^{\prime}}[n,Z^{J^{\prime\prime}}]B|\lesssim(1+|u|)^{-1}\sum_{|J^{\prime\prime\prime}|\leq|J^{\prime}|+|J^{\prime\prime}|}|Z^{J^{\prime\prime\prime}}B|$,
and the claim (9.19) follows.

Having proven (9.19), to conclude the proof of (9.15) it remains to convert
the $Z$ derivatives into $Z_{T}$ derivatives. For this, we use the bound (9.6)
and the fact that $nB=0$ to get

$|Z^{J}B|\lesssim|Z_{T}^{J}B|+\sum_{r\geq
1}\sum_{\begin{subarray}{c}|J_{1}|+\cdots|J_{r+1}|=|J|+1,\\\ |J_{k}|\geq
1,|J_{r+1}|\geq
2\end{subarray}}|\widehat{Z}^{J_{1}}\xi|\cdots|\widehat{Z}^{J_{r}}\xi||\widehat{Z}^{J_{r+1}}B|,$
(9.21)

where there are $r$ factors of $n$ present in the collection
$\widehat{Z}^{I_{1}},\dots\widehat{Z}^{I_{r+1}}$ and at least one factor of
$n$ present in $\widehat{Z}^{J_{r+1}}$. Using the bound (9.19), we find that

$|Z^{J}B|\lesssim|Z_{T}^{J}B|+\sum_{\begin{subarray}{c}|J_{1}|+\cdots|J_{r+1}|=|J|+1,\\\
|J_{k}|\geq 1,|J|-1\geq|J_{r+1}|\geq
2\end{subarray}}(1+|u|)^{-r}(1+|u|+|{Z}^{J_{1}}B|)\cdots(1+|u|+|{Z}^{J_{r}}B|)|{Z}^{J_{r+1}}B|,$
(9.22)

where the fact that $|J_{r+1}|\leq|J|-1$ follows from the fact that in (9.21),
$|J_{r+1}|\leq|J|$ and that we are using (9.19) with $j=1$. Since we also have
$|J_{k}|\leq|J|-1$ for all $k=1,...,r$ in the sum, the bound (9.15) now
follows from induction. ∎

As a result, we have the following bound.

######  Lemma 9.3.

Under the hypotheses of Lemma 9.2, we have

$|Z^{I}q-Z_{T}^{I}q|\leq
C(M)(1+|u|)\sum_{|J|\leq|I|-1}\left(|Z^{J}nq|+(1+|u|)^{-1}|Z^{J}q|\right)\\\
+C(M)(1+|u|)|B|_{I,\mathcal{Z}_{m}}\sum_{|K|\leq|I|/2+1}\left(|Z^{K}nq|+(1+|u|)^{-1}|Z^{K}q|\right).$
(9.23)

where $|B|_{I,\mathcal{Z}_{m}}$ is defined as in (6.45), and where the term
$|Z^{J}q|$ is not present when $|I|=1$.

######  Remark 5.

We will be applying this with $q$ replaced by ${\ell^{m}}q$ plus nonlinear
terms and in that case we expect the quantity $Z^{J}nq$ to be well-behaved.

######  Remark 6.

We also note that if we have the bound $|B|_{I,\mathcal{Z}_{m}}\leq M$, then
(9.23) implies that

$|Z_{T}^{I}q|\lesssim C(M)\sum_{|J|\leq|I|}|Z^{J}q|,\qquad|Z^{I}q|\lesssim
C(M)\sum_{|J|\leq|I|}|Z_{T}^{J}q|,$ (9.24)

which follows after using that $(1+|u|)|nq^{\prime}|\lesssim|Zq^{\prime}|$ and
standard properties of the fields $Z$. More generally, we have

$\displaystyle|Z_{T}^{I}q|$ $\displaystyle\lesssim
C(M)\sum_{|J|\leq|I|}|Z^{J}q|+C(M)|B|_{I,\mathcal{Z}_{m}}\sum_{|K|\leq|I|/2+1}|Z^{K}q|,$
(9.25) $\displaystyle|Z^{I}q|$ $\displaystyle\lesssim
C(M)\sum_{|J|\leq|I|}|Z_{T}^{J}q|+C(M)|B|_{I,\mathcal{Z}_{m}}\sum_{|K|\leq|I|/2+1}|Z_{T}^{K}q|,$
(9.26)

if $\sum_{|I^{\prime}|\leq|I|/2+1}|B|_{I^{\prime},\mathcal{Z}_{m}}\leq M$.

###### Proof.

By (9.6) we have

$|Z^{I}q-Z_{T}^{I}q|\lesssim|Z^{I}\xi||nq|+\sum_{r\geq
1}\sum_{\begin{subarray}{c}|I_{1}|+\cdots+|I_{r+1}|\leq|I|+1,\\\ |I_{r+1}|\geq
2\end{subarray}}|\widehat{Z}^{I_{1}}\xi|\cdots|\widehat{Z}^{I_{r}}\xi||\widehat{Z}^{I_{r+1}}q|,$
(9.27)

where there are $r$ factors of $n$ present in the collection
$\widehat{Z}^{I_{1}},...\widehat{Z}^{I_{r+1}}$ with at least one factor of $n$
in $\widehat{Z}^{I_{r+1}}$. We now re-write the last factor in terms of the
vector fields $Z\in\mathcal{Z}_{m}$, the quantity $nq$, and lower-order terms.

We claim that if there are $j\geq 1$ factors of $n$ present in
$\widehat{Z}^{J}$ then

$(1+|u|)^{j-1}|\widehat{Z}^{J}q|\lesssim\sum_{|J^{\prime}|\leq|J|-1}|Z^{J^{\prime}}nq|+(1+|u|)^{-1}|Z^{J^{\prime}}q|.$
(9.28)

This follows in a similar way to how we proved (9.19). Since
$\widehat{Z}^{J}=n^{j_{1}}Z^{J_{1}}\cdots n^{j_{k}}Z^{J_{k}}$ with
$\sum_{s=1}^{k}j_{s}=j$, we just use (9.17) to re-write $j-1$ factors of $n$
in terms of the fields $(1+|u|)^{-1}Z$ and then repeatedly use (9.18) to bound

$(1+|u|)^{j-1}|\widehat{Z}^{J}q|\lesssim\sum_{|K|\leq|J|}|\widehat{Z}^{K}q|,$
(9.29)

where the sum is over multi-indices $K$ satisfying the condition that there is
exactly one factor of $n$ present in $\widehat{Z}^{K}$. Now we write

$\widehat{Z}^{K}=Z^{K_{1}}nZ^{K_{2}}=Z^{K_{1}}Z^{K_{2}}+Z^{K_{1}}[n,Z^{K_{2}}],\qquad|K_{1}|+|K_{2}|=|K|-1,$
(9.30)

and again use (9.17)-(9.18) to bound

$|Z^{K_{1}}[n,Z^{K_{2}}]q|\lesssim(1+|u|)^{-1}\sum_{|K^{\prime}|\leq|K|-2}|Z^{K^{\prime}}q|.$
(9.31)

Combining the above, we get (9.28).

By (9.28), have

$\displaystyle\sum_{r\geq 1}$
$\displaystyle\sum_{\begin{subarray}{c}|I_{1}|+\cdots|I_{r+1}|\leq|I|+1,\\\
|I_{r+1}|\geq
2\end{subarray}}|\widehat{Z}^{I_{1}}\xi|\cdots|\widehat{Z}^{I_{r}}\xi||\widehat{Z}^{I_{r+1}}q|$
(9.32) $\displaystyle\lesssim\sum_{r\geq
1}\sum_{\begin{subarray}{c}|I_{1}|+\cdots|I_{r+1}|\leq|I|+1,\\\ |I_{r+1}|\geq
2\end{subarray}}(1+|u|)^{-r+1}|\widehat{Z}^{I_{1}}\xi|\cdots|\widehat{Z}^{I_{r}}\xi|\left(\sum_{|I^{\prime}|\leq|I_{r+1}|-1}\left(|{Z}^{I^{\prime}}nq|+(1+|u|)^{-1}|Z^{I^{\prime}}q|\right)\right)$
(9.33) $\displaystyle\leq
C(M)(1+|u|)\sum_{|J|\leq|I|-1}|{Z}^{J}nq|+(1+|u|)^{-1}|Z^{J}q|$ (9.34)
$\displaystyle\qquad+C(M)(1+|u|)|B|_{I,\mathcal{Z}}\sum_{|K|\leq|I|/2+1}|{Z}^{K}nq|+(1+|u|)^{-1}|Z^{K}q|$
(9.35)

where we used Lemma 9.2 to handle the contributions from $\xi$. ∎

####  9.1.2 Estimates for $Z^{I}-Z_{T}^{I}$ when
$\mathcal{Z}=\mathcal{Z}_{m_{B}}$

We now want a result analogous to Lemma 9.3. This is somewhat simpler than the
result in the previous section because $n$ commutes with all the fields in
$\mathcal{Z}_{m_{B}}$.

The first step is the following.

######  Lemma 9.4.

Fix a multi-index $J$ and suppose that

$\sum_{|L|\leq|J|/2+1}\frac{|Z_{m_{B},T}^{L}B|}{1+|u|}\leq M.$ (9.36)

With $\xi=u-B$, we have

$|\widehat{Z}_{m_{B}}^{J}\xi|\leq
C(M)\left(1+s+\sum_{|J^{\prime}|\leq|J|}|Z_{m_{B},T}^{J^{\prime}}B|\right)$
(9.37)

###### Proof.

First, since $[n,Z_{m_{B}}]=0$ for all $Z_{m_{B}}\in\mathcal{Z}_{m_{B}}$, it
enough to prove this bound when $Z_{m_{B}}^{J}=Z_{m_{B}}^{K}n^{j}$ for
$|K|+j=|J|$. Since $n(u-B)=1$, if $j\geq 1$ we clearly have
$|Z_{m_{B}}^{K}n^{j}\xi|\lesssim 1$ (in fact if $j\geq 1$ this is only nonzero
when $|K|=0$) so we have the simple bound
$|\widehat{Z}_{m_{B}}^{J}\xi|\lesssim
1+\sum_{|J^{\prime}|\leq|J|}|Z_{m_{B}}^{J}\xi|$, and so it is enough to bound
$|Z_{m_{B}}^{J}\xi|$. We clearly have $|Z_{m_{B}}^{J}u|\lesssim(1+s)$ and so
$|Z_{m_{B}}^{J}\xi|\lesssim 1+s+|Z_{m_{B}}^{J}B|$. It remains to handle this
last term. For this, we use (9.6), which, in light of what we have just
proved, gives

$|Z^{J}_{m_{B}}B-Z_{T,m_{B}}^{J}B|\lesssim 1+\sum_{r\geq
1}\sum_{\begin{subarray}{c}|J_{1}|+\cdots+|J_{r+1}|\leq|J|+1,\\\ |J_{r+1}|\geq
2\end{subarray}}(1+s+|\widehat{Z}^{J_{1}}_{m_{B}}B|)\cdots(1+s+|\widehat{Z}^{J_{r}}_{m_{B}}B|)|\widehat{Z}^{J_{r+1}}_{m_{B}}B|,$
(9.38)

where there are $r$ factors of $n$ present in the collection
$\widehat{Z}^{J_{1}},\dots\widehat{Z}^{J_{r+1}}_{m_{B}}$ and at least one
factor of $n$ present in $\widehat{Z}^{J_{r+1}}$. Again using that
$[n,Z_{m_{B}}]=0$ and that $nB=0$, it follows that the right-hand side is
zero, and the result now follows. ∎

Recalling that
$|B|_{I,m_{B}}=(1+s)^{-1/2}\sum_{|J|\leq|I|}|Z_{m_{B},T}^{J}B|$, we have the
following analogue of Lemma 9.3.

######  Lemma 9.5.

Under the hypotheses of Lemma 9.4, we have

$|Z^{I}_{m_{B}}q-Z_{m_{B},T}^{I}q|\leq
C(M)(1+s)\sum_{|J|\leq|I|-1}|Z_{m_{B}}^{J}nq|\\\
+C(M)(1+s)\left(1+(1+s)^{-1/2}|B|_{I,\mathcal{Z}_{m_{B}}}\right)\sum_{|L|\leq|I|/2+1}|Z_{m_{B}}^{L}nq|.$
(9.39)

######  Remark 7.

For some of our applications, it is better to write the above in the forms

$\displaystyle|Z^{I}_{m_{B}}q|$ $\displaystyle\leq
C(M)\sum_{|J|\leq|I|}|Z_{m_{B},T}^{J}q|+C(M)\left(1+(1+s)^{-1/2}|B|_{I,\mathcal{Z}_{m_{B}}}\right)\sum_{|L|\leq|I|/2+2}|Z_{m_{B},T}^{L}q|,$
(9.40) $\displaystyle|Z^{I}_{m_{B},T}q|$ $\displaystyle\leq
C(M)\sum_{|J|\leq|I|}|Z_{m_{B}}^{J}q|+C(M)\left(1+(1+s)^{-1/2}|B|_{I,\mathcal{Z}_{m_{B}}}\right)\sum_{|L|\leq|I|/2+2}|Z_{m_{B}}^{L}q|$
(9.41)

which follow from (9.39) and induction since $\mathcal{Z}_{m_{B}}$ includes
$X_{1}=s\partial_{u}=sn$ so
$(1+s)|nq|\lesssim\sum_{Z_{m_{B}}\in\mathcal{Z}_{m_{B}}}|Z_{m_{B}}q|$.

