RM_{1}:=\int_{0}^{t}\overline{A(s)}\int_{{\mathbb{R}}^{3}}e^{-is(\langle\xi\rangle-\langle\eta\rangle+\lambda)}\langle\eta\rangle^{-1}\widetilde{f}(s,\eta)\nu(\xi,\eta)d\eta\,ds,$
(4.9a) $\displaystyle
RM_{2}:=-\int_{0}^{t}a(s)\int_{{\mathbb{R}}^{3}}e^{-is(\langle\xi\rangle+\langle\eta\rangle)}\langle\eta\rangle^{-1}\overline{\widetilde{f}}(s,\eta)\nu(\xi,\eta)d\eta\,ds$
(4.9b) $\displaystyle
RM_{3}:=\int_{0}^{t}(A(s)-B(s))\int_{{\mathbb{R}}^{3}}e^{-is(\langle\xi\rangle-\langle\eta\rangle-\lambda)}\langle\eta\rangle^{-1}\widetilde{f}(s,\eta)\nu(\xi,\eta)d\eta\,ds.$
(4.9c)

  * •

The field self-interaction terms are defined by

$\displaystyle\begin{split}&F:=-\frac{1}{4}\sum_{{\epsilon_{1},\epsilon_{2}}}{\epsilon}_{1}{\epsilon}_{2}F_{{\epsilon_{1},\epsilon_{2}}}\big{(}f_{{\epsilon}_{1}},f_{{\epsilon}_{2}}\big{)},\qquad
f_{+}=f,\,\,\,f_{-}=\bar{f},\\\
&F_{{\epsilon_{1},\epsilon_{2}}}(a,b):=\int_{0}^{t}\int_{{\mathbb{R}}^{6}}\frac{e^{-is\Phi_{\epsilon_{1},\epsilon_{2}}(\xi,\eta,\sigma)}}{\langle\eta\rangle\langle\sigma\rangle}\widetilde{a}(s,\eta)\widetilde{b}(s,\sigma)\mu(\xi,\eta,\sigma)d\eta
d\sigma\,\tau_{m}(s)ds,\end{split}$ (4.10)

where the phases are defined by

$\displaystyle\Phi_{{\epsilon_{1},\epsilon_{2}}}(\xi,\eta,\sigma)=\langle\xi\rangle-{\epsilon}_{1}\langle\eta\rangle-{\epsilon}_{2}\langle\sigma\rangle.$
(4.11)

###### Proof.

We just need to verify that the terms on the right-hand side (4.6) equal all
the terms on the right-hand side of (4.4) plus $\widetilde{g}$ as defined in
(4.5). This is easily checked since the terms in (4.7) added to
$\widetilde{g}$ match the first four integrals on the right-hand side of
(4.4), the term $F$ in (4.10) matches the last integral in (4.4), and the
terms (4.8) and (4.9) match the remaining four integrals in (4.4) (after using
$f=h-g$). ∎

###  4.2. The main bootstrap

Here is our main bootstrap Proposition for the radiation term:

######  Proposition 4.2.

Let $u=a(t)\phi+v(t)$ be a solution of (1.6), let
$f=e^{-itL}(\partial_{t}v+iLv)$, and recall the definition of $h$ in (4.5).
Let $\rho(t)=|A(t)|^{2}$, where $A$ is defined as in (1.7), and assume that
(3.17) holds, that is, for $T>0,$

$\displaystyle\sup_{t\in[0,T]}\Big{|}\big{(}1+\rho_{0}\frac{\Gamma}{\lambda}t\big{)}\frac{\rho(t)}{\rho_{0}}-1\Big{|}<\frac{1}{2}.$
(4.12)

Then, there exists $\bar{{\varepsilon}}>0$ such that, if
$C_{0}^{-1}{\varepsilon}={\varepsilon}_{0}\leq\bar{{\varepsilon}}$, with
$C_{0}>1$ (sufficiently large) and

$\displaystyle\begin{split}\sup_{t\in[0,T]}\langle
t\sqrt{\rho(t)}\rangle^{-1}\rho^{\frac{2\beta-1}{2}}(t)\|\partial_{\xi}\widetilde{h}(t)\|_{L^{2}}\leqslant
2\varepsilon^{\beta},\\\ \sup_{t\in[0,T]}\|h(t)\|_{H^{N}}\leqslant
2\varepsilon,\end{split}$ (4.13)

for some small $\beta>0$, then

$\displaystyle\begin{split}\sup_{t\in[0,T]}\langle
t\sqrt{\rho(t)}\rangle^{-1}\rho^{\frac{2\beta-1}{2}}(t)\|\partial_{\xi}\widetilde{h}(t)\|_{L^{2}}\leqslant\varepsilon^{\beta},\\\
\sup_{t\in[0,T]}\|h(t)\|_{H^{N}}\leqslant\varepsilon.\end{split}$ (4.14)

Let us now explain the structure of the proof of Proposition 4.2.

###### Proof of Proposition 4.2.

First, when the time is smaller than the natural local time of existence
$1/{\varepsilon}_{0}$ it is not hard to show the following local-in-time
bootstrap: assume that $T\leqslant 1/{\varepsilon}_{0}$, that (1.46) holds and
that

$\displaystyle\sup_{t\in[0,T]}\|\partial_{\xi}\widetilde{h}(t)\|_{L^{2}}\leqslant
2{\varepsilon}^{1-\beta},\qquad\sup_{t\in[0,T]}\|h(t)\|_{H^{N}}\leqslant
2{\varepsilon};$ (4.15)

then we have (3.19) and

$\displaystyle\sup_{t\in[0,T]}\|\partial_{\xi}\widetilde{h}(t)\|_{L^{2}}\leqslant{\varepsilon}^{1-\beta},\qquad\sup_{t\in[0,T]}\|h(t)\|_{H^{N}}\leqslant{\varepsilon}.$
(4.16)

The proof of this is much simpler than for large times. In particular no
precise information about the measure $\mu$ is needed and the crude Lemma A.3
suffices. See Subsection (9.2).

The rest of the paper is then dedicated to proving the result for the
remaining times. Note that in the regime $t>\varepsilon_{0}^{-1},$ the
bootstrap assumptions (4.13) read

$\displaystyle\sup_{t\in[{\varepsilon}_{0}^{-1},T]}t^{-1}\rho^{\beta-1}(t)\|\partial_{\xi}\widetilde{h}\|_{L^{2}_{\xi}}\leqslant
2{\varepsilon}^{\beta},\qquad\sup_{t\in[{\varepsilon}_{0}^{-1},T]}\|h(t)\|_{H^{N}}\leqslant
2{\varepsilon}.$

The weighted estimates for the three types of nonlinear terms on the right-
hand side of (4.6), that is, $S$, $M$ and $F$, are performed, respectively, in
§4.3.3, Section 8 and Section 5. Together with (1.42), the main inequalities
(4.29), (5.2) and (8.3), show that there exists a constant $C_{1}>0$ and some
$\delta>0$ such that

$\displaystyle\|\partial_{\xi}\widetilde{h}\|_{L^{2}}\leqslant{\varepsilon}_{0}+C_{1}t\rho^{1-\beta+\delta}(t){\varepsilon}^{\beta}\leqslant
C_{1}\bigg{(}{\varepsilon}_{0}+{\varepsilon}_{0}^{2\delta}\big{(}t\rho^{1-\beta}(t){\varepsilon}^{\beta}\big{)}\bigg{)}.$

Therefore, since
$t\rho^{1-\beta}(t)\geqslant(1/2){\varepsilon}_{0}^{1-2\beta}$, (for
${\varepsilon}_{0}$ small enough) it suffices to choose $\bar{{\varepsilon}}$
such that
$C_{1}\big{(}(\overline{{\varepsilon}}/C_{0})^{\beta}+\overline{{\varepsilon}}^{2\delta})<1,$
to conclude that the first inequality in (4.14) holds.

Finally, the bound on the $H^{N}$ norm in (4.14) is a consequence of
Proposition 9.1. ∎

######  Remark 4.3.

In view of the local bootstrap (4.15)-(4.16), from now on we will always
assume that $t\geqslant{\varepsilon}_{0}^{-1}\approx\rho_{0}^{-1/2}.$ Note
that past this local time of existence, the first estimate in (4.14) is
implied by $\|\partial_{\xi}\widetilde{h}\|_{L^{2}}\leqslant
t\rho^{1-\beta}(t){\varepsilon}^{\beta}$, and it suffices to show that there
exists $\delta>0$ such that $\|\partial_{\xi}\widetilde{h}\|_{L^{2}}\lesssim
t\rho^{1-\beta+\delta}(t){\varepsilon}^{\beta}$ (since
$\rho(t)\lesssim{\varepsilon}_{0}^{2}$).

###  4.3. Preliminary estimates

####  4.3.1. Consequences of the a priori bounds

We now summarize the main dispersive estimates that are a consequence of the a
priori assumption (4.13).

######  Lemma 4.4.

We have

$\displaystyle\begin{split}&\|e^{itL}g(t)\|_{L^{6}_{x}}\lesssim\rho(t)\ln(1+t)\big{(}\|(1-\Delta)\phi\|_{L^{12/5}}^{2}+\|\phi\|_{H^{3}_{x}}^{2}\big{)},\\\
&\|e^{itL}g(t)\|_{L^{3}_{x}}\lesssim\rho(t)\ln(1+t)\langle\sqrt{t}\rangle\big{(}\|(1-\Delta)\phi\|_{L^{12/5}}^{2}+\|\phi\|_{H^{3}_{x}}^{2}\big{)},\end{split}$
(4.17)

More generally,

$\displaystyle\|e^{itL}g(t)\|_{L^{p}_{x}}\lesssim\rho(t)t^{1-3(1/2-1/p)}\ln(1+t),$
(4.18)

and

$\displaystyle\begin{split}&\|e^{itL}h(t)\|_{L^{q}_{x}},\quad\|e^{itL}f(t)\|_{L^{q}_{x}}\lesssim\rho(t)^{1-\beta}t^{\frac{5(q-2)}{2q}\delta_{N}+\frac{6-q}{2q}}\varepsilon^{\beta},\end{split}$
(4.19)

for $2\leqslant q\leqslant 6.$

###### Proof.

The proofs are standard. For (4.17), splitting the integral
$\int_{0}^{t}=\int_{0}^{t-1}+\int_{t-1}^{t}$ we treat the main piece using
Lemma 2.5:

$\displaystyle\|e^{itL}g(t)\|_{L^{6}_{x}}$
$\displaystyle\lesssim\int_{0}^{t-1}\frac{1}{t-s}\rho(s)ds\|(1-\Delta)\phi\|_{12/5}^{2}$
$\displaystyle\lesssim\bigg{[}\frac{1}{t}\int_{0}^{t/2}\rho(s)ds+\rho(t)\int_{t/2}^{t-1}\frac{1}{t-s}ds\bigg{]}\|(1-\Delta)\phi\|_{12/5}^{2}.$

For the other piece we use Sobolev embedding to obtain the result. The proof
is the same for the second inequality on $g$.

The inequality on $e^{itL}h$ in (4.19) in the case $q=6$ is a consequence of
the second decay estimate from Lemma 2.5 for frequencies less than
$t^{\delta_{N}}$. The inequality for the remaining frequencies is obtained
using Sobolev’s embedding and Bernstein’s inequality with the a priori bound
on the Sobolev norm in (4.13). The estimate for $f$ is obtained from summing
the estimates for $h$ and $g$.

Finally, to prove the $L^{q}$ estimates, we use the interpolation inequality

$\|f\|_{L^{q}}\lesssim\|f\|_{L^{6}}^{\frac{3(q-2)}{2q}}\|f\|_{L^{2}}^{\frac{6-q}{2q}},$

and the conclusion follows. ∎

The above bootstrap assumptions also imply a decay bound on
$\|\partial_{s}\widetilde{f}\|_{L^{2}}:$

######  Lemma 4.5.

We have

$\displaystyle\|\partial_{s}h\|_{L^{2}_{x}}\lesssim\rho(s).$ (4.20)

Also,

$\displaystyle{\big{\|}e^{isL}\partial_{s}g\big{\|}}_{L^{p}_{x}}\lesssim\rho(s),\qquad
1\leq p\leq\infty.$ (4.21)

###### Proof.

From the definition (4.5), and using the equation
$\partial_{s}f=e^{-isL}(\partial_{s}-iL)w$, we have

$\displaystyle\begin{split}\partial_{s}\widetilde{h}(s,\xi)&=\partial_{s}\widetilde{f}-\chi_{C}(\xi)B^{2}(s)e^{-is(\langle\xi\rangle-2\lambda)}\,\widetilde{\phi^{2}}(\xi)\\\
&=e^{-is\langle\xi\rangle}\widetilde{\mathcal{F}}\Big{(}\big{(}a\phi+\dfrac{1}{L}\Im
w\big{)}^{2}\Big{)}-\chi_{C}(\xi)B^{2}(s)e^{-is(\langle\xi\rangle-2\lambda)}\,\widetilde{\phi^{2}}(\xi).\end{split}$
(4.22)

Then, from Hölder, the apriori bounds (4.13) and Lemma 4.4 we obtain

$\displaystyle\|\partial_{s}\widetilde{h}\|_{L^{2}}\lesssim\rho(s)+\rho(s)^{\frac{3-\beta}{2}}s^{3\delta_{N}}\varepsilon^{\beta}+\rho(s)^{2-\beta}s^{1/2+3\delta_{N}}\varepsilon^{2\beta},$
(4.23)

from which (4.20) directly follows.

(4.21) follows directly from the definition (4.5) and the a priori bound
(4.12) transferred to $B$ via (3.34). ∎

####  4.3.2. Basic bounds on the Fermi/bad component

We collect basic properties satisfied by $g$:

######  Lemma 4.6.

For $s\approx 2^{m}$, $m=1,2,\dots$, we have

$\displaystyle{\|\widetilde{g}(s)\|}_{L^{\infty}_{\xi}}\lesssim
m\rho(2^{m})2^{m},$ (4.24)

and, for integers $n\leq N$,

$\displaystyle{\|D^{n}\widehat{\mathcal{F}}^{-1}\widetilde{g}(s)\|}_{L^{2}}\lesssim{\varepsilon}^{1-},\qquad{\|D^{n}\widehat{\mathcal{F}}^{-1}\partial_{\xi}\widetilde{g}(s)\|}_{L^{2}}\lesssim
2^{m/2}$ (4.25)

Moreover

$\displaystyle{\|\partial_{\xi}\widetilde{g}(s,\xi)\|}_{L^{2}}\lesssim
m^{2}2^{3m/2}\,\rho(2^{m}),$ (4.26)

and

$\displaystyle{\|\partial_{\xi}\widetilde{g}(s,\xi)\|}_{L^{1}_{\xi}}\lesssim
2^{m}\rho(2^{m})m^{2}.$ (4.27)

Finally

$\displaystyle\sup_{s\approx
2^{m}}{\|\partial_{\xi}^{2}\widetilde{g}(s)\|}_{L^{2}}\lesssim\rho(2^{m})2^{3m}.$
(4.28)

###### Proof.

The property (4.24) is straightforward from the definition of $g$ in (4.5) and
using (3.17).

The first inequality in (4.25) follows from (1.55), and the second can be
proved similarly.

To prove (4.26) we split, for some $A<-C-1$ to be determined later,

$\displaystyle\partial_{\xi}\widetilde{g}(s,\xi)$
$\displaystyle=I_{A}(s,\xi)+\sum_{k\geqslant
A+1}^{-C}II_{k}(s,\xi)+\\{\textrm{easier terms}\\},$ $\displaystyle
I_{A}(s,\xi)$ $\displaystyle:=\frac{\xi}{\langle\xi\rangle}\varphi_{\leq
A}(\langle\xi\rangle-2\lambda)\int_{0}^{s}\tau
B^{2}(\tau)e^{-i\tau(\langle\xi\rangle-2\lambda)}d\tau\widetilde{\theta}(\xi)$
$\displaystyle II_{k}(s,\xi)$
$\displaystyle:=\frac{\xi}{\langle\xi\rangle}\varphi_{k}(\langle\xi\rangle-2\lambda)\int_{0}^{s}\tau
B^{2}(\tau)e^{-i\tau(\langle\xi\rangle-2\lambda)}d\tau\widetilde{\theta}(\xi),$

where easier terms refers to cases where the derivative falls on either the
$\varphi_{\leqslant-C}$ cut-off or on $\widetilde{\theta}.$ For the first
piece we have the estimate $\|I(s,\xi)\|_{L^{2}_{\xi}}\lesssim 2^{A/2+m}.$ For
the second we integrate by parts in time and bound the two resulting pieces
using Lemma 3.5 and (3.17):

$\displaystyle\|II_{k}(s,\xi)\|_{L^{2}_{\xi}}$
$\displaystyle\lesssim\bigg{\|}\frac{e^{-is(\langle\xi\rangle-2\lambda)}}{\langle\xi\rangle-2\lambda}\frac{\xi}{\langle\xi\rangle}sB^{2}(s)\widetilde{\theta}(\xi)\varphi_{k}(\langle\xi\rangle-2\lambda)\bigg{\|}_{L^{2}_{\xi}}$
$\displaystyle+\bigg{\|}\frac{\xi}{\langle\xi\rangle}\varphi_{k}(\langle\xi\rangle-2\lambda)\int_{0}^{s}\tau
B^{\prime}(\tau)B(\tau)e^{-i\tau(\langle\xi\rangle-2\lambda)}d\tau\frac{\widetilde{\theta}(\xi)}{\langle\xi\rangle-2\lambda}\bigg{\|}_{L^{2}_{\xi}}$
$\displaystyle\lesssim\bigg{(}\int_{|\langle\xi\rangle-2\lambda|\approx
2^{k}}d\xi\bigg{)}^{1/2}2^{-k}\bigg{[}s\rho(s)+\ln(s\rho_{0}+1)\bigg{]}.$

Optimizing, we choose $A=-m,$ and obtain (4.26).

