$\psi$ in our region $\mathcal{R}(\tau_{0},\tau_{1})$ under consideration. In
the context of the proof of the main theorem, this will appear as a bootstrap
assumption. This will ensure that higher order nonlinear terms are indeed
absorbable.

We first explain some additional notation. We define the following summed
quantities for $\delta\leq p\leq 2-\delta$:

$\displaystyle\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}(\tau_{0},\tau_{1})$
$\displaystyle:=$
$\displaystyle\sup_{\tau^{\prime}\in[\tau_{0},\tau_{1}]}\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}(\tau^{\prime})+\sup_{v:\tau_{1}\leq
v(\tau)}\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{F}}}(v,\tau_{0},\tau_{1})+\int_{\tau_{0}}^{\tau_{1}}\,\,\accentset{\scalebox{0.6}{\mbox{\tiny$(p\mkern-6.0mu-\mkern-6.0mu1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}^{\prime}(\tau^{\prime})d\tau^{\prime},$
$\displaystyle{}^{\rho}\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}(\tau_{0},\tau_{1})$
$\displaystyle:=$
$\displaystyle\sup_{\tau^{\prime}\in[\tau_{0},\tau_{1}]}\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}(\tau^{\prime})+\sup_{v:\tau_{1}\leq
v(\tau)}\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{F}}}(v,\tau_{0},\tau_{1})+\int_{\tau_{0}}^{\tau_{1}}\left(\,\,{}^{\rho}\accentset{\scalebox{0.6}{\mbox{\tiny$(p\mkern-6.0mu-\mkern-6.0mu1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}^{\prime}(\tau^{\prime})+{}^{\tilde{\rho}}\,\accentset{\scalebox{0.6}{\mbox{\tiny$(-3\mkern-6.0mu-\mkern-6.0mu\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu1$}}}{\mathcal{E}}}^{\prime}(\tau^{\prime})\right)d\tau^{\prime},$
$\displaystyle{}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}(\tau_{0},\tau_{1})$
$\displaystyle:=$
$\displaystyle\sup_{\tau^{\prime}\in[\tau_{0},\tau_{1}]}\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}(\tau^{\prime})+\sup_{v:\tau_{1}\leq
v(\tau)}\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{F}}}(v,\tau_{0},\tau_{1})+\int_{\tau_{0}}^{\tau_{1}}\left(\,\,{}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(p\mkern-6.0mu-\mkern-6.0mu1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}^{\prime}(\tau^{\prime})+\accentset{\scalebox{0.6}{\mbox{\tiny$(p\mkern-6.0mu-\mkern-6.0mu1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu1$}}}{\mathcal{E}}}^{\prime}(\tau^{\prime})\right)d\tau^{\prime}.$

For $p=0$ we first define the analogous quantities, where $p-1$ is replaced by
$-1-\delta$:

$\displaystyle\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}(\tau_{0},\tau_{1})$
$\displaystyle:=$
$\displaystyle\sup_{\tau^{\prime}\in[\tau_{0},\tau_{1}]}\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}(\tau^{\prime})+\sup_{v:\tau_{1}\leq
v(\tau)}\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{F}}}(v,\tau_{0},\tau_{1})+\int_{\tau_{0}}^{\tau_{1}}\,\accentset{\scalebox{0.6}{\mbox{\tiny$(-1\mkern-6.0mu-\mkern-6.0mu\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}^{\prime}(\tau^{\prime})d\tau^{\prime},$
$\displaystyle{}^{\rho}\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}(\tau_{0},\tau_{1})$
$\displaystyle:=$
$\displaystyle\sup_{\tau^{\prime}\in[\tau_{0},\tau_{1}]}\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}(\tau^{\prime})+\sup_{v:\tau_{1}\leq
v(\tau)}\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{F}}}(v,\tau_{0},\tau_{1})+\int_{\tau_{0}}^{\tau_{1}}\left(\,\,{}^{\rho}\accentset{\scalebox{0.6}{\mbox{\tiny$(-1\mkern-6.0mu-\mkern-6.0mu\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}^{\prime}(\tau^{\prime})+{}^{\tilde{\rho}}\,\accentset{\scalebox{0.6}{\mbox{\tiny$(-3\mkern-6.0mu-\mkern-6.0mu\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu1$}}}{\mathcal{E}}}^{\prime}(\tau^{\prime})\right)d\tau^{\prime},$
$\displaystyle{}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}(\tau_{0},\tau_{1})$
$\displaystyle:=$
$\displaystyle\sup_{\tau^{\prime}\in[\tau_{0},\tau_{1}]}\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}(\tau^{\prime})+\sup_{v:\tau_{1}\leq
v(\tau)}\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{F}}}(v,\tau_{0},\tau_{1})+\int_{\tau_{0}}^{\tau_{1}}\left(\,\,{}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(-1\mkern-6.0mu-\mkern-6.0mu\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}^{\prime}(\tau^{\prime})+\accentset{\scalebox{0.6}{\mbox{\tiny$(-1\mkern-6.0mu-\mkern-6.0mu\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu1$}}}{\mathcal{E}}}^{\prime}(\tau^{\prime})\right)d\tau^{\prime}.$

Because $p=0$ is anomalous, however, we will need in addition the following
stronger energies which will appear _on the right hand side_ of $p=0$
estimates:

$\displaystyle\accentset{\scalebox{0.6}{\mbox{\tiny$(0+)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}(\tau_{0},\tau_{1})$
$\displaystyle:=$
$\displaystyle\sup_{\tau^{\prime}\in[\tau_{0},\tau_{1}]}\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}(\tau^{\prime})+\sup_{v:\tau_{1}\leq
v(\tau)}\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{F}}}(v,\tau_{0},\tau_{1})+\int_{\tau_{0}}^{\tau_{1}}\,\accentset{\scalebox{0.6}{\mbox{\tiny$(\delta-1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}^{\prime}(\tau^{\prime})d\tau^{\prime},$
$\displaystyle{}^{\rho}\accentset{\scalebox{0.6}{\mbox{\tiny$(0+)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}(\tau_{0},\tau_{1})$
$\displaystyle:=$
$\displaystyle\sup_{\tau^{\prime}\in[\tau_{0},\tau_{1}]}\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}(\tau^{\prime})+\sup_{v:\tau_{1}\leq
v(\tau)}\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{F}}}(v,\tau_{0},\tau_{1})+\int_{\tau_{0}}^{\tau_{1}}\left({}^{\rho}\accentset{\scalebox{0.6}{\mbox{\tiny$(\delta-1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}^{\prime}(\tau^{\prime})+{}^{\tilde{\rho}}\,\accentset{\scalebox{0.6}{\mbox{\tiny$(-3\mkern-6.0mu-\mkern-6.0mu\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu1$}}}{\mathcal{E}}}^{\prime}(\tau^{\prime})\right)d\tau^{\prime},$
$\displaystyle{}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(0+)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}(\tau_{0},\tau_{1})$
$\displaystyle:=$
$\displaystyle\sup_{\tau^{\prime}\in[\tau_{0},\tau_{1}]}\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}(\tau^{\prime})+\sup_{v:\tau_{1}\leq
v(\tau)}\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{F}}}(v,\tau_{0},\tau_{1})+\int_{\tau_{0}}^{\tau_{1}}\left({}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(\delta-1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}^{\prime}(\tau^{\prime})+\accentset{\scalebox{0.6}{\mbox{\tiny$(\delta-1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu1$}}}{\mathcal{E}}}^{\prime}(\tau^{\prime})\right)d\tau^{\prime}.$

Note the general properties that $p^{\prime}\geq p$, $k^{\prime}\geq k$
implies
$\accentset{\scalebox{0.6}{\mbox{\tiny$(p^{\prime})$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny{$k^{\prime}$}}}}{\mathcal{X}}}\gtrsim\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}$,
${}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(p^{\prime})$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny{$k^{\prime}$}}}}{\mathcal{X}}}\gtrsim{}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}$,
${}^{\rho}\accentset{\scalebox{0.6}{\mbox{\tiny$(p^{\prime})$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny{$k^{\prime}$}}}}{\mathcal{X}}}\gtrsim{}^{\rho}\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}$,
while

$\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu1$}}}{\mathcal{X}}}\lesssim{}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}.$
(4.22)

We will use the notation $\ll\mkern-6.0muk$ to denote some particular positive
integer, depending on $k$, which may be different on different instances of
our use of the notation, such that $\ll\mkern-6.0muk\leq k$, and in fact,
$\ll\mkern-6.0muk$ is “much less than $k$”, provided $k$ is sufficiently
large. In particular, for all positive integers $n$, we assume there exists a
$k(n)$ for which $k\geq k(n)$ implies $\ll\mkern-6.0muk\leq k-n$.

We have the following:

######  Proposition 4.5.1.

Let $k\geq 4$ be sufficiently large. There exists an $\varepsilon_{\rm
prelim}>0$ such that for all $0<\varepsilon\leq\varepsilon_{\rm prelim}$, the
following holds.

Let $\psi$ satisfy

$\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{X}}}\leq\varepsilon$
(4.23)

in $\mathcal{R}(\tau_{0},\tau_{1})$, for $p=0$ (or some $\delta\leq p\leq
2-\delta$), where $\ll\mkern-6.0muk\geq 4$. Then the following improved
version of (4.5) holds:

$\sup_{r\leq R}\sum_{|{\bf k}|\leq 1}(\widetilde{\mathfrak{D}}^{\bf
k}\psi)^{2}\lesssim\varepsilon\leq\sqrt{\varepsilon}$ (4.24)

###### Proof.

This follows of course immediately from the Sobolev inequality (3.111). ∎

In the context of the proof of the main theorem, inequality (4.23) will be
introduced as a bootstrap assumption, and Proposition 4.5.1 will be applied to
retrieve the assumption (4.5), necessary for the results of Sections 4.3–4.4.
Note that with the assumption (4.23), we may replace the boxed
$\varepsilon^{1/4}$ in (4.16) with $\sqrt{\varepsilon}$.

We note that in what follows, restrictions on $k$ sufficiently large will
always include the condition $\ll\mkern-6.0muk\geq 4$.

####  4.5.1 Comparability of $\mathfrak{E}$ and $\mathcal{E}$ energies

We note first the following:

######  Proposition 4.5.2.

Let $k\geq 4$ be sufficiently large. There exists an $\varepsilon_{\rm
prelim}>0$ such that the following holds.

Let $\psi$ be a solution of (4.1) in $\mathcal{R}(\tau_{0},\tau_{1})$
satisfying (4.23) for $p=0$ and some $0<\varepsilon\leq\varepsilon_{\rm
prelim}$. Then setting
$F=N^{\mu\nu}(\psi,x)\partial_{\mu}\psi\,\partial_{\nu}\psi+(\Box_{g_{0}}-\Box_{g(\psi,x)})\psi$,
we have

$\int_{\Sigma(\tau^{\prime})\cap\\{r\leq R\\}}\sum_{|{\bf k}|\leq
k-1}(\widetilde{\mathfrak{D}}^{\bf
k}F)^{2}\lesssim\min\left\\{\,\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{E}}}(\tau^{\prime})\,,\,\,\accentset{\scalebox{0.6}{\mbox{\tiny$(-1\mkern-6.0mu-\mkern-6.0mu\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{E}}}^{\prime}(\tau^{\prime})\right\\}\,\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}(\tau^{\prime})\lesssim\varepsilon\,\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}(\tau^{\prime}).$
(4.25)

###### Proof.

This is standard in view of our assumptions on $N$ and $g(\psi,x)$ from
Section 4.1, and can be proven using the Sobolev inequality (3.111) on
$\Sigma(\tau^{\prime})\cap\\{r\leq R\\}$. ∎

We may now obtain the following result, which can be viewed as a corollary of
Proposition 3.6.6.

######  Corollary 4.5.3.

Let $k\geq 4$ be sufficiently large. Then there exists an $\varepsilon_{\rm
prelim}>0$ such that the following statement holds.

Let $\psi$ be a solution of (4.1) in $\mathcal{R}(\tau_{0},\tau_{1})$
satisfying (4.23) for $p=0$ and some $0<\varepsilon\leq\varepsilon_{\rm
prelim}$. We have the analogue of Corollary 3.6.7: The calligraphic and
fraktur (when applicable) energies are comparable, i.e.

$\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}(\tau)\sim\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathfrak{E}}}(\tau)(g),$

for all $\tau_{0}\leq\tau\leq\tau_{1}$. Moreover, if $\chi=1$ and
$\tilde{\rho}=1$ identically (as in case (i)), then

$\accentset{\scalebox{0.6}{\mbox{\tiny$(p\mkern-6.0mu-\mkern-6.0mu1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}^{\prime}(\tau)\sim{}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(p\mkern-6.0mu-\mkern-6.0mu1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}^{\prime}(\tau)+{}^{\tilde{\rho}}\,\accentset{\scalebox{0.6}{\mbox{\tiny$(-3\mkern-6.0mu-\mkern-6.0mu\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu1$}}}{\mathcal{E}}}^{\prime}(\tau),\qquad\accentset{\scalebox{0.6}{\mbox{\tiny$(-1\mkern-6.0mu-\mkern-6.0mu\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}^{\prime}(\tau)\sim{}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(-1\mkern-6.0mu-\mkern-6.0mu\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}^{\prime}(\tau)+{}^{\tilde{\rho}}\,\accentset{\scalebox{0.6}{\mbox{\tiny$(-3\mkern-6.0mu-\mkern-6.0mu\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu1$}}}{\mathcal{E}}}^{\prime}(\tau)$

and thus

${}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}(\tau_{0},\tau_{1})\sim{}^{\rho}\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}(\tau_{0},\tau_{1})\sim\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}(\tau_{0},\tau_{1}).$

If $\tilde{\rho}=1$ identically, $\rho=\chi$, (i.e. as in cases (i) and (ii)),
then

${}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(p\mkern-6.0mu-\mkern-6.0mu1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}^{\prime}(\tau)+\accentset{\scalebox{0.6}{\mbox{\tiny$(p\mkern-6.0mu-\mkern-6.0mu1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu1$}}}{\mathcal{E}}}^{\prime}(\tau)\sim{}^{\rho}\accentset{\scalebox{0.6}{\mbox{\tiny$(p\mkern-6.0mu-\mkern-6.0mu1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}^{\prime}(\tau)+{}^{\tilde{\rho}}\accentset{\scalebox{0.6}{\mbox{\tiny$(-3\mkern-6.0mu-\mkern-6.0mu\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu1$}}}{\mathcal{E}}}^{\prime}(\tau),\qquad{}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(-1\mkern-6.0mu-\mkern-6.0mu\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}^{\prime}(\tau)+\accentset{\scalebox{0.6}{\mbox{\tiny$(-1\mkern-6.0mu-\mkern-6.0mu\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu1$}}}{\mathcal{E}}}^{\prime}(\tau)\sim{}^{\rho}\accentset{\scalebox{0.6}{\mbox{\tiny$(-1\mkern-6.0mu-\mkern-6.0mu\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}^{\prime}(\tau)+{}^{\tilde{\rho}}\accentset{\scalebox{0.6}{\mbox{\tiny$(-3\mkern-6.0mu-\mkern-6.0mu\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu1$}}}{\mathcal{E}}}^{\prime}(\tau)$

and thus

${}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}(\tau_{0},\tau_{1})\sim{}^{\rho}\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}(\tau_{0},\tau_{1}).$

###### Proof.

This follows from (3.103)–(3.107), with $F$ defined as in Proposition 4.5.2,
absorbing the inhomogeneous term using the estimate (4.25), for sufficiently
small $\varepsilon_{\rm prelim}$. ∎

###  4.6 Estimates on the nonlinear terms in the near region

######  Proposition 4.6.1.

Let $k\geq 4$ be sufficiently large. There exists an $\varepsilon_{\rm
prelim}>0$ such that the following holds.