###### Proof.

By (9.6) and (9.37)

$\displaystyle|Z^{I}_{m_{B}}q-$ $\displaystyle Z_{m_{B},T}^{I}q|$ (9.42)
$\displaystyle\lesssim|Z_{m_{B}}^{I}\xi||nq|+\sum_{r\geq
1}\sum_{\begin{subarray}{c}|I_{1}|+\cdots|I_{r+1}|\leq|I|+1,\\\ |I_{r+1}|\geq
2\end{subarray}}|\widehat{Z}_{m_{B}}^{I_{1}}\xi|\cdots|\widehat{Z}_{m_{B}}^{I_{r}}\xi||\widehat{Z}_{m_{B}}^{I_{r+1}}q|$
(9.43) $\displaystyle\leq
C(M)(1+s)(1+s^{-\frac{1}{2}})|B|_{I,\mathcal{Z}_{m_{B}}}|nq|$ (9.44)
$\displaystyle\qquad+C(M)\sum_{r\geq
1}\sum_{\begin{subarray}{c}|I_{1}|+\cdots|I_{r+1}|\leq|I|+1,\\\ |I_{r+1}|\geq
2\end{subarray}}(1+s)^{r}(1+s^{-\frac{1}{2}}|B|_{I_{1},\mathcal{Z}_{m_{B}}})\cdots(1+s^{-\frac{1}{2}}|B|_{I_{r},\mathcal{Z}_{m_{B}}}|)|\widehat{Z}_{m_{B}}^{I_{r+1}}q|,$
(9.45)

where there are $r$ factors of $n$ present in the collection
$\widehat{Z}_{m_{B}}^{I_{1}},...\widehat{Z}_{m_{B}}^{I_{r+1}}$ with at least
one present in $\widehat{Z}_{m_{B}}^{I_{r+1}}$. Since $[n,Z_{m_{B}}]=0$, we
have $\widehat{Z}_{m_{B}}^{I_{r+1}}q=n^{r}Z_{m_{B}}^{K}$ where
$|K|+r=|I_{r+1}|$ and since $n=\frac{1}{s}X_{1}$ we have
$|\widehat{Z}_{m_{B}}^{I_{r+1}}q|\lesssim(1+s)^{-r+1}\sum_{|K|\leq|I_{r+1}|}|Z_{m_{B}}^{K}nq|$.
∎

###  9.2 Proof of Proposition 9.1

The result is a consequence of the upcoming Lemmas 9.7, 9.8 and Proposition
9.3, as follows. Define the quantities

$\Upsilon_{I,{L}}^{+}(t_{1})=\int_{t_{0}}^{t_{1}}\int_{\Gamma^{L}_{t}}X^{\ell}|Z_{T}^{I}Y_{L}^{+}\psi_{C}|^{2}\,dSdt,$
(9.46)

where $Y_{L}^{+}$ is as in (6.11).

Combining Lemmas 9.7 and 9.8, for $|I|\leq N_{L}$, under our hypotheses we
have the bounds

$B^{L}_{I}(t_{1})\lesssim\Upsilon_{I,L}^{+}(t_{1})+\sum_{|J|\leq|I|-1}B^{L}_{J}(t_{1})+(c_{0}(\epsilon_{0})+\epsilon_{L}^{2})\epsilon_{L}^{2}+c_{0}(\epsilon_{0}),$
(9.47)

where $c_{0}$ is a continuous function with $c_{0}(0)=0$. Taking
$\epsilon_{0}$ sufficiently small and using induction we get

$B^{L}_{I}(t_{1})\lesssim\Upsilon_{I,L}^{+}(t_{1})+\epsilon_{L}^{3},$ (9.48)

and the result now follows from the upcoming Proposition 9.3 and the fact that
$\epsilon_{C}^{2}\lesssim\epsilon_{L}^{4}$ by (6.19). ∎

In the remainder of this section we prove Lemmas 9.7, 9.8, and Proposition
9.3.

####  9.2.1 Supporting lemmas for the proof of Proposition 9.1

We start with a product estimate that we will use the handle the nonlinear
terms we encounter. For this result it is important that we take $\alpha<3/2$
in the definitions of the vector fields $X_{L},X_{M}$.

######  Lemma 9.6.

Let
$Q(\partial\psi,\partial\psi)=Q^{\alpha\beta}\partial_{\alpha}\psi\partial_{\beta}\psi$
be a quadratic nonlinearity where the coefficients $Q^{\alpha\beta}$ are
smooth functions satisfying the symbol-type condition (A.16). With $X=X_{L}$
or $X=X_{M}$, writing $X=X^{\ell}\partial_{v}+X^{n}\partial_{u}$, under the
hypotheses of Proposition 9.1 we have

$\int_{t_{0}}^{t_{1}}\int_{\Gamma^{L}_{t}}|X^{\ell}||Z^{I}\left((1+v)^{-1}Q(\partial\psi_{L},\partial\psi_{L})\right)|^{2}\,dSdt\lesssim\epsilon_{L}^{4}.$
(9.49)

###### Proof.

We first claim that under our hypotheses and by our choice of the field
$X_{L}$ we have

$(1+v)^{-1}|X^{\ell}_{L}||\partial
Z^{K}\psi_{L}|\lesssim\epsilon_{L}|X^{n}_{L}|,\qquad|K|\leq N_{L}/2+1.$ (9.50)

Indeed, by the pointwise estimates from Lemma 7.1, we have

$\displaystyle\frac{|X^{\ell}|}{1+v}|\partial Z^{K}\psi_{L}|$
$\displaystyle\lesssim\epsilon_{L}\frac{|X_{L}^{\ell}|}{1+v}\frac{1}{(1+|u|)^{1/2}}\frac{1}{|X^{n}|^{1/2}}$
(9.51) $\displaystyle\lesssim\epsilon_{L}\frac{(1+s)(\log
s)^{\alpha}}{(1+s)^{1/2}(\log s)^{1/2}(\log\log s)^{\alpha/2}}$ (9.52)
$\displaystyle\lesssim\epsilon_{L}\frac{(1+s)^{1/2}(\log
s)^{\alpha-1/2}}{(\log\log s)^{\alpha/2}}$ (9.53)
$\displaystyle\lesssim\epsilon_{L}(1+s)^{1/2}(\log s)(\log\log
s)^{\alpha},\qquad|K|\leq N_{L}/2+1,$ (9.54)

since $\alpha<3/2$, and this is bounded by the right-hand side of (9.50).

In particular, (9.50) and the pointwise decay bound (7.4) imply

$\frac{|X^{\ell}_{L}|}{(1+v)^{2}}|\partial
Z^{K}\psi_{L}|^{2}|nZ^{J}\psi_{L}|^{2}\lesssim\epsilon_{L}^{2}\frac{|X^{n}_{L}|}{(1+v)(1+s)^{1/2}}|nZ^{J}\psi_{L}|^{2},$
(9.55)

and so bounding $|\partial q|\lesssim|{\ell^{m}}q|+|nq|+|\not{\nabla}q|$ and
using the simpler estimates

$\displaystyle\frac{|X^{\ell}|}{(1+v)^{2}}|\partial
Z^{K}\psi_{L}|^{2}|\partial_{v}Z^{J}\psi_{L}|^{2}$
$\displaystyle\lesssim\epsilon_{L}^{2}|X^{\ell}||{\ell^{m}}Z^{J}\psi|^{2},$
$\displaystyle\qquad|K|\leq N_{L}/2+1,$ (9.56)
$\displaystyle\frac{|X_{L}^{\ell}|}{(1+v)^{2}}|\partial
Z^{K}\psi_{L}|^{2}|\not{\nabla}Z^{J}\psi_{L}|^{2}$
$\displaystyle\lesssim\epsilon_{L}^{2}\frac{|X_{L}^{\ell}|}{(1+v)(1+s)^{1/2}}|\not{\nabla}Z^{J}\psi_{L}|^{2},$
$\displaystyle\qquad|K|\leq N_{L}/2+1$ (9.57)

we therefore have

$\sum_{|K|\leq N_{L}/2+1}\sum_{|J|\leq
N_{L}}\int_{t_{0}}^{t_{1}}\int_{\Gamma^{L}_{t}}\frac{|X_{L}^{\ell}|}{(1+v)^{2}}|\partial
Z^{K}\psi_{L}|^{2}|\partial Z^{J}\psi_{L}|^{2}\\\
\lesssim\epsilon_{L}^{2}\sum_{|J|\leq
N_{L}}\int_{t_{0}}^{t_{1}}\int_{\Gamma^{L}_{t}}\left(|X_{L}^{\ell}||\partial_{v}Z^{J}\psi_{L}|^{2}+\frac{|X_{L}^{n}|}{(1+v)(1+s)^{1/2}}|nZ^{J}\psi_{L}|^{2}+\frac{|X_{L}^{\ell}|}{(1+v)(1+s)^{1/2}}|\not{\nabla}Z^{J}\psi_{L}|^{2}\right)\\\
\lesssim\epsilon_{L}^{2}\sum_{|J|\leq
N_{L}}\int_{t_{0}}^{t_{1}}\int_{\Gamma^{L}_{t}}|X_{L}^{\ell}||\partial_{v}Z^{J}\psi_{L}|^{2}\,dSdt+\epsilon_{L}^{4}.$
(9.58)

which gives (9.49) after bounding
$(1+v)|Z^{I}(1+v)^{-1}Q(\partial\psi_{L},\partial\psi_{L})|\lesssim\sum_{|J|\leq|I|}\sum_{|K|\leq|I|/2+1}|\partial
Z^{J}\psi_{L}||\partial Z^{K}\psi_{L}|$ and using the bootstrap assumption
(9.2). ∎

The first step in the proof of Proposition 9.1 is to commute ${\ell^{m}}$ with
$Z^{I}$ and write the result in terms of the nonlinear boundary operator
$Y_{L}^{-}$ (recall the definition (6.10)).

######  Lemma 9.7.

Under the hypotheses of Proposition 9.1, we have

$\int_{t_{0}}^{t_{1}}\int_{\Gamma^{L}_{t}}vf(v)|\partial_{v}\psi_{L}^{I}|^{2}\,dSdt\lesssim\int_{t_{0}}^{t_{1}}\int_{\Gamma^{L}_{t}}vf(v)|Z^{I}Y_{L}^{-}\psi_{L}|^{2}\,dSdt+(c_{0}(\epsilon_{0})+\epsilon_{L}^{2})\epsilon_{L}^{2}\\\
+\sum_{|K|\leq|I|-1}\int_{t_{0}}^{t_{1}}\int_{\Gamma^{L}_{t}}vf(v)|\partial_{v}\psi_{L}^{J}|^{2}\,dSdt.$
(9.59)

###### Proof.

We just prove the result with $\psi_{L}^{I}=rZ^{I}\phi_{L}$ replaced with
$Z^{I}(r\phi_{L})$, the difference being straightforward to handle using
arguments we have by now used many times.

Recalling the definition of $Y_{L}^{-}$ from (6.10), along $\Gamma^{L}$ we
have

$\displaystyle|{\ell^{m}}Z^{I}\psi_{L}|$
$\displaystyle\lesssim|Z^{I}{\ell^{m}}\psi_{L}|+|[Z^{I},{\ell^{m}}]\psi_{L}|$
(9.60)
$\displaystyle\lesssim|Z^{I}Y_{L}^{-}\psi_{L}|+|Z^{I}\left((1+v)^{-1}Q(\partial\psi_{L},\partial\psi_{L})\right)|+|[Z^{I},{\ell^{m}}]\psi_{L}|$
(9.61)
$\displaystyle\lesssim|Z^{I}Y^{L}_{-}\psi_{L}|+|Z^{I}\left((1+v)^{-1}Q(\partial\psi_{L},\partial\psi_{L})\right)|$
(9.62)
$\displaystyle+\sum_{|J|\leq|I|-1}|{\ell^{m}}Z^{J}\psi_{L}|+\frac{1}{(1+v)^{3/2}}|nZ^{J}\psi_{L}|+\frac{1}{(1+v)^{3/4}}|\not{\nabla}Z^{J}\psi_{L}|$
(9.63)

where we used (A.21) and the fact that $|u|\lesssim s^{1/2}$ along
$\Gamma^{L}$ to bound $|[Z^{I},{\ell^{m}}]\psi_{L}|$. Using Lemma 9.6 to
control the quadratic term here, it remains only to handle the contribution
from the last line of (9.63).