For the (4.27) we have similarly

$\displaystyle\|I_{A}(s,\xi)\|_{L^{1}_{\xi}}\lesssim
2^{A+m},\qquad\|II_{k}(s,\xi)\|_{L^{1}_{\xi}}\lesssim\big{[}s\rho(s)+\ln(s\rho_{0}+1)\big{]}(-A),$

and the result follows with the same choice of $A$ as above.

The last bound is obtained more crudely just by differentiating the definition
of $\widetilde{g}$ in (4.5). ∎

####  4.3.3. Source terms

We have the following bound on the ‘good’ source terms (4.7):

######  Lemma 4.7.

Under the a priori assumptions (1.46), we have, for some
$\delta\in(0,\beta/2)$,

$\displaystyle\|\partial_{\xi}S\|_{L^{2}}\lesssim
t\rho(t)^{1-\beta+\delta}{\varepsilon}^{\beta}.$ (4.29)

###### Proof.

We start with the easier term $S_{2}$. We localize in time using (1.74) and
treat the main contribution integrating by parts in $s$ and using Lemma 3.5:

$\displaystyle\bigg{\|}(1-\chi_{C}(\xi))\int_{0}^{t}\tau_{m}(s)s\frac{\xi}{\langle\xi\rangle}\,e^{-is(\langle\xi\rangle-2\lambda)}B^{2}(s)\,\widetilde{\theta(\xi)}ds\bigg{\|}_{L^{2}}$
$\displaystyle\lesssim\bigg{\|}\tau_{m}(t)tB^{2}(t)\frac{\xi}{\langle\xi\rangle}\frac{e^{-it(\langle\xi\rangle-2\lambda)}}{\langle\xi\rangle-2\lambda}(1-\chi_{C}(\xi))\,\widetilde{\theta(\xi)}\bigg{\|}_{L^{2}}$
$\displaystyle+\bigg{\|}\frac{1-\chi_{C}(\xi)}{\langle\xi\rangle-2\lambda}\frac{\xi}{\langle\xi\rangle}\int_{0}^{t}e^{-is(\langle\xi\rangle-2\lambda)}\big{(}2sB^{\prime}(s)B(s)\tau_{m}(s)+sB^{2}(s)2^{-m}\tau^{\prime}_{m}(s)+B^{2}(s)\tau_{m}(s)\big{)}ds\,\widetilde{\theta(\xi)}\bigg{\|}_{L^{2}}$
$\displaystyle\lesssim\int_{s\approx 2^{m}}\rho(s)ds\lesssim\ln\langle
2^{m}\rho_{0}\rangle.$

Summing over $m$ gives a bound by $\log\langle t\rho_{0}^{2}\rangle$ which is
acceptable for the right-hand side of (4.29). The above integration by parts
in time is the typical strategy that we can use to treat non-resonant terms
with non-vanishing phases.

For the term $S_{1}$ in (4.7a) we can estimate

$\displaystyle\|\partial_{\xi}S_{1}\|_{L^{2}}\lesssim\sum_{m}\Bigg{\|}\frac{\xi}{\langle\xi\rangle}\int_{0}^{t}\tau_{m}(s)se^{-is(\langle\xi\rangle-2\lambda)}\big{(}A^{2}(s)-B^{2}(s)\big{)}\widetilde{\theta}(\xi)ds\Bigg{\|}_{L^{2}},$

having disregarded the term where the derivative falls on $\widetilde{\theta}$
which is faster decaying and easy to estimate. Now we write
$A^{2}-B^{2}=2B(A-B)+(A-B)^{2}.$ Using (3.34), we see that
$(A-B)^{2}=O(A^{4}),$ hence

$\displaystyle\Bigg{\|}\frac{\xi}{\langle\xi\rangle}\int_{0}^{t}s\tau_{m}(s)e^{-is(\langle\xi\rangle-2\lambda)}\big{(}A(s)-B(s)\big{)}^{2}\widetilde{\theta}(\xi)ds\Bigg{\|}_{L^{2}}\lesssim
2^{2m}\rho^{2}(2^{m}),$ (4.30)

which is sufficient. For the other term with $B(A-B)$, we also use (3.34) to
expand $A-B$ and obtain three terms. The cases of the terms
$B\overline{A}^{2}$ and $B|A|^{2}$ are non-resonant, and the associated phase
does not vanish and can be handled as $S_{2}$ above. We focus on the more
difficult term with $BA^{2}$:

$\displaystyle\|\partial_{\xi}S_{1}\|_{L^{2}}\lesssim\sum_{m}\Bigg{\|}\frac{\xi}{\langle\xi\rangle}\int_{0}^{t}s\tau_{m}(s)e^{-is(\langle\xi\rangle-3\lambda)}B(s)A^{2}(s)\widetilde{\theta}(\xi)ds\Bigg{\|}_{L^{2}}.$

As we did above, we can write $BA^{2}=B(A^{2}-B^{2})+B^{3},$ the first term
having enough decay to be treated as (4.30) above. For the remaining piece, we
can proceed exactly as in the proof of (4.26) above, replacing $2\lambda$ by
$3\lambda$, and obtain the desired bound.

For the remaining term $S_{3}$ in (4.7c), we start by replacing the $A$ terms
by $B$ terms. The error terms are then of the same type as $S_{1},$ but easier
to treat since the phases are not resonant. The terms that remain are then of
the same type as $S_{2},$ in the sense that they are quadratic in $B$ with a
non-resonant phase. Therefore, the same argument applies and we can skip the
details. ∎

##  5\. Field self-interactions

Our main goal is to prove the main weighted bound for the terms that are
quadratic in the continuous component $f$ of the solution. In (distorted)
Fourier space this is the term $F$ from (4.10) (or equivalently the last term
of (4.3)) which, recall, is given by

$\displaystyle\begin{split}&F=F(t,\xi)=\widetilde{\mathcal{F}}\Big{(}\int_{0}^{t}e^{-isL}\Big{(}\frac{\Im
w(s)}{L}\Big{)}^{2}ds\Big{)}(\xi),\qquad w=e^{itL}f.\\\ \end{split}$ (5.1)

The main result we want to prove is:

######  Proposition 5.1.

Under the a priori assumptions (4.12)-(4.13) we have

$\displaystyle{\|\partial_{\xi}F(t)\|}_{L^{2}}\lesssim{\varepsilon}^{\beta}t\rho^{1-\beta+\delta}(t),$
(5.2)

for some $\delta\in(0,\beta/2).$

Moreover, as mentioned in Section 3, we also need to prove the estimate
(3.27), which was used to obtain Lemma 3.4, and whose proof was postponed to
the current section. Expressed in terms of $F$, see (5.1), the estimate we
need is contained in the following:

######  Proposition 5.2.

Under the a priori assumptions (4.12)-(4.13) there exists $\frac{1}{2}<a<1$
such that

$\displaystyle{\big{\|}e^{itL}\widetilde{\mathcal{F}}^{-1}(F)(t)\big{\|}}_{L^{\infty}}\lesssim\frac{\rho_{0}^{a}}{\langle
t\rangle^{1+a}}.$ (5.3)

The proof of the main Propositions 5.1 and 5.2 occupies all of the current
section and Sections 6 and 7.

######  Remark 5.3.

Note that using the sharp linear estimate (2.15) (and Bernstein’s inequality),
together with (5.2), we could obtain a bound similar to (5.3) but with a decay
rate slightly slower than $\langle t\rangle^{-1}$. Unfortunately, this is
insufficient for the arguments in Section 3 where the rate needs to be
integrable. As it turns out, proving (5.3) requires additional arguments from
those in the proof of (5.2).

###  5.1. Set-up

In this set-up section we first explain how to deal with high and (very) low
frequencies. More precisely, we state results that prove (5.2) and (5.3) when
the input and output frequencies are larger than a small power of time or
smaller than a large negative power of time. This is convenient as it will
reduce the analysis to the case when all the input and output frequencies
involved are bounded above by a small power of the time variable, and are
lower bounded as well, so that summations over their respective dyadic scales
can then be done at the expense of innocuous logarithmic factors in time.

At the end of this subsection we discuss the different types of quadratic
interactions that we are going to need to estimate, and the strategy for the
proofs of the main Propositions 5.1 and 5.2.

####  5.1.1. High frequencies

We distinguish between (very) high frequency and mid-low frequencies by
introducing a parameter $M_{0}\in\mathbb{N}$ such that
$2^{M_{0}}\approx\langle s\rangle^{\delta_{N}}$

$\displaystyle M_{0}=\lfloor\delta_{N}\log_{2}\langle
s\rangle\rfloor,\qquad\delta_{N}=5/N,$ (5.4)

and splitting $F$ as follows:

$\displaystyle\begin{split}&-4F:=F_{H}+F_{L},\\\
&F_{L}(t,\xi):=\int_{0}^{t}\varphi_{\leqslant
M_{0}}(\xi)\,e^{-is\langle\xi\rangle}\widetilde{\mathcal{F}}\Big{(}\frac{P_{\leqslant
M_{0}}\Im w(s)}{L}\Big{)}^{2}ds\end{split}$ (5.5)

with the natural definition of $F_{H}$. Recall that $P_{\leqslant k}$ denotes
the distorted Littlewood-Paley projector with symbol $\varphi_{\leqslant
k}=\sum_{k^{\prime}\leqslant k}\varphi_{k^{\prime}}$; see §1.8. Using the
Fourier inversion (Proposition 2.1) as done before, we can write the above
expression in distorted Fourier space as

$\displaystyle\begin{split}&(2\pi)^{9/2}F_{\ast}=\sum_{{\epsilon_{1},\epsilon_{2}}\in\\{+,-\\}}{\epsilon}_{1}{\epsilon}_{2}F_{\ast,{\epsilon_{1},\epsilon_{2}}}\big{(}f_{{\epsilon}_{1}},f_{{\epsilon}_{2}}\big{)},\qquad
f_{+}=f,\,\,f_{-}=\bar{f},\qquad\ast\in\\{L,H\\},\end{split}$ (5.6)

where

$\displaystyle\begin{split}&F_{\ast,{\epsilon_{1},\epsilon_{2}}}(a,b):=\int_{0}^{t}\int_{{\mathbb{R}}^{6}}\frac{e^{-is\Phi_{\epsilon_{1},\epsilon_{2}}(\xi,\eta,\sigma)}}{\langle\eta\rangle\langle\sigma\rangle}\widetilde{a}(s,\eta)\widetilde{b}(s,\sigma)\mu_{\ast}(\xi,\eta,\sigma)d\eta
d\sigma\,ds,\\\
&\qquad\Phi_{\epsilon_{1},\epsilon_{2}}(\xi,\eta,\sigma):=\langle\xi\rangle-{\epsilon}_{1}\langle\eta\rangle-{\epsilon}_{2}\langle\sigma\rangle,\end{split}$
(5.7)

with

$\displaystyle\begin{split}\mu_{L}(\xi,\eta,\sigma):=\mu(\xi,\eta,\sigma)\varphi_{\leqslant
M_{0}}(\langle\xi\rangle)\varphi_{\leqslant
M_{0}}(\langle\eta\rangle)\varphi_{\leqslant
M_{0}}(\langle\sigma\rangle),\qquad\mu_{H}:=\mu-\mu_{L},\end{split}$ (5.8)

recall (4.1). Note that we are suppressing the dependence of
$F_{\ast,{\epsilon_{1},\epsilon_{2}}}(a,b)$ from its independent variables
$(t,\xi)$; we may sometimes indicate this dependence but will usually omit it,
as it will be clear from the context. Similarly, we may omit the dependence on
the inputs $(f_{{\epsilon}_{1}},f_{{\epsilon}_{2}})$. We remark that in what
follows the signs ${\epsilon_{1},\epsilon_{2}}$ will not play an essential
role, since the phases $\Phi_{{\epsilon_{1},\epsilon_{2}}}$ will have similar
properties (in particular we will show they are suitably lower bounded on a
certain region of frequency space, see Lemma 5.10), and the conjugation on $f$
(or $h$ or $g$) is irrelevant in view of Lemma 2.3.

The estimate for the high frequency part is the following:

######  Lemma 5.4.

Under the a priori assumptions (4.12)-(4.13), there exists
$\delta^{\prime}\in(0,\beta/2)$ such that

$\displaystyle\|\partial_{\xi}F_{H}(t)\|_{L^{2}}\lesssim{\varepsilon}^{\beta}t\rho^{1-\beta+\delta^{\prime}}(t),$
(5.9)

and $a\in(0,1)$ such that

$\displaystyle{\big{\|}e^{itL}\widetilde{\mathcal{F}}^{-1}(F_{H})(t)\big{\|}}_{L^{\infty}}\lesssim\frac{\rho_{0}^{a}}{\langle
t\rangle^{1+a}}.$ (5.10)

The proof of this lemma is postponed to Subsection 9.2.

####  5.1.2. Low frequencies

In view of Lemma 5.4, from (5.5)-(5.8), we see that for (5.2) it suffices to
show that for all ${\epsilon_{1},\epsilon_{2}}=\\{+,-\\}$ we have

$\displaystyle{\big{\|}\partial_{\xi}F_{L,{\epsilon_{1},\epsilon_{2}}}(t)\|}_{L^{2}}\lesssim{\varepsilon}^{\beta}t\rho^{1-\beta+\delta^{\prime}}(t);$
(5.11)

similarly, (5.3) would follow from

$\displaystyle{\big{\|}e^{itL}\widetilde{\mathcal{F}}^{-1}\left(F_{L,{\epsilon_{1},\epsilon_{2}}}\right)(t)\big{\|}}_{L^{\infty}}\lesssim\frac{\rho_{0}^{a}}{\langle
t\rangle^{1+a}}.$ (5.12)

The next result, whose proof is postponed to Subsection A.1, reduces matters
to profiles that are supported at frequencies that are not too small; more
precisely, we have:

######  Lemma 5.5.

Under the a priori assumptions (4.12)-(4.13), there exists
$\delta^{\prime}\in(0,\beta/2)$ and $a\in(0,1)$ such that

$\displaystyle\|\partial_{\xi}F_{L,{\epsilon_{1},\epsilon_{2}}}(A_{{\epsilon}_{1}},B_{{\epsilon}_{2}})(t)\|_{L^{2}}\lesssim\varepsilon^{\beta}t\rho^{1-\beta+\delta^{\prime}}(t),$
$\displaystyle{\big{\|}e^{itL}\widetilde{\mathcal{F}}^{-1}\left(F_{L,{\epsilon_{1},\epsilon_{2}}}(A_{{\epsilon}_{1}},B_{{\epsilon}_{2}})\right)(t)\big{\|}}_{L^{\infty}}\lesssim\frac{\rho_{0}^{a}}{\langle
t\rangle^{1+a}},$ $\displaystyle\mbox{for
all}\quad(A,B)\in\big{\\{}(f_{\leqslant\langle
s\rangle^{-5}},f_{\leqslant\langle s\rangle^{-5}}),\,(f_{\leqslant\langle
s\rangle^{-5}},f_{\geqslant\langle s\rangle^{-5}}),\,(f_{\geqslant\langle
s\rangle^{-5}},f_{\leqslant\langle s\rangle^{-5}})\big{\\}}.$

Here $f_{\ast}:=P_{\ast}f$, where $P$ is the distorted Littlewood-Paley
projection (see §1.8).

As a consequence of Lemma 5.5 the profiles we consider from now on carry a
localization to frequencies larger than $\langle s\rangle^{-5}.$ Furthermore,
since

$\displaystyle{\big{\|}\partial_{\xi}\big{(}\widetilde{P_{\geqslant\langle
s\rangle^{-5}}f}\big{)}\big{\|}}_{L^{2}}\lesssim{\|\partial_{\xi}\widetilde{f}\|}_{L^{2}},$

which is obtained as a direct consequence of Hardy’s inequality, we can safely
suppress the notation $P_{\geqslant\langle s\rangle^{-5}}$ on the profiles,
and still denote by $F_{L,{\epsilon_{1},\epsilon_{2}}}(t)$ the bilinear
operator as in (5.7)-(5.8) with this additional restriction to
$|\eta|,|\sigma|\gtrsim\langle s\rangle^{-5}.$

####  5.1.3. Types of quadratic interactions and organization of the proof

Since $f=h-g$, we need to estimate the quadratic terms

$\displaystyle\begin{split}F_{L,{\epsilon_{1},\epsilon_{2}}}(a,b)\qquad\mbox{with}\qquad
a\in\\{h_{{\epsilon}_{1}},g_{{\epsilon}_{1}}\\},\,\,b\in\\{h_{{\epsilon}_{2}},g_{{\epsilon}_{2}}\\}.\end{split}$
(5.13)

We then have three main kinds of interactions to analyze which we denote as
follows:

  * •

Purely Fermi-type interactions when $a,b=g_{\pm}$;

  * •

Fermi-continuous interactions when $\\{a,b\\}=\\{h_{\pm},g_{\pm}\\}$;

  * •

Purely continuous interactions when $a,b=h_{\pm}$.

These three types of interactions require different treatments and we dedicate
to each of them separate subsections within Sections 6 and 7. One important
aspect of our analysis will be the splitting of the bilinear terms (5.13) \-
more precisely of their leading order contribution, corresponding to the
leading order contribution of $\mu$ \- into a singular and a regular part; see
Proposition 5.6 below, and the consequent definition of a singular part
$F^{S}$ and a regular part $F^{R}$ in (5.46). The singular, resp. regular,
parts are treated in Section 6, resp. 7.