Let $\psi$ be a solution of (4.1) in $\mathcal{R}(\tau_{0},\tau_{1})$
satisfying (4.23) for $p=0$ and some $0<\varepsilon\leq\varepsilon_{\rm
prelim}$. Then all integrals on lines (4.13)–(4.16), restricted to
$\mathcal{R}(\tau_{0},\tau_{1})\cap\\{r\leq R\\}$, may be estimated by

$\displaystyle\ldots$
$\displaystyle\lesssim\int_{\tau_{0}}^{\tau}\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}(\tau^{\prime})\sqrt{\,\,\accentset{\scalebox{0.6}{\mbox{\tiny$(-1\mkern-6.0mu-\mkern-6.0mu\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{E}}}^{\prime}(\tau^{\prime})\,\,}d\tau^{\prime},$
(4.26)

while the integrals on lines (4.18)–(4.19), again restricted to
$\mathcal{R}(\tau_{0},\tau_{1})\cap\\{r\leq R\\}$, may be similarly estimated
by

$\displaystyle\ldots$
$\displaystyle\lesssim\int_{\tau_{0}}^{\tau}\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}(\tau^{\prime})\sqrt{\,\,\accentset{\scalebox{0.6}{\mbox{\tiny$(-1\mkern-6.0mu-\mkern-6.0mu\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{E}}}^{\prime}(\tau^{\prime})\,\,}d\tau^{\prime}.$
(4.27)

###### Proof.

This is standard given the assumptions in Section 4.1 and can be proven using
the Sobolev inequality (3.111) on $\Sigma(\tau)\cap\\{r\leq R\\}$, cf. the
proof of (4.25) which in fact already bounds several of the terms. (Note that
the boxed $\varepsilon^{1/4}$ in (4.16) may clearly now be replaced by the
quantity
$\sqrt{\,\,\accentset{\scalebox{0.6}{\mbox{\tiny$(-1\mkern-6.0mu-\mkern-6.0mu\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{E}}}^{\prime}(\tau^{\prime})}$
within the integral.) ∎

We remark again that, as opposed to (4.26), the control (4.27) loses a
derivative, as the energy on the left hand side of (4.18) is $k-1$’th order
while the right hand side of (4.27) is of $k$’th order.

###  4.7 Assumptions for the nonlinearity in the far region: the “null”
condition

We may now state our specific assumption capturing the null condition for the
semilinear terms.

Rather than formulate algebraically the null condition in terms of the form of
$N$, for maximum generality we will make our basic assumption directly at the
level of an estimate for terms on the right hand side of the inequalities of
Proposition 4.4.1.

Our assumptions will only depend on the region $r\geq 8R/9$, so we will need
some notation to denote the restriction of energies, etc., to this region.
Given $r_{*}\geq r_{0}$ and $v$, by

$\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}_{r_{*},v},\qquad\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}_{r_{*},v},\qquad{}^{\rho}\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}_{r_{*},v},\qquad\ldots$

we shall mean the expressions defined as before, but where all domains of
integration are restricted to the region
$\mathcal{R}(\tau_{0},\tau_{1},v)\cap\\{r\geq r_{*}\\}$. We may thus define
these expressions for functions
$\psi:\mathcal{R}(\tau_{0},\tau_{1},v)\cap\\{r\geq r_{*}\\}\to\mathbb{R}$.
Similarly, the above expressions without the $v$ subscript, e.g.
$\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}_{r_{*}}$,
will be defined where all domains of intergation are restricted to
$\mathcal{R}(\tau_{0},\tau_{1})\cap\\{r\geq r_{*}\\}$. These can then be
defined for functions $\psi:\mathcal{R}(\tau_{0},\tau_{1})\cap\\{r\geq
r_{*}\\}\to\mathbb{R}$ . Let us note that

$r_{*}\geq
8R/9\implies\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}_{r_{*},v}={}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}_{r_{*},v}={}^{\rho}\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}_{r_{*},v}$
(4.28)

and similarly for the quantities without the $v$ subscript.

######  Assumption 4.7.1 (Null condition for semilinear terms).

There exists a $k_{\rm null}$, and for all $k\geq k_{\rm null}$, an
$\varepsilon_{\rm null}>0$, such that the following holds for all
$\tau_{0}\leq\tau_{1}$, $v$.

Let $\psi$ be a smooth function defined on
$\mathcal{R}(\tau_{0},\tau_{1},v)\cap\\{r\geq 8R/9\\}$ satisfying the bound

$\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{X}}}_{8R/9,v}\leq\varepsilon$
(4.29)

for some $0<\varepsilon\leq\varepsilon_{null}$. Then for all $\delta\leq p\leq
2-\delta$,

$\displaystyle\int_{\mathcal{R}(\tau_{0},\tau_{1},v)\cap\\{r\geq R\\}}$
$\displaystyle\sum_{|{\bf k}|\leq k}(|r^{p}r^{-1}L(r{\mathfrak{D}}^{\bf
k}\psi)|+|L{\mathfrak{D}}^{\bf k}|+|\underline{L}{\mathfrak{D}}^{\bf
k}\psi|+|r^{-1}\not{\nabla}{\mathfrak{D}}^{\bf
k}\psi|+|r^{-1}{\mathfrak{D}}^{\bf k}\psi|)|{\mathfrak{D}}^{\bf
k}(N^{\alpha\beta}(\psi,x)\partial_{\alpha}\psi\,\partial_{\beta}\psi)|$
$\displaystyle\qquad\qquad\qquad\qquad\qquad+(\mathfrak{D}^{\bf
k}(N^{\alpha\beta}(\psi,x)\partial_{\alpha}\psi\,\partial_{\beta}\psi))^{2}$
$\displaystyle\lesssim\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}_{\frac{8R}{9},v}(\tau_{0},\tau_{1})\sqrt{\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{X}}}_{\frac{8R}{9},v}(\tau_{0},\tau_{1})}+\sqrt{\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}_{\frac{8R}{9},v}(\tau_{0},\tau_{1})}\sqrt{\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}_{\frac{8R}{9},v}(\tau_{0},\tau_{1})}\sqrt{\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{X}}}_{\frac{8R}{9},v}(\tau_{0},\tau_{1})},$
(4.30)

while, corresponding to $p=0$ we have

$\displaystyle\int_{\mathcal{R}(\tau_{0},\tau_{1},v)\cap\\{r\geq R\\}}$
$\displaystyle\sum_{|{\bf k}|\leq k}(|r^{-1}L(r{\mathfrak{D}}^{\bf
k}\psi)|+|L{\mathfrak{D}}^{\bf k}|+|\underline{L}{\mathfrak{D}}^{\bf
k}\psi|+|r^{-1}\not{\nabla}{\mathfrak{D}}^{\bf
k}\psi|+|r^{-1}{\mathfrak{D}}^{\bf k}\psi|)|{\mathfrak{D}}^{\bf
k}(N^{\alpha\beta}(\psi,x)\partial_{\alpha}\psi\,\partial_{\beta}\psi)|$
$\displaystyle\qquad\qquad\qquad\qquad\qquad+(\mathfrak{D}^{\bf
k}(N^{\alpha\beta}(\psi,x)\partial_{\alpha}\psi\,\partial_{\beta}\psi))^{2}$
$\displaystyle\lesssim\accentset{\scalebox{0.6}{\mbox{\tiny$(0+)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}_{\frac{8R}{9},v}(\tau_{0},\tau_{1})\sqrt{\accentset{\scalebox{0.6}{\mbox{\tiny$(0+)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{X}}}_{\frac{8R}{9},v}(\tau_{0},\tau_{1})}.$
(4.31)

We note the following

######  Proposition 4.7.2.

Assumption 4.7.1 holds for equations (4.1) where $g_{0}$ is Minkowski and
where the semilinear terms $N$ satisfy the classical null condition of
Klainerman [Kla86], and more generally when $g_{0}$ is the Kerr metric and the
semilinear terms $N$ belong to the class considered by Luk [Luk13].

###### Proof.

See Appendix C. ∎

######  Remark 4.7.3.

Let us note that for $\psi$ defined on the slab
$\mathcal{R}(\tau_{1},\tau_{2})\cap\\{r\geq 8R/9\\}$, from the trivial bound
$\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}_{r_{*},v}\leq\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}_{r_{*}}$
we may drop the $v$ subscripts. For $\psi$ defined globally on the slab
$\mathcal{R}(\tau_{1},\tau_{2})$, then in view of (4.28) and the trivial
bounds
${}^{\rho}\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}_{r_{*}}\leq{}^{\rho}\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}$,
etc., we may replace the $\frac{8M}{9},v$ subscripts in all the
$\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}$,
etc., energies on the right hand side with the $\rho$ superscript, i.e.
replace
$\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}_{\frac{8M}{9},v}$,
etc., with
${}^{\rho}\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}$,
etc. Also, for such $\psi$, we may clearly replace (4.29) with assumption
(4.23).

###  4.8 The final estimates

Putting everything together we have the following:

######  Proposition 4.8.1.

Consider $(\mathcal{M},g_{0})$ satisfying the assumptions of Sections 2 and 3
(for cases (i), (ii) or (iii)) and equation (4.1) satisfying the assumptions
of Section 4.1 and 4.7.

Then for all $k\geq 4$ sufficiently large, there exist constants $C>0$ (also
implicit in the $\lesssim$ below), $c>0$ and an $\varepsilon_{\rm prelim}>0$,
such that the following is true.

Let $\tau_{0}\leq\tau_{1}$ and let $\psi$ be a solution of (4.1) in
$\mathcal{R}(\tau_{0},\tau_{1})$ satisfying (4.23) with $p=0$ for some
$0<\varepsilon\leq\varepsilon_{\rm prelim}$. Then for $\delta\leq p\leq
2-\delta$, one has the estimates

$\displaystyle\sup_{v:\tau\leq\tau(v)}\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\mathfrak{F}}(v,\tau_{0},\tau),$
$\displaystyle\quad
c\,{}^{\rho}\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}(\tau_{0},\tau_{1}),\quad\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathfrak{E}}}(\tau_{1})$
$\displaystyle\leq\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathfrak{E}}}(\tau_{0})+A\int_{\tau_{0}}^{\tau_{1}}\accentset{\scalebox{0.8}{\mbox{\tiny$\xi$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu1$}}}{\mathcal{E}}}^{\prime}(\tau^{\prime})d\tau^{\prime}$
$\displaystyle\qquad+C\,\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}_{\frac{8R}{9}}(\tau_{0},\tau_{1})\sqrt{\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{X}}}_{\frac{8R}{9}}(\tau_{0},\tau_{1})}+C\,\sqrt{\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}_{\frac{8R}{9}}(\tau_{0},\tau_{1})}\sqrt{\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}_{\frac{8R}{9}}(\tau_{0},\tau_{1})}\sqrt{\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{X}}}_{\frac{8R}{9}}(\tau_{0},\tau_{1})}$
$\displaystyle\qquad+C\int_{\tau_{0}}^{\tau_{1}}\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}(\tau^{\prime})\sqrt{\,\,\accentset{\scalebox{0.6}{\mbox{\tiny$(-1\mkern-6.0mu-\mkern-6.0mu\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{E}}}^{\prime}(\tau^{\prime})\,\,}d\tau^{\prime}$
(4.32)
$\displaystyle{}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu1$}}}{\mathcal{X}}}(\tau_{0},\tau_{1})$
$\displaystyle\lesssim\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu1$}}}{\mathcal{E}}}(\tau_{0})+\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu1$}}}{\mathcal{X}}}_{\frac{8R}{9}}(\tau_{0},\tau_{1})\sqrt{\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{X}}}_{\frac{8R}{9}}(\tau_{0},\tau_{1})}+\sqrt{\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu1$}}}{\mathcal{X}}}_{\frac{8R}{9}}(\tau_{0},\tau_{1})}\sqrt{\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu1$}}}{\mathcal{X}}}_{\frac{8R}{9}}(\tau_{0},\tau_{1})}\sqrt{\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{X}}}_{\frac{8R}{9}}(\tau_{0},\tau_{1})}$
$\displaystyle\qquad+\int_{\tau_{0}}^{\tau_{1}}\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}(\tau^{\prime})\sqrt{\,\,\accentset{\scalebox{0.6}{\mbox{\tiny$(-1\mkern-6.0mu-\mkern-6.0mu\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{E}}}^{\prime}(\tau^{\prime})\,\,}d\tau^{\prime}$
(4.33)

while for $p=0$, one has the estimates

$\displaystyle\sup_{v:\tau\leq\tau(v)}\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\mathfrak{F}}(v,\tau_{0},\tau),\quad$
$\displaystyle
c\,{}^{\rho}\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}(\tau_{0},\tau_{1}),\quad\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathfrak{E}}}(\tau_{1})\leq\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathfrak{E}}}(\tau_{0})+A\int_{\tau_{0}}^{\tau_{1}}\accentset{\scalebox{0.8}{\mbox{\tiny$\xi$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu1$}}}{\mathcal{E}}}^{\prime}(\tau^{\prime})d\tau^{\prime}+C\,\accentset{\scalebox{0.6}{\mbox{\tiny$(0+)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}_{\frac{8R}{9}}(\tau_{0},\tau_{1})\sqrt{\accentset{\scalebox{0.6}{\mbox{\tiny$(0+)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{X}}}_{\frac{8R}{9}}(\tau_{0},\tau_{1})}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad+C\int_{\tau_{0}}^{\tau_{1}}\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}(\tau^{\prime})\sqrt{\,\,\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{E}}}(\tau^{\prime})\,\,}d\tau^{\prime},$
(4.34)
$\displaystyle{}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu1$}}}{\mathcal{X}}}(\tau_{0},\tau_{1})$
$\displaystyle\lesssim\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu1$}}}{\mathcal{E}}}(\tau_{0})+\accentset{\scalebox{0.6}{\mbox{\tiny$(0+)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}_{\frac{8R}{9}}(\tau_{0},\tau_{1})\sqrt{\accentset{\scalebox{0.6}{\mbox{\tiny$(0+)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{X}}}_{\frac{8R}{9}}(\tau_{0},\tau_{1})}+\int_{\tau_{0}}^{\tau_{1}}\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}(\tau^{\prime})\sqrt{\,\,\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{E}}}(\tau^{\prime})\,\,}d\tau^{\prime}.$
(4.35)

Finally, in both the case $p=0$ and the case $\delta\leq p\leq 2-\delta$, one
has the alternative estimate

$\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}(\tau_{0},\tau_{1})\lesssim\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}(\tau_{0})+(1+\tau_{1}-\tau_{0})\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}(\tau_{0},\tau_{1})\sqrt{{}\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{X}}}(\tau_{0},\tau_{1})},$
(4.36)

depending on $\tau_{1}-\tau_{0}$.