The contribution from the first term there is bounded by the last term in
(9.59). For the other two terms, we note that along $\Gamma^{L}$, we have

$\displaystyle\frac{1}{(1+v)^{3}}vf(v)$ $\displaystyle\lesssim
c_{0}(\epsilon_{0})\frac{1}{1+v}\log(1+s)(\log\log(1+s))^{\alpha},$ (9.64)
$\displaystyle\frac{1}{(1+v)^{3/2}}vf(v)$ $\displaystyle\lesssim
c_{0}(\epsilon_{0})(1+s)^{1/2}\log(1+s)(\log\log(1+s))^{\alpha},$ (9.65)

for a continuous function $c_{0}(\epsilon_{0})$ with $c_{0}(0)=0$, where
recall $v\gtrsim\frac{1}{\epsilon_{0}}$ along $\Gamma^{L}$. As a result, using
(7.53) for $|J|\leq N_{L}$ we have the bounds

$\int_{t_{0}}^{t_{1}}\int_{\Gamma^{L}_{t}}vf(v)\frac{1}{(1+v)^{3}}|nZ^{J}\psi_{L}|^{2}\,dSdt\\\
\lesssim
c_{0}(\epsilon_{0})\int_{t_{0}}^{t_{1}}\int_{\Gamma^{L}_{t}}\frac{1}{1+v}\log(1+s)(\log\log(1+s))^{\alpha}|nZ^{J}\psi_{L}|^{2}\,dSdt\lesssim
c_{0}(\epsilon_{0})\epsilon_{L}^{2},$ (9.66)

and

$\int_{t_{0}}^{t_{1}}\int_{\Gamma^{L}_{t}}vf(v)\frac{1}{(1+v)^{3/2}}|\not{\nabla}Z^{J}\psi_{L}|^{2}\,dSdt\\\
\lesssim
c_{0}(\epsilon_{0})\int_{t_{0}}^{t_{1}}\int_{\Gamma^{L}_{t}}(1+s)^{1/2}\log(1+s)(\log\log(1+s))^{\alpha}|\not{\nabla}Z^{J}\psi_{L}|^{2}\lesssim
c_{0}(\epsilon_{0})\epsilon_{L}^{2},$ (9.67)

as needed.

∎

We now want to handle the nonlinear boundary operator $Y_{L}^{-}$ appearing on
the right-hand side of (9.59). For this we use Lemma 9.3 to replace the fields
$Z^{I}$ with the tangential fields $Z_{T}^{I}$. This generates error terms
involving the function $B^{L}$ which defines the boundary and which are
bounded using our assumptions on the geometry of the shocks.

######  Lemma 9.8.

With $B^{L}_{K}$ defined as in (9.1), under the hypotheses of Proposition 9.1,
for $|I|\leq N_{L}$ we have

$\int_{t_{0}}^{t_{1}}\int_{\Gamma^{L}_{t}}X^{\ell}|Z^{I}Y_{L}^{-}\psi_{L}|^{2}\,dSdt\lesssim\Upsilon_{L}^{+}(t_{1})+(c_{0}(\epsilon_{0})+\epsilon_{L}^{2})\epsilon_{L}^{2}+\sum_{|K|\leq|I|-1}B^{L}_{K}(t_{1})+c_{0}(\epsilon_{0}),$
(9.68)

for a continuous function $c_{0}$ with $c_{0}(0)=0$.

###### Proof.

We first use Lemma 9.3 to convert the fields $Z$ into tangential fields
$Z_{T}$,

$|Z^{I}Y_{L}^{-}\psi_{L}|\lesssim|Z_{T}^{I}Y_{L}^{-}\psi_{L}|+C(M)(1+|u|)\sum_{|J|\leq|I|-1}\left(|Z^{J}nY_{L}^{-}\psi_{L}|+(1+|u|)^{-1}|Z^{J}Y_{L}^{-}\psi_{L}|\right)\\\
+C(M)|B|_{I,\mathcal{Z}_{m}}(1+|u|)\sum_{|K|\leq|I|/2+1}\left(|Z^{K}nY_{L}^{-}\psi_{L}|+(1+|u|)^{-1}|Z^{K}Y_{L}^{-}\psi_{L}|\right).$
(9.69)

Since the fields $Z_{T}$ are tangent to the shock, by the boundary condition
(6.9) we have

$Z_{T}^{I}Y_{L}^{-}\psi_{L}=Z_{T}^{I}Y_{L}^{+}\psi_{C}+Z_{T}^{I}G,\qquad\text{
at }\Gamma^{L}$

and so recalling the definition of $\Upsilon_{L}^{+}$ from (9.46), to conclude
it is enough to prove that for $|I|\leq N_{L}$ we have the following
estimates,

$\int_{t_{0}}^{t_{1}}\int_{\Gamma^{L}_{t}}X^{\ell}(1+|u|)^{2}|Z^{J}nY_{L}^{-}\psi_{L}|^{2}\,dSdt\lesssim\sum_{|K|\leq|I|-1}B^{L}_{K}(t_{1})+(c_{0}(\epsilon_{0})+\epsilon_{L}^{2})\epsilon_{L}^{2}\qquad|J|\leq|I|-1$
(9.70)
$\int_{t_{0}}^{t_{1}}\int_{\Gamma^{L}_{t}}X^{\ell}|Z^{J}Y_{L}^{-}\psi_{L}|^{2}\,dSdt\lesssim\sum_{|K|\leq|I|-1}B^{L}_{K}(t_{1})+(c_{0}(\epsilon_{0})+\epsilon_{L}^{2})\epsilon_{L}^{2}\qquad|J|\leq|I|-1$
(9.71)
$\int_{t_{0}}^{t_{1}}\int_{\Gamma^{L}_{t}}X^{\ell}|B^{L}|_{I,\mathcal{Z}_{m}}^{2}\left((1+|u|)^{2}|Z^{K}nY_{L}^{-}\psi_{L}|^{2}+|Z^{K}Y_{L}^{-}\psi_{L}|^{2}\right)\\\
\lesssim\sum_{|K|\leq|I|-1}B^{L}_{K}(t_{1})+(c_{0}(\epsilon_{0})+\epsilon_{L}^{2})\epsilon_{L}^{2}\qquad|K|\leq|I|/2+1,$
(9.72)
$\displaystyle\int_{t_{0}}^{t_{1}}\int_{\Gamma^{L}_{t}}X^{\ell}|Z_{T}^{I}G|^{2}\,dSdt$
$\displaystyle\lesssim\sum_{|K|\leq|I|-1}B^{L}_{K}(t_{1})+(c_{0}(\epsilon_{0})+\epsilon_{L}^{2})\epsilon_{L}^{2}+c_{0}(\epsilon_{0}),$
(9.73)

where the implicit constants depend on $M$.

To prove (9.70), we recall the definition of $Y_{L}^{-}$ from (6.10) and use
that $(1+|u|)|\partial q|\lesssim\sum_{Z\in\mathcal{Z}_{m}}|Zq|$, which gives

$(1+|u|)|Z^{J}nY_{L}^{-}\psi_{L}|\lesssim(1+|u|)|Z^{J}n{\ell^{m}}\psi_{L}|+\sum_{|J^{\prime}|\leq|J|+1}|Z^{J^{\prime}}\left((1+v)^{-1}Q(\partial\psi_{L},\partial\psi_{L})\right)|.$
(9.74)

By Lemma 9.6, for $|J|\leq N_{L}-1$, the contribution from the nonlinear term
here is bounded by the right-hand side of (9.68). For the first term we use
the equation (6.7) for $\psi_{L}$ and bound

$(1+|u|)|Z^{J}n{\ell^{m}}\psi_{L}|\lesssim(1+|u|)|Z^{J}\not{\Delta}\psi_{L}|+\sum_{|J^{\prime}|\leq|J|+1}|Z^{J^{\prime}}\left((1+v)^{-1}Q(\partial\psi_{L},\partial\psi_{L})\right)|+(1+|u|)|Z^{J}F^{\prime}|,$
(9.75)

where we again used that $(1+|u)|\partial
q|\lesssim\sum_{Z\in\mathcal{Z}_{m}}|Zq|$. Just as above, the contribution
from the quadratic term here is bounded by the right-hand side of (9.68). The
contribution from $F^{\prime}$ is simpler to deal with so we skip it. To
handle the first term on the right-hand side of (9.75), we write
$\not{\Delta}=\frac{1}{r^{2}}\Omega^{2}$ and bound

$(1+|u|)|Z^{J}\not{\Delta}\psi_{L}|\lesssim\sum_{|J^{\prime}|\leq|J|+2}\frac{1+|u|}{(1+v)^{2}}|Z^{J^{\prime}}\psi_{L}|\\\
\lesssim\sum_{|J^{\prime}|\leq|J|+1}\frac{1+|u|}{1+v}\left(|{\ell^{m}}Z^{J^{\prime}}\psi_{L}|+|\not{\nabla}Z^{J^{\prime}}\psi_{L}|\right)+\frac{(1+|u|)^{2}}{(1+v)^{2}}|nZ^{J^{\prime}}\psi_{L}|.$
(9.76)

Now we bound $(1+|u|)(1+v)^{-1}\lesssim(1+v)^{-3/4}$ and
$(1+|u|)^{2}(1+v)^{-2}\lesssim(1+v)^{-3/2}$. We now recall that $|J|\leq|I|-1$
and argue as in (9.64)-(9.67) to get (9.70) after additionally bounding
$(1+|u|)(1+v)^{-1}\lesssim c_{0}(\epsilon_{0})$ to handle the first term here.

To prove (9.71), we just note that $\sum_{|K|\leq|I|-1}B^{L}_{K}(t_{1})$
appears on the right-hand side and use the product estimate from Lemma 9.6,
the definition of $Y_{L}^{-}$, along with the argument of Lemma 9.7 applied in
reverse to deal with $[\ell^{m},Z^{k}]$ (see (9.63)).

To prove (9.72), we bound

$\int_{t_{0}}^{t_{1}}\int_{\Gamma^{L}_{t}}X^{\ell}|B|_{I,\mathcal{Z}_{m}}^{2}\left((1+|u|)^{2}|Z^{K}nY_{L}^{-}\psi_{L}|^{2}+|Z^{K}Y_{L}^{-}\psi_{L}|^{2}\right)\,dSdt\\\
\lesssim\left(\int_{t_{0}}^{t_{1}}\sup_{\Gamma^{L}_{t}}|X^{\ell}|\left[(1+|u|)^{2}|Z^{K}nY_{-}^{L}\psi_{L}|^{2}+|Z^{K}Y^{L}_{-}\psi_{L}|^{2}\,dt\right]\right)\left(\sup_{t_{0}\leq
t\leq t_{1}}\int_{\Gamma^{L}_{t}}|B|_{I,\mathcal{Z}_{m}}^{2}\,dS\right)\\\
\lesssim
C(M)\int_{t_{0}}^{t_{1}}\sup_{\Gamma^{L}_{t}}|X^{\ell}|\left[(1+|u|)^{2}|Z^{K}nY_{-}^{L}\psi_{L}|^{2}+|Z^{K}Y^{L}_{-}\psi_{L}|^{2}\,dt\right].$
(9.77)

To handle this last term, we just use Sobolev embedding on
$\Gamma^{L}_{t}\sim\mathbb{S}^{2}$ and the bounds (9.70)-(9.71) that we just
proved. Specifically, we use that the fields $\Omega_{T}$ span the tangent
space to $\Gamma^{L}_{t}$ at each point and bound
$\sup_{\Gamma^{L}_{t}}|q|\lesssim\sum_{|R|\leq
2}\|\Omega_{T}^{R}q\|_{L^{2}(\Gamma^{L}_{t})}$ and using (9.24) to bound this
in terms of vector fields applied to $q$ gives
$\sup_{\Gamma^{L}_{t}}|q|\lesssim C(M)\sum_{|R|\leq 2}|Z^{R}q|$. Applying this
with $q=Z^{K}nY_{-}^{L}\psi_{L}$ and $q=Z^{K}Y_{-}^{L}\psi_{L}$ where $|K|\leq
N_{L}/2+1$ and applying (9.70)-(9.71) gives the result.