For the singular part of the bilinear interactions we can perform a normal
form transformation. The (time)-boundary terms are estimates in Subsection
6.2, and the three types of interactions described above are dealt with in
Subsections 6.3, 6.4 and 6.5. For the regular part, the three different kinds
of interactions are estimated in Subsections 7.2, 7.3 and 7.4.

###  5.2. Decomposition of the NSD

In the previous subsection we have reduced the problem to the mid-low
frequencies interaction term $F_{L}$; in particular, all the bilinear
operators that we are going to estimate will be supported (before time
integration) in a region where all input and output frequencies are less than
$2^{M_{0}}\approx\langle s\rangle^{\delta_{N}}$, with an additional
restriction on the inputs being projected to not-very-low frequencies via
implicit $P_{\geqslant s^{-5}}$ operators. Then, our goal is to prove (5.11)
and (5.12) for $F_{L}$, which, together with Lemmas 5.4 and 5.5, will imply
Propositions 5.1 and 5.2.

To prove our main results we first need a precise description of the
interaction of three distorted complex exponentials, encoded by the
distribution $\mu(\xi,\eta,\sigma)$, and a splitting of it into a singular and
regular part, plus a suitable remainder. This is contained in the next
proposition, which is mostly based on results from [41] (see Proposition B.1
in Appendix A):

######  Proposition 5.6.

Let $\mu$ be the distribution defined in (4.1), with $\psi$ satisfying (2.1)
for a generic potential777As already mentioned, a bound on sufficiently many
Schwartz semi-norms of $V$ is enough. For example, we can assume that $V$ is
in a weighted Sobolev space, say $\langle x\rangle^{N}V\in H^{N}$, where $N$
is the Sobolev regularity parameter in (1.45); see also Theorem 2.2.
$V\in\mathcal{S}$. Let888In our application the parameter $M_{0}\geqslant 1$
that appears here will coincide with the one in (5.4). $M_{0}$ be a fixed
parameter, and let999The parameter $N_{2}$ can be fixed large enough as a
fraction of $N$. $N_{2}$ be a sufficiently large integer.

We can decompose $\mu$ into a singular part (apex $S$), a regular part (apex
$R$) and a remainder part (apex $Re$)

$\displaystyle\mu=\mu^{S}+\mu^{R}+\mu^{Re}.$ (5.14)

These three components can be further decomposed as follows, up to irrelevant
multiplicative constants:

$\displaystyle\begin{split}&\mu^{S}(\xi,\eta,\sigma):=\delta_{0}(\xi-\eta-\sigma)+\mu_{1}^{S}(\xi,\eta,\sigma)+\mu_{2}^{S}(\xi,\eta,\sigma)+\mu_{3}^{S}(\xi,\eta,\sigma),\\\
&\mu^{R}(\xi,\eta,\sigma):=\mu_{1}^{R}(\xi,\eta,\sigma),\end{split}$ (5.15)

where the right-hand sides are defined as follows:

  * •

$\mu_{1}^{S}$ and $\mu_{1}^{R}$ are given by the formulas

$\displaystyle\mu_{1}^{\ast}(\xi,\eta,\sigma)$
$\displaystyle:=\nu_{1}^{\ast}(-\xi+\eta,\sigma)+\nu_{1}^{\ast}(-\xi+\sigma,\eta)+\overline{\nu_{1}^{\ast}(-\eta-\sigma,\xi)},\qquad\ast\in\\{S,R\\}$
(5.16)

with

$\displaystyle\begin{split}\nu_{1}^{S}(p,q)&:=\varphi_{\leqslant-
M_{0}-5}(|p|-|q|)\Big{[}\nu_{0}(p,q)\\\
&+\frac{1}{|p|}\sum_{a=1}^{N_{2}}\sum_{J\in{\mathbb{Z}}}b_{a,J}(p,q)\cdot
2^{J}K_{a}\big{(}2^{J}(|p|-|q|)\big{)}\Big{]}\end{split}$ (5.17)

with

$\displaystyle\nu_{0}(p,q):=\frac{b_{0}(p,q)}{|p|}\Big{[}i\pi\,\delta(|p|-|q|)+\mathrm{p.v.}\frac{1}{|p|-|q|}\Big{]},$
(5.18)

where $b_{0}$ is a symbol satisfying (B.5), $K_{a}$ are Schwartz functions,
$b_{a,J}$ are symbols satisfying the bounds stated in (B.7), and $\nu_{1}^{R}$
satisfies, for all $|p|\approx 2^{P}$, $|q|\approx 2^{Q}$, with $P,Q\leqslant
M_{0}$ and for all $|a|+|b|\leq N_{2}$,

$\displaystyle\begin{split}&|\nabla_{p}^{a}\nabla_{q}^{b}\nu_{1}^{R}(p,q)|\\\
&\lesssim\begin{cases}2^{-2P}\,2^{-(|a|+|b|)P}\cdot
2^{(2+|a|+|b|)5M_{0}}&\textrm{if}\quad|P-Q|<5\\\ 2^{-2(P\vee Q)}\cdot
2^{-|a|(P\vee Q)}\max\\{1,2^{(1-|b|)Q^{-}}\\}\cdot
2^{(|a|+|b|+2)5M_{0}}&\textrm{if}\quad|P-Q|\geqslant 5.\end{cases}\end{split}$
(5.19)

  * •

$\mu_{2}^{S}$ is given by the formulas

$\displaystyle\mu_{2}^{S}(\xi,\eta,\sigma)$
$\displaystyle:=\nu_{2}^{S,1}(\xi,\eta,\sigma)+\nu_{2}^{S,2}(\xi,\eta,\sigma)+\nu_{2}^{S,2}(\xi,\sigma,\eta),$
(5.20)

with

$\displaystyle\begin{split}\nu_{2}^{S,1}(\xi,\eta,\sigma):=\frac{1}{|\xi|}\sum_{i=1}^{N_{2}}\sum_{J\in{\mathbb{Z}}}\varphi_{\leqslant-
M_{0}-5}(|\xi|-|\eta|-|\sigma|)\\\ \times b_{i,J}(\xi,\eta,\sigma)\cdot
K_{i}\big{(}2^{J}(|\xi|-|\eta|-|\sigma|)\big{)},\end{split}$ (5.21)

and

$\displaystyle\begin{split}\nu_{2}^{S,2}(\xi,\eta,\sigma)=\frac{1}{|\eta|}\sum_{\epsilon\in\\{1,-1\\}}\sum_{i=1}^{N_{2}}\sum_{J\in{\mathbb{Z}}}\varphi_{\leqslant-
M_{0}-5}(|\xi|+\epsilon|\eta|-|\sigma|)\\\ \times
b_{i,J}^{\epsilon}(\xi,\eta,\sigma)\cdot
K_{i}\big{(}2^{J}(|\xi|+\epsilon|\eta|-|\sigma|)\big{)},\end{split}$ (5.22)

where $K_{i}$ are Schwartz functions and $b_{i,J}$ and $b_{i,J}^{\epsilon}$
are symbols satisfying the bounds stated in (B.13) and (B.17).

  * •

$\mu_{3}^{S}$ is given by the formula

$\displaystyle\begin{split}\mu_{3}^{S}(\xi,\eta,\sigma):=\sum_{i=0}^{N_{2}}\sum_{J\in{\mathbb{Z}}}\varphi_{\leqslant-
M_{0}-5}(|\xi|-|\eta|-|\sigma|)\\\ \times b_{i,J}(\xi,\eta,\sigma)\cdot
K_{i}\big{(}2^{J}(|\xi|-|\eta|-|\sigma|)\big{)},\end{split}$ (5.23)

where $K_{i}$ are Schwartz functions and $b_{i,J}$ are symbols satisfying the
bounds stated in (B.7).

  * •

$\mu^{Re}$ satisfies the following estimate: for all $|\xi|\approx 2^{k_{1}}$,
$|\eta|\approx 2^{k_{2}}$ and $|\sigma|\approx 2^{k_{3}}$ with
$k_{1},k_{2},k_{3}\leqslant M_{0}$ and for all $|a|+|\alpha|+|\beta|\leq
N_{2}$,

$\displaystyle|\nabla_{\xi}^{a}\nabla_{\eta}^{\alpha}\nabla_{\sigma}^{\beta}\mu^{Re}(\xi,\eta,\sigma)|$
$\displaystyle\lesssim 2^{-2\max\\{k_{1},k_{2},k_{3}\\}}\cdot
2^{-(|\alpha_{a}|+|\alpha_{e}|)\max\\{k_{1},k_{2},k_{3}\\}}$ (5.24)
$\displaystyle\cdot\max\big{(}1,2^{(1-|\alpha_{i}|)\min\\{k_{1},k_{2},k_{3}\\}}\big{)}\cdot
2^{5M_{0}(|a|+|\alpha|+|\beta|+1)},$

where $\alpha_{i}$ denotes the order of differentiation of the variable that
is located at the smallest frequency, $\alpha_{e}$ is similarly defined for
the middle frequency and $\alpha_{a}$ for the largest one.

The above proposition essentially follows from Proposition B.1, whose content
is taken from [41], and which we rewrote in the Appendix for the convenience
of the reader. Proposition B.1 gives a decomposition of $\mu(\xi,\eta,\sigma)$
in terms of the singularities in $|\xi-\eta|-|\sigma|$, $|\xi-\sigma|-|\eta|$
and so on. Understanding the precise structure of these singularities is one
of the keys to extracting oscillations in distorted frequency space from the
integrals (5.7).

Here are a few comments on the statement of Proposition 5.6.

  * •

Proposition 5.6 goes a step further compared to Proposition B.1, and adapts
the geometry of $\mu$ and its various components to the nonlinear structure of
the Klein-Gordon flow. In particular, the various pieces are decomposed into
singular and regular parts. There is also an even more regular remainder part
which we omit for the sake of this discussion.

The distinction between singular and regular parts of $\mu$, which also gives
an analogous splitting for the nonlinear terms (see (5.46)) is made so that,
on the support of the singular parts, we will be able to lower bound the
nonlinear Klein-Gordon phase $\Phi_{{\epsilon_{1},\epsilon_{2}}}$ (see (5.7));
this is done in Lemma 5.10.

  * •

The estimate (5.19), implies that, up to small losses for large frequencies
(which will be innocuous for our arguments), the regular measure
$\nu_{1}^{R}(p,q)$ behaves almost like a Coifman-Meyer multiplier times a
factor of $(|p|\vee|q|)^{-2}$, and better (upon localization) than a
multiplier in $\widehat{\mathcal{F}}^{-1}_{(x,y)\rightarrow(p,q)}L^{1}_{x,y}$
times a factor of $(|p|\vee|q|)^{-2}$. More importantly, it is not very
singular so that we can also use these estimates pointwise.

  * •

The distributions $\nu_{1}^{S}$ are the most singular and arise from the
interaction (in the definition of $\mu$) of two plane waves and one
‘perturbed’ wave $\psi(x,\zeta)-e^{ix\cdot\zeta}$; see (2.1). The
distributions (measures, in fact) $\nu_{2}^{S,1}$ and $\nu_{2}^{S,2}$ are a
little more regular since they arise from the interaction of at least two
perturbed waves. Estimates on the bilinear operators associated to these
distributions are given in Theorem 5.7. The statement also contains bilinear
estimates for the operators associated to $\mu^{R}$ and $\mu^{Re}$.

###### Proof.

We look at Proposition B.1 and extract all the distributions in the statement
from the formulas there.

We first inspect the formula for $\mu_{1}$ in (B.2), which gives

$\displaystyle\mu_{1}(\xi,\eta,\sigma)$
$\displaystyle=\nu_{1}(-\xi+\eta,\sigma)+\nu_{1}(-\xi+\sigma,\eta)+\overline{\nu_{1}(-\eta-\sigma,\xi)}$

with the decomposition of $\nu_{1}$ in (B.3), with (B.4) and (B.5), (B.6) and
(B.7), and the bounds (B.8)-(B.9). Then, with the definition (5.17)-(5.18) for
$\nu_{1}^{S}$ and letting

$\displaystyle\begin{split}\nu_{1}^{R}(p,q)&:=\varphi_{>-M_{0}-5}(|p|-|q|)\Big{[}\nu_{0}(p,q)\\\
&+\frac{1}{|p|}\sum_{a=1}^{N_{2}}\sum_{J\in{\mathbb{Z}}}b_{a,J}(p,q)\cdot
2^{J}K_{a}\big{(}2^{J}(|p|-|q|)\big{)}\Big{]}+\nu_{R}(p,q),\end{split}$ (5.25)

where $\nu_{R}(p,q)$ is the symbol satisfying (B.8)-(B.9), we have that the
sum of (5.17) and (5.25) gives us the term $\nu_{1}$ in (B.3), and therefore

$\displaystyle\mu_{1}(\xi,\eta,\sigma)=\mu_{1}^{S}(\xi,\eta,\sigma)+\mu_{1}^{R}(\xi,\eta,\sigma).$
(5.26)

The bound (5.19) for the expression in (5.25) holds as a result of (B.8)-(B.9)
(which is the desired bound for $\nu_{R}$) and since the same can be verified
directly for the first expression on the right-hand side of (5.25).

To verify the properties in the second bullet, we start by inspecting the
formulas (B.11)-(B.12) for $\nu_{2}^{1}$, with the bounds (B.13) and (B.14),
and the formulas (B.15)-(B.16) for $\nu_{2}^{2}$, with the bounds
(B.17)-(B.18). We then set

$\displaystyle\begin{split}\nu_{2}^{R,1}(\xi,\eta,\sigma):=\frac{1}{|\xi|}\sum_{i=1}^{N_{2}}\sum_{J\in{\mathbb{Z}}}\varphi_{>-M_{0}-5}(|\xi|-|\eta|-|\sigma|)\\\
\times b_{i,J}(\xi,\eta,\sigma)\cdot
K_{i}\big{(}2^{J}(|\xi|-|\eta|-|\sigma|)\big{)},\end{split}$ (5.27)
$\displaystyle\begin{split}\nu_{2}^{R,2}(\xi,\eta,\sigma):=\frac{1}{|\eta|}\sum_{\epsilon\in\\{1,-1\\}}\sum_{i=1}^{N_{2}}\sum_{J\in{\mathbb{Z}}}\varphi_{>-M_{0}-5}(|\xi|+\epsilon|\eta|-|\sigma|)\\\
\times b_{i,J}^{\epsilon}(\xi,\eta,\sigma)\cdot
K_{i}\big{(}2^{J}(|\xi|+\epsilon|\eta|-|\sigma|)\big{)},\end{split}$ (5.28)

and see that we can write

$\displaystyle\nu_{2}^{1}(\xi,\eta,\sigma)=\nu_{2}^{S,1}(\xi,\eta,\sigma)+\nu_{2}^{R,1}(\xi,\eta,\sigma)+\nu_{2,R}^{1}(\xi,\eta,\sigma)$
(5.29)

where (note the different placing of the indexes) $\nu_{2,R}^{1}$ is the
remainder in (B.11) satisfying the estimate (B.14). Similarly,

$\displaystyle\nu_{2}^{2}(\xi,\eta,\sigma)=\nu_{2}^{S,2}(\xi,\eta,\sigma)+\nu_{2}^{R,2}(\xi,\eta,\sigma)+\nu_{2,R}^{2}(\xi,\eta,\sigma)$
(5.30)

where (note again the different placing of the indexes) $\nu_{2,R}^{2}$ is the
remainder appearing at the end of the formula (B.15) which satisfies the
estimates (B.18). From (B.10) and the definitions in (5.20), we then deduce
that the $\mu_{2}$ component in (B.10) is given by

$\displaystyle\begin{split}\mu_{2}(\xi,\eta,\sigma)&=\mu_{2}^{S}(\xi,\eta,\sigma)\\\
&+\nu_{2}^{R,1}(\xi,\eta,\sigma)+\nu_{2}^{R,2}(\xi,\eta,\sigma)+\nu_{2}^{R,2}(\xi,\sigma,\eta)\\\
&+\nu_{2,R}^{1}(\xi,\eta,\sigma)+\nu_{2,R}^{2}(\xi,\eta,\sigma)+\nu_{2,R}^{2}(\xi,\sigma,\eta).\end{split}$
(5.31)

Similarly, by letting

$\displaystyle\mu_{3}^{R}(\xi,\eta,\sigma):=\sum_{i=0}^{N_{2}}\sum_{J\in{\mathbb{Z}}}\varphi_{>-M_{0}-5}(|\xi|+|\eta|-|\sigma|)b_{i,J}(\xi,\eta,\sigma)\cdot
K_{i}\big{(}2^{J}(|\xi|-|\eta|-|\sigma|)\big{)},$

we can see that the term $\mu_{3}$ in (B.19) can be written as

$\displaystyle\begin{split}\mu_{3}(\xi,\eta,\sigma)&=\mu_{3}^{S}(\xi,\eta,\sigma)+\mu_{3}^{R}(\xi,\eta,\sigma)+\mu_{3,R}(\xi,\eta,\sigma),\end{split}$
(5.32)

where $\mu_{3,R}$ (note again the different placing of the indexes) is the
last term in (B.19) and satisfies (B.22).

We then let

$\displaystyle\begin{split}\mu^{Re}(\xi,\eta,\sigma)&:=\nu_{2,R}^{1}(\xi,\eta,\sigma)+\nu_{2,R}^{2}(\xi,\eta,\sigma)+\nu_{2,R}^{2}(\xi,\sigma,\eta)\\\
&+\nu_{2}^{R,1}(\xi,\eta,\sigma)+\nu_{2}^{R,2}(\xi,\eta,\sigma)+\nu_{2}^{R,2}(\xi,\sigma,\eta)\\\
&+\mu_{3}^{R}(\xi,\eta,\sigma)+\mu_{3,R}(\xi,\eta,\sigma)\end{split}$ (5.33)

so that the identities (5.14)-(5.15) hold true in view of (B.1) and (5.26),
(5.31) and (5.32).