###### Proof.

This follows from Proposition 4.4.1, Corollary 4.5.3, Proposition 4.6.1 and
Assumption 4.7.1. ∎

######  Remark 4.8.2.

Note how the case $p=0$ is anomalous in that one sees $0+$ on the right hand
side of (4.34)–(4.35). It is this fact which will require us to use $p>0$
weights to close the global estimates (4.32)–(4.33), even in the case (i). On
the other hand, (4.36) is sufficient to show local existence even using only
the $p=0$ weight.

######  Remark 4.8.3.

In using the above we shall always simply replace terms like
$\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}_{\frac{8R}{9}}(\tau_{0},\tau_{1})$,
etc., by
${}^{\rho}\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}(\tau_{0},\tau_{1})$,
etc., according to Remark 4.7.3. We have kept the terms in the above form
simply to highlight their origin in the estimate of Assumption 4.7.1.

######  Remark 4.8.4.

In the case where we only assume the black box inequality (3.3) and the
assumptions of asymptotic flatness of Sections 3.5 and 3.6.2 as interpreted in
Remark 3.6.9 (necessary to obtain (3.109)) and moreover, the equation (4.1) is
semilinear, i.e. $g(x,\psi)=g_{0}$ for all $x\in\mathcal{M}$, then we obtain
the statement of Proposition 4.8.1 without inequalities (4.32) and (4.34) and
where (4.33) and (4.35) are replaced by

$\displaystyle{}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}(\tau_{0},\tau_{1})$
$\displaystyle\lesssim\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}(\tau_{0})+\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}_{\frac{8R}{9}}(\tau_{0},\tau_{1})\sqrt{\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{X}}}_{\frac{8R}{9}}(\tau_{0},\tau_{1})}+\sqrt{\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}_{\frac{8R}{9}}(\tau_{0},\tau_{1})}\sqrt{\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}_{\frac{8R}{9}}(\tau_{0},\tau_{1})}\sqrt{\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{X}}}_{\frac{8R}{9}}(\tau_{0},\tau_{1})}$
$\displaystyle\qquad+\int_{\tau_{0}}^{\tau_{1}}\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}(\tau^{\prime})\sqrt{\,\,\accentset{\scalebox{0.6}{\mbox{\tiny$(-1\mkern-6.0mu-\mkern-6.0mu\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{E}}}^{\prime}(\tau^{\prime})\,\,}d\tau^{\prime},$
(4.37)
$\displaystyle{}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}(\tau_{0},\tau_{1})$
$\displaystyle\lesssim\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}(\tau_{0})+\accentset{\scalebox{0.6}{\mbox{\tiny$(0+)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}_{\frac{8R}{9}}(\tau_{0},\tau_{1})\sqrt{\accentset{\scalebox{0.6}{\mbox{\tiny$(0+)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{X}}}_{\frac{8R}{9}}(\tau_{0},\tau_{1})}+\int_{\tau_{0}}^{\tau_{1}}\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}(\tau^{\prime})\sqrt{\,\,\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{E}}}(\tau^{\prime})\,\,}d\tau^{\prime},$
(4.38)

respectively. (In particular, equation (4.36) also still holds as stated.)

###  4.9 Local well posedness

Finally, we give a local well posedness statement and an extension criterion.

Since our initial data hypersurfaces $\Sigma(\tau_{0})$ are in part null, it
will be convenient to assume that our initial data are smooth. We define a
smooth initial data set on $\Sigma(\tau_{0})$ to be a pair
$(\uppsi,\uppsi^{\prime})$ where $\uppsi$ is a function on $\Sigma(\tau_{0})$
smooth on $\Sigma(\tau_{0})\cap\\{r\leq R\\}$ and smooth on
$\Sigma(\tau_{0})\cap\\{r\geq R\\}$, and $\uppsi^{\prime}$ is a smooth
function on $\Sigma(\tau_{0})\cap\\{r\leq R\\}$, such that moreover there
exists a smooth a function $\Psi$ on $\mathcal{R}(\tau_{0},\tau_{1})$ for some
$\tau_{1}$ such that $\Psi|_{\Sigma(\tau_{0})}=\uppsi$, and
$n\Psi|_{\Sigma(\tau_{0})\cap\\{r\leq R\\}}=\uppsi^{\prime}$, where $n$
denotes the normal to $\Sigma(\tau_{0})\cap\\{r\leq R\\}$.

Note that given such initial data on $\Sigma(\tau_{0})$ one may define

$\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}[\uppsi,\uppsi^{\prime}]$
(4.39)

as follows:

Using the equation (4.1) we may compute along $\Sigma(\tau_{0})$ the $k+1$-jet
of any smooth solution $\psi$ of (4.1) such that

$\psi|_{\Sigma(\tau_{0})}=\uppsi,\qquad n\psi|_{\Sigma(\tau_{0})\cap\\{r\leq
R\\}}=\uppsi^{\prime}.$ (4.40)

We may define
$\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}[\uppsi,\uppsi^{\prime}]$
to equal the usual
$\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}(\tau_{0})$
where the derivatives are interpreted in terms of this formal computation.
Note of course that the expression (4.39) may well be infinite.

Alternatively, we may express this as follows. Using the above computation, we
may in fact define a smooth $\Psi$ as in the previous paragraph, such that
$\Psi$ moreover satisfies (4.1) along $\Sigma(\tau_{0})$, and, for all $|{\bf
k}|\leq k$, $\widetilde{\mathfrak{D}}^{\bf k}\Psi$ satisfies along
$\Sigma(\tau_{0})$ the equation one obtains by commuting (4.1) by
$\widetilde{\mathfrak{D}}^{\bf k}$. We may then simply define (4.39)
substituting this $\Psi$ for $\psi$ in the usual expression for
$\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}(\tau_{0})$.

With this, we can now state our local well posedness:

######  Proposition 4.9.1 (Local well posedness).

Consider $(\mathcal{M},g_{0})$ satisfying the assumptions of Sections 2 and 3
(for cases (i), (ii) or (iii)) and equation (4.1) satisfying the assumptions
of Section 4.1 and 4.7. Fix either $p=0$ or $\delta\leq p\leq 2-\delta$.

There exists a positive integer $k_{\rm loc}\geq 4$ such that the following
holds. Let $k\geq k_{\rm loc}$. Then there exists a positive real constant
$C>0$ sufficiently large, a positive real parameter $\varepsilon_{\rm loc}>0$
sufficiently small, and a decreasing positive function $\tau_{\rm
exist}:(0,\varepsilon_{\rm loc})\to\mathbb{R}$ such that for all smooth
initial data $(\uppsi,\uppsi^{\prime})$ on $\Sigma(\tau_{0})$ such that

$\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}[\uppsi,\uppsi^{\prime}]\leq\varepsilon_{0}\leq\varepsilon_{\rm
loc},$

there exists a smooth solution $\psi$ of (4.1) in
$\mathcal{R}(\tau_{0},\tau_{1})$ for $\tau_{1}:=\tau_{0}+\tau_{\rm exist}$
which satisfies

$\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}(\tau_{0},\tau_{1})\leq
C\varepsilon_{0}.$

Moreover, for all $\tau_{0}\leq\tau\leq\tau_{1}$, any other smooth
$\widetilde{\psi}$ defined on $\mathcal{R}(\tau_{0},\tau)$ satisfying (4.1) in
$\mathcal{R}(\tau_{0},\tau)$ and attaining the initial data, i.e. satisfying
(4.40) (with $\widetilde{\psi}$ replacing $\psi$), coincides with the
restriction of $\psi$, i.e.
$\widetilde{\psi}=\psi|_{\mathcal{R}(\tau_{0},\tau)}$.

We have in addition the following propagation of higher order regularity
and/or higher weighted estimates. Given $p$, $k\geq k_{\rm loc}$,
$\varepsilon_{0}\leq\varepsilon_{\rm loc}$, $\tau_{0}$, $\tau_{1}$ as above,
$2-\delta\geq p^{\prime}\geq p$ if $p\geq\delta$ and either $2-\delta\geq
p^{\prime}\geq\delta$ or $p^{\prime}=0$ if $p=0$, and $k^{\prime}\geq k$. Then
there exists a constant $C(k^{\prime},\tau_{1}-\tau_{0})$ such that given
$\psi$ as above,

$\accentset{\scalebox{0.6}{\mbox{\tiny$(p^{\prime})$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny{$k^{\prime}$}}}}{\mathcal{X}}}_{r^{\prime}_{*}}(\tau_{0},\tau_{1})\leq
C(k^{\prime},\tau_{1}-\tau_{0})\,\accentset{\scalebox{0.6}{\mbox{\tiny$(p^{\prime})$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny{$k^{\prime}$}}}}{\mathcal{E}}}_{r^{\prime}_{*}}(\tau_{0}).$

###### Proof.

This can be easily proven using estimate (4.36). We leave the details to the
reader. ∎

Note also the following easy corollary, which we will use as an extension
criterion:

######  Corollary 4.9.2 (Continuation criterion).

Fix $p=0$ or $\delta\leq p\leq 2-\delta$, and let $k\geq k_{\rm loc}$ and
$\varepsilon_{\rm loc}$ be as in Proposition 4.9.1. There exists a constant
$C>0$ and an $\epsilon>0$ such that the following is true.

Let $\tau_{f}>\tilde{\tau}_{0}$ and suppose that there exists a smooth
solution $\psi$ of (4.1) on
$\cup_{\tilde{\tau}_{0}<\tau<\tau_{f}}\mathcal{R}(\tilde{\tau}_{0},\tau)$ such
that

$\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}(\tau)\leq\varepsilon_{\rm
loc}$

for $k\geq k_{\rm loc}$ and all $\tilde{\tau}_{0}\leq\tau<\tau_{f}$. Then,
defining $\tau_{1}:=\tau_{f}+\epsilon$, $\psi$ extends uniquely to a smooth
function on $\mathcal{R}(\tilde{\tau}_{0},\tau_{1})$ satisfying the equation
(4.1) on this set and, setting
$\tau_{0}:=\max\\{\tau_{f}-\epsilon,\tilde{\tau}_{0}\\}$, satisfying the
estimate

$\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}(\tau_{0},\tau_{1})\leq
C\varepsilon_{\rm loc}.$ (4.41)

Moreover, for all $k^{\prime}\geq k$ and all $2-\delta\geq p^{\prime}\geq p$,
there exist constants $C(k^{\prime})$ such that

$\accentset{\scalebox{0.6}{\mbox{\tiny$(p^{\prime})$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny{$k^{\prime}$}}}}{\mathcal{X}}}(\tau_{0},\tau_{1})\leq
C(k^{\prime})\,\accentset{\scalebox{0.6}{\mbox{\tiny$(p^{\prime})$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny{$k^{\prime}$}}}}{\mathcal{E}}}(\tau_{0}).$
(4.42)

######  Remark 4.9.3.

Recall that by our conventions the constants $C$, $k_{\rm loc}$,
$\varepsilon_{\rm loc}$ depend in addition on $k$. We may assume that $k_{\rm
loc}$ is sufficiently large so that $k\geq k_{\rm loc}$ satisfies the
largeness constraint of all previous propositions in this Section, and that
$\varepsilon_{\rm loc}\leq\varepsilon_{\rm prelim}$.

######  Remark 4.9.4.

In view of Remark 4.8.4, and the fact that one needs only (4.36), all the
statements of this section also hold in the semilinear case, under the relaxed
assumptions described there.

##  5 The main theorem: global existence and stability

We may now state the main result of this paper.

######  Theorem 5.1 (Global existence and stability).

Consider $(\mathcal{M},g_{0})$ satisfying the assumptions of Sections 2 and 3
(for cases (i), (ii) or (iii)) and equation (4.1) satisfying the assumptions
of Section 4.1 and 4.7.

There exists a positive integer $k_{\rm global}\geq k_{\rm local}$
sufficiently large, such that, given $k\geq k_{\rm global}$, there exists a
positive $0<\varepsilon_{\rm global}<\varepsilon_{\rm loc}$ sufficiently
small, and a positive constant $C>0$ sufficiently large, so that the following
holds.

Fix $\tau_{0}=1$ and consider as in Proposition 4.9.1 initial data
$(\uppsi,\uppsi^{\prime})$ on $\Sigma(\tau_{0})$ for (4.1) satisfying

$\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}[\uppsi,\uppsi^{\prime}]\leq\varepsilon_{0}\leq\varepsilon_{\rm
global},\qquad\accentset{\scalebox{0.6}{\mbox{\tiny$(1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}[\uppsi,\uppsi^{\prime}]+\accentset{\scalebox{0.6}{\mbox{\tiny$(2-\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu1$}}}{\mathcal{E}}}[\uppsi,\uppsi^{\prime}]\leq\varepsilon_{0}\leq\varepsilon_{\rm
global},\qquad\accentset{\scalebox{0.6}{\mbox{\tiny$(1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}[\uppsi,\uppsi^{\prime}]+\accentset{\scalebox{0.6}{\mbox{\tiny$(2-\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu2$}}}{\mathcal{E}}}[\uppsi,\uppsi^{\prime}]\leq\varepsilon_{0}\leq\varepsilon_{\rm
global},$

according to case (i), (ii) and (iii) respectively, where in the former case
$\delta\leq p\leq 2-\delta$. Then the $\psi$ given by Proposition 4.9.1
extends to a unique globally defined solution of (4.1) in
$\mathcal{R}(\tau_{0},\infty)$, satisfying the estimates

${}^{\rho}\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}(\tau_{0},\tau)\leq
C\varepsilon_{0},\qquad{}^{\rho}\accentset{\scalebox{0.6}{\mbox{\tiny$(1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}(\tau_{0},\tau)+{}^{\rho}\accentset{\scalebox{0.6}{\mbox{\tiny$(2-\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu1$}}}{\mathcal{X}}}(\tau_{0},\tau)\leq
C\varepsilon_{0},\qquad{}^{\rho}\accentset{\scalebox{0.6}{\mbox{\tiny$(1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}(\tau_{0},\tau)+{}^{\rho}\accentset{\scalebox{0.6}{\mbox{\tiny$(2-\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu2$}}}{\mathcal{X}}}(\tau_{0},\tau)\leq
C\varepsilon_{0},$ (5.1)

according to cases (i), (ii), (iii), respectively, for all $\tau\geq\tau_{0}$,
in particular

$\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}(\tau)\leq
C\varepsilon_{0},\qquad\accentset{\scalebox{0.6}{\mbox{\tiny$(1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}(\tau)+\accentset{\scalebox{0.6}{\mbox{\tiny$(2-\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu1$}}}{\mathcal{E}}}(\tau)\leq
C\varepsilon_{0},\qquad\accentset{\scalebox{0.6}{\mbox{\tiny$(1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}(\tau)+\accentset{\scalebox{0.6}{\mbox{\tiny$(2-\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu2$}}}{\mathcal{E}}}(\tau)\leq
C\varepsilon_{0},$

respectively.