It remains to prove the bound (9.73) for the remainder term $G$, which is
given explicitly in (D.15). The terms in (D.13) and the last term in (D.15)
are all straightforward to handle using similar arguments to the ones we have
encountered many times by now. Note that the term $c_{0}(\epsilon_{0})$ on the
right-hand side of (9.73) is needed to control the quantities involving
$\Sigma$ in (D.13). We just show how to deal with the second term in (D.15),
and we will prove

$\int_{t_{0}}^{t_{1}}\int_{\Gamma^{L}_{t}}X^{\ell}\left|Z_{T}^{I}\frac{s}{u}[\not{\nabla}\psi]^{2}\right|\,dSdt\lesssim\sum_{|K|\leq|I|-1}B^{L}_{K}(t_{1})+(c_{0}(\epsilon_{0})+\epsilon_{L}^{2})\epsilon_{L}^{2}.$
(9.78)

Arguing as above to replace $Z_{T}$ with $Z$, bounding $\frac{s}{u}\lesssim
s^{1/2}$ and performing straightforward estimates, the main ingredients needed
for this are the bounds

$\displaystyle\sum_{|J|\leq|I|}\sum_{|K|\leq|I|/2+1}\int_{t_{0}}^{t_{1}}\int_{\Gamma^{L}_{t}}(1+s)X^{\ell}\left(|\not{\nabla}\psi^{J}_{L}|^{2}|\not{\nabla}\psi^{K}_{L}|^{2}\right)\,dSdt$
$\displaystyle\lesssim\epsilon_{L}^{2},$ (9.79)
$\displaystyle\sum_{|J|\leq|I|}\sum_{|K|\leq|I|/2+1}\int_{t_{0}}^{t_{1}}\int_{\Gamma^{L}_{t}}(1+s)X^{\ell}\left(|\not{\nabla}Z^{J}\psi_{C}|^{2}|\not{\nabla}Z^{K}\psi_{C}|^{2}\right)\,dSdt$
$\displaystyle\lesssim\epsilon_{L}^{2},$ (9.80)

which we now prove. For both estimates we handle the lower-order terms by
bounding $|\not{\nabla}q|\lesssim(1+v)^{-1}|\Omega q|$,

$|\not{\nabla}\psi^{K}_{L}|^{2}\lesssim\frac{1}{(1+v)^{2}}|\Omega\psi^{K}_{L}|^{2},\qquad|\not{\nabla}Z^{K}\psi_{C}|^{2}\lesssim\frac{1}{(1+v)^{2}}|\Omega
Z^{K}\psi_{C}|^{2}.$ (9.81)

By the Poincare inequality (F.5) from Lemma F.2 and Sobolev embedding, for
$|K|\leq|I|/2+1$ we have

$|\Omega\psi^{K}_{L}|^{2}\lesssim\epsilon_{L}^{2},$ (9.82)

at the shock, and the left-hand side of (9.79) is then bounded by

$\displaystyle\sum_{|J|\leq|I|}\sum_{|K|\leq|I|/2+1}\int_{t_{0}}^{t_{1}}\int_{\Gamma^{L}_{t}}(1+s)X^{\ell}$
$\displaystyle\left(|\not{\nabla}\psi^{J}_{L}|^{2}|\not{\nabla}\psi^{K}_{L}|^{2}\right)\,dSdt$
(9.83)
$\displaystyle\lesssim\epsilon_{L}^{2}\sum_{|J|\leq|I|}\int_{t_{0}}^{t_{1}}\int_{\Gamma^{L}_{t}}\frac{(1+s)X^{\ell}}{(1+v)^{2}}|\not{\nabla}\psi^{J}_{L}|^{2}\,dSdt$
(9.84)
$\displaystyle\lesssim\epsilon_{L}^{2}\sum_{|J|\leq|I|}\int_{t_{0}}^{t_{1}}\int_{\Gamma^{L}_{t}}\frac{(1+s)^{2}(\log
s)^{\alpha}}{1+v}|\not{\nabla}\psi^{J}_{L}|^{2}\,dSdt\lesssim\epsilon_{L}^{4},$
(9.85)

by the bound (7.53) for the energies on the timelike side of the left shock,
since $|I|\leq N_{L}$.

For (9.80), since we are below the top-order number of derivatives of
$\psi_{C}$ we can in fact use (9.81) in each factor of $\psi_{C}$ which gives

$\sum_{|J|\leq|I|}\sum_{|K|\leq|I|/2+1}\int_{t_{0}}^{t_{1}}\int_{\Gamma^{L}_{t}}(1+s)X^{\ell}\left(|\not{\nabla}Z^{J}\psi_{C}|^{2}|\not{\nabla}Z^{K}\psi_{C}|^{2}\right)\,dSdt\\\
\lesssim\sum_{|J|\leq|I|}\int_{t_{0}}^{t_{1}}\int_{\Gamma^{L}_{t}}\frac{(1+s)X^{\ell}}{(1+v)^{4}}|\Omega
Z^{J}\psi_{C}|^{4}\,dSdt\lesssim\epsilon_{C}^{4},$ (9.86)

where we used that $\frac{(1+s)X^{\ell}}{(1+v)^{4}}\lesssim\frac{s}{v^{2}}$,
the Hardy inequality (F.4), and the bound (7.47) for the energy at the right
shock for $|I|\leq N_{L}\leq N_{C}-3$. Since we have taken
$\epsilon_{C}\leq\epsilon_{L}$ this gives (9.80). ∎

To complete the proof of Proposition 9.1, we need to prove the bound for the
quantity $\Upsilon_{I,L}^{+}$.

####  9.2.2 Control of $\Upsilon_{I,L}^{+}$

If it was not for the large (relative to the weights we use in the central
region) weight $|X^{\ell}|$, it would be straightforward to control
$\Upsilon_{I,L}^{+}$ by using the bounds for $\psi_{C}$ in Lemma 7.7. Instead,
we are going to handle $\Upsilon_{I,L}^{+}$ by integrating to the right shock.
Schematically, this amounts to trading a factor of $|u|\sim(\log t)^{1/2}$
(the width of $D^{C}_{t}$) for a derivative of (vector fields applied to)
$Y_{L}^{+}\psi_{C}$, see (9.91). We can afford this trade in the central
region, ultimately because we control the vector field
$X_{1}=s\partial_{u}\sim(\log t)\partial_{u}$ applied to $\psi_{C}$. As a
result, this trade in fact gains us a factor of $(\log t)^{1/2}$, which is
enough to close our estimates.

######  Proposition 9.3.

Under the hypotheses of Proposition 9.1, provided $\epsilon_{0}$ is taken
sufficiently small, we have

$\sum_{|I|\leq N_{L}}\Upsilon_{I,L}^{+}(t_{1})\lesssim\epsilon_{C}^{2}$ (9.87)

The proof relies on the upcoming Lemmas 9.9 and 9.10.

###### Proof.

We start by recalling that quantities in the definition of
$\Upsilon_{I,L}^{+}$ involve tangential vector fields $Z_{T}$. For our
purposes it is simpler if we replace these with the usual vector fields $Z$.
We therefore use Lemma 9.3 and the fact that $(1+|u|)|\partial
q|\lesssim\sum_{Z\in\mathcal{Z}_{m}}$ to bound

$\Upsilon_{I,L}^{+}\leq
C(M)\sum_{|J|\leq|I|}\int_{t_{0}}^{t_{1}}\int_{\Gamma^{L}_{t}}|X^{\ell}||Z^{J}Y_{L}^{+}\psi_{C}|^{2}\,dSdt\\\
+C(M)\sum_{|K|\leq|I|/2+1}\int_{t_{0}}^{t_{1}}\int_{\Gamma^{L}_{t}}|X^{\ell}||B^{L}|_{I,m}|Z^{K}Y_{L}^{+}\psi_{C}|^{2}\,dSdt.$
(9.88)

We claim that this implies the following bound,

$\Upsilon_{I,L}^{+}(t_{1})\leq
C(M)\sum_{|J|\leq|I|}\int_{t_{0}}^{t_{1}}\int_{D^{C}_{t}}(1+t)(1+\log
t)^{3/2}(1+\log\log t)^{\alpha}|\partial_{r}Z^{J}Y_{L}^{+}\psi_{C}|^{2}\ dt\\\
+C(M)\sum_{|J|\leq|I|}\int_{t_{0}}^{t_{1}}\int_{\Gamma^{R}_{t}}(1+t)(1+\log
t)(1+\log\log t)^{\alpha}|Z^{J}Y_{L}^{+}\psi_{C}|^{2}\,dSdt.$ (9.89)

In the upcoming Lemmas 9.9 and 9.10, we bound the right-hand side here by the
right-hand side of (9.87).

The idea behind (9.89) is to control the quantities
$q^{J}=Z^{J}Y_{L}^{+}\psi_{C}|_{\Gamma^{L}_{t}}$ by integrating along the ray
$x/|x|=\omega$ at fixed time to the right shock, using the bounds for
$\psi_{C}$ in the central region to handle the interior term this generates
and the boundary condition at the right shock to handle the boundary term this
generates.

In particular, for $A=L,R$, we let $r_{A}(t^{\prime},\omega)$ denote the value
of $|x|$ of the point lying at the intersection of the sets
$\\{t=t^{\prime}\\}$, $\\{x/|x|=\omega\\}$ and $\Gamma^{A}$. That is, $r_{L}$
is defined by the property that
$t-r_{L}(t,\omega)=B^{L}(t,r_{L}(t,\omega)\omega)$ and similarly for $r_{R}$.
For any function $q$ defined in $D^{C}$, fixing $t,\omega$ and integrating
from $r=r_{L}(t,\omega)$ to $r=r_{R}(t,\omega)$ we find

$|q(t,r_{L}(t,\omega)\omega)|\leq|q(t,r_{R}(t,\omega)\omega)|+\int_{r_{L}(t,\omega)}^{r_{R}(t,\omega)}|(\partial_{r}q)(r^{\prime})|\,dr^{\prime}\\\
\leq|q(t,r_{R}(t,\omega)\omega)|+|r_{L}(t,\omega)-r_{R}(t,\omega)|^{1/2}\left(\int_{r_{L}(t,\omega)}^{r_{R}(t,\omega)}|\partial_{r}q|^{2}\,dr\right)^{1/2}\\\
\lesssim|q(t,r_{R}(t,\omega)\omega)|+(\log
t)^{1/4}\left(\int_{r_{L}(t,\omega)}^{r_{R}(t,\omega)}|\partial_{r}q|^{2}\,dr\right)^{1/2},$
(9.90)

and in particular we have the bound

$\int_{\Gamma^{L}_{t}}|q|^{2}\,dS\lesssim\int_{\mathbb{S}^{2}}|q(t,r_{L}(t,\omega)\omega)|^{2}\,dS(\omega)\lesssim\int_{\Gamma^{R}_{t}}|q|^{2}\,dS+(\log
t)^{1/2}\int_{D^{C}_{t}}|\partial_{r}q|^{2}$ (9.91)

using that by (2.59) the surface measures $dS$ and $dS(\omega)$ are
equivalent. Applying this to $q=q^{J}$ as in the above paragraph and
integrating in $t$, we find that the first term in (9.88) is bounded by

$\displaystyle\int_{t_{0}}^{t_{1}}\int_{\Gamma^{L}_{t}}|X^{\ell}||Z^{J}Y_{L}^{+}\psi_{C}|^{2}\,dSdt$
$\displaystyle\lesssim\int_{t_{0}}^{t_{1}}\int_{\Gamma^{L}_{t}}(1+t)(1+\log
t)(1+\log\log t)^{\alpha}|Z^{J}Y_{L}^{+}\psi_{C}|^{2}\,dSdt$ (9.92)
$\displaystyle\lesssim\int_{t_{0}}^{t_{1}}\int_{\Gamma^{R}_{t}}(1+t)(1+\log
t)(1+\log\log t)^{\alpha}|Z^{J}Y_{L}^{+}\psi_{C}|^{2}\,dSdt$ (9.93)
$\displaystyle+\int_{t_{0}}^{t_{1}}\int_{D^{C}_{t}}(1+t)(1+\log
t)^{3/2}(1+\log\log t)^{\alpha}|\partial_{r}Z^{J}Y_{L}^{+}\psi_{C}|^{2}\ dt.$
(9.94)

We now perform a similar manipulation to the second term in (9.88). We first
bound

$\int_{t_{0}}^{t_{1}}\int_{\Gamma^{L}_{t}}|X^{\ell}||B|_{I,m}^{2}|Z^{K}Y_{L}^{+}\psi_{C}|^{2}\,dSdt\lesssim\left(\sup_{t_{0}\leq
t\leq
t_{1}}\int_{\Gamma^{L}_{t}}|B^{L}|_{I,m}^{2}\,dS\right)\left(\int_{t_{0}}^{t_{1}}\sup_{\Gamma^{L}_{t}}|X^{\ell}||Z^{K}Y_{L}^{+}\psi_{C}|^{2}\,dt\right).$
(9.95)

Using Sobolev embedding on $\Gamma^{L}_{t}$ as in the proof of the last
result, using the bound (6.82) for the high-order derivatives of $B^{L}$, and
using (9.94) again, after returning to (9.88) we have the claim (9.89). ∎

We now prove the needed lemmas. First, we get control over certain time-
integrated weighted norms of $\partial Z^{J}Y_{L}^{+}\psi_{C}$ in $D^{C}$. The
idea behind this estimate is that as usual the most dangerous term is when
$\partial=\partial_{u}$. Since $Y_{L}^{+}=\ell^{m_{B}}$ up to nonlinear terms,
and since the equation (6.5) in the central region expresses
$\partial_{u}\ell^{m_{B}}\psi_{C}$ in terms of $\not{\Delta}\psi_{C}$ and
nonlinear terms, this term can be handled.