To conclude we need to verify the bound (5.24) for the remainder (5.33).
Notice first that (B.22) gives a stronger bound for $\mu_{3,R}$. Using the
formula (5.32) we can directly verify the bound for $\mu_{3}^{R}$. We can then
inspect the formulas (5.27) and (5.28) to check the bounds for the terms in
the second line of (5.33). Finally, for the terms on the first line of the
right-hand side of (5.33) we invoke (B.14) and (B.18). ∎

###  5.3. Bilinear operators: identities and bounds

Recall that our main term to estimate is $F_{L}$ from (5.6)-(5.7). Below we
give some useful identities and bounds for bilinear terms as in (5.7) that we
are going to use in the rest of the arguments.

First, in view of Proposition 5.6, we define several bilinear operators, which
correspond to the various pieces in which $\mu$ is decomposed. For a general
symbol $b=b(\xi,\eta,\sigma)$ let us denote

$\displaystyle T_{0}[b](g,h)(x)$
$\displaystyle:=\widehat{\mathcal{F}}^{-1}_{\xi\rightarrow
x}\iint_{{\mathbb{R}}^{3}\times{\mathbb{R}}^{3}}g(\eta)h(\sigma)\,b(\xi,\eta,\sigma)\,\delta_{0}(\xi-\eta-\sigma)\,d\eta
d\sigma,$ (5.34a) $\displaystyle T_{1}^{\ast,1}[b](g,h)(x)$
$\displaystyle:=\widehat{\mathcal{F}}^{-1}_{\xi\rightarrow
x}\iint_{{\mathbb{R}}^{3}\times{\mathbb{R}}^{3}}g(\xi-\eta)h(\sigma)\,b(\xi,\eta,\sigma)\,\nu_{1}^{\ast}(\eta,\sigma)\,d\eta
d\sigma,\quad\ast\in\\{S,R\\},$ (5.34b) $\displaystyle
T_{1}^{\ast,2}[b](g,h)(x)$
$\displaystyle:=\widehat{\mathcal{F}}^{-1}_{\xi\rightarrow
x}\iint_{{\mathbb{R}}^{3}\times{\mathbb{R}}^{3}}g(-\eta-\sigma)h(\sigma)\,b(\xi,\eta,\sigma)\,\overline{\nu_{1}^{\ast}(\eta,\xi)}\,d\eta
d\sigma,\quad\ast\in\\{S,R\\},$ (5.34c) $\displaystyle T_{2}^{S}[b](g,h)(x)$
$\displaystyle:=\widehat{\mathcal{F}}^{-1}_{\xi\rightarrow
x}\iint_{{\mathbb{R}}^{3}\times{\mathbb{R}}^{3}}g(\eta)h(\sigma)\,b(\xi,\eta,\sigma)\,\mu_{2}^{S}(\xi,\eta,\sigma)\,d\eta
d\sigma,$ (5.34d) $\displaystyle T_{3}^{S}[b](g,h)(x)$
$\displaystyle:=\widehat{\mathcal{F}}^{-1}_{\xi\rightarrow
x}\iint_{{\mathbb{R}}^{3}\times{\mathbb{R}}^{3}}g(\eta)h(\sigma)\,b(\xi,\eta,\sigma)\,\mu_{3}^{S}(\xi,\eta,\sigma)\,d\eta
d\sigma,$ (5.34e) $\displaystyle T^{Re}[b](g,h)(x)$
$\displaystyle:=\widehat{\mathcal{F}}^{-1}_{\xi\rightarrow
x}\iint_{{\mathbb{R}}^{3}\times{\mathbb{R}}^{3}}g(\eta)h(\sigma)\,b(\xi,\eta,\sigma)\,\mu^{Re}(\xi,\eta,\sigma)\,d\eta
d\sigma.$ (5.34f)

The distributions appearing above are defined in Proposition 5.6 and in its
proof; see (5.17), (5.20) and (5.23), (5.25) and (5.33).

Second, for a general symbol $b$, let us adopt the following notation for $b$
evaluated at different inputs:

$\displaystyle\begin{split}b_{1}(\xi,\eta,\sigma)&=b(\xi,\xi-\eta,\sigma),\\\
b_{1}^{\prime}(\xi,\eta,\sigma)&=b(\xi,\sigma,\xi-\eta),\\\
b_{2}(\xi,\eta,\sigma)&=b(\xi,-\eta-\sigma,\sigma).\end{split}$ (5.35)

With the definitions (5.34) and the notation (5.35), using the decomposition
from Proposition 5.6 and applying changes of variable, we see that we can
write bilinear operators of the type (5.6)-(5.7) as follows: with
$\widetilde{A_{{\epsilon}_{1}}}:=e^{is{\epsilon}_{1}\langle\xi\rangle}\widetilde{a}$
and
$\widetilde{B_{{\epsilon}_{2}}}:=e^{is{\epsilon}_{2}\langle\xi\rangle}\widetilde{b}$
we have

$\displaystyle\begin{split}&e^{is\langle\nabla\rangle}\widehat{\mathcal{F}}_{\xi\mapsto
x}^{-1}\int_{{\mathbb{R}}^{6}}e^{-is\Phi_{\epsilon_{1},\epsilon_{2}}(\xi,\eta,\sigma)}\widetilde{a}(s,\eta)\widetilde{b}(s,\sigma)\mu^{S}(\xi,\eta,\sigma)\,m(\xi,\eta,\sigma)\,d\eta
d\sigma\\\
&=T_{0}[m]\big{(}\widetilde{A_{{\epsilon}_{1}}},\widetilde{B_{{\epsilon}_{1}}}\big{)}(x)\\\
&+T_{1}^{S,1}[m_{1}]\big{(}\widetilde{A_{{\epsilon}_{1}}},\widetilde{B_{{\epsilon}_{1}}}\big{)}(x)+T_{1}^{S,1}[m_{1}^{\prime}]\big{(}\widetilde{B_{{\epsilon}_{1}}},\widetilde{A_{{\epsilon}_{1}}}\big{)}(x)+T_{1}^{S,2}[m_{2}]\big{(}\widetilde{A_{{\epsilon}_{1}}},\widetilde{B_{{\epsilon}_{1}}}\big{)}(x)\\\
&+T_{2}^{S}[m]\big{(}\widetilde{A_{{\epsilon}_{1}}},\widetilde{B_{{\epsilon}_{1}}}\big{)}(x)+T_{3}^{S}[m]\big{(}\widetilde{A_{{\epsilon}_{1}}},\widetilde{B_{{\epsilon}_{1}}}\big{)}(x),\end{split}$
(5.36)

and

$\displaystyle\begin{split}&e^{is\langle\nabla\rangle}\widehat{\mathcal{F}}_{\xi\mapsto
x}^{-1}\int_{{\mathbb{R}}^{6}}\frac{e^{-is\Phi_{\epsilon_{1},\epsilon_{2}}(\xi,\eta,\sigma)}}{\langle\eta\rangle\langle\sigma\rangle}\widetilde{a}(s,\eta)\widetilde{b}(s,\sigma)\mu^{R}(\xi,\eta,\sigma)\,m(\xi,\eta,\sigma)\,d\eta
d\sigma\\\
&=T_{1}^{R,1}[m_{1}]\big{(}\widetilde{A_{{\epsilon}_{1}}},\widetilde{B_{{\epsilon}_{1}}}\big{)}(x)+T_{1}^{R,1}[m_{1}^{\prime}]\big{(}\widetilde{B_{{\epsilon}_{1}}},\widetilde{A_{{\epsilon}_{1}}}\big{)}(x)+T_{1}^{R,2}[m_{2}]\big{(}\widetilde{A_{{\epsilon}_{1}}},\widetilde{B_{{\epsilon}_{1}}}\big{)}(x).\end{split}$
(5.37)

In particular, we can write $F_{{\epsilon_{1},\epsilon_{2}}}(a,b)$ as the time
integral of the sum of the expressions in (5.36) and (5.37) with
$m(\xi,\eta,\sigma)=\langle\eta\rangle^{-1}\langle\sigma\rangle^{-1}$, with
similar formulas for $F_{L,{\epsilon_{1},\epsilon_{2}}}$ (inserting the
cutoffs $\varphi_{\leqslant M_{0}}$).

We now state the main bilinear estimates satisfied by the operators in (5.34):

######  Theorem 5.7.

Let a symbol $b=b(\xi,\eta,\sigma)$ be given such that

$\displaystyle\textrm{supp}(b)\subseteq\big{\\{}(\xi,\eta,\sigma)\in\mathbb{R}^{9}\,:\,|\xi|+|\eta|+|\sigma|\leq
2^{A}\big{\\}},$

for some $A\geq 1$ and such that, for $|\xi|\approx 2^{K},|\eta|\approx
2^{L},|\sigma|\approx 2^{M}$,

$\displaystyle\big{|}\nabla_{\xi}^{a}\nabla_{\eta}^{\alpha}\nabla_{\sigma}^{\beta}b(\xi,\eta,\sigma)\big{|}\lesssim
2^{-|a|K-|\alpha|L-|\beta|M}\cdot
2^{(|a|+|\alpha|+|\beta|)A},\qquad|a|,|\alpha|,|\beta|\leqslant 4.$ (5.38)

Let $p,q,r\in(1,\infty),$ with

$\frac{1}{p}+\frac{1}{q}>\frac{1}{r},$

and assume there is $10A\leqslant D\leqslant 2^{A/10}$ such that

$\displaystyle\mathcal{D}(g,h):=\|g\|_{L^{2}}\|h\|_{L^{2}}+\min\big{(}\|\partial_{\xi}g\|_{L^{2}}\|h\|_{L^{2}},\|g\|_{L^{2}}\|\partial_{\xi}h\|_{L^{2}}\big{)}\leqslant
2^{D}.$ (5.39)

Then, there exists an absolute constant $C_{0}$ (for example, $C_{0}:=65$ is
suitable), such that the following bilinear bounds hold for the operators
defined in (5.34):

$\displaystyle\big{\|}T_{0}[b](g,h)\big{\|}_{L^{r}}\lesssim\|\widehat{g}\|_{L^{p}}\|\widehat{h}\|_{L^{q}}\cdot
2^{C_{0}A},$ (5.40a)
$\displaystyle\big{\|}T_{1}^{S,1}[b](g,h)\big{\|}_{L^{r}}+\big{\|}P_{K}T_{1}^{S,2}[b](g,h)\big{\|}_{L^{r}}\lesssim\|\widehat{g}\|_{L^{p}}\|\widehat{h}\|_{L^{q}}\cdot
2^{C_{0}A}+2^{-D}\mathcal{D}(g,h),$ (5.40b)
$\displaystyle\big{\|}T_{1}^{R,1}[b](g,h)\big{\|}_{L^{r}}+\big{\|}P_{K}T_{1}^{R,2}[b](g,h)\big{\|}_{L^{r}}\lesssim\|\widehat{g}\|_{L^{p}}\|\widehat{h}\|_{L^{q}}\cdot
2^{C_{0}A}$ (5.40c)
$\displaystyle\big{\|}P_{K}T_{i}^{S}[b](g,h)\big{\|}_{L^{r}}\lesssim\|\widehat{g}\|_{L^{p}}\|\widehat{h}\|_{L^{q}}\cdot
2^{C_{0}A},\qquad i=2,3,$ (5.40d)
$\displaystyle\big{\|}P_{K}T^{Re}[b](g,h)\big{\|}_{L^{r}}\lesssim\|\widehat{g}\|_{L^{p}}\|\widehat{h}\|_{L^{q}}\cdot
2^{C_{0}A}.$ (5.40e)

###### Proof.

For the convenience of the reader we include a short explanation on how these
estimates follow from (the proof of) Theorem B.2, which is in [41].

Consider first the operators $T_{1}^{i}$, $i=1,2$, as they appear in
(B.24)-(B.25). The proof of the estimates (B.32)-(B.33) is based on a
decomposition made according to the size/distance from the singularity (e.g.
depending on the size of $|\eta|-|\sigma|$ in the case of $T_{1}^{1}$). The
only difference with the operators $T_{1}^{S,i}$, $i=1,2$, is the presence of
an additional cut-off $\varphi_{\leqslant-M_{0}-5}$ in (5.17); compare with
the definition of $\nu_{1}$ in (B.3). These slight modifications do not change
the estimate for the corresponding bilinear operators, and only restrict some
indices of summation in the proof of Theorem B.2. The same holds true for the
(complementary) operators $T_{1}^{R,i}$ associated to $\nu_{1}^{R}$; see
(5.16) and (5.25).

The proofs for the other operators $T_{2}^{S}$ and $T_{3}^{S}$ are also
identical to those for the similar operators associated to (B.10) and (B.20),
since the extra cutoffs are again harmless.

Finally, note that the $\mathcal{D}$ factor is only introduced to deal with
the principal value singularity. and is absent in (5.40c), (5.40d) and
(5.40e). ∎

######  Remark 5.8.

Using Theorem 5.7 in combination with the identity (5.36), and the boundedness
of the wave operator $\widehat{\mathcal{F}}^{-1}\widetilde{\mathcal{F}}$, see
Theorem 2.2, will let us estimate the bilinear expressions that arise from
$\partial_{\xi}F_{L}$, and establish the desired bound (5.11), as well as
(5.12).

In our argument we will also need to deal with derivatives of the components
of the measure $\mu$ and use the following remark.

######  Remark 5.9 (Derivative of $\mu_{1}^{S}$).

When estimating $\partial_{\xi}$ of the expressions in (5.6) we also need to
take care of the differentiation of the measure $\mu$, and, in particular, of
the leading order singular component $\mu_{1}^{S}$, see (5.16). Notice that
derivatives of the regular component are not so harmful, essentially by
definition; see, for example, (5.19).

Note first that, for a function $b$, we can write

$\displaystyle\nabla_{\xi}b(|-\xi+\sigma|-|\eta|)$
$\displaystyle=\frac{\sigma-\xi}{|\sigma-\xi|}\,\frac{\eta}{|\eta|}\cdot\nabla_{\eta}b(|-\xi+\sigma|-|\eta|),$
$\displaystyle\nabla_{\xi}b(|\eta+\sigma|-|\xi|)$
$\displaystyle=-\frac{\xi}{|\xi|}\,\frac{\eta+\sigma}{|\eta+\sigma|}\cdot\nabla_{\eta}b(|\eta+\sigma|-|\xi|).$

Second, from (5.17) and (5.18) we see that $\mu_{1}^{S}$ is a linear
combination of terms of the form

$\displaystyle\nu_{S}(-\xi+\eta,\sigma)=\frac{1}{|-\xi+\eta|}\nu(|-\xi+\eta|-|\sigma|)m_{0}(-\xi+\eta,\sigma),$
$\displaystyle\nu_{S}(-\xi+\sigma,\eta)=\frac{1}{|-\xi+\sigma|}\nu(|-\xi+\sigma|-|\eta|)m_{0}(-\xi+\sigma,\eta),$
$\displaystyle\nu_{S}(-\eta-\sigma,\xi)=\frac{1}{|-\eta-\sigma|}\nu(|-\eta-\sigma|-|\xi|)m_{0}(-\eta-\sigma,\xi),$

where $\nu$ is some distribution (either $\delta$ or $\mathrm{p.v.}$) or a
Schwartz function, and $m_{0}$ denote suitable symbols, as described in
Proposition 5.6 above; for example, $m_{0}$ may be the coefficient/symbol
$b_{0}$ in (5.18) satisfying (B.5), or any of the other rather harmless
coefficients/symbols appearing in the statement.

Then, we have the following distributional identities for the derivatives of
these singular expressions:

$\displaystyle\begin{split}\nabla_{\xi}\nu_{S}(-\xi+\eta,\sigma)&=-\nabla_{\eta}\nu_{S}(-\xi+\eta,\sigma),\\\
\nabla_{\xi}\nu_{S}(-\xi+\sigma,\eta)&=\frac{\xi-\sigma}{|\xi-\sigma|}\,\frac{\eta}{|\eta|}\cdot\nabla_{\eta}\Big{[}\nu(|-\xi+\sigma|-|\eta|)\Big{]}\frac{m_{0}(-\xi+\sigma,\eta)}{|-\xi+\sigma|}\\\
&+\nu(|-\xi+\sigma|-|\eta|)\nabla_{\xi}\Big{[}\frac{m_{0}(-\xi+\sigma,\eta)}{|-\xi+\sigma|}\Big{]},\\\
\nabla_{\xi}\nu_{S}(-\eta-\sigma,\xi)&=-\frac{\xi}{|\xi|}\,\frac{\eta+\sigma}{|\eta+\sigma|}\cdot\nabla_{\eta}\Big{[}\nu(|\eta+\sigma|-|\xi|)\Big{]}\,\frac{m_{0}(-\eta-\sigma,\xi)}{|\eta+\sigma|}\\\
&+\nu(|\eta-\sigma|-|\xi|)\frac{\nabla_{\xi}m_{0}(-\eta-\sigma,\xi)}{|\eta+\sigma|}.\end{split}$
(5.41)

The first formula above can be used directly to integrate by parts, thus
removing the $\nabla_{\xi}$ on the distribution by transforming it into a
$\nabla_{\eta}$ derivative.