The solution will satisfy moreover estimates (6.8) in case (i), estimates
(6.62) and (6.64)–(6.68) in case (ii) and estimates (6.116)–(6.121) in case
(iii).

We shall give the proof of this theorem in Section 6.

######  Remark 5.1.

Our theorem also applies under the relaxed assumptions of Remark 4.8.4, where
(4.1) is however required to be semilinear. Here we can again distinguish case
(i) where $\chi=1$ and case (ii) where $\chi$ is allowed to degenerate. In
case (i), the theorem holds as stated except for (6.8), which no longer
applies. Similarly, in case (ii), the theorem holds as stated except for
(6.62) which no longer applies. The necessary modifications will be collected
in a series of remarks in Sections 6.1 and 6.2. (See already Section 6.2.5 for
the completion of case (ii).)

We reiterate (cf. Propositions 4.7.2 and Theorem A.1) that the above theorem
in particular holds for equations (4.1) on the Kerr spacetime
$(\mathcal{M},g_{a,M})$ for $|a|\ll M$, with quasilinear terms as described in
Section 4.1 and the general semilinear terms considered in [Luk13], and, by
the above remark, restricted to the purely semilinear case, for the full range
of parameters $|a|<M$.

##  6 The estimate hierarchies and global existence and decay

We proceed with the proof of Theorem 5.1 in cases (i), (ii) and (iii). These
will be treated in Sections 6.1, 6.2 and 6.3 below, respectively.

Note that we will now suppress some of the cumbersome notation with the
following conventions:

In this section, the fraktur energies $\mathfrak{E}$, $\mathfrak{F}$ will
everywhere denote $\mathfrak{E}(g)$, $\mathfrak{F}(g)$, so we will drop
explicit reference to $g$ here. When the region
$\mathcal{R}(\tau_{0},\tau_{1})$ is assumed, we will omit the $\tau_{0}$ or
$(\tau_{0},\tau_{1})$ arguments in various energies, etc., and denote
$\mathcal{X}(\tau_{0},\tau_{1})$ by $\mathcal{X}$.

The smallness parameter $\varepsilon_{\rm global}$ will be determined in the
context of the proof. We will in fact introduce other parameters on the way;
these will be in the relation

$\varepsilon_{\rm global}\leq\hat{\varepsilon}_{\rm slab}\leq\varepsilon_{\rm
slab}\leq\varepsilon_{\rm boot}\leq\varepsilon_{\rm loc},$ (6.1)

where parameters $\hat{\varepsilon}_{\rm slab}$, $\varepsilon_{\rm slab}$ will
only appear for cases (ii) and (iii). The parameter $\varepsilon_{\rm boot}$
will be used in the context of a continuity or bootstrap argument. Let us
remark already that if the basic bootstrap estimate (4.23) holds for an
$\varepsilon\leq\varepsilon_{\rm boot}$, then by Remark 4.9.3, $\varepsilon$
will satisfy the smallness requirements of all propositions of Section 4, for
the regions under consideration.

###  6.1 Case (i)

Case (i) is the most elementary case, where, moreover only the minimal
nontrivial $r^{p}$ weights for $2-\delta\geq p\geq\delta$ are required on
data. (For instance, one may fix $p=\delta$.) We present first the fundamental
estimate in Section 6.1.1 that holds under the bootstrap assumption (4.23) and
then carry out the global existence proof in Section 6.1.2.

The proof will also hold for the semilinear case with the relaxed assumptions
of Remark 4.8.4, if we are in the analogue of case (i), i.e. if $\chi=1$. The
modifications necessary to treat this case are described in a series of
remarks (see already Remarks 6.1.2 and 6.1.3).

####  6.1.1 The fundamental estimate

The fundamental estimate is given simply by the following:

######  Proposition 6.1.1.

Let $\delta\leq p\leq 2-\delta$ and let $k$ be sufficiently large, and let us
assume the case (i) assumptions. There exists an $\varepsilon_{\rm boot}>0$
sufficiently small so that the following is true.

Let $\psi$ solve (4.1) on $\mathcal{R}(\tau_{0},\tau_{1})$, satisfying
moreover (4.23) with our chosen $p$ and with
$0<\varepsilon\leq\varepsilon_{\rm boot}$. Then we have:

$\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}\lesssim\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}(\tau_{0}),\qquad\sup_{\tau_{0}\leq\tau\leq\tau_{1}}\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathfrak{E}}}(\tau)\leq\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathfrak{E}}}(\tau_{0})(1+C\varepsilon^{1/2}).$
(6.2)

###### Proof.

Let $\varepsilon_{\rm boot}\leq\varepsilon_{\rm loc}$. The assumption of
Proposition 4.8.1 is satisfied in view of our discussion following (6.1). From
estimate (4.32) of Proposition 4.8.1, since in case (i) we have $\rho=1$,
$A=0$, $\tilde{A}=0$, it follows that

$\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}\lesssim\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}(\tau_{0})+\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}\sqrt{\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{X}}}},\qquad\sup_{\tau_{0}\leq\tau\leq\tau_{1}}\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathfrak{E}}}(\tau)\leq\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathfrak{E}}}(\tau_{0})+C\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}\sqrt{\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{X}}}}.$
(6.3)

By our bootstrap assumption (4.23), we have

$\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{X}}}\leq\varepsilon.$
(6.4)

Thus, possibly requiring $\varepsilon_{\rm boot}$ to be even smaller, we may
absorb the nonlinear term in the first inequality of (6.3) to obtain the first
inequality of (6.2).

To obtain the second inequality of (6.2), we remark that Corollary 4.5.3
applies to yield in particular

$\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}(\tau_{0})\sim\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathfrak{E}}}(\tau_{0}).$
(6.5)

The second inequality of (6.2) now immediately follows from the the second
inequality of (6.3), plugging in the first inequality of (6.2) just
established, (6.5) and (6.4). ∎

######  Remark 6.1.2.

Under the relaxed assumptions of Remark 4.8.4, where (4.1) is however required
to be semilinear and we are in the analogue of case (i), i.e. $\chi=1$, then
from (4.38) we obtain the first inequality of (6.3). Thus, the above
proposition holds as stated for the first inequality of (6.2).

####  6.1.2 Proof of Theorem 5.1 in case (i)

We now carry out the proof of Theorem 5.1 proper in case (i).

Let $\delta\leq p\leq 2-\delta$ be as in the statement and recall the
assumption

$\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}(\tau_{0})\leq\varepsilon_{0}\leq\varepsilon_{\rm
global}$

on initial data, for a sufficiently small $\varepsilon_{\rm global}$ to be
determined.

Consider the set $\mathfrak{B}$ consisting of all $\tau_{f}>\tau_{0}$ such
that a solution $\psi$ of (4.1) obtaining the data exists on
$\mathcal{R}(\tau_{0},\tau_{f})$, and such that moreover the energy bootstrap
assumption (4.23) holds on $\mathcal{R}(\tau_{0},\tau_{1}:=\tau_{f})$ with our
chosen $p$ and where $0<\varepsilon\leq\varepsilon_{\rm boot}$ is chosen to
satisfy

$1\gg\varepsilon\gg\varepsilon_{\rm global}.$ (6.6)

(The above relation in particular constrains $\varepsilon_{\rm global}$ to be
small.) By the local well posedness statement Proposition 4.9.1, it follows
that since $k\geq k_{\rm loc}$ and $\varepsilon_{0}\leq\varepsilon_{\rm
global}\leq\varepsilon_{\rm loc}$, we have that $\tau_{0}+\tau_{\rm
exist}\in\mathfrak{B}$ and thus $\mathfrak{B}\neq\emptyset$. Also note that a
fortiori, if $\tau_{f}\in\mathfrak{B}$, then
$(\tau_{0},\tau_{f}]\subset\mathfrak{B}$ and thus $\mathfrak{B}$ is manifestly
a connected subset of $(\tau_{0},\infty)$.

For any $\tau_{1}\in\mathfrak{B}$, Proposition 6.1.1 applies in
$\mathcal{R}(\tau_{0},\tau_{1})$. The first estimate of (6.2) then yields

$\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}\lesssim\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}(\tau_{0})\lesssim\varepsilon_{0}.$
(6.7)

If follows that for sufficiently small $\varepsilon_{\rm glob}$, and all
$\varepsilon_{0}\leq\varepsilon_{\rm glob}$, the continuation criterion
Corollary 4.9.2 applies to obtain that $\psi$ extends as a solution to (4.1)
on some $\mathcal{R}(\tau_{0},\tau_{f}+\epsilon)$, where $\epsilon$ is
independent of $\tau_{f}$, and that, considering now
$\tau_{1}:=\tau_{f}+\epsilon$, inequality (6.7) holds also in
$\mathcal{R}(\tau_{0},\tau_{1}:=\tau_{f}+\epsilon)$ (with a slightly different
implicit constant than that of (6.7)). (To infer this from (4.41), which was
an estimate on
$\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}(\tau_{f},\tau_{f}+\epsilon)$,
note that
$\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}(\tau_{0},\tau_{1})\lesssim\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}(\tau_{0},\tau_{f})+\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}(\tau_{f},\tau_{f}+\epsilon)$.)

Finally we note that for $\varepsilon$ satisfying (6.6), inequality (6.7) in
$\mathcal{R}(\tau_{0},\tau_{1}:=\tau_{f}+\epsilon)$ shows in particular that
inequalities (4.23) hold in this set. It follows by the definition of
$\mathfrak{B}$ that $\tau_{f}+\epsilon\in\mathfrak{B}$ and thus, given also
its connectedness, the set $\mathfrak{B}$ is open. Since $\epsilon$ does not
depend on $\tau_{f}$, it follows that $\mathfrak{B}$ is also closed, as any
limit point $\tau_{\rm limit}$ of $\mathfrak{B}$ in $(\tau_{0},\infty)$
satisfies $\tau_{\rm limit}\leq\tau_{f}+\epsilon$ for some
$\tau_{f}\in\mathfrak{B}$. We have shown that $\mathfrak{B}$ is a non-empty
open and closed subset of $(\tau_{0},\infty)$ and thus,
$\mathfrak{B}=(\tau_{0},\infty)$. Hence the solution $\psi$ exists globally in
$\mathcal{R}(\tau_{0},\infty)$ and satisfies (6.7) in
$\mathcal{R}(\tau_{0},\tau_{1})$ where $\tau_{1}$ is now any
$\tau_{1}>\tau_{0}$. This gives (5.1).

Revisiting (6.2) we see finally that we have the following more precise global
estimate:

$\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathfrak{E}}}(\tau)\leq\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathfrak{E}}}(\tau_{0})(1+C\varepsilon_{0}^{1/2}).$
(6.8)

This completes the proof.

######  Remark 6.1.3.

Note that in view of Remark 6.1.2, the above proof goes through, except for
the last paragraph involving (6.8), also under the relaxed assumptions of
Remark 4.8.4, where (4.1) is however required to be semilinear and we are in
the analogue of case (i), i.e. $\chi=1$. Thus, in this case, one indeed
obtains the statement as given in Remark 5.1.

###  6.2 Case (ii)

Case (ii) is the second simplified case which we shall consider. Unlike case
(i), we shall require proving actual $\tau$-decay above a certain threshold in
order to close, and this in turn will require raising $p$ to $p=2-\delta$ in
our initial energy assumptions.

The logic of the proof is a simplified version of the scheme we shall use for
the general case and which has already been summarised in the introduction:

  1. 1.

Rather than bootstrap directly $t$-weighted estimates, we formulate the
fundamental estimates, depending only on the bootstrap assumption (4.23), as a
hierarchy of estimates on a spacetime slab of $\tau$-length at most $L$. This
is the content of Section 6.2.1.

  2. 2.

We shall then show by a bootstrap argument, _restricted to such a slab_ , how
global existence and estimates on the slab can be proven, with suitable
assumptions on the initial data of the slab, which now involve $L$. This is
the content of Section 6.2.2.

  3. 3.

Still restricted to a given slab of length $L$, we will show that one can in
fact a posteriori improve the above estimates under suitable additional
assumptions on the initial data, and, using a pigeonhole argument, show
moreover an improved estimate for any $\Sigma(\tau^{\prime})$ slice near the
top of the slab. This is the content of Section 6.2.3.

  4. 4.

Finally, global existence and $\tau$ decay now follow by iterating the above
estimates on a consecutive sequence of spacetime slabs of dyadic time-length
$L_{i}=2^{i}$. This is the content of Section 6.2.4.

We shall in addition give an alternative iteration proof in Section 6.2.5,
which does not use the exact boundedness statement of the fraktur energies.
The advantage of this alternative proof is that it allows us to treat the
semilinear case with the relaxed assumptions of Remark 4.8.4. The
modifications necessary to treat this case are described in a series of
remarks (see already Remarks 6.2.2, 6.2.4, 6.2.7 and 6.2.8).

####  6.2.1 The hierarchy of inequalities

######  Proposition 6.2.1.

Let $k$ be sufficiently large and let us assume the case (ii) assumptions.
There exist constants $C>0$, $c>0$ and an $\varepsilon_{\rm boot}>0$
sufficiently small such that the following is true.