######  Lemma 9.9.

Under the hypotheses of Proposition 9.1, provided $\epsilon_{0}$ is taken
sufficiently small, for $|J|\leq N_{L}\leq N_{C}-4$, we have

$\int_{t_{0}}^{t_{1}}\int_{D^{C}_{t}}(1+t)(1+\log t)^{3/2}(1+\log\log
t)^{\alpha}|\partial_{r}Z^{J}Y_{L}^{+}\psi_{C}|^{2}\
dt\lesssim\epsilon_{C}^{2}.$ (9.96)

###### Proof.

We recall the definition of $Y_{L}^{+}$ from (6.11) and bound

$\displaystyle|\partial_{r}Z^{J}Y_{L}^{+}\psi_{C}|$
$\displaystyle\lesssim|\partial_{r}Z^{J}\ell^{m_{B}}\psi_{C}|+|\partial
Z^{J}\left((1+v)^{-1}Q(\partial\psi_{C},\partial\psi_{C})\right)|.$ (9.97)

To handle the contribution from the nonlinear term here, we use Lemma A.7 and
the fact that we have $(1+s)|\partial
q|\lesssim\sum_{Z_{m_{B}}\in\mathcal{Z}_{m_{B}}}|Z_{m_{B}}q|$ to get that for
$|J|\leq N_{C}-3$,

$|\partial
Z^{J}\left((1+v)^{-1}Q(\partial\psi_{C},\partial\psi_{C})\right)|\lesssim\frac{1}{1+v}\frac{1}{1+s}\sum_{|J^{\prime}|\leq|J|+1}\sum_{|K|\leq|J|/2+1}|\partial
Z_{m_{B}}^{J^{\prime}}\psi_{C}||\partial Z_{m_{B}}^{K}\psi_{C}|\\\
\lesssim\epsilon_{C}\frac{1}{1+v}\frac{1}{(1+s)^{3/2}}\sum_{|J^{\prime}|\leq|J|+1}|\partial
Z_{m_{B}}^{J^{\prime}}\psi_{C}|,$ (9.98)

by the pointwise bound (7.2). Therefore, the contribution from this term into
(9.96) is bounded by

$\int_{t_{0}}^{t_{1}}\int_{D^{C}_{t}}(1+t)(1+\log t)^{3/2}(1+\log\log
t)^{\alpha}|\partial
Z^{J}\left((1+v)^{-1}Q(\partial\psi_{C},\partial\psi_{C})\right)|^{2}\ dt\\\
\lesssim\epsilon_{C}^{2}\sum_{|J^{\prime}|\leq|J|+1}\int_{t_{0}}^{t_{1}}\int_{D^{C}_{t}}\frac{1}{1+t}\frac{1}{(1+\log
t)^{3/2}}(1+\log\log t)^{\alpha}|\partial
Z_{m_{B}}^{J^{\prime}}\psi_{C}|^{2}\,dt\lesssim\epsilon_{C}^{4},$ (9.99)

using the bound for the lower-order energy (6.37) and the fact that
$1/((1+t)(1+\log(1+t))^{3/2})$ is time-integrable.

We now handle the contribution from the first term in (9.97). Recalling that
$\partial_{r}=\partial_{v}-\partial_{u}$, we first bound the contribution from
$\partial_{v}$ as follows,

$\int_{t_{0}}^{t_{1}}\int_{D^{C}_{t}}(1+t)(1+\log t)^{3/2}(1+\log\log
t)^{\alpha}|\partial_{v}Z^{J}\ell^{m_{B}}\psi_{C}|^{2}\,dt\\\
\lesssim\int_{t_{0}}^{t_{1}}\int_{D^{C}_{t}}\left(\frac{1}{(1+t)^{2}}(1+\log
t)^{3/2}(1+\log\log
t)^{\alpha}\right)(1+v)|ZZ^{J}\ell^{m_{B}}\psi_{C}|^{2}\,dt\lesssim\epsilon_{C}^{2},$
(9.100)

after using the bound for the energy (6.32), (A.81) to convert $Z$ derivatives
into $Z_{m_{B}}$ derivatives, and the bound (A.84) to handle the commutators
between the $Z_{m_{B}}$ and $\ell^{m_{B}}$.

It remains to prove the analogous bound for $\partial_{u}$, namely

$\int_{t_{0}}^{t_{1}}\int_{D^{C}_{t}}(1+t)(1+\log t)^{3/2}(1+\log\log
t)^{\alpha}|\partial_{u}Z^{J}\ell^{m_{B}}\psi_{C}|^{2}\,dt\lesssim\epsilon_{C}^{2}.$
(9.101)

By the bound (A.22) for the commutator $[\partial_{u},Z^{J}]$, we have

$|\partial_{u}Z^{J}\ell^{m_{B}}\psi_{C}|\lesssim|Z^{J}\partial_{u}\ell^{m_{B}}\psi_{C}|+\sum_{|J^{\prime}|\leq|J|-1}|\partial_{u}Z^{J^{\prime}}\ell^{m_{B}}\psi_{C}|+|\not{\nabla}Z^{J^{\prime}}\ell^{m_{B}}\psi_{C}|+|{\ell^{m}}Z^{J^{\prime}}\ell^{m_{B}}\psi_{C}|\\\
\lesssim|Z^{J}\partial_{u}\ell^{m_{B}}\psi_{C}|+\sum_{|J^{\prime}|\leq|J|-1}|\partial_{u}Z^{J^{\prime}}\ell^{m_{B}}\psi_{C}|+\frac{1}{1+v}|ZZ^{J^{\prime}}\ell^{m_{B}}\psi_{C}|.$
(9.102)

To handle the first term here, we use the equation (6.5) which gives

$\displaystyle|Z^{J}$ $\displaystyle\partial_{u}\ell^{m_{B}}\psi_{C}|$ (9.103)
$\displaystyle\lesssim|Z^{J}\not{\Delta}\psi_{C}|+|Z^{J}\partial_{\mu}(\tfrac{u}{vs}a^{\mu\nu}\partial_{\nu}\psi_{C})|+|Z^{J}\partial_{\mu}(\gamma^{\mu\nu}\partial_{\nu}\psi_{C})|+|Z^{J}\partial_{\mu}P^{\mu}|+|Z^{J}F|+|Z^{J}F_{\Sigma}|$
(9.104)
$\displaystyle\lesssim\sum_{|J^{\prime}|\leq|J|}\left(\frac{1}{(1+v)^{2}}|Z^{J^{\prime}}\Omega^{2}\psi_{C}|+\frac{1}{(1+v)(1+s)^{1/2}}|\partial^{2}Z^{J^{\prime}}\psi_{C}|+\frac{1}{(1+v)(1+s)}|\partial
Z^{J^{\prime}}\psi_{C}|\right)$ (9.105)
$\displaystyle+|Z^{J}\partial_{\mu}(\gamma^{\mu\nu}\partial_{\nu}\psi_{C})|+|Z^{J}\partial_{\mu}P^{\mu}|+|Z^{J}F|+|Z^{J}F_{\Sigma}|.$
(9.106)

In getting the above bound we used that
$|\partial\tfrac{u}{vs}|\lesssim\tfrac{1}{vs}$, that
$Z\frac{u}{vs}=c_{Z}\frac{u}{vs}$ for smooth functions $c_{Z}$ satisfying the
symbol condition (A.37) and ignored the structure of the coefficients
$a^{\mu\nu}$ which is not important for this part of the argument. We just
show how to handle the terms on the first line of (9.105), the terms on the
second line being very similar after noting that $\gamma$ behaves like
$(1+v)^{-1}\partial\psi_{C}$ and that we have the pointwise bound
$|\partial\psi_{C}|\lesssim(1+s)^{-1/2}$, and that the terms
$\partial_{\mu}P^{\mu},F,F_{\Sigma}$ are better-behaved (recall that these
quantities are given explicitly in (C.19)-(C.23)).

The contribution from the third term on the first line of (9.105) into (9.101)
is actually the most dangerous one, and it is bounded by

$\displaystyle\int_{t_{0}}^{t_{1}}\int_{D^{C}_{t}}(1+t)(1+\log t)^{3/2}$
$\displaystyle(1+\log\log t)^{\alpha}\left(\frac{1}{1+v}\frac{1}{1+s}|\partial
Z^{J^{\prime}}\psi_{C}|\right)^{2}\,dt$ (9.107)
$\displaystyle\lesssim\int_{t_{0}}^{t_{1}}\int_{D^{C}_{t}}\frac{1}{1+t}\frac{1}{(1+\log
t)^{1/2}}(1+\log\log t)^{\alpha}|\partial Z^{J^{\prime}}\psi_{C}|^{2}\,dt$
(9.108)
$\displaystyle\lesssim\int_{t_{0}}^{t_{1}}\int_{D^{C}_{t}}\frac{1}{1+t}\frac{1}{(1+\log
t)^{3/2}}(1+\log\log t)^{\alpha}\left(s|\partial
Z^{J^{\prime}}\psi_{C}|^{2}\right)\,dt$ (9.109)
$\displaystyle\lesssim\sum_{|J^{\prime\prime}|\leq|J^{\prime}|}\int_{t_{0}}^{t_{1}}\int_{D^{C}_{t}}\frac{1}{1+t}\frac{1}{(1+\log
t)^{5/4}}\left(s|\partial
Z_{m_{B}}^{J^{\prime\prime}}\psi_{C}|^{2}\right)\,dt$ (9.110)
$\displaystyle\lesssim\epsilon_{C}^{2},$ (9.111)

for $|J^{\prime}|\leq N_{C}-3$. Here, we have used (A.81) to convert the
fields $Z$ into the fields $Z_{m_{B}}$, the definition of the energy in the
central region, the fact that $1/((1+t)(1+\log t)^{5/4})$ is time-integrable,
and bounded $(\log\log t)^{\alpha}\lesssim(\log t)^{1/4}$. As for the second
term on the first line of (9.105), its contribution is bounded by

$\displaystyle\int_{t_{0}}^{t_{1}}\int_{D^{C}_{t}}\frac{1}{1+t}$
$\displaystyle(1+\log t)^{1/2}(1+\log\log
t)^{\alpha}|\partial^{2}Z^{J^{\prime}}\psi_{C}|^{2}\,dt$ (9.112)
$\displaystyle\lesssim\int_{t_{0}}^{t_{1}}\int_{D^{C}_{t}}\frac{1}{1+t}(1+\log
t)^{1/2}(1+\log\log t)^{\alpha}|\partial^{2}Z^{J^{\prime}}\psi_{C}|^{2}\,dt$
(9.113)
$\displaystyle\lesssim\int_{t_{0}}^{t_{1}}\int_{D^{C}_{t}}\frac{1}{1+t}\frac{1}{(1+\log
t)^{5/4}}(|\partial X_{1}Z^{J^{\prime}}\psi_{C}|^{2}+|\partial
Z^{J^{\prime}}\psi_{C}|^{2})\,dt$ (9.114)
$\displaystyle\lesssim\epsilon_{C}^{2},$ (9.115)

recalling the definition $X_{1}=s\partial_{u}$ and then arguing as above to
bound this quantity by the energy in the central region.