Next, let us inspect the second formula in (5.41); the analysis for the third
one is identical. We have

$\displaystyle\nabla_{\xi}\nu_{S}(-\xi+\sigma,\eta)$
$\displaystyle=\frac{\xi-\sigma}{|\xi-\sigma|}\frac{\eta}{|\eta|}\cdot\nabla_{\eta}\nu_{S}(-\xi+\sigma,\eta)+\nu_{S}^{\prime}(\xi,\eta,\sigma)$
(5.42)

where

$\displaystyle\begin{split}\nu_{S}^{\prime}(\xi,\eta,\sigma):=&-\frac{\xi-\sigma}{|\xi-\sigma|}\,\nu(|-\xi+\sigma|-|\eta|)\,\frac{\eta}{|\eta|}\cdot\frac{\nabla_{\eta}m_{0}(-\xi+\sigma,\eta)}{|-\xi+\sigma|}\\\
&+\nu(|-\xi+\sigma|-|\eta|)\nabla_{\xi}\Big{[}\frac{m_{0}(-\xi+\sigma,\eta)}{|-\xi+\sigma|}\Big{]}\end{split}$
(5.43)

In other words, $\nabla_{\xi}\nu_{S}(-\xi+\sigma,\eta)$ can be written as the
sum of two terms:

1\. $\nabla_{\eta}\nu_{S}(-\xi+\sigma,\eta)$ times harmless factors involving
symbols of Riesz transforms, and

2\. A distribution, $\nu^{\prime}_{S}$, that looks exactly like $\nu_{S}$
itself (it has the same singular structure) up to possible extra singular
factor of $1/|\eta|$ or $1/|-\xi+\sigma|$ (these have the same size on the
support of the $\nu_{S}$-type distributions under consideration).

The first term will be handled by integrating by parts in $\eta$, and we will
detail how to bound the resulting contributions. The presence of the symbols
of the Riesz transforms is harmless since we will always be in a situation
where the respective variables are localized at fixed dyadic scales. Also, the
differentiation of the factor $\eta/|\eta|$ upon integration by parts can be
handled via Hardy’s inequality; in particular it has the same effect of
differentiating the input profile, which is already included in our treatment
in view of the integration by parts in $\eta$ just described above.

The operators associated to the distribution $\nu_{S}^{\prime}$ satisfy the
same estimates of the operators associated to $\nu_{S}$ up possible extra
singular factors of $2^{-K}$ where $2^{K}$ is the size of the input variables
$|-\xi+\sigma|\approx|\eta|$. Once again, this extra factor can be handled
through Hardy’s inequality. Therefore, we can systematically disregard the
remainder-type distributions $\nu^{\prime}_{S}$. The above considerations show
that we can effectively utilize the identity

$\displaystyle\begin{split}&\nabla_{\xi}\mu_{1}^{S}(\xi,\eta,\sigma)\approx\nabla_{\eta}\mu_{1}^{S}(\xi,\eta,\sigma)\approx\nabla_{\sigma}\mu_{1}^{S}(\xi,\eta,\sigma),\end{split}$
(5.44)

where $\approx$ is up to harmless terms as described above.

Finally, since we can write

$\displaystyle\nabla_{\xi}b(|\xi|+\epsilon_{1}|\eta|+\epsilon_{2}|\sigma|)$
$\displaystyle=\epsilon_{1}\frac{\xi}{|\xi|}\frac{\eta}{|\eta|}\cdot\nabla_{\eta}b(|\xi|+\epsilon_{1}|\eta|+\epsilon_{2}|\sigma|)$
$\displaystyle=\epsilon_{2}\frac{\xi}{|\xi|}\frac{\sigma}{|\sigma|}\cdot\nabla_{\sigma}b(|\xi|+\epsilon_{1}|\eta|+\epsilon_{2}|\sigma|),$

we deduce, in the same sense of (5.44), that

$\displaystyle\begin{split}&\nabla_{\xi}\mu_{i}^{S}(\xi,\eta,\sigma)\approx\nabla_{\eta}\mu_{i}^{S}(\xi,\eta,\sigma)\approx\nabla_{\sigma}\mu_{i}^{S}(\xi,\eta,\sigma),\qquad
i=2,3.\end{split}$ (5.45)

###  5.4. Singular vs. Regular decomposition of nonlinear terms

Consistently with the decomposition in Proposition 5.6, we now split the
nonlinear terms $F_{L}$ (which we will often just call $F$ for better
legibility). We write, see (5.6)-(5.8), and recall the mid-low frequencies
localization,

$\displaystyle\begin{split}&F_{L}=F^{S}+F^{R}+F^{Re},\qquad
F^{\ast}=\sum_{{\epsilon_{1},\epsilon_{2}}}{\epsilon}_{1}{\epsilon}_{2}F_{{\epsilon_{1},\epsilon_{2}}}^{\ast}\big{(}f_{{\epsilon}_{1}},f_{{\epsilon}_{2}}\big{)},\\\
&F_{{\epsilon_{1},\epsilon_{2}}}^{\ast}(a,b):=\int_{0}^{t}\int_{{\mathbb{R}}^{6}}\frac{e^{-is\Phi_{\epsilon_{1},\epsilon_{2}}(\xi,\eta,\sigma)}}{\langle\eta\rangle\langle\sigma\rangle}\widetilde{a}(s,\eta)\widetilde{b}(s,\sigma)\\\
&\qquad\qquad\qquad\times\varphi_{\leqslant
M_{0}}(\langle\xi\rangle)\varphi_{\leqslant
M_{0}}(\langle\eta\rangle)\varphi_{\leqslant
M_{0}}(\langle\sigma\rangle)\,\mu^{\ast}(\xi,\eta,\sigma)\,d\eta d\sigma
ds,\qquad\ast\in\\{S,R,Re\\}.\end{split}$ (5.46)

The next lemma shows that, on the support of the singular distribution
$\mu^{S}$ the phases of the operators (5.13) are lower bounded, up to a small
loss depending on the high-low frequencies cutoff at scale $2^{M_{0}}$.

######  Lemma 5.10.

On the support of the integral defining $F^{S}$, see (5.46) and the definition
of $\mu^{S}$ in Proposition 5.6, we have (recall also the notation (5.7))

$\displaystyle\big{|}\Phi_{\epsilon_{1},\epsilon_{2}}(\xi,\eta,\sigma)\big{|}\geqslant\frac{1}{8}\frac{1}{\langle\xi\rangle+\langle\eta\rangle+\langle\sigma\rangle},$
(5.47)

for all ${\epsilon}_{1},{\epsilon}_{2}\in\\{+,-\\}$. Moreover we have the
(weak) symbol type bounds

$\displaystyle\bigg{|}\varphi_{\leqslant
M_{0}}(\langle\xi\rangle)\varphi_{\leqslant
M_{0}}(\langle\eta\rangle)\varphi_{\leqslant
M_{0}}(\langle\sigma\rangle)\nabla_{\xi}^{a}\nabla_{\eta}^{\alpha}\nabla_{\sigma}^{\beta}\frac{1}{\Phi_{\epsilon_{1},\epsilon_{2}}(\xi,\eta,\sigma)}\bigg{|}\lesssim
2^{M_{0}(|a|+|\alpha|+|\beta|+1)}.$ (5.48)

######  Remark 5.11.

The bound (5.48) is far from optimal since we allow growth for large
frequencies, but it suffices for our purposes since frequencies are not very
large compared to the time variable. Moreover, we need to allow this type of
weak symbol bounds in any case when we differentiate the coefficients
appearing in the distribution $\mu^{S}$.

###### Proof.

The estimate (5.47) in the case $({\epsilon_{1},\epsilon_{2}})=(-,-)$ is
obvious. To prove (5.47) on the support of $\delta_{0}(\xi-\eta-\sigma)$ for
the case $({\epsilon_{1},\epsilon_{2}})=(-,+)$ we write

$\displaystyle\Phi_{-,+}(\xi,\eta,\xi-\eta)$
$\displaystyle=\langle\xi\rangle+\langle\xi-\eta\rangle-\langle\eta\rangle\geqslant\frac{1+2\big{(}-|\xi||\xi-\eta|+\langle\xi-\eta\rangle\langle\xi\rangle\big{)}}{\langle\xi\rangle+\langle\xi-\eta\rangle+\langle\eta\rangle}$
$\displaystyle=$
$\displaystyle\frac{1}{\langle\xi\rangle+\langle\xi-\eta\rangle+\langle\eta\rangle}\bigg{(}1+2\frac{1+|\xi-\eta|^{2}+|\xi|^{2}}{|\xi||\xi-\eta|+\langle\xi\rangle\langle\xi-\eta\rangle}\bigg{)}$
$\displaystyle\geqslant$
$\displaystyle\frac{1}{\langle\xi\rangle+\langle\xi-\eta\rangle+\langle\eta\rangle}.$

(5.47) for the case $({\epsilon_{1},\epsilon_{2}})=(+,-)$ follows by
exchanging the roles of $\eta$ and $\xi-\eta$, and for the case
$({\epsilon_{1},\epsilon_{2}})=(+,+)$ by exchanging the role of $\xi$ and
$\eta$.

Let us now look at the case of the principal value distribution $\mu_{1}^{S}$,
see (5.16)-(5.18). We begin with the estimate on the support of
$\nu_{1}^{S}(-\xi+\eta,\sigma)$. In this region we have

$\displaystyle|\langle\xi-\eta\rangle-\langle\sigma\rangle|\leqslant||\eta-\xi|-|\sigma||\leqslant
c2^{-M_{0}}\leqslant c\langle\max(|\xi|,|\eta|,|\sigma|)\rangle^{-1},$

for some $c\leqslant 1/8$, so that

$\displaystyle|\langle\xi\rangle+\langle\xi-\eta\rangle+\langle\eta\rangle|\leqslant|\langle\xi\rangle+\langle\eta\rangle+\langle\sigma\rangle|+c\langle\max(|\xi|,|\eta|,|\sigma|)\rangle^{-1}\leqslant
2|\langle\xi\rangle+\langle\eta\rangle+\langle\sigma\rangle|.$

For the case $({\epsilon_{1},\epsilon_{2}})=(-,+)$, using the inequalities
above, we have

$\displaystyle\big{|}\Phi_{-,+}(\xi,\eta,\sigma)\big{|}$
$\displaystyle\geqslant\big{|}\langle\xi\rangle+\langle\eta\rangle-\langle\xi-\eta\rangle\big{|}-\big{|}\langle\xi-\eta\rangle-\langle\sigma\rangle\big{|}$
$\displaystyle\geqslant\frac{1}{\langle\xi\rangle+\langle\xi-\eta\rangle+\langle\eta\rangle}-c\langle\max(|\xi|,|\eta|,|\sigma|)\rangle^{-1}$
$\displaystyle\geqslant\frac{1}{4}\frac{1}{\langle\xi\rangle+\langle\eta\rangle+\langle\sigma\rangle}.$

Since $\Phi_{+,+}(\xi,\eta,\sigma)=-\Phi_{+,-}(\eta,\xi,\sigma)$, we are left
with the case $({\epsilon_{1},\epsilon_{2}})=(+,+)$. We can use an argument
similar to the one above by writing
$\Phi_{+,+}(\xi,\eta,\sigma)=\Phi_{+,+}(\xi,\eta,\xi-\eta)-\langle\sigma\rangle+\langle\xi-\eta\rangle$,
and using the lower bound (5.47) with $\sigma=\xi-\eta$ just proved above.

The proof of (5.47) on the support of $\nu_{S}(-\xi+\sigma,\eta)$ follows
similarly by writing

$\displaystyle\big{|}\Phi_{-,+}(\xi,\eta,\sigma)\big{|}$
$\displaystyle\geqslant\big{|}\Phi_{-,+}(\xi,\xi-\sigma,\sigma)\big{|}-\big{|}\langle\eta\rangle-\langle\xi-\sigma\rangle\big{|}$

and using the bound (5.47) with $\eta=\xi-\sigma$ and
$\big{|}\langle\eta\rangle-\langle\xi-\sigma\rangle\big{|}\leqslant
c\langle\max(|\xi|,|\eta|,|\sigma|)\rangle^{-1}$ on the support of the
distribution. Exchanging the roles of $\xi$ and $\sigma$ proves the desired
lower bound for $\Phi_{+,+}(\xi,\eta,\sigma)=-\Phi_{-,+}(\sigma,\eta,\xi)$.
The remaining case of $\Phi_{+,-}$ can be treated relying again on the lower
bound for $\Phi_{+,-}(\xi,\xi-\sigma,\sigma)$. Also on the support of
$\nu_{S}(-\eta-\sigma,\xi)$ analogous arguments apply.

Now we prove (5.47) on the support of $\mu_{i}^{S},i=2,3$; see (5.20) and
(5.23). We start with the basic inequality

$\displaystyle\langle\eta\rangle+\langle\sigma\rangle-\langle|\eta|+|\sigma|\rangle$
$\displaystyle=\langle\sigma\rangle-\frac{|\sigma|^{2}+2|\eta||\sigma|}{\langle\eta\rangle+\langle|\eta|+|\sigma|\rangle}$
$\displaystyle=\frac{1}{\langle\eta\rangle+\langle|\eta|+|\sigma|\rangle}\Bigg{(}\frac{1+|\sigma|^{2}+|\eta|^{2}}{\langle\sigma\rangle\langle\eta\rangle+|\sigma||\eta|}+\frac{1+|\sigma|^{2}+||\eta|+|\sigma||^{2}}{\langle\sigma\rangle\langle|\eta|+|\sigma|\rangle+|\sigma|(|\sigma|+|\eta|)}\Bigg{)}$
$\displaystyle\geqslant\frac{1}{4}\frac{1}{\langle\eta\rangle+\langle\sigma\rangle+\langle|\eta|+|\sigma|\rangle}.$

Next, note that on the support of $\mu_{i}^{S}$ there exist
$\epsilon_{1}^{\prime},\epsilon_{2}^{\prime}\in\\{+,-\\},(\epsilon_{1}^{\prime},\epsilon_{2}^{\prime})\neq(+,+)$
such that
$\big{|}|\xi|+\epsilon_{1}^{\prime}|\eta|+\epsilon_{2}^{\prime}|\sigma|\big{|}\leqslant
c2^{-M_{0}}$ for some small constant $c$. Starting with the case
$\epsilon_{1}^{\prime}=\epsilon_{2}^{\prime}=-$, we write that

$\displaystyle|\Phi_{+,+}(\xi,\eta,\sigma)|$
$\displaystyle\geqslant\big{|}\langle|\eta|+|\sigma|\rangle+\epsilon_{1}\langle\eta\rangle+\epsilon_{2}\langle\sigma\rangle\big{|}-\big{|}\big{\langle}\xi\rangle-\langle|\eta|+|\sigma|\rangle\big{|}$
$\displaystyle\geqslant\frac{1}{4}\frac{1}{\langle\eta\rangle+\langle\sigma\rangle+\langle|\eta|+|\sigma|\rangle}-c2^{-M_{0}}$
$\displaystyle\geqslant\frac{1}{8}\frac{1}{\langle\xi\rangle+\langle\eta\rangle+\langle\sigma\rangle}.$

The other cases are deduced from this computation after permutation of the
variables. To treat the case $({\epsilon}_{1},{\epsilon}_{2})=(-,+)$ we
exchange the roles of variables $\xi$ and $\sigma$ and write, using the
previous inequality,

$\displaystyle|\Phi_{-,+}(\xi,\eta,\sigma)|=|\Phi_{+,+}(\sigma,\eta,\xi)|\geqslant\frac{1}{8}\frac{1}{\langle\xi\rangle+\langle\eta\rangle+\langle\sigma\rangle}.$

To treat the case $({\epsilon}_{1},{\epsilon}_{2})=(+,-)$ we write
$|\Phi_{+-}(\xi,\eta,\sigma)|=|\Phi_{++}(\eta,\xi,\sigma)|$ and obtain the
result exchanging the roles of $\xi$ and $\eta.$ The other combinations of
signs ${\epsilon}_{1}^{\prime},{\epsilon}_{2}^{\prime}$ are obtained
similarly.

The weak symbol bound (5.48) directly follows from (5.47), simply by noting
the basic fact that $\nabla_{x}^{a}\langle x\rangle$ is uniformly bounded. ∎

##  6\. Singular quadratic terms

The goal of this section is to estimate as in (5.11) and (5.12) the singular
terms $F^{S}$ defined in (5.46).

###  6.1. Set-up

We begin by decomposing the integrals inserting a time localization (see the
notation in §1.8):

$\displaystyle\begin{split}F^{S}_{{\epsilon_{1},\epsilon_{2}}}(G,H)&=\sum_{m=0}^{L+1}F^{S}_{{\epsilon_{1},\epsilon_{2}},m}(G,H)\\\
F^{S}_{{\epsilon_{1},\epsilon_{2}},m}(G,H)&:=\int_{0}^{t}\tau_{m}(s)\int_{{\mathbb{R}}^{6}}\frac{e^{-is\Phi_{\epsilon_{1},\epsilon_{2}}(\xi,\eta,\sigma)}}{\langle\eta\rangle\langle\sigma\rangle}\widetilde{G}(s,\eta)\widetilde{H}(s,\sigma)\\\
&\qquad\times\varphi_{\leqslant M_{0}}(\langle\eta\rangle)\varphi_{\leqslant
M_{0}}(\langle\sigma\rangle)\varphi_{\leqslant
M_{0}}(\langle\xi\rangle)\mu^{S}(\xi,\eta,\sigma)d\eta d\sigma ds,\end{split}$
(6.1)

recall (5.4).

######  Remark 6.1 (Notation simplification).

We may sometime dispense of some of the indexes, such as the ‘$m$’ indicating
the dyadic time scale in our notation for the quadratic terms (6.1), and for
similar ones, in order to improve the legibility. Also, since all bounds are
obvious for $m=0$ (or finite fixed $m$), we may assume without loss of
generality that $m\geqslant 1$.

Since the signs ${\epsilon_{1},\epsilon_{2}}$ also play a minor role we may
sometimes omit them too. We are also often omitting the dependence on
$(t,\xi)$ when this is obvious, for example in the case of expressions like
the one on the left-hand side of (6.1). We may adopt other similar
simplifications in the course of our argument to lighten up the notation.