Consider a region $\mathcal{R}(\tau_{0},\tau_{1})$ and a $\psi$ solving (4.1)
on $\mathcal{R}(\tau_{0},\tau_{1})$, satisfying moreover (4.23) with $p=0$ and
with $0<\varepsilon\leq\varepsilon_{\rm boot}$. Let us assume moreover that

$\tau_{1}\leq\tau_{0}+L$

for some arbitrary $L>0$. We have the following hierarchy of inequalities on
$\mathcal{R}(\tau_{0},\tau_{1})$:

$\displaystyle\accentset{\scalebox{0.6}{\mbox{\tiny$(2-\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathfrak{F}}}(v,\tau_{1}),\quad\accentset{\scalebox{0.6}{\mbox{\tiny$(2-\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathfrak{E}}}(\tau_{1}),\quad
c\,{}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(2-\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}$
$\displaystyle\leq$
$\displaystyle\accentset{\scalebox{0.6}{\mbox{\tiny$(2-\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathfrak{E}}}(\tau_{0})+C\,\left({}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(2-\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}\sqrt{\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{X}}}}+\sqrt{{}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(2-\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}}\sqrt{{}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}}\sqrt{\accentset{\scalebox{0.6}{\mbox{\tiny$(2\mkern-6.0mu-\mkern-6.0mu\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{X}}}}\right)+C\,{}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}\sqrt{\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{X}}}}\sqrt{L},$
(6.9)
$\displaystyle\accentset{\scalebox{0.6}{\mbox{\tiny$(1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathfrak{F}}}(v,\tau_{1}),\quad\accentset{\scalebox{0.6}{\mbox{\tiny$(1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathfrak{E}}}(\tau_{1}),\quad
c\,{}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}$
$\displaystyle\leq$
$\displaystyle\accentset{\scalebox{0.6}{\mbox{\tiny$(1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathfrak{E}}}(\tau_{0})+C\,{}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}\sqrt{\accentset{\scalebox{0.6}{\mbox{\tiny$(1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{X}}}}+C\,{}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}\sqrt{\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{X}}}}\sqrt{L},$
(6.10)
$\displaystyle\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathfrak{F}}}(v,\tau_{1}),\quad\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathfrak{E}}}(\tau_{1}),\quad
c\,{}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}$
$\displaystyle\leq$
$\displaystyle\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathfrak{E}}}(\tau_{0})+C\left(\,{}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}+(\,{}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}})^{\frac{1-\delta}{1+\delta}}(\,{}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}})^{\frac{2\delta}{1+\delta}}\right)\sqrt{\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{X}}}+(\,\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{X}}})^{\frac{1-\delta}{1+\delta}}(\,\accentset{\scalebox{0.6}{\mbox{\tiny$(1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{X}}})^{\frac{2\delta}{1+\delta}}}$
(6.11)
$\displaystyle\qquad+C\,{}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}\sqrt{\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{X}}}}\sqrt{L}.$

###### Proof.

Again we recall that if $\varepsilon_{\rm boot}\leq\varepsilon_{\rm loc}$,
then the assumption of Proposition 4.8.1 holds. The first two inequalities
follow from the estimate (4.32) of Proposition 4.8.1 applied to $p=2-\delta$,
$p=1$ in view of the fact that $\rho=\chi$, where we have used the relations
(4.22) and (4.28) to replace the far-away supported nonlinear terms with those
displayed above, together with the estimate

$\int_{\tau_{0}}^{\tau_{1}}\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}(\tau^{\prime})\sqrt{\,\,\accentset{\scalebox{0.6}{\mbox{\tiny$(-1\mkern-6.0mu-\mkern-6.0mu\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{E}}}^{\prime}(\tau^{\prime})}d\tau^{\prime}\lesssim\sup_{\tau_{0}\leq\tau\leq\tau_{1}}\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}(\tau)\sqrt{\int_{\tau_{0}}^{\tau_{1}}\accentset{\scalebox{0.6}{\mbox{\tiny$(-1\mkern-6.0mu-\mkern-6.0mu\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{E}}}^{\prime}(\tau^{\prime})d\tau^{\prime}}\,\cdot\,\sqrt{L}\lesssim{}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}\sqrt{\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{X}}}}\sqrt{L}.$

Note that for the $p=1$ estimate we retain only the less precise expression
${}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}\sqrt{\accentset{\scalebox{0.6}{\mbox{\tiny$(1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{X}}}}$
which will be sufficient for our purposes.

The third inequality follows similarly using estimate (4.34), where we use
also the interpolation inequality (3.113) to obtain

$\accentset{\scalebox{0.6}{\mbox{\tiny$(0+)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}_{\frac{8R}{9}}(\tau_{0},\tau_{1})\sqrt{\accentset{\scalebox{0.6}{\mbox{\tiny$(0+)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{X}}}_{\frac{8R}{9}}(\tau_{0},\tau_{1})}\lesssim\left(\,{}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}+(\,{}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}})^{\frac{1-\delta}{1+\delta}}(\,{}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}})^{\frac{2\delta}{1+\delta}}\right)\sqrt{\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{X}}}+(\,\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{X}}})^{\frac{1-\delta}{1+\delta}}(\,\accentset{\scalebox{0.6}{\mbox{\tiny$(1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{X}}})^{\frac{2\delta}{1+\delta}}}.$

∎

######  Remark 6.2.2.

We note that inequalities (6.9)–(6.11) imply of course

$\displaystyle{}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(2-\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}$
$\displaystyle\lesssim$
$\displaystyle\accentset{\scalebox{0.6}{\mbox{\tiny$(2-\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}(\tau_{0})+\left({}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(2-\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}\sqrt{\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{X}}}}+\sqrt{{}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(2-\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}}\sqrt{{}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}}\sqrt{\accentset{\scalebox{0.6}{\mbox{\tiny$(2\mkern-6.0mu-\mkern-6.0mu\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{X}}}}\right)+{}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}\sqrt{\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{X}}}}\sqrt{L},$
(6.12)
$\displaystyle{}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}$
$\displaystyle\lesssim$
$\displaystyle\accentset{\scalebox{0.6}{\mbox{\tiny$(1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}(\tau_{0})+{}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}\sqrt{\accentset{\scalebox{0.6}{\mbox{\tiny$(1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{X}}}}+{}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}\sqrt{\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{X}}}}\sqrt{L},$
(6.13)
$\displaystyle{}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}$
$\displaystyle\lesssim$
$\displaystyle\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}(\tau_{0})+\left(\,{}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}+(\,{}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}})^{\frac{1-\delta}{1+\delta}}(\,{}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}})^{\frac{2\delta}{1+\delta}}\right)\sqrt{\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{X}}}+(\,\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{X}}})^{\frac{1-\delta}{1+\delta}}(\,\accentset{\scalebox{0.6}{\mbox{\tiny$(1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{X}}})^{\frac{2\delta}{1+\delta}}}+{}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}\sqrt{\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{X}}}}\sqrt{L}.$
(6.14)

Following the above proof but now using equations (4.37) and (4.38), we may in
fact directly deduce the inequalities (6.12)–(6.14) under the relaxed
assumptions of Remark 4.8.4, where (4.1) is however required to be semilinear
and we are in the analogue of case (ii). Thus, the analogue of Proposition
6.2.1 holds in that case where (6.9)–(6.11) are replaced by (6.12)–(6.14).

####  6.2.2 Global existence in $L$-slabs

The presence of positive powers of $L$ in (6.9)–(6.11) means that our
smallness assumptions must involve negative powers of $L$ in order for the
estimates to close.

######  Proposition 6.2.3.

Let $k-1\geq k_{\rm loc}$ be sufficiently large and let us assume the case
(ii) assumptions. Then there exists an $0<\varepsilon_{\rm
slab}\leq\varepsilon_{\rm loc}$ and a constant $C>0$ implicit in the sign
$\lesssim$ below such that the following is true.

Given arbitrary $L\geq 1$, $\tau_{0}\geq 0$,
$0<\varepsilon_{0}\leq\varepsilon_{\rm slab}$ and initial data
$(\uppsi,\uppsi^{\prime})$ on $\Sigma(\tau_{0})$ as in Proposition 4.9.1,
satisfying moreover

$\qquad\accentset{\scalebox{0.6}{\mbox{\tiny$(1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}(\tau_{0})\leq\varepsilon_{0},\qquad\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu1$}}}{\mathcal{E}}}(\tau_{0})\leq\varepsilon_{0}L^{-1},$
(6.15)

then the unique solution of Proposition 4.9.1 obtaining the data can be
extended to a $\psi$ defined on the entire spacetime slab
$\mathcal{R}(\tau_{0},\tau_{0}+L)$ satisfying the equation (4.1) and the
estimates

$\qquad{}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}\lesssim\varepsilon_{0},\qquad{}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu1$}}}{\mathcal{X}}}\lesssim\varepsilon_{0}L^{-1}.$
(6.16)

###### Proof.

Consider the set $\mathfrak{B}\subset(\tau_{0},\tau_{0}+L]$ consisting of all
$\tau_{0}+L\geq\tau_{f}\geq\tau_{0}$ such that a solution $\psi$ of (4.1)
obtaining the data exists on $\mathcal{R}(\tau_{0},\tau_{f})$ and such that
the boostrap assumption (4.23) with $p=1$ and also the additional bootstrap
assumption

$\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{X}}}\leq\varepsilon
L^{-1}$ (6.17)

hold in $\mathcal{R}(\tau_{0},\tau_{1}:=\tau_{f})$, where
$0<\varepsilon\leq\varepsilon_{\rm boot}$ is a small constant satisfying

$1\gg\varepsilon\gg\varepsilon_{\rm slab}.$ (6.18)

(The above relation in particular already constrains $\varepsilon_{\rm slab}$
to be sufficiently small.)

By the local well posedness statement Proposition 4.9.1, it follows that since
$k-1\geq k_{\rm loc}$ and $\varepsilon_{0}\leq\varepsilon_{\rm
slab}\leq\varepsilon_{\rm loc}$, we have that $\tau_{0}+\tau_{\rm
exist}\in\mathfrak{B}$ and thus $\mathfrak{B}\neq\emptyset$, provided that
$\varepsilon$ satisfies (6.18). Also note that a fortiori, if
$\tau_{f}\in\mathfrak{B}$, then $(\tau_{0},\tau_{f}]\in\mathfrak{B}$ and thus
$\mathfrak{B}$ is manifestly a connected subset of $(\tau_{0},\infty)$.

For $\tau_{1}:=\tau_{f}\in\mathcal{B}$, Proposition 6.2.1 applies in
$\mathcal{R}(\tau_{0},\tau_{1})$. From (6.10) and (6.15) we obtain

${}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}\lesssim\varepsilon_{0}+\varepsilon^{1/2}\,{}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}},$
(6.19)

where we have used the bootstrap assumptions (4.23) and (6.17).

From (6.11) for $k-1$ and (6.15), we obtain

${}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu1$}}}{\mathcal{X}}}\lesssim\varepsilon_{0}L^{-1}+\varepsilon^{1/2}(\,{}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu1$}}}{\mathcal{X}}})^{\frac{1-\delta}{1+\delta}}(\,{}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu1$}}}{\mathcal{X}}})^{\frac{2\delta}{1+\delta}}L^{-\frac{1}{2}\frac{1-\delta}{1+\delta}}+\varepsilon^{1/2}\,{}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu1$}}}{\mathcal{X}}}.$
(6.20)

It follows that for $\varepsilon$ satisfying (6.18), we obtain

${}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(2-\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}\lesssim\varepsilon_{0},\qquad{}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu1$}}}{\mathcal{X}}}\lesssim\varepsilon_{0}L^{-\frac{1}{2}\frac{1-\delta}{1+\delta}}$
(6.21)

and plugging the second inequality of (6.21) into (6.20) we obtain

${}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu1$}}}{\mathcal{X}}}\lesssim\varepsilon_{0}L^{-1}+\varepsilon^{1/2}\varepsilon_{0}L^{-\frac{1}{2}(\frac{1-\delta}{1+\delta})^{2}}L^{-\frac{1}{2}\frac{1-\delta}{1+\delta}}.$
(6.22)

Now defining

$\gamma_{0}:=\frac{1}{2}\left(\frac{1-\delta}{1+\delta}\right)^{2}+\frac{1}{2}\frac{1-\delta}{1+\delta},$

we have

${}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu1$}}}{\mathcal{X}}}\lesssim\varepsilon_{0}L^{-\gamma_{0}},$
(6.23)

whence, plugging (6.23) into (6.20) we improve to

${}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu1$}}}{\mathcal{X}}}\lesssim\varepsilon_{0}L^{-1}+\varepsilon^{1/2}\varepsilon_{0}L^{-\gamma_{0}(\frac{1-\delta}{1+\delta})^{2}}L^{-\frac{1}{2}\frac{1-\delta}{1+\delta}}.$
(6.24)

Defining $\gamma_{i}$ iteratively by
$\gamma_{i}=\gamma_{i-1}\left(\frac{1-\delta}{1+\delta}\right)^{2}+\frac{1}{2}\frac{1-\delta}{1+\delta}$,
then, in view of the restriction (3.1), there exists a first $i$ such that
$\gamma_{i}\geq 1$, whence we obtain

${}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu1$}}}{\mathcal{X}}}\lesssim\varepsilon_{0}L^{-1}.$
(6.25)

We may now already apply our continuation criterion Corollary 4.9.2, applied
with $p=0$ and $k-1$, to assert the existence of an $\epsilon$, independent of
$\tau_{1}$, such that now defining
$\tau_{1}:=\min\\{\tau_{f}+\epsilon,\tau_{0}+L\\}$, the solution $\psi$
extends to a smooth solution of (4.1) on $\mathcal{R}(\tau_{0},\tau_{1})$, and
moreover, from (4.41), that

${}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu1$}}}{\mathcal{X}}}\lesssim\varepsilon_{0}L^{-1}$
(6.26)

holds on $\mathcal{R}(\tau_{0},\tau_{1})$, and from (4.42), that

${}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}\lesssim\varepsilon_{0}$
(6.27)

holds on $\mathcal{R}(\tau_{0},\tau_{1})$. It follows that (6.17) and (4.23)
(with $p=1$) hold on $\mathcal{R}(\tau_{0},\tau_{1})$ and thus
$\tau_{1}=\min\\{\tau_{f}+\epsilon,\tau_{0}+L\\}\in\mathfrak{B}$.

Since $\epsilon$ is independent of $\tau_{f}$, and in view also of the
connectivity of $\mathfrak{B}$, it follows that $\mathfrak{B}$ is a nonempty
open and closed subset of $(\tau_{0},\tau_{0}+L]$ and thus
$\mathfrak{B}=(\tau_{0},\tau_{0}+L]$ and hence the solution exists in the
entire spacetime slab $\mathcal{R}(\tau_{0},\tau_{0}+L)$.

The estimates (6.21) and (6.25) thus hold in the entire spacetime slab. This
gives (6.16). ∎

######  Remark 6.2.4.

Examining the proof, it is clear that we have only used (6.12)–(6.14) and not
the full (6.9)–(6.11) of Proposition 6.2.1. Thus, in view of the comments of
Remark 6.2.2, the proof holds also for the semilinear case under the relaxed
assumptions described in Remark 4.8.4. In general, examining the proof, we may
in fact relax the assumption of the first inequality of (6.15) to the
assumption

$\accentset{\scalebox{0.6}{\mbox{\tiny$(1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}(\tau_{0})\leq\varepsilon_{0}L^{\beta}$
(6.28)

for a sufficiently small $\beta$, in which case the first inequality of (6.16)
is replaced by

${}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}(\tau_{0})\leq\varepsilon_{0}L^{\beta}.$
(6.29)

This will again be useful for the semilinear case.

####  6.2.3 The pigeonhole argument

The above assumptions on initial data are sufficient for global existence in
the slab, but are not sufficient to iterate. For this we shall need
strengthened assumptions.

######  Proposition 6.2.5.

Under the assumptions of Proposition 6.2.3, there exists a constant $C>0$,
implicit in the inequalities $\lesssim$ below, a parameter $\alpha_{0}\gg 1$
and, for all $\alpha\geq\alpha_{0}$, a parameter $\hat{\varepsilon}_{\rm
slab}(\alpha)$ such that for all
$0<\hat{\varepsilon}_{0}\leq\hat{\varepsilon}_{\rm slab}(\alpha)$ the
following holds.