Finally, to deal with the first term in (9.105) we use the identity (A.15) to
bound

$\frac{1}{(1+v)^{2}}|Z^{J^{\prime}}\Omega^{2}\psi_{C}|\lesssim\frac{1}{1+v}\sum_{|J^{\prime\prime}|\leq|J^{\prime}|+1}\left(|\partial_{v}Z^{J^{\prime\prime}}\psi_{C}|+|\not{\nabla}Z^{J^{\prime\prime}}\psi_{C}|+\frac{(1+|u|)}{(1+v)}|\partial_{u}Z^{J^{\prime\prime}}\psi_{C}|\right),$
(9.116)

and the contribution from these terms into (9.101) are easily handled.

Combining the above estimates we have

$\int_{t_{0}}^{t_{1}}\int_{D^{C}_{t}}(1+t)(1+\log t)^{3/2}(1+\log\log
t)^{\alpha}|\partial_{u}Z^{J}\ell^{m_{B}}\psi_{C}|^{2}\,dt\\\
\lesssim\epsilon_{C}^{2}+\sum_{|J^{\prime}|\leq|J|-1}\int_{t_{0}}^{t_{1}}\int_{D^{C}_{t}}(1+t)(1+\log
t)^{3/2}(1+\log\log
t)^{\alpha}|\partial_{u}Z^{J^{\prime}}\ell^{m_{B}}\psi_{C}|^{2}\,dt\\\
+\sum_{|J^{\prime}|\leq|J|}\int_{t_{0}}^{t_{1}}\int_{D^{C}_{t}}\frac{1}{1+t}(1+\log
t)^{3/2}(1+\log\log t)^{\alpha}|Z^{J^{\prime}}\ell^{m_{B}}\psi_{C}|^{2}\,dt.$
(9.117)

For the last term here, we bound

$\int_{t_{0}}^{t_{1}}\int_{D^{C}_{t}}\frac{1}{1+t}(1+\log t)^{3/2}(1+\log\log
t)^{\alpha}|Z^{J^{\prime}}\ell^{m_{B}}\psi_{C}|^{2}\,dt\\\
\lesssim\int_{t_{0}}^{t_{1}}\int_{D^{C}_{t}}\left(\frac{1}{(1+t)^{2}}(1+\log
t)^{3/2}(1+\log\log
t)^{\alpha}\right)(1+v)|Z^{J^{\prime}}\ell^{m_{B}}\psi_{C}|^{2}\,dt\lesssim\epsilon_{C}^{2},$
(9.118)

as in (9.100). The result now follows from this bound, (9.117), and induction.

∎

We now control the boundary term from (9.93).

######  Lemma 9.10.

Under the hypotheses of Proposition 9.1, for $|J|\leq N_{L}\leq N_{C}-4$,
provided $\epsilon_{0}$ is taken sufficiently small, we have

$\int_{t_{0}}^{t_{1}}\int_{\Gamma^{R}_{t}}(1+t)(1+\log t)(1+\log\log
t)^{\alpha}|Z^{J}Y_{L}^{+}\psi_{C}|^{2}\,dSdt\lesssim\epsilon_{C}^{2}$ (9.119)

###### Proof.

We start by using Lemma 9.3 to convert the vector fields $Z$ into tangential
vector fields $Z_{T}$ at the right shock, which gives

$|Z^{J}Y_{L}^{+}\psi_{C}|\leq\sum_{|J^{\prime}|\leq|J|}|Z_{T}^{J^{\prime}}Y_{L}^{+}\psi_{C}|+C(M)\sum_{|J^{\prime}|\leq|J|-1}(1+|u|)|Z^{J^{\prime}}nY_{L}^{+}\psi_{C}|+|Z^{J^{\prime}}Y_{L}^{+}\psi_{C}|\\\
+C(M)|B^{R}|_{J,m_{B}}\sum_{|K|\leq|J|/2+1}(1+|u|)|Z^{K}nY_{L}^{+}\psi_{C}|+|Z^{K}Y_{L}^{+}\psi_{C}|,$
(9.120)

where we used (A.81) to bound the norm of $B^{R}$ Since the vector fields
$Z_{T}^{J}$ are tangent to $\Gamma^{R}$, at $\Gamma^{R}$ by the boundary
condition (6.12) we have

$|Z_{T}^{J}Y_{L}^{+}\psi_{C}|=|Z_{T}^{J}Y_{R}^{-}\psi_{C}|\lesssim|Z_{T}^{J}Y_{R}^{+}\psi_{R}|+|Z_{T}^{J}G|.$
(9.121)

We now use Lemma 9.3 again to convert the tangential fields $Z_{T}^{I}$ into
the usual fields $Z^{I}$ and bound the first term here by

$|Z_{T}^{J}Y_{R}^{+}\psi_{R}|\leq\sum_{|J^{\prime}|\leq|J|}|Z^{J}Y_{R}^{+}\psi_{R}|+C(M)\sum_{|J^{\prime}|\leq|J|-1}\left((1+|u|)|nZ^{J^{\prime}}Y_{R}^{+}\psi_{R}|+|Z^{J^{\prime}}Y_{R}^{+}\psi_{R}|\right)\\\
+C(M)|B^{R}|_{J,m}\sum_{|K|\leq|J|/2+1}\left((1+|u|)|nZ^{J^{\prime}}Y_{R}^{+}\psi_{R}|+|Z^{J^{\prime}}Y_{R}^{+}\psi_{R}|\right)$
(9.122)

Thanks to the large weights appearing in the bounds (7.1) and (7.50) for the
potential in the rightmost region, it is straightforward to deal with these
terms using arguments very similar but considerably simpler than ones we have
encountered many times by now, and the result is that

$\int_{t_{0}}^{t_{1}}\int_{\Gamma^{R}_{t}}|X^{\ell}||Z_{T}^{J}Y_{R}^{+}\psi_{R}|^{2}\,dSdt\lesssim\epsilon_{R}^{2}\leq\epsilon_{C}^{4},$
(9.123)

by (6.19). See also Lemma 9.3 for a nearly identical but slightly more
delicate estimate. The contribution from the nonlinear error terms $G^{L}$ can
be handled as in the proof of (9.73).

We now handle the terms involving $nY_{L}^{+}\psi_{C}$ (9.120). We claim that

$\int_{t_{0}}^{t_{1}}\int_{\Gamma^{R}_{t}}(1+t)(1+\log t)(1+\log\log
t)^{\alpha}\left((1+|u|)|nZ^{J^{\prime}}Y_{L}^{+}\psi_{C}|\right)^{2}\,dSdt\lesssim\epsilon_{C}^{2}.$
(9.124)

In the same way that (9.101) implied the previous result, this bound implies
(9.119). To prove this, we use (9.105) again. As in the proof of the previous
result, we will just consider the terms on the first line of (9.105), the
remaining terms being simpler.

For the third term on the first line of (9.105), using that $|u|\sim s^{1/2}$
along $\Gamma^{R}$, we have

$\displaystyle\int_{t_{0}}^{t_{1}}\int_{\Gamma^{R}_{t}}(1+t)(1+\log
t)(1+\log\log t)^{\alpha}$
$\displaystyle\left(\frac{(1+|u|)}{(1+v)(1+s)}|\partial
Z^{J^{\prime}}\psi_{C}|\right)^{2}\,dSdt$ (9.125)
$\displaystyle\lesssim\int_{t_{0}}^{t_{1}}\int_{\Gamma^{R}_{t}}\frac{1}{1+t}(1+\log\log
t)^{\alpha}|\partial Z^{J^{\prime}}\psi_{C}|^{2}\,dSdt$ (9.126)
$\displaystyle\lesssim\int_{t_{0}}^{t_{1}}\int_{\Gamma^{R}_{t}}\frac{(1+s)^{1/2}}{1+v}|\partial
Z^{J^{\prime}}\psi_{C}|^{2}\,dSdt$ (9.127)
$\displaystyle\lesssim\epsilon_{C}^{2},$ (9.128)

in light of (7.50) and after using the bound (A.81) to relate norms involving
the $Z$ fields to those involving the $Z_{m_{B}}$ fields. Similarly, for the
second term in (9.105) we have

$\displaystyle\int_{t_{0}}^{t_{1}}\int_{\Gamma^{R}_{t}}(1+t)$
$\displaystyle(1+\log t)(1+\log\log
t)^{\alpha}\left(\frac{(1+|u|)}{(1+v)(1+s)^{1/2}}|\partial^{2}Z^{J^{\prime}}\psi_{C}|\right)^{2}\,dSdt$
(9.129)
$\displaystyle\lesssim\int_{t_{0}}^{t_{1}}\int_{\Gamma^{R}_{t}}\frac{1}{1+t}(1+\log
t)(1+\log\log t)^{\alpha}|\partial^{2}Z^{J^{\prime}}\psi_{C}|^{2}\,dSdt$
(9.130)
$\displaystyle\lesssim\int_{t_{0}}^{t_{1}}\int_{\Gamma^{R}_{t}}\frac{1}{1+t}\frac{1}{1+\log
t}(1+\log\log t)^{\alpha}(|\partial X_{1}Z^{J^{\prime}}\psi_{C}|^{2}+|\partial
Z^{J^{\prime}}\psi_{C}|^{2})\,dt$ (9.131)
$\displaystyle\lesssim\epsilon_{C}^{2}.$ (9.132)

Finally, using (9.116) again it is straightforward to handle the contribution
from the first term in (9.105) into (9.124). It remains to handle the lower-
order terms (the last term on the first line and those on the second line)
from (9.120)). The terms $Z^{J^{\prime}}Y_{L}^{+}\psi_{C}$ for
$|J^{\prime}|\leq|J|$ can be handled using induction. As for the lower-order
term involving $nY_{L}^{+}$, we just use the equation (9.105) again to express
$nY_{L}^{+}\psi_{C}$ in terms of nonlinear terms and linear terms involving
$\not{\Delta}$ .

∎

###  9.3 Proof of Proposition 9.2

We use a similar strategy as in the previous section. The bounds in this
section are simpler because in this section we only need to consider the
weight $X^{\ell}=1+v$ which is smaller than the weight we needed to consider
in the previous section and there is more room in all of our estimates. On the
other hand, the estimates are somewhat more cumbersome because we have worse
control over the solution at top order than we did in the central region
(recall the definition of the energies (6.26)).

The bound (9.4) is a consequence of the upcoming Lemmas 9.12, 9.13 and 9.14,
as follows. Define the quantities

$\Upsilon_{I,R}^{+}(t_{1})=\int_{t_{0}}^{t_{1}}\int_{\Gamma^{R}_{t}}(1+v)|Z_{m_{B},T}^{I}Y_{L}^{+}\psi_{R}|^{2}\,dSdt,$
(9.133)

where $Y_{R}^{+}$ is as in (6.14).

By Lemmas 9.12 and 9.13, for $|I|\leq N_{C}$, we have the bound

$B^{C}_{I}(t_{1})\lesssim\Upsilon_{I,R}^{+}(t_{1})+\sum_{|K|\leq|I|-1}B^{C}_{K}(t_{1})+(c_{0}(\epsilon_{0})+\epsilon_{C}^{2})\epsilon_{C}^{2}+c_{0}(\epsilon_{0}),$
(9.134)

where $c_{0}$ is a continuous function with $c_{0}(0)=0$, so taking
$\epsilon_{0}$ sufficiently small and using induction we find

$B^{C}_{I}(t_{1})\lesssim\Upsilon_{I,R}^{+}(t_{1})+\epsilon_{C}^{3},$ (9.135)

and the result follows after using Lemma 9.14 to control the quantities
$\Upsilon_{I,R}^{+}$. ∎

In the rest of this section, we prove the cited Lemmas 9.12-9.14.

####  9.3.1 Supporting lemmas for the proof of Proposition 9.2

As in the last section, we start by recording a product estimate that we will
use to handle the nonlinear terms we encounter. Fortunately this bound is less
delicate than Lemma 9.6.