We learn from the inequalities in Lemma 5.10 that the phases in the operators
$F^{S}_{{\epsilon_{1},\epsilon_{2}},m}$ are lower bounded; more precisely they
are lower bounded by $C2^{-M_{0}}$. Therefore, we can integrate by parts in
$s$ to see that

$\displaystyle
F^{S}_{{\epsilon_{1},\epsilon_{2}},m}(G,H)=T_{m}^{{\epsilon_{1},\epsilon_{2}},(1)}(G,H)+T_{m}^{{\epsilon_{1},\epsilon_{2}},(2)}(G,H)+\mbox{`symmetric'},$
(6.2)

where we defined

$\displaystyle T^{{\epsilon_{1},\epsilon_{2}},(1)}_{m}(G,H)$
$\displaystyle:=\int_{0}^{t}\tau_{m}(s)\int_{{\mathbb{R}}^{6}}\frac{e^{-is\Phi_{\epsilon_{1},\epsilon_{2}}(\xi,\eta,\sigma)}}{i\Phi_{\epsilon_{1},\epsilon_{2}}(\xi,\eta,\sigma)\langle\eta\rangle\langle\sigma\rangle}\varphi_{\leqslant
M_{0}}(\langle\eta\rangle)\varphi_{\leqslant M_{0}}(\langle\sigma\rangle)$
(6.3)
$\displaystyle\qquad\times\partial_{s}\widetilde{G}(s,\eta)\,\widetilde{H}(s,\sigma)\,\varphi_{\leqslant
M_{0}}(\langle\xi\rangle)\,\mu^{S}(\xi,\eta,\sigma)d\eta d\sigma ds,$

‘symmetric’ denotes an analogous term where $\partial_{s}$ hits
$\widetilde{H}$ instead of $\widetilde{G}$, and

$\displaystyle T^{{\epsilon_{1},\epsilon_{2}},(2)}_{m}(G,H)$
$\displaystyle:=\tau_{m}(t)\int_{{\mathbb{R}}^{6}}\frac{e^{-it\Phi_{\epsilon_{1},\epsilon_{2}}(\xi,\eta,\sigma)}}{-i\Phi_{\epsilon_{1},\epsilon_{2}}(\xi,\eta,\sigma)\langle\eta\rangle\langle\sigma\rangle}\varphi_{\leqslant
M_{0}}(\langle\eta\rangle)\varphi_{\leqslant M_{0}}(\langle\sigma\rangle)$
(6.4)
$\displaystyle\qquad\times\widetilde{G}(t,\eta)\widetilde{H}(t,\sigma)\,\varphi_{\leqslant
M_{0}}(\langle\xi\rangle)\,\mu^{S}(\xi,\eta,\sigma)d\eta d\sigma.$

The first operator is a bulk term (an integral over space and time), while the
second is a (time-)boundary term. We can obviously disregard the ‘symmetric’
term without loss of generality.

To obtain the desired bounds in Propositions 5.1 and 5.2 for the singular
bilinear terms $F^{S}$ in (5.46), also according to (6.1) and (6.2)-(6.4), it
then suffices to prove

######  Proposition 6.2 (Estimates for the Singular part).

Under the a priori assumptions (4.13), for all $m=0,1,\dots,L+1$, for all
${\epsilon}_{1},{\epsilon}_{2}\in\\{+,-\\}$, and for $j=1,2$, we have

$\displaystyle{\big{\|}\partial_{\xi}T^{{\epsilon_{1},\epsilon_{2}},(j)}_{m}(G_{{\epsilon}_{1}},H_{{\epsilon}_{2}})\big{\|}}_{L^{2}}\lesssim{\varepsilon}^{\beta}2^{m}\rho^{1-\beta+\delta^{\prime}}(2^{m}),\qquad
G,H\in\\{g,h\\},$ (6.5)

for some $\delta^{\prime}\in(0,\beta/2).$

Moreover, there exists $0<a<1$ such that, for $j=1,2$,

$\displaystyle{\left\|e^{itL}\widetilde{\mathcal{F}}^{-1}_{\xi\mapsto
x}\big{[}T^{{\epsilon_{1},\epsilon_{2}},(j)}_{m}(G_{{\epsilon}_{1}},H_{{\epsilon}_{2}})\big{]}\right\|}_{L^{\infty}_{x}}\lesssim\frac{\rho_{0}^{a}}{\langle
t\rangle^{1+a+\delta^{\prime}}},\qquad G,H\in\\{g,h\\}.$ (6.6)

Let us introduce the following convenient quantity:

$\displaystyle
Z=Z_{\beta}({\varepsilon},m):=\big{(}2^{m}\rho(2^{m})\big{)}^{1-\beta}\big{(}2^{m}\varepsilon\big{)}^{\beta}.$
(6.7)

Recall that $\rho(2^{m})\approx\min(2^{-m},{\varepsilon}^{2})$ so that we have
that $Z\approx{\varepsilon}^{\beta}t^{\beta}$ if $2^{m}\approx
t\gtrsim{\varepsilon}^{-2}$, while $Z\approx{\varepsilon}^{2-\beta}t$ if
$t\lesssim{\varepsilon}^{-2}$.

######  Remark 6.3 (About (6.7), (6.5) and “acceptable” upperbounds).

First note that the a priori assumption (1.47) can be written as follows: for
all $t\approx 2^{m}$, with
$2^{m}\gtrsim\rho_{0}^{-1/2}\approx{\varepsilon}^{-1}$ (i.e., past the trivial
local-existence time)

$\displaystyle{\big{\|}\nabla_{\xi}\widetilde{h}(t)\big{\|}}_{L^{2}}\leqslant
C{\varepsilon}^{\beta}2^{m}\rho^{1-\beta}(2^{m})=CZ_{\beta}({\varepsilon},m).$
(6.8)

For $t\approx 2^{m}$, the right-hand side of (6.5) in terms of $Z$ reads

$\displaystyle{\varepsilon}^{\beta}t\rho^{1-\beta+\delta^{\prime}}(t)\approx
Z\cdot\rho^{\delta^{\prime}}(t)$ (6.9)

In the sequel we will encounter two types of “acceptable” upperbounds for the
left-hand side of (6.5):

  * •

Bounds by $2^{-am}Z$ for some $a>0,$ which are stronger than the right-hand
side of (5.11) for $a>2\delta^{\prime}$, see (6.9), and since we will only
consider times past $\rho_{0}^{-1/2}$; see Remark 4.3 which deals with the
local existence times $t\lesssim\rho_{0}^{-1/2}\approx{\varepsilon}^{-1}$.

  * •

Bounds like $2^{-am}Z^{2}$ which are again stronger than the right-hand side
of (6.5) provided $a>\beta+\delta^{\prime}$, since $Z\lesssim 2^{\beta
m}{\varepsilon}^{\beta}$, and we can choose $2\delta^{\prime}<\beta$.

Notice that we can pick $\beta$ and $\delta^{\prime}$ arbitrarily small, and,
in particular, small enough depending on the exponent ‘$a$’ appearing in the
two types of bounds described above, which we will obtain from the nonlinear
estimates. A convenient choice is to let $2\delta^{\prime}<\beta<a/2$.

To prove (6.6) in the case of $j=1$, that is, for the bulk operator (6.3), we
will use the following result:

######  Lemma 6.4.

Assume that for some $a\ll 1$ such that $a>5\delta_{N}$ we have

$\displaystyle{\left\|\widehat{\mathcal{F}}^{-1}_{\xi\mapsto
x}I(s)\right\|}_{L^{8/7}_{x}}\lesssim\frac{\rho_{0}^{a}}{\langle
s\rangle^{1+2a}},$ (6.10)

then

$\displaystyle{\Big{\|}e^{itL}\widetilde{\mathcal{F}}^{-1}_{\xi\mapsto
x}\int_{0}^{t}\varphi_{\leqslant\delta_{N}m+5}(\xi)e^{-is\langle\xi\rangle}I(s,\xi)\,\tau_{m}(s)ds\Big{\|}}_{L^{\infty}_{x}}\lesssim\frac{\rho_{0}^{a}}{\langle
t\rangle^{1+a+\delta^{\prime}}},$ (6.11)

for $\delta^{\prime}<\delta_{N}$.

We will apply this Lemma with

$\displaystyle I(t)$
$\displaystyle=e^{it\langle\xi\rangle}\partial_{t}T^{{\epsilon_{1},\epsilon_{2}},(1)}_{m}(G_{{\epsilon}_{1}},H_{{\epsilon}_{2}})(t),$
(6.12)

so that, after verifying the bound (6.10) for this choice of $I$, and using
the fact that for $s\approx 2^{m}$ the frequency integral in (6.3) is already
localized in $|\xi|\leqslant 2^{\delta_{N}m+3}$, i.e.

$\partial_{t}T^{{\epsilon_{1},\epsilon_{2}},(1)}_{m}(s)=\varphi_{\leqslant\delta_{N}m+5}(\xi)\partial_{t}T^{{\epsilon_{1},\epsilon_{2}},(1)}_{m}(s),$

we will obtain the desired (6.6).

###### Proof of Lemma 6.4.

Using the (distorted) Bernstein’s inequality (with $|\xi|\lesssim\langle
t\rangle^{\delta_{N}}$), the $L^{8}$ decay estimate from Lemma 2.5, and
Bernstein again, we find that

$\displaystyle{\Big{\|}e^{itL}\int_{0}^{t}\widetilde{\mathcal{F}}_{\xi\mapsto
x}^{-1}\big{(}\varphi_{\leqslant\delta_{N}m+5}(\xi)e^{-is\langle\xi\rangle}I(s,\xi)\big{)}\,\tau_{m}(s)ds\Big{\|}}_{L^{\infty}_{x}}$
$\displaystyle\lesssim\langle
t\rangle^{3\delta_{N}/8}{\Big{\|}\int_{0}^{t}e^{i(t-s)L}P_{\leqslant\delta_{N}m+5}\widetilde{\mathcal{F}}_{\xi\mapsto
x}^{-1}\big{(}I(\xi,s)\big{)}\,\tau_{m}(s)ds\Big{\|}}_{L^{8}_{x}}$
$\displaystyle\lesssim\langle
t\rangle^{2\delta_{N}}\int_{0}^{t}\frac{1}{\langle t-s\rangle^{9/8}}\langle
s\rangle^{2\delta_{N}}{\big{\|}\widetilde{\mathcal{F}}_{\xi\mapsto
x}^{-1}I(\xi,s)\big{\|}}_{L^{8/7}_{x}}\,\tau_{m}(s)ds.$

Notice that in the inequality above we have also used Bernstein to deal with
the integral from $t-1$ to $t$, passing from $L^{8}_{x}$ to $L_{x}^{2}$, using
Minkowski, the unitarity of $e^{itL}$ and then Bernstein again from
$L^{2}_{x}$ to $L^{8/7}_{x}$. Then we conclude using the boundedness of wave
operators, the assumption (6.10) and the fact that

$\displaystyle\langle t\rangle^{2\delta_{N}}\int_{0}^{t}\frac{1}{\langle
t-s\rangle^{9/8}}\langle s\rangle^{2\delta_{N}}\frac{1}{\langle
s\rangle^{1+2a}}ds\lesssim\frac{1}{\langle
t\rangle^{1+2a-4\delta_{N}}}\lesssim\frac{1}{\langle
t\rangle^{1+a+\delta^{\prime}}},$

for $\delta^{\prime}+4\delta_{N}<a$. ∎

Lemma 6.4 allows us to systematically reduce the proof of (6.6) to
$L^{8/7}_{x}$ bounds for the inverse transform of the right-hand side of
(6.12), that is:

$\displaystyle\begin{split}\tau_{m}(s)\widehat{\mathcal{F}}^{-1}&\int_{{\mathbb{R}}^{6}}\frac{\varphi_{\leqslant
M_{0}}(\langle\xi\rangle)\varphi_{\leqslant
M_{0}}(\langle\eta\rangle)\varphi_{\leqslant
M_{0}}(\langle\sigma\rangle)}{\Phi_{\epsilon_{1},\epsilon_{2}}(\xi,\eta,\sigma)\langle\eta\rangle\langle\sigma\rangle}\,e^{i{\epsilon}_{1}s\langle\eta\rangle}e^{i{\epsilon}_{2}s\langle\sigma\rangle}\\\
&\times\partial_{s}\widetilde{G_{{\epsilon}_{1}}}(s,\eta)\,\widetilde{H_{{\epsilon}_{2}}}(s,\sigma)\mu^{S}(\xi,\eta,\sigma)d\eta
d\sigma,\qquad G,H\in\\{g,h\\}.\end{split}$ (6.13)

Note that this expression gives bilinear terms like those appearing in (5.34),
where the inputs are (up to complex conjugation) of the form
$(\widetilde{\mathcal{F}}(e^{-isL}\partial_{s}G),\widetilde{\mathcal{F}}(e^{-isL}H))$.
Therefore, since $e^{-isL}\partial_{s}g$ and $e^{-isL}\partial_{s}h$ are
quadratic in $A$ and $f$, and in view of the product estimates of Theorem 5.7
the terms (6.13) should be essentially thought of as cubic terms in $A$ and
$f$.

A similar lemma that we will use to prove (6.6) for the boundary-type
operators (6.4) is the following:

######  Lemma 6.5.

Assume that

$\displaystyle{\left\|\widehat{\mathcal{F}}^{-1}_{\xi\mapsto
x}J(t)\right\|}_{L^{8/7}_{x}}\lesssim\rho_{0}^{a},$ (6.14)

then, for $\delta_{N}$ small enough, one has

$\displaystyle{\Big{\|}e^{itL}\widetilde{\mathcal{F}}^{-1}_{\xi\mapsto
x}\varphi_{\leqslant\delta_{N}\log_{2}\langle
t\rangle+10}(\xi)J(t,\xi)\Big{\|}}_{L^{\infty}_{x}}\lesssim\frac{\rho_{0}^{a}}{\langle
t\rangle^{10/9}}.$ (6.15)

The proof follows the same steps as in the proof of Lemma 6.4 so we skip it.

In the rest of this subsection, the goal is to prove Proposition 6.2, which
will then imply the main bounds (5.2) and (5.3) for the terms $F^{S}$ through
(6.2).

###  6.2. Estimates for the boundary terms

We start by treating the easier terms that involve the operators (6.4).

######  Lemma 6.6.

Under the assumptions of Proposition 6.2, we have

$\displaystyle{\big{\|}\partial_{\xi}T^{{\epsilon_{1},\epsilon_{2}},(2)}_{m}(G_{{\epsilon}_{1}},H_{{\epsilon}_{2}})\big{\|}}_{L^{2}_{\xi}}\lesssim{\varepsilon}^{\beta}2^{m}\rho^{1-\beta+\delta^{\prime}}(2^{m}),\qquad
G,H\in\\{g,h\\},$ (6.16)

for some $0<\delta^{\prime}<\beta/2$.

Moreover, there exists $a>\delta^{\prime}$, such that

$\displaystyle{\left\|e^{itL}\widetilde{\mathcal{F}}^{-1}_{\xi\mapsto
x}T^{{\epsilon_{1},\epsilon_{2}},(2)}_{m}(G_{{\epsilon}_{1}},H_{{\epsilon}_{2}})\right\|}_{L^{\infty}_{x}}\lesssim\frac{\rho_{0}^{a}}{\langle
t\rangle^{1+a+\delta^{\prime}}},\qquad G,H\in\\{h,g\\}.$ (6.17)

###### Proof.

We first prove the more difficult estimate (6.16), and then the simpler
(6.17).

Proof of (6.16). We subdivide the proof into three steps.

Step 1. When applying $\partial_{\xi}$ to
$T^{{\epsilon_{1},\epsilon_{2}},(2)}$, we claim that is suffices to focus on
the worse case when the derivative hits the exponential. In fact, the terms
where the frequency derivative hits the symbol $1/\Phi$ or the cutoff
$\varphi_{\leqslant M_{0}}$ are clearly much easier to treat, also in view of
(5.47). When instead $\partial_{\xi}$ hits the measure $\mu$ we can use Remark
5.9 to argue as follows:

1\. For the $\delta$ contribution we integrate by parts using the
distributional identity
$\partial_{\xi}\delta(\xi-\eta-\sigma)=-\partial_{\eta}\delta(\xi-\eta-\sigma)$.
This generates:

(a) a term where $\partial_{\eta}$ hits the exponential, which is essentially
identical to the leading order term that we will treat,

(b) some other easier terms where the symbols and cutoff are differentiated,
and

(c) a term where $\partial_{\eta}$ hits $\widetilde{f}$. This latter is the
only non-trivial term, although it is still easier to estimate than the term
where the differentiation falls on the exponential; we will treat this term in
details together with the main one.

2\. For the $\mu_{1}^{S}$ parts, we can use the identities (5.41) to do
similar integration by parts and obtain terms like those described above. The
analogues of (a) and (b) above are all substantially easier to bound than the
main term (where the derivative hits the phase) and the analogue of the term
in (c); below we are going to treat in full details these two types of terms.