Let us assume that in addition to (6.15) we have

$\accentset{\scalebox{0.6}{\mbox{\tiny$(1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathfrak{E}}}(\tau_{0})\leq\hat{\varepsilon}_{0},\qquad\accentset{\scalebox{0.6}{\mbox{\tiny$(2-\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu1$}}}{\mathfrak{E}}}(\tau_{0})\leq\hat{\varepsilon}_{0},\qquad\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu1$}}}{\mathfrak{E}}}(\tau_{0})\leq\hat{\varepsilon}_{0}\alpha
L^{-1},\qquad\accentset{\scalebox{0.6}{\mbox{\tiny$(1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu2$}}}{\mathfrak{E}}}(\tau_{0})\leq\hat{\varepsilon}_{0}\alpha
L^{-1+\delta},\qquad\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu3$}}}{\mathfrak{E}}}(\tau_{0})\leq\hat{\varepsilon}_{0}\alpha^{2}L^{-2+\delta}.$
(6.30)

Then the solution $\psi$ of (4.1) on $\mathcal{R}(\tau_{0},\tau_{0}+L)$ given
by Proposition 6.2.3 satisfies the additional estimates

$\sup_{\tau_{0}\leq\tau\leq\tau_{0}+L}\accentset{\scalebox{0.6}{\mbox{\tiny$(1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}(\tau)\leq\alpha\hat{\varepsilon}_{0},$
(6.31)
$\sup_{\tau_{0}\leq\tau\leq\tau_{0}+L}\accentset{\scalebox{0.6}{\mbox{\tiny$(1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathfrak{E}}}(\tau)\leq\hat{\varepsilon}_{0}(1+\hat{\varepsilon}_{0}^{1/4}L^{(-1+\delta)/2}),$
(6.32)
$\sup_{\tau_{0}\leq\tau\leq\tau_{0}+L}\accentset{\scalebox{0.6}{\mbox{\tiny$(2-\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu1$}}}{\mathfrak{E}}}(\tau)\leq\hat{\varepsilon}_{0}(1+\hat{\varepsilon}_{0}^{1/4}L^{-1/2}),$
(6.33)
${}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}\lesssim\hat{\varepsilon}_{0},\qquad{}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(2-\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu1$}}}{\mathcal{X}}}\lesssim\hat{\varepsilon}_{0},\qquad{}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu1$}}}{\mathcal{X}}}\lesssim\hat{\varepsilon}_{0}\alpha
L^{-1},\qquad{}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu2$}}}{\mathcal{X}}}\lesssim\hat{\varepsilon}_{0}\alpha
L^{-1+\delta},\qquad{}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu3$}}}{\mathcal{X}}}\lesssim\hat{\varepsilon}_{0}\alpha^{2}L^{-2+\delta}.$
(6.34)

Moreover, for all times $\tau^{\prime}$ with $L\geq\tau^{\prime}-\tau_{0}\geq
L/2$, we have that

$\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu1$}}}{\mathfrak{E}}}(\tau^{\prime})\leq\frac{1}{2}\hat{\varepsilon}_{0}\alpha
L^{-1},$ (6.35)
$\accentset{\scalebox{0.6}{\mbox{\tiny$(1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu1$}}}{\mathfrak{E}}}(\tau^{\prime})\leq\frac{1}{2}\hat{\varepsilon}_{0}\alpha
L^{-1+\delta},$ (6.36)
$\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu2$}}}{\mathfrak{E}}}(\tau^{\prime})\leq\frac{1}{4}\hat{\varepsilon}_{0}\alpha^{2}L^{-2+\delta}.$
(6.37)

###### Proof.

Note that by the statement of the proposition, we are in particular also
assuming a priori (6.15) for _some_ $\varepsilon_{0}\leq\varepsilon_{\rm
slab}$, since this is included in the assumptions of Proposition 6.2.3. In
view now of Corollary 4.5.3, however, for $\alpha_{0}$ sufficiently large, if
say $\hat{\varepsilon}_{\rm slab}(\alpha)\ll\alpha^{-3}\varepsilon_{\rm
slab}$, it follows from the additional assumptions (6.30) that the estimates
(6.15) in fact hold with the specific constant
$\varepsilon_{0}:=\hat{\varepsilon}_{0}\alpha^{3}$ for all
$\alpha\geq\alpha_{0}$.

To obtain (6.34) we argue as follows. We note from the proof of Proposition
6.2.3 that we have the following system of inequalities

${}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}\lesssim\hat{\varepsilon}_{0}+\varepsilon_{0}^{1/2}\,{}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}},$
(6.38)
${}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(2-\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu1$}}}{\mathcal{X}}}\lesssim\hat{\varepsilon}_{0}+\varepsilon_{0}^{1/2}\,{}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(2-\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu1$}}}{\mathcal{X}}},$
(6.39)
${}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu1$}}}{\mathcal{X}}}\lesssim\hat{\varepsilon}_{0}\alpha
L^{-1}+(\,{}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu1$}}}{\mathcal{X}}})^{\frac{1-\delta}{1+\delta}}(\,{}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu1$}}}{\mathcal{X}}})^{\frac{2\delta}{1+\delta}}\sqrt{\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{X}}}+(\,\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{X}}})^{\frac{1-\delta}{1+\delta}}(\,\accentset{\scalebox{0.6}{\mbox{\tiny$(1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{X}}})^{\frac{2\delta}{1+\delta}}}+\varepsilon_{0}^{1/2}\,{}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu1$}}}{\mathcal{X}}},$
(6.40)
${}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu2$}}}{\mathcal{X}}}\lesssim\hat{\varepsilon}_{0}\alpha
L^{-1+\delta}+\varepsilon_{0}^{1/2}\,{}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu2$}}}{\mathcal{X}}},$
(6.41)
${}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu3$}}}{\mathcal{X}}}\lesssim\hat{\varepsilon}_{0}\alpha^{2}L^{-2+\delta}+(\,{}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu3$}}}{\mathcal{X}}})^{\frac{1-\delta}{1+\delta}}(\,{}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu3$}}}{\mathcal{X}}})^{\frac{2\delta}{1+\delta}}\sqrt{\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{X}}}+(\,\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{X}}})^{\frac{1-\delta}{1+\delta}}(\,\accentset{\scalebox{0.6}{\mbox{\tiny$(1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{X}}})^{\frac{2\delta}{1+\delta}}}+\varepsilon_{0}^{1/2}\,{}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu3$}}}{\mathcal{X}}},$
(6.42)

where we have used also the initial data assumptions (6.30).

For $\hat{\varepsilon}_{\rm slab}(\alpha)$ sufficiently small, we have
$\varepsilon_{0}\ll 1$ and thus we obtain immediately that

$\displaystyle{}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}\lesssim\hat{\varepsilon}_{0},\qquad{}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(2-\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu1$}}}{\mathcal{X}}}\lesssim\hat{\varepsilon}_{0},\qquad{}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu2$}}}{\mathcal{X}}}\lesssim\hat{\varepsilon}_{0}\alpha
L^{-1+\delta},$ (6.43)
$\displaystyle{}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu1$}}}{\mathcal{X}}}\lesssim\hat{\varepsilon}_{0}\alpha
L^{-1}+(\,{}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu1$}}}{\mathcal{X}}})^{\frac{1-\delta}{1+\delta}}(\,{}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu1$}}}{\mathcal{X}}})^{\frac{2\delta}{1+\delta}}\sqrt{\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{X}}}+(\,\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{X}}})^{\frac{1-\delta}{1+\delta}}(\,\accentset{\scalebox{0.6}{\mbox{\tiny$(1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{X}}})^{\frac{2\delta}{1+\delta}}},$
$\displaystyle{}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu3$}}}{\mathcal{X}}}\lesssim\hat{\varepsilon}_{0}\alpha^{2}L^{-2+\delta}+(\,{}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu3$}}}{\mathcal{X}}})^{\frac{1-\delta}{1+\delta}}(\,{}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu3$}}}{\mathcal{X}}})^{\frac{2\delta}{1+\delta}}\sqrt{\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{X}}}+(\,\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{X}}})^{\frac{1-\delta}{1+\delta}}(\,\accentset{\scalebox{0.6}{\mbox{\tiny$(1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{X}}})^{\frac{2\delta}{1+\delta}}}.$

The first two inequalities of (6.43) yield the first two inequalities of
(6.34), while the third inequality of (6.43) yields the fourth inequality of
(6.34). On the other hand, from the latter inequality of (6.43) we obtain

${}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu3$}}}{\mathcal{X}}}\lesssim\hat{\varepsilon}_{0}\alpha^{2}L^{-2+\delta}+\hat{\varepsilon}_{0}\varepsilon_{0}^{1/2}L^{-\frac{3}{2}\frac{(1-\delta)(1+2\delta)}{1+\delta}}.$
(6.44)

Note that $-\frac{3}{2}\frac{(1-\delta)(1+2\delta)}{1+\delta}>-2+\delta$. We
may however improve (6.44) iteratively as follows: If

${}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu3$}}}{\mathcal{X}}}\lesssim\hat{\varepsilon}_{0}\alpha^{2}L^{-\gamma},$

for $\gamma<2-\delta$, then

$\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{X}}}\lesssim\hat{\varepsilon}_{0}\alpha^{2}L^{-\gamma}\lesssim\varepsilon_{0}L^{-\gamma},$

hence, plugging this again into the final inequality of (6.43) to estimate the
nonlinear term, we obtain

${}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu3$}}}{\mathcal{X}}}\lesssim\hat{\varepsilon}_{0}\alpha^{2}L^{-2+\delta}+\hat{\varepsilon}_{0}\varepsilon_{0}^{1/2}L^{-\frac{3}{2}\frac{(1-\delta)(\gamma+2\delta)}{1+\delta}}.$
(6.45)

Setting $\gamma_{0}=\frac{3}{2}\frac{(1-\delta)(1+2\delta)}{1+\delta}$, and
defining inductively
$\gamma_{i}=\frac{3}{2}\frac{(1-\delta)(\gamma_{i-1}+2\delta)}{1+\delta}$, we
have that there exists a first $i\geq 1$ such that
$\frac{3}{2}\frac{(1-\delta)(\gamma_{i}+2\delta)}{1+\delta}>2-\delta$. It
follows that (6.45) holds for $\gamma=\gamma_{i}$, hence

${}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu3$}}}{\mathcal{X}}}\lesssim\hat{\varepsilon}_{0}\alpha^{2}L^{-2+\delta}.$

This yields the fifth inequality of (6.34). Note that this implies

$\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{X}}}\lesssim\hat{\varepsilon}_{0}\alpha^{2}L^{-2+\delta}\lesssim\varepsilon_{0}L^{-2+\delta}.$
(6.46)

Note also that third inequality of (6.43) implies

$\accentset{\scalebox{0.6}{\mbox{\tiny$(1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{X}}}\lesssim\hat{\varepsilon}_{0}\alpha
L^{-1+\delta}\lesssim\varepsilon_{0}L^{-1+\delta}.$ (6.47)

Finally, plugging these bounds into the fourth inequality of (6.43) yields the
third inequality of (6.34), completing the proof of (6.34).

To obtain (6.32), we recall (6.10) from Proposition 6.2.1. This gives, for
sufficiently small $\hat{\varepsilon}_{\rm slab}$,

$\displaystyle\sup_{\tau_{0}\leq\tau\leq\tau_{0}+L}\accentset{\scalebox{0.6}{\mbox{\tiny$(1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathfrak{E}}}(\tau)$
$\displaystyle\leq$
$\displaystyle\accentset{\scalebox{0.6}{\mbox{\tiny$(1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathfrak{E}}}(\tau_{0})+C\,{}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}\sqrt{\accentset{\scalebox{0.6}{\mbox{\tiny$(1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{X}}}}+C\,{}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}\sqrt{\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{X}}}}\sqrt{L},$
$\displaystyle\leq$
$\displaystyle\hat{\varepsilon}_{0}+C\hat{\varepsilon}_{0}\varepsilon_{0}^{1/2}L^{(-1+\delta)/2}+C\hat{\varepsilon}_{0}\varepsilon_{0}^{1/2}L^{(-2+\delta)/2}L^{1/2}$
$\displaystyle\leq$
$\displaystyle\hat{\varepsilon}_{0}(1+\hat{\varepsilon}_{0}^{1/4}L^{(-1+\delta)/2})$

where we have used the first inequality of (6.30), the first inequality of
(6.43), the estimate (6.47) and the estimate (6.46) and we have replaced
$\varepsilon_{0}^{1/2}$ by $\hat{\varepsilon}_{0}^{1/4}$ in the penultimate
line, sacrificing a quarter power to absorb the resulting $\alpha$ term and
the constant $C$.

To obtain (6.33), we now recall (6.9) from Proposition 6.2.1. This gives, for
sufficiently small $\hat{\varepsilon}_{\rm slab}$,

$\displaystyle\sup_{\tau_{0}\leq\tau\leq\tau_{0}+L}\accentset{\scalebox{0.6}{\mbox{\tiny$(2-\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu1$}}}{\mathfrak{E}}}(\tau)$
$\displaystyle\leq$
$\displaystyle\accentset{\scalebox{0.6}{\mbox{\tiny$(2-\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu1$}}}{\mathfrak{E}}}(\tau_{0})+C\,\left({}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(2-\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu1$}}}{\mathcal{X}}}\sqrt{\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{X}}}}+\sqrt{{}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(2-\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu1$}}}{\mathcal{X}}}}\sqrt{{}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu1$}}}{\mathcal{X}}}}\sqrt{\accentset{\scalebox{0.6}{\mbox{\tiny$(2\mkern-6.0mu-\mkern-6.0mu\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{X}}}}\right)+C\,{}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu1$}}}{\mathcal{X}}}\sqrt{\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{X}}}}\sqrt{L}$
$\displaystyle\leq$
$\displaystyle\hat{\varepsilon}_{0}+C\hat{\varepsilon}_{0}\varepsilon_{0}^{1/2}L^{(-2+\delta)/2}+C\hat{\varepsilon}_{0}\alpha^{1/2}\varepsilon_{0}^{1/2}L^{-1/2}+C\hat{\varepsilon}_{0}\alpha\varepsilon_{0}^{1/2}L^{(-2+\delta)/2}$
$\displaystyle\leq$
$\displaystyle\hat{\varepsilon}_{0}(1+\hat{\varepsilon}_{0}^{1/4}L^{-1/2}),$

where we have used the second inequality of (6.30), the second and third
inequalities of (6.34), the estimate (6.46) and the first inequality of
(6.44), and we have replaced $\varepsilon_{0}^{1/2}$ by
$\hat{\varepsilon}_{0}^{1/4}$ in the penultimate line, sacrificing a quarter
power to absorb the resulting $\alpha$ term and the constant $C$.