######  Lemma 9.11.

Let
$Q(\partial\psi,\partial\psi)=Q^{\alpha\beta}\partial_{\alpha}\psi\partial_{\beta}\psi$
be a quadratic nonlinearity where the coefficients $Q^{\alpha\beta}(\omega)$
are smooth functions satisfying the symbol condition (A.37). Under the
hypotheses of Proposition 9.2, we have

$\int_{t_{0}}^{t_{1}}\int_{\Gamma^{R}_{t}}(1+v)|Z_{m_{B}}^{I}\left((1+v)^{-1}Q(\partial\psi_{C},\partial\psi_{C})\right)|^{2}\,dSdt\lesssim\epsilon_{C}^{4}.$
(9.136)

###### Proof.

We follow the same strategy as in the proof of Lemma 9.6. We first note that
by the decay estimates (7.2) we have
$|\partial\psi^{K}_{C}|\lesssim\epsilon_{C}(1+s)^{-1/2}$ for $|K|\leq
N_{C}/2+1$, and since both of the multipliers $X_{C}$ and $X_{T}$ in the
central region satisfy the bounds $|X_{m_{B}}^{n}|\gtrsim(1+s)^{-1/2}$ along
the shocks we have in particular

$|\partial\psi^{K}_{C}|\lesssim\epsilon_{C}|X^{n}|,\qquad|K|\leq N_{C}/2+1.$
(9.137)

As a result, using $|\partial\psi^{K}_{C}|\lesssim\epsilon_{C}(1+s)^{-1/2}$
again we find

$\frac{1}{1+v}|\partial\psi^{K}_{C}|^{2}|n\psi^{J}_{C}|^{2}\lesssim\epsilon_{C}^{2}\frac{1}{(1+v)(1+s)^{1/2}}|X^{n}||n\psi^{J}_{C}|^{2}.$
(9.138)

Bounding $|\partial q|\lesssim|\ell^{m_{B}}q|+|nq|+|\not{\nabla}q|$ and using
the simpler estimates

$\displaystyle\frac{1}{1+v}|\partial\psi_{C}^{K}|^{2}|\ell^{m_{B}}\psi_{C}^{J}|^{2}\lesssim\epsilon_{C}^{2}|\ell^{m_{B}}\psi_{C}^{J}|^{2},\qquad|K|\leq
N_{C}/2+1$ (9.139)
$\displaystyle\frac{1}{1+v}|\partial\psi_{C}^{K}|^{2}|\not{\nabla}\psi_{C}^{J}|^{2}\lesssim\epsilon_{C}^{2}\frac{1}{(1+s)^{1/2}}|\not{\nabla}\psi_{C}^{J}|^{2},\qquad|K|\leq
N_{C}/2+1,$ (9.140)

we therefore have the needed bound,

$\sum_{|K|\leq N_{C}/2+1}\sum_{|J|\leq
N_{C}}\int_{t_{0}}^{t_{1}}\int_{\Gamma^{R}_{t}}\frac{1}{1+v}|\partial\psi_{C}^{K}|^{2}|\partial\psi_{C}^{I}|^{2}\,dSdt\\\
\lesssim\epsilon_{C}^{2}\sum_{|J|\leq
N_{C}}\int_{t_{0}}^{t_{1}}\int_{\Gamma^{R}_{t}}\left((1+v)|\ell^{m_{B}}\psi_{C}^{J}|^{2}+\frac{|X^{n}|}{(1+v)(1+s)^{1/2}}|n\psi_{C}^{J}|^{2}+\frac{1}{(1+s)^{1/2}}|\not{\nabla}\psi_{C}^{J}|^{2}\right)\,dSdt\\\
\lesssim\epsilon_{C}^{2}\sum_{|J|\leq
N_{C}}\int_{t_{0}}^{t_{1}}\int_{\Gamma^{R}_{t}}(1+v)|\ell^{m_{B}}\psi_{C}^{J}|^{2}\,dSdt+\epsilon_{C}^{4}\lesssim\epsilon_{C}^{4}.$
(9.141)

∎

We now prove the analogue of Lemma 9.7, where we commute $\ell^{m_{B}}$ with
$Z_{m_{B}}^{I}$.

######  Lemma 9.12.

Under the hypotheses of Proposition 9.2, we have

$B^{C}_{I}(t_{1})\lesssim\int_{t_{0}}^{t_{1}}\int_{\Gamma^{R}_{t}}(1+v)|Z^{I}_{m_{B}}Y_{R}^{-}\psi_{C}|^{2}\,dSdt+\sum_{|K|\leq|I|-1}B^{C}_{K}(t_{1})+(c_{0}(\epsilon_{0})+\epsilon_{C}^{2})\epsilon_{C}^{2}+c_{0}(\epsilon_{0})$
(9.142)

for a continuous function $c_{0}$ with $c_{0}(0)=0$.

###### Proof.

Recalling the definition of $Y_{R}^{-}$ from (6.13), along $\Gamma^{R}$ we
have

$\displaystyle|\ell^{m_{B}}Z^{I}_{m_{B}}\psi_{C}|$
$\displaystyle\lesssim|Z_{m_{B}}^{I}\ell^{m_{B}}\psi_{C}|+|[Z_{m_{B}}^{I},\ell^{m_{B}}]\psi_{C}|$
(9.143)
$\displaystyle\lesssim|Z_{m_{B}}^{I}Y_{L}^{-}\psi_{C}|+|Z^{I}_{m_{B}}\left((1+v)^{-1}Q(\partial\psi_{C},\partial\psi_{C})\right)|+|[Z_{m_{B}}^{I},\ell^{m_{B}}]\psi_{C}|$
(9.144)
$\displaystyle\lesssim|Z_{m_{B}}^{I}Y_{L}^{-}\psi_{C}|+|Z^{I}_{m_{B}}\left((1+v)^{-1}Q(\partial\psi_{C},\partial\psi_{C})\right)|$
(9.145)
$\displaystyle\qquad+\sum_{|J|\leq|I|-1}|\ell^{m_{B}}Z^{J}_{m_{B}}\psi_{C}|+\frac{1}{(1+v)(1+s)}|\partial
Z^{J}_{m_{B}}\psi_{C}|.$ (9.146)

By Lemma 9.11 and the definition of $B^{C}_{K}$, the contribution from the
terms on the first line here are bounded by the right-hand side of (9.142),
and the terms on the second line are easily handled after bounding

$\int_{t_{0}}^{t_{1}}\int_{\Gamma^{R}_{t}}(1+v)\left(\frac{1}{(1+v)(1+s)}|\partial
Z^{J}_{m_{B}}\psi_{C}|\right)^{2}\,dSdt\\\
\lesssim\int_{t_{0}}^{t_{1}}\int_{\Gamma^{R}_{t}}\frac{1}{1+v}\frac{1}{(1+s)^{2}}|\partial
Z^{J}_{m_{B}}\psi_{C}|^{2}\,dSdt\lesssim c_{0}(\epsilon_{0})\epsilon_{C}^{2},$
(9.147)

recalling the bounds from Lemma 7.7 for $\psi_{C}$ along the right shock. ∎

We now prove the analogue of Lemma 9.8.

######  Lemma 9.13.

Under the hypotheses of Proposition 9.2, we have

$\int_{t_{0}}^{t_{1}}\int_{\Gamma^{R}_{t}}(1+v)|Z_{m_{B}}^{I}Y_{R}^{-}\psi_{C}|^{2}\,dSdt\lesssim\Upsilon_{R}^{+}(t_{1})+\sum_{|K|\leq|I|-1}B_{K}^{C}(t_{1})+(c_{0}(\epsilon_{0})+\epsilon_{C}^{2})\epsilon_{C}^{2}+c_{0}(\epsilon_{0})$
(9.148)

for a continuous function $c_{0}$ with $c_{0}(0)=0$.

###### Proof.

By Lemma 9.5, we have the bound

$|Z_{m_{B}}^{I}Y_{R}^{-}\psi_{C}|\lesssim|Z_{m_{B},T}^{I}Y_{R}^{-}\psi_{C}|+C(M)(1+s)\sum_{|J|\leq|I|-1}|Z_{m_{B}}^{J}nY_{R}^{-}\psi_{C}|\\\
+C(M)(1+s)(1+(1+s)^{-1/2}|B^{R}|_{I,m_{B}})\sum_{|K|\leq|I|/2+1}|Z^{K}_{m_{B}}nY_{R}^{-}\psi_{C}|.$
(9.149)

Since the fields $Z_{m_{B},T}$ are tangent to the shock, by the boundary
condition (6.12) we have
$Z_{m_{B},T}^{I}Y_{R}^{-}\psi_{C}=Z_{m_{B},T}^{I}Y_{R}^{+}\psi_{R}+G$ at the
shock, so to conclude it is enough to prove that for $|I|\leq N_{C}$, we have
the following estimates,

$\int_{t_{0}}^{t_{1}}\int_{\Gamma^{R}_{t}}(1+v)(1+s)^{2}|Z^{J}_{m_{B}}nY_{R}^{-}\psi_{C}|^{2}\,dSdt\lesssim\sum_{|K|\leq|I|-1}B^{C}_{K}(t_{1})(c_{0}(\epsilon_{0})+\epsilon_{C}^{2})\epsilon_{C}^{2}\qquad|J|\leq|I|-1,$
(9.150)
$\int_{t_{0}}^{t_{1}}\int_{\Gamma^{R}_{t}}(1+v)|B^{R}|_{I,m_{B}}^{2}(1+s)|Z^{K}_{m_{B}}nY_{R}^{-}\psi_{C}|^{2}\,dSdt\\\
\lesssim\sum_{|K|\leq|I|-1}B^{C}_{K}(t_{1})+(c_{0}(\epsilon_{0})+\epsilon_{C}^{2})\epsilon_{C}^{2}\qquad|K|\leq|I|/2+1,$
(9.151)
$\int_{t_{0}}^{t_{1}}\int_{\Gamma^{R}_{t}}(1+v)|Z_{m_{B},T}^{I}G|\,dSdt\lesssim\sum_{|K|\leq|I|-1}B^{C}_{K}(t_{1})+(c_{0}(\epsilon_{0})+\epsilon_{C}^{2})\epsilon_{C}^{2}+c_{0}(\epsilon_{0})\qquad|K|\leq|I|/2+1.$
(9.152)

As in the proof of Lemma 9.8 we just prove the first two bounds here, the
bound for the remainder term $G$ being similar.

To prove (9.150), we recall the definition of $Y_{R}^{-}$ from (D.18) and use
that $(1+s)|\partial
q|\lesssim\sum_{Z_{m_{B}}\in\mathbb{Z}_{m_{B}}}|Z_{m_{B}}q|$ to gain an extra
power of $s$ and bound

$(1+s)|Z^{J}_{m_{B}}nY_{R}^{-}\psi_{C}|\lesssim(1+s)|Z^{J}_{m_{B}}n\ell^{m_{B}}\psi_{C}|+\sum_{|J^{\prime}|\leq|J|+1}|Z_{m_{B}}^{J^{\prime}}\left((1+v)^{-1}Q(\partial\psi_{C},\partial\psi_{C})\right)|.$
(9.153)

By (9.11), the term contributed from the second term here into (9.150)
satisfies the needed bound. For the first term here, we use the equation (6.8)
to bound

$(1+s)|Z^{J}_{m_{B}}n\ell^{m_{B}}\psi_{C}|\lesssim(1+s)|Z_{m_{B}}^{J}\not{\Delta}\psi_{C}|+(1+s)|Z_{m_{B}}^{J}\left(\partial_{\mu}(\tfrac{u}{vs}a^{\mu\nu}\partial_{\nu}\psi_{C})\right)|\\\
+\sum_{|J^{\prime}|\leq|J|+1}|Z_{m_{B}}^{J^{\prime}}\left((1+v)^{-1}Q(\partial\psi_{C},\partial\psi_{C})\right)|+(1+s)|Z^{J}F^{\prime}|.$
(9.154)

The quadratic term here can be handled by Lemma 9.11, the first term here can
be handled by writing $\not{\Delta}=\frac{1}{r^{2}}\Omega^{2}$ and making
straightforward estimates, and the last term as usual is easier to handle then
either of these terms. We will therefore just prove the bound for the term
involving $a$. Unlike in (9.105) where we did not need to worry about the
structure of this term, here we will need to use the fact that that term
verifies the null condition (3.6). This is because this term appears linearly
here and we have very weak control over the solution at top order along the
shocks, whereas in (9.105) we could afford to treat this term as an error term
because we only needed to consider lower-order derivatives and because we took
$\epsilon_{C}\leq\epsilon_{L}^{2}$.