According to the discussion above, the main terms to estimate from
$\partial_{\xi}T^{{\epsilon_{1},\epsilon_{2}},(2)}_{m}(G_{{\epsilon}_{1}},H_{{\epsilon}_{2}})$
are

$\displaystyle\begin{split}B_{1}(t,\xi)&:=t\,\tau_{m}(t)\int_{{\mathbb{R}}^{6}}e^{-it\Phi_{\epsilon_{1},\epsilon_{2}}(\xi,\eta,\sigma)}\widetilde{G}(t,\eta)\widetilde{H}(t,\sigma)\,\,b(\xi,\eta,\sigma)\,\mu^{S}(\xi,\eta,\sigma)d\eta
d\sigma,\\\
&b(\xi,\eta,\sigma):=\frac{\xi}{\langle\xi\rangle}\frac{\varphi_{\leqslant
M_{0}}(\langle\eta\rangle)\varphi_{\leqslant
M_{0}}(\langle\sigma\rangle)\varphi_{\leqslant
M_{0}}(\langle\xi\rangle)}{\langle\eta\rangle\langle\sigma\rangle\Phi_{\epsilon_{1},\epsilon_{2}}(\xi,\eta,\sigma)},\end{split}$
(6.18)

and

$\displaystyle\begin{split}B_{2}(t,\xi)&:=\tau_{m}(t)\int_{{\mathbb{R}}^{6}}e^{-it\Phi_{\epsilon_{1},\epsilon_{2}}(\xi,\eta,\sigma)}\partial_{\eta}\widetilde{G}(t,\eta)\widetilde{H}(t,\sigma)\,\,b^{\prime}(\xi,\eta,\sigma)\,\mu^{S}(\xi,\eta,\sigma)d\eta
d\sigma,\end{split}$ (6.19)

where $b^{\prime}$ denotes any of the symbols that are obtained when
performing the integration by parts manipulation described above; see Remark
5.9 and (5.44), and the exact definitions in (6.26) below.

Step 2. Next, we compare the formula in (6.18) and the identity (5.36) with
the definitions of the operators in (5.34) and insert additional localization
in $|\eta|\approx 2^{k_{1}}$ and $|\sigma|\approx 2^{k_{2}}$ in the integrals
in their definitions; we can then estimate

$\displaystyle\begin{split}&\big{\|}\widehat{\mathcal{F}}^{-1}_{\xi\mapsto
x}\big{(}e^{it\langle\xi\rangle}B_{1}(t)\big{)}\big{\|}_{L^{2}}\lesssim\sum_{k_{1},k_{2}\leqslant
M_{0}}\big{\|}t\,T_{0}[b_{0}]\big{(}e^{{\epsilon}_{1}it\langle\xi\rangle}\widetilde{G_{{\epsilon}_{1}}},e^{{\epsilon}_{2}it\langle\xi\rangle}\widetilde{H_{{\epsilon}_{2}}}\big{)}\big{\|}_{L^{2}}\\\
&+\sum_{|k_{1}-k_{2}|<5,k_{2}\leqslant
M_{0}+5}\Big{(}\big{\|}t\,T_{1}^{S,1}[b_{1}]\big{(}e^{{\epsilon}_{1}it\langle\xi\rangle}\widetilde{G_{{\epsilon}_{1}}},e^{{\epsilon}_{2}it\langle\xi\rangle}\widetilde{H_{{\epsilon}_{2}}}\big{)}\big{\|}_{L^{2}}+\big{\|}t\,T_{1}^{S,1}[b_{1}^{\prime}]\big{(}e^{{\epsilon}_{2}it\langle\xi\rangle}\widetilde{H_{{\epsilon}_{2}}},e^{{\epsilon}_{1}it\langle\xi\rangle}\widetilde{G_{{\epsilon}_{1}}}\big{)}\big{\|}_{L^{2}}\Big{)}\\\
&+\sum_{k_{1},k_{2}\leqslant
M_{0}}\Big{(}\big{\|}t\,T_{1}^{S,2}[b_{2}]\big{(}e^{{\epsilon}_{1}it\langle\xi\rangle}\widetilde{G_{{\epsilon}_{1}}},e^{{\epsilon}_{2}it\langle\xi\rangle}\widetilde{H_{{\epsilon}_{2}}}\big{)}\big{\|}_{L^{2}}\\\
&+\big{\|}t\,T_{2}^{S}[b_{0}]\big{(}e^{{\epsilon}_{1}it\langle\xi\rangle}\widetilde{G_{{\epsilon}_{1}}},e^{{\epsilon}_{2}it\langle\xi\rangle}\widetilde{H_{{\epsilon}_{2}}}\big{)}\big{\|}_{L^{2}}+\big{\|}t\,T_{3}^{S}[b_{0}]\big{(}e^{{\epsilon}_{1}it\langle\xi\rangle}\widetilde{G_{{\epsilon}_{1}}},e^{{\epsilon}_{2}it\langle\xi\rangle}\widetilde{H_{{\epsilon}_{2}}}\big{)}\big{\|}_{L^{2}}\Big{)}\end{split}$
(6.20)

where, using a notation similar to that of (5.35) (but with the additional
frequency localization) we let

$\displaystyle\begin{split}b_{0}(\xi,\eta,\sigma)&:=b(\xi,\eta,\sigma)\varphi_{k_{1}}(\eta)\varphi_{k_{2}}(\sigma),\\\
b_{1}(\xi,\eta,\sigma)&:=b(\xi,\xi-\eta,\sigma)\varphi_{k_{1}}(\eta)\varphi_{k_{2}}(\sigma),\\\
b_{1}^{\prime}(\xi,\eta,\sigma)&:=b(\xi,\sigma,\xi-\eta)\varphi_{k_{1}}(\eta)\varphi_{k_{2}}(\sigma),\\\
b_{2}(\xi,\eta,\sigma)&:=b(\xi,-\eta-\sigma,\sigma)\varphi_{k_{1}}(\eta)\varphi_{k_{2}}(\sigma).\end{split}$
(6.21)

Note the following:

  * 1.

We are not indicating explicitly the dependence of the symbols in (6.21) on
$k_{1},k_{2}$, but we are dealing with integrals in $d\eta d\sigma$ (as in
(5.34)) with localization to $|\eta|\approx 2^{k_{1}}$ and $|\sigma|\approx
2^{k_{2}}$.

  * 2.

The restriction $|k_{1}-k_{2}|<5$ for the $T_{1}^{S,1}$ operators comes from
the formula (5.17), and the indicator functions in the bounds (B.5) and (B.7).

  * 3.

The restriction $k_{2}\leqslant M_{0}$ (and, similarly, the restriction
$k_{1}\leqslant M_{0}$ for the other terms) comes from the restriction to
inputs and outputs in the symbol of (6.18). Note that, to be precise, we
should write $M_{0}+3$ instead of $M_{0}$ in some cases, but we do not make
such a distinction as it is clearly not relevant.

Using (5.48), we see that $b$ as in (6.18) satisfies (B.29) in Theorem B.2
with the choice of $A=M_{0}$, and so do all the symbols (6.21). In particular,
we can apply Theorem 5.7 to the terms in (6.20).

More precisely, using (5.40b) we can write, for $|t|\approx 2^{m}$,

$\displaystyle\begin{split}&t\,{\big{\|}T_{1}^{S,1}[b_{1}]\big{(}e^{{\epsilon}_{1}it\langle\xi\rangle}\widetilde{G_{{\epsilon}_{1}}},e^{{\epsilon}_{2}it\langle\xi\rangle}\widetilde{H_{{\epsilon}_{2}}}\big{)}\big{\|}}_{L^{2}}\\\
&\lesssim
2^{m}\cdot{\big{\|}e^{itL}G\big{\|}}_{L^{3}}{\|e^{itL}P_{k_{2}}H\|}_{L^{6-}}\cdot
2^{C_{0}M_{0}}+2^{-D}\cdot\mathcal{D}\big{(}e^{{\epsilon}_{1}it\langle\xi\rangle}\widetilde{G_{{\epsilon}_{1}}},e^{{\epsilon}_{2}it\langle\xi\rangle}\widetilde{H_{{\epsilon}_{2}}}\big{)},\end{split}$
(6.22)

having used the boundedness of wave operators as well. To estimate the first
term on the right-hand side of (6.22) for any $G,H\in\\{g,h\\}$, we use the
bounds (4.19) from Lemma 4.4 to get, for $|t|\approx 2^{m}$,

$\displaystyle\begin{split}{\big{\|}e^{itL}G\big{\|}}_{L^{3}}{\|e^{itL}P_{k_{2}}H\|}_{L^{6-}}&\lesssim\rho(2^{m})^{1-\beta}2^{m/2+2\delta_{N}m}{\varepsilon}^{\beta}\cdot\rho(2^{m})^{1-\beta}2^{2\delta_{N}m}{\varepsilon}^{\beta}\\\
&\lesssim\rho(2^{m})^{2-2\beta}2^{3m/4}\varepsilon^{2\beta}=2^{-5m/4}Z^{2},\end{split}$
(6.23)

where $Z$ is defined in (6.7). Note that this bound is consistent with a bound
by $2^{-am}Z^{2}$ for the term in (6.22), with $a=1/4$, which is an acceptable
bound (see Remark 6.3).

######  Remark 6.7.

On the support of $T^{S,1}_{1}[b_{1}]$ we have that either $k_{2}\sim 1$ when
$H=g$, or that $-5m\leqslant k_{2}\leqslant M_{0}$ when $H=h$ (see Lemma 5.5).
In all cases the sum has at most $O(m)$ terms since $|k_{1}-k_{2}|<5.$ An
acceptable bound like those in Remark 6.3 for each fixed pair $k_{1},k_{2}$,
will then yield the desired estimates. The same reasoning holds for
$T_{1}^{S,1}[b_{1}^{\prime}]$ with $k_{1}$ instead of $k_{2}$ and $G$ instead
of $H.$

For the $\mathcal{D}$ term in (6.23), recall the definition (5.39), we use the
a priori bound on the weighted norm of $h$ from (4.13), and the larger bound
on the weighted norm of $g$ from (4.26), as well as (4.24) and the $L^{2}$
bound on $h$ in (4.13), to see that, for all $G,H\in\\{g,h\\}$ and $|t|\approx
2^{m}$,

$\displaystyle\begin{split}\mathcal{D}\big{(}e^{{\epsilon}_{1}it\langle\xi\rangle}P_{k_{1}}\widetilde{G_{{\epsilon}_{1}}},e^{{\epsilon}_{2}it\langle\xi\rangle}P_{k_{2}}\widetilde{H_{{\epsilon}_{2}}}\big{)}&\lesssim\Big{(}{\big{\|}\partial_{\xi}\big{(}\varphi_{k_{1}}\widetilde{G_{{\epsilon}_{1}}}\big{)}\big{\|}}_{L^{2}}+{\|\varphi_{k_{1}}\widetilde{G_{{\epsilon}_{1}}}\|}_{L^{2}}\Big{)}{\big{\|}\varphi_{k_{2}}\widetilde{H_{{\epsilon}_{2}}}\big{\|}}_{L^{2}}\\\
&\lesssim
m^{2}2^{3m/2}\rho(2^{m})\cdot\big{[}m2^{m}\rho(2^{m})+{\varepsilon}^{\beta}2^{m}\rho(2^{m})^{1-\beta}\big{]}\\\
&\lesssim\rho(2^{m})^{2-2\beta}2^{5m/2+}\varepsilon^{2\beta}\lesssim
2^{m/2+}Z^{2}.\end{split}$ (6.24)

This bound is consistent with the assumption on $D$ (see above (5.39)) if we
choose, for example, $D=5m$; indeed, our current choice is $A=M_{0}$, where
$2^{M_{0}}\approx 2^{\delta_{N}m}$ (see (5.4)) with $\delta_{N}\ll 1$ that can
be chosen sufficiently small so to ensure $20M_{0}\leq 5m\leq 2^{M_{0}/5}$
which suffices. The second term on the right-hand side of (6.22) is then
bounded by $2^{-4m}Z^{2}$.

The same argument used for $T_{1}^{S,1}[b_{1}]$ above applies verbatim to
$T_{1}^{S,1}[b_{1}^{\prime}]$, and to the simpler case of $T_{0}[b_{0}]$. A
similar argument can be applied to the operators $T_{i}^{S}[b_{0}],i=2,3$ as
well. The only difference is that there is no additional restriction on
indices in the sum for these terms, except for $-5m\leqslant
k_{1},k_{2}\leqslant M_{0}$; see the formulas (5.34d)-(5.34e), the definition
of $b_{0}$ in (6.21) and Lemmas 5.4 and 5.5. We can then use (5.40d) for each
fixed pair $k_{1}$, $k_{2}$ and sum over these indexes at the expense of a
harmless $O(m^{2})$ factor. Note that there are no $\mathcal{D}$ terms in
(5.40d).

For the last term $T_{1}^{S,2}$, we can use an almost identical argument based
on the bilinear bound (5.40b); the only minor difference is the need to
additionally localize in $|\xi|\approx 2^{K}$ to be able to use (5.40b). But
this can also be easily taken care of applying Bernstein for $K\leqslant 0$
and estimating the $L^{2-}$ norm of $P_{K}T_{2}[b_{2}]$, as well as using that
$K\leqslant M_{0}$; the (arbitrarily) small loss of $2^{M_{0}}$ resulting from
this can be absorbed by the bounds (6.23) and (6.24).

In conclusion, putting together (6.23) and (6.24), and the similar bounds for
all the other terms on the right-hand side of (6.20), we have obtained

$\displaystyle\begin{split}{\|B_{1}(t)\|}_{L^{2}}&\lesssim\big{(}2^{m}2^{C_{0}M_{0}}\cdot
2^{-5m/4}Z^{2}+2^{-4m}Z^{2}\big{)}2^{(0+)m}\lesssim 2^{-m/5}Z^{2},\end{split}$

provided $C_{0}M_{0}$ is small enough (that is, $\delta_{N}$ is small enough),
which is sufficient.

Step 3. To estimate the $L^{2}$ norm of (6.19) we can use very similar
arguments. Note that this terms does not carry a factor of $t$ in front, and
is therefore easier to handle. We begin with the analogue of (6.20), that is,

$\displaystyle\begin{split}&\big{\|}\widehat{\mathcal{F}}^{-1}_{\xi\mapsto
x}\big{(}e^{it\langle\xi\rangle}B_{2}(t)\big{)}\big{\|}_{L^{2}}\lesssim\sum_{k_{1},k_{2}\leqslant
M_{0}}\big{\|}\,T_{0}[b_{0}]\big{(}e^{{\epsilon}_{1}it\langle\xi\rangle}\partial_{\xi}\widetilde{G_{{\epsilon}_{1}}},e^{{\epsilon}_{2}it\langle\xi\rangle}\widetilde{H_{{\epsilon}_{2}}}\big{)}\big{\|}_{L^{2}}\\\
&+\big{\|}T_{1}^{S,2}[m_{2}^{j}]\big{(}e^{{\epsilon}_{1}it\langle\xi\rangle}\partial_{\xi_{j}}\widetilde{G_{{\epsilon}_{1}}},e^{{\epsilon}_{2}it\langle\xi\rangle}\widetilde{H_{{\epsilon}_{2}}}\big{)}\big{\|}_{L^{2}},\\\
&+\big{\|}T_{2}^{S}[m_{0}^{j}]\big{(}e^{{\epsilon}_{1}it\langle\xi\rangle}\partial_{\xi_{j}}\widetilde{G_{{\epsilon}_{1}}},e^{{\epsilon}_{2}it\langle\xi\rangle}\widetilde{H_{{\epsilon}_{2}}}\big{)}\big{\|}_{L^{2}}+\big{\|}T_{3}^{S}[m_{0}^{j}]\big{(}e^{{\epsilon}_{1}it\langle\xi\rangle}\partial_{\xi_{j}}\widetilde{G_{{\epsilon}_{1}}},e^{{\epsilon}_{2}it\langle\xi\rangle}\widetilde{H_{{\epsilon}_{2}}}\big{)}\big{\|}_{L^{2}}\\\
&+\sum_{|k_{1}-k_{2}|<5,k_{2}\leqslant
M_{0}}\Big{(}\big{\|}T_{1}^{S,1}[b_{1}]\big{(}e^{{\epsilon}_{1}it\langle\xi\rangle}\partial_{\xi}\widetilde{G_{{\epsilon}_{1}}},e^{{\epsilon}_{2}it\langle\xi\rangle}\widetilde{H_{{\epsilon}_{2}}}\big{)}\big{\|}_{L^{2}}\\\
&+\big{\|}\,T_{1}^{S,1}[m_{1}^{j}]\big{(}e^{{\epsilon}_{2}it\langle\xi\rangle}\widetilde{H_{{\epsilon}_{2}}},e^{{\epsilon}_{1}it\langle\xi\rangle}\partial_{\xi_{j}}\widetilde{G_{{\epsilon}_{1}}}\big{)}\big{\|}_{L^{2}}\Big{)}\end{split}$
(6.25)

where we are implicitly summing over $j=1,2,3$ in the last three lines, and
the symbols are

$\displaystyle\begin{split}m_{0}^{j}(\xi,\eta,\sigma)&=\frac{\xi}{|\xi|}\frac{\eta_{j}}{|\eta|}b_{0}(\xi,\eta,\sigma),\qquad
m_{1}^{j}(\xi,\eta,\sigma)=\frac{\eta}{|\eta|}\frac{\sigma_{j}}{|\sigma|}b_{1}^{\prime}(\xi,\eta,\sigma),\\\
m_{2}^{j}(\xi,\eta,\sigma)&=\frac{\xi}{|\xi|}\frac{\eta_{j}}{|\eta|}b_{2}(\xi,\eta,\sigma);\end{split}$
(6.26)

see Remark 5.9 and the definitions (6.21).

To estimate the terms in (6.25) we use the bilinear bounds (5.40) proceeding
as follows. For the case when $G=h$ in (6.25) we do an $L^{2}\times L^{6}$
estimate using the a priori bound on $\partial_{\xi}\widetilde{h}$ from (4.13)
and the $L^{6}$ decay bounds in Lemma 4.4; for the case when $G=g$ in (6.25)
we can use instead (4.26) and again the $L^{6}$ estimates in Lemma 4.4. The
term $\mathcal{D}$ that appears on the right-hand side of (5.40b) is a fast
decaying remainder that can be handled similarly to (6.24); exploiting the
minimum in the definition (5.39) we only differentiate the input that does not
already carry the weight, so to never have the appearance of
$\partial_{\xi}^{2}\widetilde{h}$.