To show (6.35)–(6.37), we first apply the pigeonhole principle as in [DR09] to
the inequality

$\int_{\tau_{0}}^{\tau_{0}+L}\left(\,\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu1$}}}{\mathcal{E}}}^{\prime}(\tau^{\prime})+\accentset{\scalebox{0.6}{\mbox{\tiny$(1-\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu2$}}}{\mathcal{E}}}^{\prime}(\tau^{\prime})+\alpha^{-1}L^{1-\delta}\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu3$}}}{\mathcal{E}}}^{\prime}(\tau^{\prime})\right)d\tau^{\prime}\lesssim\hat{\varepsilon}_{0},$

which, upon addition, follows from the first, second and fourth inequalities
of the estimate (6.34) already shown. Recalling from (3.92) that for both
$p=2-\delta$ and $p=1$, we have

$\accentset{\scalebox{0.6}{\mbox{\tiny$(p\mkern-6.0mu-\mkern-6.0mu1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu2$}}}{\mathcal{E}}}^{\prime}\gtrsim\accentset{\scalebox{0.6}{\mbox{\tiny$(p\mkern-6.0mu-\mkern-6.0mu1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu2$}}}{\mathcal{E}}},\qquad\accentset{\scalebox{0.6}{\mbox{\tiny$(p\mkern-6.0mu-\mkern-6.0mu1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu3$}}}{\mathcal{E}}}^{\prime}\gtrsim\accentset{\scalebox{0.6}{\mbox{\tiny$(p\mkern-6.0mu-\mkern-6.0mu1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu3$}}}{\mathcal{E}}},$

we obtain that there exists $\tau^{\prime\prime}\in[\tau_{0},\tau_{0}+L/2]$,
whose precise value depends on the solution, such that

$\displaystyle\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu1$}}}{\mathcal{E}}}(\tau^{\prime\prime})$
$\displaystyle\lesssim$ $\displaystyle\hat{\varepsilon}_{0}\cdot L^{-1},$
(6.48)
$\displaystyle\accentset{\scalebox{0.6}{\mbox{\tiny$(1-\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu2$}}}{\mathcal{E}}}(\tau^{\prime\prime})$
$\displaystyle\lesssim$ $\displaystyle\hat{\varepsilon}_{0}\cdot L^{-1},$
(6.49)
$\displaystyle\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu3$}}}{\mathcal{E}}}(\tau^{\prime\prime})$
$\displaystyle\lesssim$ $\displaystyle\hat{\varepsilon}_{0}\alpha
L^{-1+\delta}\cdot L^{-1}.$ (6.50)

Now in view of the interpolation estimate (3.112) of Proposition 3.6.11 we
have

$\accentset{\scalebox{0.6}{\mbox{\tiny$(1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu2$}}}{\mathcal{E}}}(\tau^{\prime\prime})\lesssim\left(\,\,\accentset{\scalebox{0.6}{\mbox{\tiny$(1-\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu2$}}}{\mathcal{E}}}(\tau^{\prime\prime})\right)^{1-\delta}\left(\,\,\accentset{\scalebox{0.6}{\mbox{\tiny$(2-\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu2$}}}{\mathcal{E}}}(\tau^{\prime\prime})\right)^{\delta}\leq\left(\,\,\accentset{\scalebox{0.6}{\mbox{\tiny$(1-\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu2$}}}{\mathcal{E}}}(\tau^{\prime\prime})\right)^{1-\delta}\left(\sup_{\tau\in[\tau_{0},\tau_{0}+L]}\accentset{\scalebox{0.6}{\mbox{\tiny$(2-\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu2$}}}{\mathcal{E}}}(\tau)\right)^{\delta}$

and thus

$\accentset{\scalebox{0.6}{\mbox{\tiny$(1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu2$}}}{\mathcal{E}}}(\tau^{\prime\prime})\lesssim\hat{\varepsilon}_{0}L^{-1+\delta}$
(6.51)

where we have used (6.49) and the estimate for
$\accentset{\scalebox{0.6}{\mbox{\tiny$(2-\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu2$}}}{\mathcal{E}}}$
contained in (6.34).

Now we apply (6.10) and (6.11) again, with $\tau^{\prime\prime}$ in place of
$\tau_{0}$, using (6.48), (6.51) and (6.50) to bound the initial data, to
obtain that for all $\tau_{0}+L\geq\tau^{\prime}\geq\tau_{0}+L/2$, we have

$\displaystyle\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu1$}}}{\mathfrak{E}}}(\tau^{\prime})$
$\displaystyle\sim$
$\displaystyle\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu1$}}}{\mathcal{E}}}(\tau^{\prime})\lesssim\hat{\varepsilon}_{0}L^{-1},$
(6.52)
$\displaystyle\accentset{\scalebox{0.6}{\mbox{\tiny$(1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu2$}}}{\mathfrak{E}}}(\tau^{\prime})$
$\displaystyle\sim$
$\displaystyle\accentset{\scalebox{0.6}{\mbox{\tiny$(1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu2$}}}{\mathcal{E}}}(\tau^{\prime})\lesssim\hat{\varepsilon}_{0}L^{-1+\delta},$
(6.53)
$\displaystyle\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu3$}}}{\mathfrak{E}}}(\tau^{\prime})$
$\displaystyle\sim$
$\displaystyle\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu3$}}}{\mathcal{E}}}(\tau^{\prime})\lesssim\hat{\varepsilon}_{0}\alpha
L^{-2+\delta}.$ (6.54)

Thus, in view of the requirement $\alpha\geq\alpha_{0}\gg 1$, for sufficiently
large $\alpha_{0}$ we may absorb the constants implicit in $\lesssim$ by
explicit constants of our choice by adding extra positive $\alpha$ powers to
the right hand side of (6.53) and (6.54). In this way, we obtain the specific
estimates (6.36) and (6.37) for all $\tau^{\prime}\geq\tau^{\prime\prime}$ and
thus in particular for all $L\geq\tau^{\prime}-\tau_{0}\geq L/2$. In the same
way, we also obtain the specific constant of the estimate of (6.31) which will
be convenient in our scheme. ∎

Let us state an alternative “relaxed” version of the above proposition.

######  Proposition 6.2.6.

Under the relaxed assumptions for Proposition 6.2.3 described in Remark 6.2.4,
there exists a constant $C>0$, implicit in the inequalities $\lesssim$ below,
a parameter $\beta>0$ sufficiently small, a parameter $\alpha_{0}\gg 1$ and,
for all $\alpha\geq\alpha_{0}$, a parameter $\hat{\varepsilon}_{\rm
slab}(\alpha)$ such that the following holds.

Given $0<\hat{\varepsilon}_{0}\leq\hat{\varepsilon}_{\rm slab}$, let us assume
in addition that

$\accentset{\scalebox{0.6}{\mbox{\tiny$(1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}(\tau_{0})\leq\hat{\varepsilon}_{0}L^{\beta},\quad\accentset{\scalebox{0.6}{\mbox{\tiny$(2-\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu1$}}}{\mathcal{E}}}(\tau_{0})\leq\hat{\varepsilon}_{0}L^{\beta},\quad\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu1$}}}{\mathcal{E}}}(\tau_{0})\leq\hat{\varepsilon}_{0}\alpha^{1/2}L^{-1+\beta},\quad\accentset{\scalebox{0.6}{\mbox{\tiny$(1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu2$}}}{\mathcal{E}}}(\tau_{0})\leq\hat{\varepsilon}_{0}\alpha
L^{-1+\delta+\beta},\quad\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu3$}}}{\mathcal{E}}}(\tau_{0})\leq\hat{\varepsilon}_{0}\alpha^{2}L^{-2+\delta+\beta}.$
(6.55)

Then the solution $\psi$ of (4.1) on $\mathcal{R}(\tau_{0},\tau_{0}+L)$ given
by Proposition 6.2.3 satisfies the additional estimates

$\sup_{\tau\leq\tau\leq\tau_{0}+L}\accentset{\scalebox{0.6}{\mbox{\tiny$(1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}(\tau)\leq\alpha^{\beta}\hat{\varepsilon}_{0}L^{\beta},\qquad\sup_{\tau_{0}\leq\tau\leq\tau_{0}+L}\accentset{\scalebox{0.6}{\mbox{\tiny$(2-\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu1$}}}{\mathcal{E}}}(\tau)\leq\alpha^{\beta}\hat{\varepsilon}_{0}L^{\beta},$
(6.56)
${}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}\lesssim\hat{\varepsilon}_{0}L^{\beta},\quad{}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(2-\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu1$}}}{\mathcal{X}}}\lesssim\hat{\varepsilon}_{0}L^{\beta},\quad{}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu1$}}}{\mathcal{X}}}\lesssim\hat{\varepsilon}_{0}\alpha
L^{-1+\beta},\quad{}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu2$}}}{\mathcal{X}}}\lesssim\hat{\varepsilon}_{0}\alpha
L^{-1+\delta+\beta},\quad{}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu3$}}}{\mathcal{X}}}\lesssim\hat{\varepsilon}_{0}\alpha^{2}L^{-2+\delta+\beta}.$
(6.57)

Moreover, for all times $\tau^{\prime}$ with $L\geq\tau^{\prime}-\tau_{0}\geq
L/2$, we have that

$\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu1$}}}{\mathcal{E}}}(\tau^{\prime})\leq\hat{\varepsilon}_{0}\alpha^{2\beta}L^{-1+\beta},\qquad\accentset{\scalebox{0.6}{\mbox{\tiny$(1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu1$}}}{\mathcal{E}}}(\tau^{\prime})\leq\hat{\varepsilon}_{0}\alpha^{2\beta}L^{-1+\delta+\beta},\qquad\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu2$}}}{\mathcal{E}}}(\tau^{\prime})\leq\hat{\varepsilon}_{0}\alpha^{1+2\beta}L^{-2+\delta+\beta}.$
(6.58)

###### Proof.

The proof is similar to that of Proposition 6.2.5 and is left to the reader. ∎

######  Remark 6.2.7.

The above proposition can now immediately be seen to also hold in the
semilinear case, with the relaxed assumptions described in Remark 4.8.4.

####  6.2.4 The iteration: proof of Theorem 5.1 in case (ii)

We may now prove Theorem 5.1 in case (ii).

We shall proceed iteratively.

We define $\tau_{0}=1$, $L_{0}=1$, $L_{i}=2^{i}$, $\tau_{i+1}=\tau_{i}+L_{i}$,
and fix $\alpha\geq\alpha_{0}$ so that the statement of Proposition 6.2.5
holds. (Note that we shall no longer track the dependence of constants and
parameters on $\alpha$ since it is now fixed; one could simply take
$\alpha=\alpha_{0}$.)

For $0<\varepsilon_{0}\leq\varepsilon_{\rm global}$ and $\varepsilon_{\rm
global}$ sufficiently small, since

$\accentset{\scalebox{0.6}{\mbox{\tiny$(1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}(\tau_{0})+\accentset{\scalebox{0.6}{\mbox{\tiny$(2-\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu1$}}}{\mathcal{E}}}(\tau_{0})\leq\varepsilon_{0},$

then in view of Corollary 4.5.3, we have

$\accentset{\scalebox{0.6}{\mbox{\tiny$(1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathfrak{E}}}(\tau_{0})\lesssim\varepsilon_{0},\qquad\accentset{\scalebox{0.6}{\mbox{\tiny$(2-\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu1$}}}{\mathfrak{E}}}(\tau_{0})\lesssim\varepsilon_{0}.$

Thus, for sufficiently small $\varepsilon_{\rm
global}\ll\hat{\varepsilon}_{\rm slab}$, it follows that there exists a
$\hat{\varepsilon}_{0}(\varepsilon_{0})\sim\varepsilon_{0}$, satisfying
moreover $2\alpha\hat{\varepsilon}_{0}\leq\varepsilon_{\rm slab}$, such that

$\hat{\varepsilon}_{0}\leq\frac{1}{2}\hat{\varepsilon}_{\rm slab}$

and

$\accentset{\scalebox{0.6}{\mbox{\tiny$(1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathfrak{E}}}(\tau_{0})\leq\hat{\varepsilon}_{0},\qquad\accentset{\scalebox{0.6}{\mbox{\tiny$(2-\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu1$}}}{\mathfrak{E}}}(\tau_{0})\leq\hat{\varepsilon}_{0},\qquad\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu1$}}}{\mathfrak{E}}}(\tau_{0})\leq\hat{\varepsilon}_{0}\alpha
L^{-1},\qquad\accentset{\scalebox{0.6}{\mbox{\tiny$(1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu2$}}}{\mathfrak{E}}}(\tau_{0})\leq\hat{\varepsilon}_{0}\alpha
L_{0}^{-1+\delta},\qquad\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu3$}}}{\mathfrak{E}}}(\tau_{0})\leq\hat{\varepsilon}_{0}\alpha^{2}L^{-2+\delta}.$

Finally, with our restriction on the definition of $\varepsilon_{0}$, we may
also write

$\accentset{\scalebox{0.6}{\mbox{\tiny$(1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}(\tau_{0})\leq\alpha\hat{\varepsilon}_{0}.$

In general, given $\tau_{i}\geq 1$ defined above,
$\hat{\varepsilon}_{i}\leq\hat{\varepsilon}_{\rm slab}$, and a solution $\psi$
of (4.1) on $\mathcal{R}(\tau_{0},\tau_{i})$ such that

$\accentset{\scalebox{0.6}{\mbox{\tiny$(1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}(\tau_{i})\leq\alpha\hat{\varepsilon}_{i}$
(6.59)

and

$\accentset{\scalebox{0.6}{\mbox{\tiny$(1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathfrak{E}}}(\tau_{i})\leq\hat{\varepsilon}_{i},\qquad\accentset{\scalebox{0.6}{\mbox{\tiny$(2-\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu1$}}}{\mathfrak{E}}}(\tau_{i})\leq\hat{\varepsilon}_{i},\qquad\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu1$}}}{\mathfrak{E}}}(\tau_{i})\leq\hat{\varepsilon}_{i}\alpha
L^{-1},\qquad\accentset{\scalebox{0.6}{\mbox{\tiny$(1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu1$}}}{\mathfrak{E}}}(\tau_{i})\leq\hat{\varepsilon}_{i}\alpha
L_{i}^{-1+\delta},\qquad\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu2$}}}{\mathfrak{E}}}(\tau_{i})\leq\hat{\varepsilon}_{i}\alpha^{2}L_{i}^{-2+\delta},$
(6.60)

note that by our restrictions on $\hat{\varepsilon}_{\rm slab}$, the
assumptions of Proposition 6.2.3 hold with $\alpha^{2}\hat{\varepsilon}_{i}$
in place of $\varepsilon_{0}$ (here we are using (6.59) to invoke Corollary
4.5.3 to rewrite (6.60) in terms of the calligraphic energies; cf. the first
lines of the proof of Proposition 6.2.5) and the assumptions of Proposition
6.2.5 then apply with $\hat{\varepsilon}_{i}$ in place of
$\hat{\varepsilon}_{0}$, where both propositions are understood now with
$\tau_{i}$, $\tau_{i+1}$ in place of $\tau_{0}$, $\tau_{1}$. It follows that
the solution $\psi$ extends to a solution defined also in
$\mathcal{R}_{i}:=\mathcal{R}(\tau_{i},\tau_{i}+L_{i})$ satisfying the
estimates (6.31)–(6.34) while for $\tau^{\prime}=\tau_{i+1}=\tau_{i}+L_{i}$ we
have in addition (6.36)–(6.37). We have thus