Noting that
$a^{\mu\nu}\partial_{\mu}\partial_{\nu}\psi_{C}=a^{\mu\nu}\overline{\partial}_{\mu}\partial_{\nu}\psi_{C}$,
we bound

$(1+s)|Z_{m_{B}}^{I}\left(\tfrac{u}{vs}a^{\mu\nu}\partial_{\mu}\partial_{\nu}\psi_{C})\right)|\lesssim\frac{1+s}{1+v}\sum_{|J|\leq|I|+1}|\overline{\partial}Z_{m_{B}}^{I}\psi_{C}|+\frac{1+s}{(1+v)^{2}}\sum_{|J|\leq|I|+1}|\partial
Z_{m_{B}}^{I}\psi_{C}|,$ (9.155)

after using (A.64) to commute our fields with
$\overline{\partial}\in\\{\partial_{v},\not{\nabla}\\}$. As for the term where
the derivative falls on the factor $u/vs$, thanks to the null condition (3.6),
we can write

$\displaystyle
a^{\mu\nu}\left(\partial_{\mu}\frac{u}{vs}\right)\partial_{\nu}\psi_{C}$
$\displaystyle=a_{1}^{\mu\nu}\left(\overline{\partial}_{\mu}\frac{u}{vs}\right)\partial_{\nu}\psi_{C}+a_{2}^{\mu\nu}\left(\partial_{\mu}\frac{u}{vs}\right)\overline{\partial}_{\nu}\psi_{C}$
(9.156)
$\displaystyle=\frac{1+|u|}{(1+v)^{2}(1+s)}b_{1}^{\mu}\partial_{\mu}\psi_{C}+\frac{1}{(1+v)(1+s)}b_{2}^{\mu}\overline{\partial}_{\mu}\psi_{C},$
(9.157)

where the coefficients above satisfy the symbol condition (A.37). Since
$Z_{m_{B}}^{J}|u|\lesssim(1+s)$ for any $J$ it follows from this observation
and the fact that the $a^{\mu\nu}$ satisfy (A.37) that

$(1+s)|Z_{m_{B}}^{I}\left(a^{\mu\nu}\partial_{\mu}(\tfrac{u}{vs})\partial_{\nu}\psi_{C})\right)|\lesssim\frac{1+s}{(1+v)^{2}}\sum_{|I^{\prime}|\leq|I|}|\partial
Z_{m_{B}}^{I^{\prime}}\psi_{C}|+\frac{1}{1+v}\sum_{|I^{\prime}|\leq|I|}|\overline{\partial}Z_{m_{B}}^{I^{\prime}}\psi_{C}|.$
(9.158)

We note at this point that if we had not made use of the structure of $a$, for
the last term we would only have $(1+v)^{-1}|\partial
Z_{m_{B}}^{I^{\prime}}\psi_{C}|$, and the contribution from this term into
(9.150) would be too large for us to handle when $|I|=N_{C}$ since at top-
order we can only hope to control
$\int_{t_{0}}^{t_{1}}\int_{\Gamma^{R}_{t}}(1+v)^{-1}(1+s)^{-1}|\partial
Z_{m_{B}}^{I^{\prime}}\psi_{C}|^{2}$. Combining the last two bounds and using
that $a^{\mu\nu}$ satisfy the symbol condition (A.37), we find

$\int_{t_{0}}^{t_{1}}\int_{\Gamma^{R}_{t}}(1+v)(1+s)^{2}|Z_{m_{B}}^{I}\partial_{\mu}\left(\tfrac{u}{vs}a^{\mu\nu}\partial_{\nu}\psi_{C})\right)|^{2}\,dSdt\\\
\lesssim\sum_{|J|\leq|I|+1}\int_{t_{0}}^{t_{1}}\int_{\Gamma^{R}_{t}}\left(\frac{(1+s)^{2}}{1+v}(|\partial_{v}Z_{m_{B}}^{J}\psi_{C}|^{2}+|\not{\nabla}Z_{m_{B}}^{J}\psi_{C}|^{2})+\frac{(1+s)^{2}}{(1+v)^{3}}|\partial
Z_{m_{B}}^{J}\psi_{C}|^{2}\right)\,dSdt\\\ \lesssim
c_{0}(\epsilon_{0})\epsilon_{C}^{2},$ (9.159)

which is of the correct form for (9.150) for $|I|\leq N_{C}$.

We now move on to proving (9.151). For this we bound

$\int_{t_{0}}^{t_{1}}\int_{\Gamma^{R}_{t}}(1+v)|B^{R}|_{I,m_{B}}^{2}(1+s)|Z^{K}_{m_{B}}nY_{R}^{-}\psi_{C}|^{2}\,dSdt\\\
\lesssim\left(\sup_{t_{0}\leq t\leq
t_{1}}\int_{\Gamma^{R}_{t}}\frac{|B^{R}|_{I,m_{B}}^{2}}{1+s}\,dS\right)\left(\int_{t_{0}}^{t_{1}}\sup_{\Gamma^{R}_{t}}(1+v)(1+s)^{2}|Z^{K}_{m_{B}}nY_{R}^{-}\psi_{C}|^{2}\right),$
(9.160)

and using Sobolev embedding on $\Gamma^{R}_{t}$ to handle the second factor as
in the proof of Lemma 9.8 and using the above bound (9.150) again and the
bound (6.83) for $G^{R}_{N_{C}}$ (defined in(6.47)), we get the result. Note
that here we are able to close the estimate even though we only control a
relatively weak norm of $B^{R}$ at top order in light of the strong decay
estimates we have for $nY_{R}^{-}\psi_{C}$ at low order. ∎

We now prove the analogue of Proposition 9.3. This is fortunately much simpler
than that result since we have extremely good control over the solution on the
spacelike side of the right shock.

######  Lemma 9.14.

With $\Upsilon_{I,R}^{+}$ defined as in (9.133), under the hypotheses of
Proposition 9.2, we have

$\sum_{|I|\leq N_{C}}\Upsilon_{I,R}^{+}(t_{1})\lesssim\epsilon_{R}^{2}.$
(9.161)

###### Proof.

We will need to replace the vector fields $Z_{m_{B}}$ with the Minkowskian
fields $Z$. For this, we start with the observation that the fields
$Z_{m_{B}}$ and the fields $Z$ satisfy

$Z_{m_{B}}=\sum_{Z\in\mathcal{Z}_{m}}c_{Z_{m_{B}}}^{Z}\frac{1+s}{1+|u|}Z+d_{Z_{m_{B}}}^{Z}Z$
(9.162)

where the coefficients satisfy the symbol condition (A.37). This follows
easily from the well-known identity (A.17) which expresses
$\partial_{u},\partial_{v}$ in terms of the Minkowskian fields. Repeatedly
applying this formula and using basic properties of the fields $Z$ gives the
bound

$|Z_{m_{B}}^{I}q|\lesssim(1+s)^{|I|/2}\sum_{|J|\leq|I|}|Z^{J}q|,\qquad\text{
along }\Gamma^{R}.$ (9.163)

We now prove (9.161). We start by using Lemma 9.5 to convert the tangential
fields to the fields $Z_{m_{B}}$ at the right shock, which gives

$\displaystyle|Z_{m_{B},T}^{I}Y^{+}_{R}\psi_{R}|$
$\displaystyle\leq|Z_{m_{B}}^{I}Y^{+}_{R}\psi_{R}|+C(M)(1+s)\sum_{|J|\leq|I|-1}|Z_{m_{B}}^{J}nY^{+}_{R}\psi_{R}|$
(9.164)
$\displaystyle\qquad+C(M)(1+s)\left(1+(1+s)^{-1/2}|B^{R}|_{I,\mathcal{Z}_{m_{B}}}\right)\sum_{|L|\leq|I|/2+2}|Z_{m_{B}}^{L}nY^{+}_{R}\psi_{R}|$
(9.165) $\displaystyle\leq
C(1+s)^{|I|/2}\sum_{|J|\leq|I|}|Z^{J}Y^{+}_{R}\psi_{R}|+C(M)(1+s)^{|I|/2+1}\sum_{|J|\leq|I|-1}|Z^{J}nY^{+}_{R}\psi_{R}|$
(9.166)
$\displaystyle\qquad+C(M)(1+s)^{|I|/2+1}\left(1+(1+s)^{-1/2}|B^{R}|_{I,\mathcal{Z}_{m_{B}}}\right)\sum_{|L|\leq|I|/2+2}|Z^{L}nY^{+}_{R}\psi_{R}|,$
(9.167)

where we used (9.163) in the second step. We now handle these terms in the
usual way. Recalling the definition of $Y^{+}_{R}$ from (6.14), we first bound

$|Z^{J}Y^{+}_{R}\psi_{R}|\lesssim|{\ell^{m}}Z^{J}\psi_{R}|+|[{\ell^{m}},Z^{J}]\psi_{-}^{R}|+|Z^{J}\left((1+v)^{-1}Q(\partial\psi_{R},\partial\psi_{R})\right)|$
(9.168)

Inserting this into the right-hand side of (9.167), for $|I|\leq N_{C}$, the
contribution from the first term here into $\Upsilon_{I,R}^{+}$ is bounded by

$\sum_{|J|\leq|I|}\int_{t_{0}}^{t_{1}}\int_{\Gamma^{R}_{t}}(1+v)(1+s)^{|I|}|{\ell^{m}}Z^{J}\psi_{R}|^{2}\,dSdt\lesssim\sum_{|J|\leq|I|}\int_{t_{0}}^{t_{1}}\int_{\Gamma^{R}_{t}}r(\log
r)^{\nu}|{\ell^{m}}Z^{J}\psi_{R}|^{2}\,dSdt\lesssim\epsilon_{R}^{2},$ (9.169)

where we used that $r\sim v$ along $\Gamma^{R}$, the bound (6.77) for the
boundary term in the definition of the energy $\mathcal{E}_{R}$ in
(LABEL:ERdef), and the fact that by our choice of parameters (6.20), $|I|\leq
N_{C}\leq\nu$.

To handle the contribution from the nonlinear term in (9.168) into
$\Upsilon_{I,R}^{+}$, we bound

$\int_{t_{0}}^{t_{1}}\int_{\Gamma^{R}_{t}}(1+v)(1+s)^{|I|+2}|Z^{J}\left((1+v)^{-1}Q(\partial\psi_{R},\partial\psi_{R})\right)|^{2}\,dSdt\\\
\lesssim\sum_{|J^{\prime}|\leq|J|}\sum_{|K|\leq|J|/2+1}\int_{t_{0}}^{t_{1}}\int_{\Gamma^{R}_{t}}\frac{1}{1+v}(1+s)^{N_{C}+2}|\partial
Z^{J^{\prime}}\psi_{R}|^{2}|\partial Z^{K}\psi_{R}|^{2}\,dSdt\\\
\lesssim\sum_{|J^{\prime}|\leq|J|}\int_{t_{0}}^{t_{1}}\int_{\Gamma^{R}_{t}}\frac{1}{1+v}(1+s)^{N_{C}+2-\mu}|\partial
Z^{J^{\prime}}\psi_{R}|^{2}\,dSdt\lesssim\epsilon_{R}^{2},$ (9.170)

where we used that $N_{C}+2-\mu\leq\mu-1/2$ by our choice of $\mu$ in (6.20).
The contribution from $[Z^{J},{\ell^{m}}]$ from (9.168) into our estimates is
straightforward to handle using (A.21) so we skip it.

It remains only to handle the terms involving $nY_{R}+$ from (9.167). We just
show how to handle the term on the first line of (9.167) since the term on the
second line can be handled using the same idea. For both of these terms, the
idea is to write
$nY^{+}_{R}\psi_{R}=n{\ell^{m}}\psi_{R}+(1+v)^{-1}n(Q(\partial\psi_{R},\partial\psi_{R}))$,
and to handle the first term by using the equation (6.7) for $\psi_{R}$. This
gives

$\displaystyle\int_{t_{0}}^{t_{1}}\int_{\Gamma^{R}_{t}}(1+v)(1+s)^{|I|+2}$
$\displaystyle|Z^{J}n\ell\psi_{R}|^{2}\,dSdt$ (9.171)
$\displaystyle\quad\lesssim\int_{t_{0}}^{t_{1}}\int_{\Gamma^{R}_{t}}(1+v)(1+s)^{|I|+2}|Z^{J}\not{\Delta}\psi_{R}|^{2}\,dSdt$
(9.172)
$\displaystyle\quad+\int_{t_{0}}^{t_{1}}\int_{\Gamma^{R}_{t}}(1+v)(1+s)^{|I|+2}|Z^{J}\left((1+v)^{-1}Q(\partial\psi_{R},\partial\psi_{R})\right)|^{2}\,dSdt$
(9.173)
$\displaystyle\quad+\int_{t_{0}}^{t_{1}}\int_{\Gamma^{R}_{t}}(1+v)(1+s)^{|I|+2}|Z^{J}F|^{2}\,dSdt.$
(9.174)

As usual we skip the bounds for the last term. The second term here is bounded
exactly as in (9.170). The first term can be handled after writing
$\not{\Delta}=\frac{1}{r}\Omega\not{\nabla}$ and using straightforward
estimates along with the bounds (7.44) along the spacelike side of the right
shock. The nonlinear term contributed by using the above formula for
$nY^{+}_{R}$ can be handled exactly as in (9.170).

∎

##  10 The transport equation for the boundary-defining function

In the last three sections, we showed that provided the shocks
$\Gamma^{L},\Gamma^{R}$ were close to the model shocks (in the sense that
(6.85) holds, with $K^{R},K^{L}$ as in (6.49)-(6.51)), and provided that we
have bounds for high-order derivatives of the boundary-defining functions