$\displaystyle\accentset{\scalebox{0.6}{\mbox{\tiny$(1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}(\tau_{i+1})$
$\displaystyle\leq\alpha\hat{\varepsilon}_{i}$ $\displaystyle\leq$
$\displaystyle\alpha\hat{\varepsilon}_{i+1},$
$\displaystyle\accentset{\scalebox{0.6}{\mbox{\tiny$(1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathfrak{E}}}(\tau_{i+1})$
$\displaystyle\leq\hat{\varepsilon}_{i}(1+\hat{\varepsilon}_{i}^{1/4}L_{i}^{-(1+\delta)/2})$
$\displaystyle\leq$ $\displaystyle\hat{\varepsilon}_{i+1}$
$\displaystyle\accentset{\scalebox{0.6}{\mbox{\tiny$(2-\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu1$}}}{\mathfrak{E}}}(\tau_{i+1})$
$\displaystyle\leq\hat{\varepsilon}_{i}(1+\hat{\varepsilon}_{i}^{1/4}L_{i}^{-1/2})$
$\displaystyle\leq$ $\displaystyle\hat{\varepsilon}_{i+1},$
$\displaystyle\accentset{\scalebox{0.6}{\mbox{\tiny$(1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu1$}}}{\mathfrak{E}}}(\tau_{i+1})$
$\displaystyle\leq\frac{1}{2}\hat{\varepsilon}_{i}\alpha L_{i}^{-1}$
$\displaystyle\leq$ $\displaystyle\hat{\varepsilon}_{i+1}\alpha L_{i+1}^{-1},$
$\displaystyle\accentset{\scalebox{0.6}{\mbox{\tiny$(1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu2$}}}{\mathfrak{E}}}(\tau_{i+1})$
$\displaystyle\leq\frac{1}{2}\hat{\varepsilon}_{i}\alpha L_{i}^{-1+\delta}$
$\displaystyle\leq$ $\displaystyle\hat{\varepsilon}_{i+1}\alpha
L_{i+1}^{-1+\delta},$
$\displaystyle\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu3$}}}{\mathfrak{E}}}(\tau_{i+1})$
$\displaystyle\leq\frac{1}{4}\hat{\varepsilon}_{i}\alpha^{2}L_{i}^{-2+\delta}$
$\displaystyle\leq$
$\displaystyle\hat{\varepsilon}_{i+1}\alpha^{2}L_{i+1}^{-2+\delta},$

as long as

$\hat{\varepsilon}_{i+1}:=\hat{\varepsilon}_{i}(1+\hat{\varepsilon}_{i}^{1/4}L_{i}^{-(1+\delta)/2}).$
(6.61)

Now note that for $\varepsilon_{\rm global}$ sufficiently small, the above
inductive definition (6.61) of $\hat{\varepsilon}_{i}$ ensures that
$\hat{\varepsilon}_{i}\leq 2\hat{\varepsilon}_{0}$ for all $i$.

It follows that a solution exists in
$\mathcal{R}(\tau_{0},\infty)=\cup\mathcal{R}(\tau_{i},\tau_{i}+L_{i})$ and in
each interval the estimates (6.34) hold.

We obtain finally that for all $\tau\geq 1$ we have

$\accentset{\scalebox{0.6}{\mbox{\tiny$(1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathfrak{E}}}(\tau)\leq\hat{\varepsilon}_{0}+C\hat{\varepsilon}_{0}^{3/2},\qquad\accentset{\scalebox{0.6}{\mbox{\tiny$(2-\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu1$}}}{\mathfrak{E}}}(\tau)\leq\hat{\varepsilon}_{0}+C\hat{\varepsilon}_{0}^{3/2},$
(6.62)
$\int_{\tau_{0}}^{\tau}{}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(1-\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu1$}}}{\mathcal{E}}}^{\prime}+\accentset{\scalebox{0.6}{\mbox{\tiny$(1-\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu2$}}}{\mathcal{E}}}^{\prime}\lesssim\hat{\varepsilon}_{0}\log(\tau+1),$
(6.63)
$\accentset{\scalebox{0.6}{\mbox{\tiny$(1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu2$}}}{\mathcal{E}}}(\tau)\lesssim\hat{\varepsilon}_{0}\tau^{-1+\delta},\qquad\accentset{\scalebox{0.6}{\mbox{\tiny$(1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu2$}}}{\mathcal{F}}}(v,\tau)\lesssim\hat{\varepsilon}_{0}\tau^{-1+\delta},$
(6.64)
$\int_{\tau}^{\infty}{}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu2$}}}{\mathcal{E}}}^{\prime}+\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu3$}}}{\mathcal{E}}}^{\prime}\lesssim\hat{\varepsilon}_{0}\tau^{-1+\delta},$
(6.65)
$\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu3$}}}{\mathcal{E}}}(\tau)\lesssim\hat{\varepsilon}_{0}\tau^{-2+\delta},\qquad\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu3$}}}{\mathcal{F}}}(v,\tau)\lesssim\hat{\varepsilon}_{0}\tau^{-2+\delta},$
(6.66)
$\int_{\tau}^{\infty}{}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(-1\mkern-6.0mu-\mkern-6.0mu\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu3$}}}{\mathcal{E}}}^{\prime}+\accentset{\scalebox{0.6}{\mbox{\tiny$(-1\mkern-6.0mu-\mkern-6.0mu\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu4$}}}{\mathcal{E}}}^{\prime}\lesssim\hat{\varepsilon}_{0}\tau^{-2+\delta}.$
(6.67)

Here, by $\mathcal{F}(v,\tau)$ we denote the restriction of the flux on
$\underline{C}_{v}$ to $J^{+}(\Sigma(\tau))$.

Let us note that we may improve, a posteriori, the inequality (6.63) to

$\int_{\tau_{0}}^{\infty}{}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(1-\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu1$}}}{\mathcal{E}}}^{\prime}+\accentset{\scalebox{0.6}{\mbox{\tiny$(1-\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu2$}}}{\mathcal{E}}}^{\prime}\lesssim\hat{\varepsilon}_{0}.$
(6.68)

To show (6.68), for all $\tau\geq\tau_{0}$, we reapply the estimates of
Proposition 4.8.1 globally on $\mathcal{R}(\tau_{0},\tau)$. To control the
nonlinear error bulk integrals, redecompose the domain in dyadic time slabs
and apply the estimates proven and sum. In view of the estimates, all these
error bulk integrals can be controlled by $C\hat{\varepsilon}_{0}^{3/2}$.

####  6.2.5 Alternative proof using Proposition 6.2.6 and the semilinear case

We note that we may prove the theorem alternatively using Proposition 6.2.6.
We first fix $\alpha\geq\alpha_{0}$, and now define $L_{i}:=\alpha^{i}$. Here
our iterative assumption is

$\accentset{\scalebox{0.6}{\mbox{\tiny$(1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}(\tau_{0})\leq\hat{\varepsilon}_{0}L_{i}^{\beta},\quad\accentset{\scalebox{0.6}{\mbox{\tiny$(2-\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu1$}}}{\mathcal{E}}}(\tau_{0})\leq\hat{\varepsilon}_{0}L_{i}^{\beta},\quad\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu1$}}}{\mathcal{E}}}(\tau_{0})\leq\hat{\varepsilon}_{0}\alpha
L_{i}^{-1+\beta},\quad\accentset{\scalebox{0.6}{\mbox{\tiny$(1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu2$}}}{\mathcal{E}}}(\tau_{0})\leq\hat{\varepsilon}_{0}\alpha
L_{i}^{-1+\delta+\beta},\quad\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu3$}}}{\mathcal{E}}}(\tau_{0})\leq\hat{\varepsilon}_{0}\alpha^{2}L_{i}^{-2+\delta+\beta}.$

In view of (6.56) and (6.58) and the definition of $L_{i}$, we see that this
is closed under iteration. We obtain thus existence in
$\mathcal{R}(\tau_{0},\infty)$, and at first instance the bounds

$\accentset{\scalebox{0.6}{\mbox{\tiny$(1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}(\tau)\lesssim\hat{\varepsilon}_{0}\tau^{\beta},\qquad\accentset{\scalebox{0.6}{\mbox{\tiny$(2-\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu1$}}}{\mathcal{E}}}(\tau)\lesssim\hat{\varepsilon}_{0}\tau^{\beta},$
$\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu1$}}}{\mathcal{E}}}(\tau)\lesssim\hat{\varepsilon}_{0}\tau^{-1+\beta},\qquad\accentset{\scalebox{0.6}{\mbox{\tiny$(1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu2$}}}{\mathcal{E}}}(\tau)\lesssim\hat{\varepsilon}_{0}\tau^{-1+\delta+\beta},\qquad\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu3$}}}{\mathcal{E}}}(\tau_{0})\lesssim\hat{\varepsilon}_{0}\tau^{-2+\delta+\beta}.$

These bounds are of course weaker than those of (6.62)–(6.67). The
$\tau^{\beta}$ factors terms may be, however, removed a posteriori, as in the
last paragraph of Section 6.2.4, by revisiting the estimates globally on
$\mathcal{R}(\tau_{0},\tau)$, controlling the nonlinear error bulk integrals
by redecomposing into dyadic time slabs, applying the estimates already
proven, and summing.

Thus this proof in the end yields finally the same estimates as before, but
where (6.62) is replaced by

$\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}\lesssim\hat{\varepsilon}_{0},\qquad\accentset{\scalebox{0.6}{\mbox{\tiny$(1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu2$}}}{\mathcal{E}}}(\tau)\lesssim\hat{\varepsilon}_{0}.$
(6.69)

######  Remark 6.2.8.

In view of Remark 6.2.7, this proof in particular applies to the semilinear
case with the relaxed assumptions of Remark 4.8.4, giving the result for case
(ii) as stated in Remark 5.1.

######  Remark 6.2.9.

The disadvantage of this alternative proof over the proof given in Section
6.2.4 is that to obtain the fundamental top-order orbital stability statement
(6.69), one must revisit the estimates globally. We thus believe that the
proof in Section 6.2.4 better reflects the dyadically localised philosophy of
our method.

###  6.3 Case (iii)

We now turn to the most general case we consider, case (iii), where we do not
assume that (3.3) follows from a physical space identity (3.14), but only
assume (3.3) as a black box, together with the weaker physical space identity
described in Section 3.4.3.

This case is slightly more involved because we must combine the estimate
originating from Section 3.4.3 with the estimate originating in Section 3.2.
As we shall see, however, the difference with case (ii) is in fact quite
minor.

####  6.3.1 The hierarchy of inequalities

######  Proposition 6.3.1.

Let $k$ be sufficiently large and let us assume the case (iii) assumptions.
There exist constants $C>0$, $c>0$ and an $\varepsilon_{\rm boot}>0$
sufficiently small such that the following is true.

Consider a region $\mathcal{R}(\tau_{0},\tau_{1})$ and a $\psi$ solving (4.1)
on $\mathcal{R}(\tau_{0},\tau_{1})$, satisfying moreover (4.23) for $p=0$ and
$0<\varepsilon\leq\varepsilon_{\rm boot}$. Let us assume moreover that

$\tau_{1}\leq\tau_{0}+L$

for some arbitrary $L>0$. We have the following hierarchy of inequalities:

$\displaystyle\accentset{\scalebox{0.6}{\mbox{\tiny$(2-\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathfrak{F}}}(v,\tau_{1}),\quad\accentset{\scalebox{0.6}{\mbox{\tiny$(2-\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathfrak{E}}}(\tau_{1}),\quad
c\,{}^{\rho}\accentset{\scalebox{0.6}{\mbox{\tiny$(2-\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}$
$\displaystyle\leq$
$\displaystyle\accentset{\scalebox{0.6}{\mbox{\tiny$(2-\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathfrak{E}}}(\tau_{0})+AC\,{}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu1$}}}{\mathcal{X}}}+C\,\left({}^{\rho}\accentset{\scalebox{0.6}{\mbox{\tiny$(2-\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}\sqrt{\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{X}}}}+\sqrt{{}^{\rho}\accentset{\scalebox{0.6}{\mbox{\tiny$(2-\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}}\sqrt{{}^{\rho}\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}}\sqrt{\accentset{\scalebox{0.6}{\mbox{\tiny$(2\mkern-6.0mu-\mkern-6.0mu\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{X}}}}\right)+C\,{}^{\rho}\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}\sqrt{\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{X}}}}\sqrt{L}\,,$
(6.70)
$\displaystyle{}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(2-\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu1$}}}{\mathcal{X}}}$
$\displaystyle\lesssim$
$\displaystyle\accentset{\scalebox{0.6}{\mbox{\tiny$(2-\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu1$}}}{\mathcal{E}}}(\tau_{0})+{}^{\rho}\accentset{\scalebox{0.6}{\mbox{\tiny$(2-\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu1$}}}{\mathcal{X}}}\sqrt{\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{X}}}}+\sqrt{{}^{\rho}\accentset{\scalebox{0.6}{\mbox{\tiny$(2-\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu1$}}}{\mathcal{X}}}}\sqrt{{}^{\rho}\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu1$}}}{\mathcal{X}}}}\sqrt{\accentset{\scalebox{0.6}{\mbox{\tiny$(2\mkern-6.0mu-\mkern-6.0mu\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{X}}}}+{}^{\rho}\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}\sqrt{\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{X}}}}\sqrt{L}\,,$
(6.71)
$\displaystyle\accentset{\scalebox{0.6}{\mbox{\tiny$(1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathfrak{F}}}(v,\tau_{1}),\quad\accentset{\scalebox{0.6}{\mbox{\tiny$(1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathfrak{E}}}(\tau_{1}),\quad
c\,{}^{\rho}\accentset{\scalebox{0.6}{\mbox{\tiny$(1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}$
$\displaystyle\leq$
$\displaystyle\accentset{\scalebox{0.6}{\mbox{\tiny$(1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathfrak{E}}}(\tau_{0})+AC\,{}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu1$}}}{\mathcal{X}}}+C\,{}^{\rho}\accentset{\scalebox{0.6}{\mbox{\tiny$(1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}\sqrt{\accentset{\scalebox{0.6}{\mbox{\tiny$(1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{X}}}}+C\,{}^{\rho}\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}\sqrt{\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{X}}}}\sqrt{L}\,,$
(6.72)
$\displaystyle{}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu1$}}}{\mathcal{X}}}$
$\displaystyle\lesssim$