Proof of (6.17). In view of Lemma 6.5 it suffice to prove, see the formula
(6.4), that

$\displaystyle{\big{\|}\widehat{\mathcal{F}}^{-1}_{\xi\mapsto
x}J(t)\big{\|}}_{L^{8/7}_{x}}\lesssim\rho_{0}^{a},$ (6.27)

where

$\displaystyle\begin{split}J(t)&=\tau_{m}(t)\int_{{\mathbb{R}}^{6}}\frac{\varphi_{\leqslant
M_{0}}(\langle\xi\rangle)\varphi_{\leqslant
M_{0}}(\langle\eta\rangle)\varphi_{\leqslant
M_{0}}(\langle\sigma\rangle)}{\Phi_{\epsilon_{1},\epsilon_{2}}(\xi,\eta,\sigma)\langle\eta\rangle\langle\sigma\rangle}\\\
&\qquad\times
e^{i{\epsilon}_{1}t\langle\eta\rangle}\widetilde{G}(t,\eta)e^{i{\epsilon}_{1}t\langle\sigma\rangle}\widetilde{H}(t,\sigma)\,\mu^{S}(\xi,\eta,\sigma)d\eta
d\sigma.\end{split}$ (6.28)

Choosing $b$ as in (6.18), but without the $\xi/\langle\xi\rangle$ factor, and
with the same definitions (6.21), we have identities analogous to (6.20). It
follows that

$\displaystyle\begin{split}&\big{\|}\widehat{\mathcal{F}}^{-1}_{\xi\mapsto
x}J(t)\big{\|}_{L^{8/7}_{x}}\lesssim\\\ &\sum_{k_{1},k_{2}\leqslant
M_{0}}\big{\|}T_{0}[b_{0}]\big{(}e^{{\epsilon}_{1}it\langle\xi\rangle}\widetilde{G_{{\epsilon}_{1}}},e^{{\epsilon}_{2}it\langle\xi\rangle}\widetilde{H_{{\epsilon}_{2}}}\big{)}\big{\|}_{L^{8/7}_{x}}+\big{\|}T_{2}^{S}[b_{0}]\big{(}e^{{\epsilon}_{1}it\langle\xi\rangle}\widetilde{G_{{\epsilon}_{1}}},e^{{\epsilon}_{2}it\langle\xi\rangle}\widetilde{H_{{\epsilon}_{2}}}\big{)}\big{\|}_{L^{8/7}_{x}}\\\
&+\big{\|}T_{3}^{S}[b_{0}]\big{(}e^{{\epsilon}_{1}it\langle\xi\rangle}\widetilde{G_{{\epsilon}_{1}}},e^{{\epsilon}_{2}it\langle\xi\rangle}\widetilde{H_{{\epsilon}_{2}}}\big{)}\big{\|}_{L^{8/7}_{x}}+\big{\|}T_{1}^{S,2}[b_{2}]\big{(}e^{{\epsilon}_{1}it\langle\xi\rangle}\widetilde{G_{{\epsilon}_{1}}},e^{{\epsilon}_{2}it\langle\xi\rangle}\widetilde{H_{{\epsilon}_{2}}}\big{)}\big{\|}_{L^{8/7}_{x}}\\\
&+\sum_{|k_{1}-k_{2}|<5,k_{2}\leqslant
M_{0}}\big{\|}T_{1}^{S,1}[b_{1}]\big{(}e^{{\epsilon}_{1}it\langle\xi\rangle}\widetilde{G_{{\epsilon}_{1}}},e^{{\epsilon}_{2}it\langle\xi\rangle}\widetilde{H_{{\epsilon}_{2}}}\big{)}\big{\|}_{L^{8/7}_{x}}+\big{\|}T_{1}^{S,1}[b_{1}^{\prime}]\big{(}e^{{\epsilon}_{2}it\langle\xi\rangle}\widetilde{H_{{\epsilon}_{2}}},e^{{\epsilon}_{1}it\langle\xi\rangle}\widetilde{G_{{\epsilon}_{1}}}\big{)}\big{\|}_{L^{8/7}_{x}}.\end{split}$
(6.29)

The estimate (5.40b) gives us

$\displaystyle\begin{split}&\big{\|}T_{1}^{S,1}[b_{1}]\big{(}e^{{\epsilon}_{1}it\langle\xi\rangle}\widetilde{G_{{\epsilon}_{1}}},e^{{\epsilon}_{2}it\langle\xi\rangle}\widetilde{H_{{\epsilon}_{2}}}\big{)}\big{\|}_{L^{8/7}_{x}}\\\
&\lesssim{\big{\|}e^{itL}G\big{\|}}_{L^{8/3-}}{\|e^{itL}P_{k_{2}}H\|}_{L^{2}}\cdot
2^{(C_{0}+1)M_{0}}+2^{-D}\cdot\mathcal{D}\big{(}e^{{\epsilon}_{1}it\langle\xi\rangle}\widetilde{G_{{\epsilon}_{1}}},e^{{\epsilon}_{2}it\langle\xi\rangle}\varphi_{k_{2}}\widetilde{H_{{\epsilon}_{2}}}\big{)}.\end{split}$
(6.30)

Then we can use the $L^{q}$ dispersive estimate in Lemma 4.4 to obtain (recall
Remark 6.7), for all $t\approx 2^{m}$,

$\displaystyle{\big{\|}e^{itL}G\big{\|}}_{L^{8/3-}}{\|e^{itL}P_{k_{2}}H\|}_{L^{2}}$
$\displaystyle\lesssim\rho(2^{m})^{1-\beta}2^{5m/8+10\delta_{N}m}{\varepsilon}^{\beta}\cdot\rho(2^{m})^{1-\beta}2^{m}{\varepsilon}^{\beta}$
$\displaystyle\lesssim\rho(2^{m})^{2-2\beta}2^{7m/4}{\varepsilon}^{2\beta}\lesssim{\varepsilon}^{2a},$

for some $a>2\beta$, which is consistent with (6.27). All the other terms in
(6.29) can be estimated similarly as above, using (5.40a)-(5.40d). The
$\mathcal{D}$ factor is estimated by (6.24). These bounds and (6.30) give us
(6.27) for $a$ small enough, provided $M_{0}$ is small enough. ∎

In the remainder of this section we estimate the integrated terms (6.3) by
distinguishing the three different types of interactions described in §5.1.3.
Recall the definition of the bulk operator (6.3), and that we are aiming to
prove the $L^{2}$ weighted bound (6.5) and the decay bound (6.6) (via Lemma
6.4).

###  6.3. Fermi interactions

We first look at the interactions that involve only the Fermi component $g$.

######  Lemma 6.8.

Under the assumptions of Proposition 6.2, for
${\epsilon_{1},\epsilon_{2}}\in\\{+,-\\}$, we have

$\displaystyle\big{\|}\partial_{\xi}T^{{\epsilon_{1},\epsilon_{2}},(1)}_{m,k_{1},k_{2}}(g,g)\big{\|}_{L^{2}}\lesssim
2^{m}\rho^{1-\beta+\delta^{\prime}}(2^{m}){\varepsilon}^{\beta},$ (6.31)
$\displaystyle{\big{\|}\mathcal{F}^{-1}_{\xi\mapsto
x}e^{it\langle\xi\rangle}\partial_{t}T^{{\epsilon_{1},\epsilon_{2}},(1)}_{m,k_{1},k_{2}}(g,g)\big{\|}}_{L^{8/7}_{x}}\lesssim\frac{\rho_{0}^{1/2}}{\langle
t\rangle^{11/8}},$ (6.32)

for some $\delta^{\prime}>0$.

###### Proof.

Proof of (6.31). Arguing as at the beginning of the proof of Lemma 6.6 we can
reduce to bounding the $L^{2}$-norm of the two main terms, that is those where
$\partial_{\xi}$ hits the exponential in the formula for
$T^{{\epsilon_{1},\epsilon_{2}},(1)}_{m,k_{1},k_{2}}(g,g)$, see (6.3), and
where the derivatives hits $\widetilde{g}$. Let us denote these two types of
terms by

$\displaystyle
C_{1}(t,\xi):=C_{1,{\epsilon_{1},\epsilon_{2}}}[g_{{\epsilon}_{1}},g_{{\epsilon}_{2}}](t,\xi),\qquad
C_{2}(t,\xi):=C_{2,{\epsilon_{1},\epsilon_{2}}}[g_{{\epsilon}_{1}},g_{{\epsilon}_{2}}](t,\xi),$
(6.33)

where we define

$\displaystyle\begin{split}C_{1,{\epsilon_{1},\epsilon_{2}}}[G,H](t,\xi)&:=\int_{0}^{t}s\,\tau_{m}(s)\int_{{\mathbb{R}}^{6}}e^{-is\Phi_{\epsilon_{1},\epsilon_{2}}(\xi,\eta,\sigma)}\partial_{s}\widetilde{G}(t,\eta)\,\widetilde{H}(t,\sigma)\,\,b(\xi,\eta,\sigma)\,\mu^{S}(\xi,\eta,\sigma)d\eta
d\sigma\,ds,\\\
C_{2,{\epsilon_{1},\epsilon_{2}}}[G,H](t,\xi)&:=\int_{0}^{t}\tau_{m}(s)\int_{{\mathbb{R}}^{6}}e^{-is\Phi_{\epsilon_{1},\epsilon_{2}}(\xi,\eta,\sigma)}\partial_{s}\widetilde{G}(t,\eta)\,\partial_{\sigma}\widetilde{H}(t,\sigma)\,\,b^{\prime}(\xi,\eta,\sigma)\,\mu^{S}(\xi,\eta,\sigma)d\eta
d\sigma\,ds,\end{split}$ (6.34)

and $b,b^{\prime}$ are defined as in (6.18), (6.19) (see also the more precise
(6.26)). Notice that, as done before, when reducing to the terms (6.33)-(6.34)
we have taken into account the case where $\partial_{\xi}$ hits the
distribution $\mu^{S}$; in this case the derivative can be converted into
$\partial_{\sigma}$ through formulas like (5.41)-(5.44) with the roles of
$\eta$ and $\sigma$ exchanged, and an integration by parts in $\sigma$ can be
performed without the need to differentiate $\partial_{s}\widetilde{g}$ in
$\eta$.

Estimates for the $C_{1}$ terms. We can proceed similarly to the proof of
Lemma 6.6. In particular, we let $b$ be the same symbol defined in (6.18) and
use the same notation in (6.21) so that we can estimate similarly to (6.20).
As in (6.21), we introduce frequency localizations in $\eta,\sigma$. Notice
that since $g$ is localized at frequency $\sim 1$ (see (4.5)), the sums over
$k_{1}$ and $k_{2}$ are finite for all the terms except $T_{1}^{S,2}.$ We have

$\displaystyle\begin{split}&{\Big{\|}\widehat{\mathcal{F}}^{-1}_{\xi\mapsto
x}C_{1}(t)\Big{\|}}_{L^{2}}\lesssim 2^{2m}\sup_{k_{1},k_{2}\approx
1}\sup_{s\approx
2^{m}}\Big{[}{\big{\|}T_{0}[b_{0}]\big{(}e^{{\epsilon}_{1}is\langle\xi\rangle}\partial_{s}\widetilde{g_{{\epsilon}_{1}}},e^{{\epsilon}_{2}is\langle\xi\rangle}\widetilde{g_{{\epsilon}_{2}}}\big{)}\big{\|}}_{L^{2}}\\\
&+\sum_{i=2,3}{\big{\|}T_{i}^{S}[b_{0}]\big{(}e^{{\epsilon}_{1}is\langle\xi\rangle}\partial_{s}\widetilde{g_{{\epsilon}_{1}}},e^{{\epsilon}_{2}is\langle\xi\rangle}\widetilde{g_{{\epsilon}_{2}}}\big{)}\big{\|}}_{L^{2}}\\\
&+{\big{\|}T_{1}^{S,1}[b_{1}]\big{(}e^{{\epsilon}_{1}is\langle\xi\rangle}\partial_{s}\widetilde{g_{{\epsilon}_{1}}},e^{{\epsilon}_{2}is\langle\xi\rangle}\widetilde{g_{{\epsilon}_{2}}}\big{)}\big{\|}}_{L^{2}}+{\big{\|}T_{1}^{S,1}[b_{1}^{\prime}]\big{(}e^{{\epsilon}_{2}is\langle\xi\rangle}\widetilde{g_{{\epsilon}_{2}}},e^{{\epsilon}_{1}is\langle\xi\rangle}\partial_{s}\widetilde{g_{{\epsilon}_{1}}}\big{)}\big{\|}}_{L^{2}}\\\
&+2^{2m}\sup_{s\approx 2^{m}}\sum_{k_{1}<5,k_{2}\approx
1}{\big{\|}T_{1}^{S,2}[b_{0}]\big{(}e^{{\epsilon}_{1}is\langle\xi\rangle}\partial_{s}\widetilde{g_{{\epsilon}_{1}}},e^{{\epsilon}_{2}is\langle\xi\rangle}\widetilde{g_{{\epsilon}_{2}}}\big{)}\big{\|}}_{L^{2}}\Big{]},\end{split}$
(6.35)

Let us look at the $T_{1}^{S,1}$ term on the right-hand side above first.
Using (5.40b) we have

$\displaystyle\begin{split}{\big{\|}T_{1}^{S,1}[b_{1}]\big{(}e^{{\epsilon}_{1}is\langle\xi\rangle}\partial_{s}\widetilde{g_{{\epsilon}_{1}}},e^{{\epsilon}_{2}is\langle\xi\rangle}\widetilde{g_{{\epsilon}_{2}}}\big{)}\big{\|}}_{L^{2}}&\lesssim{\big{\|}e^{isL}P_{k_{1}}\partial_{s}g\big{\|}}_{L^{3-}}{\|e^{isL}P_{k_{2}}g\|}_{L^{6}}\cdot
2^{C_{0}M_{0}}\\\
&+2^{-D}\cdot\mathcal{D}\big{(}e^{{\epsilon}_{1}is\langle\xi\rangle}\varphi_{k_{1}}\partial_{s}\widetilde{g_{{\epsilon}_{1}}},e^{{\epsilon}_{2}is\langle\xi\rangle}\varphi_{k_{2}}\widetilde{g_{{\epsilon}_{2}}}\big{)}.\end{split}$
(6.36)

From the dispersive estimates (4.17) and (4.21), we see that, for all
$s\approx 2^{m}$,

$\displaystyle\begin{split}{\big{\|}e^{isL}\partial_{s}g\big{\|}}_{L^{3-}}{\|e^{isL}g\|}_{L^{6}}\cdot
2^{C_{0}M_{0}}&\lesssim\rho(2^{m})^{2}\cdot m\cdot 2^{C_{0}\delta_{N}m}\\\
&\lesssim 2^{-2m}\cdot\rho(2^{m})^{\beta/2}m2^{C_{0}\delta_{N}m}\cdot
2^{m}\rho(2^{m})\cdot Z.\end{split}$

having used ${\varepsilon}^{-\beta}\rho(2^{m})^{\beta/2}\lesssim 1$.
Multiplied by the factor of $2^{2m}$ which appears on the right-hand side of
(6.35), the above bound is acceptable (see Remark 6.3) provided $\delta_{N}$
is small enough.

To estimate the second line of (6.36) we use (4.26) to bound
$\partial_{s}\widetilde{g}$ in $L^{2}$, and use again (4.21) to see that,
recall the definition (5.39),

$\displaystyle\mathcal{D}\big{(}e^{{\epsilon}_{1}is\langle\xi\rangle}\varphi_{k_{1}}\partial_{s}\widetilde{g_{{\epsilon}_{1}}},e^{{\epsilon}_{2}is\langle\xi\rangle}\varphi_{k_{2}}\widetilde{g_{{\epsilon}_{2}}}\big{)}\lesssim\rho(2^{m})^{2}\cdot
2^{3m/2}m^{2}.$ (6.37)

We can then pick, say, $D=5m$ (here $A=M_{0}=m\delta_{N}$) so that the second
line of (6.36) is a fast decaying remainder.

Bounds like those just proved for
$T_{1}^{S,1}[b_{1}]\big{(}e^{{\epsilon}_{1}is\langle\xi\rangle}\partial_{s}\widetilde{g_{{\epsilon}_{1}}},e^{{\epsilon}_{2}is\langle\xi\rangle}\widetilde{g_{{\epsilon}_{2}}}\big{)}$
can be obtained identically for all the other terms in (6.35), which concludes
the estimate for $C_{1}$.

Estimates for the $C_{2}$ terms. For the second integral in (6.34), using the
same notation as above (see (6.26)), we see that

$\displaystyle\begin{split}&{\Big{\|}\widehat{\mathcal{F}}^{-1}_{\xi\mapsto
x}C_{2}(t)\Big{\|}}_{L^{2}}\lesssim 2^{m}\sup_{k_{1},k_{2}\approx
1}\sup_{s\approx
2^{m}}\Big{[}\big{\|}T_{0}[b_{0}]\big{(}e^{{\epsilon}_{1}is\langle\xi\rangle}\partial_{s}\widetilde{g_{{\epsilon}_{1}}},e^{{\epsilon}_{2}is\langle\xi\rangle}\partial_{\xi}\widetilde{g_{{\epsilon}_{2}}}\big{)}\big{\|}_{L^{2}}\\\
&+\big{\|}T_{2}^{S}[m_{0}^{j}]\big{(}e^{{\epsilon}_{1}is\langle\xi\rangle}\partial_{s}\widetilde{g_{{\epsilon}_{1}}},e^{{\epsilon}_{2}is\langle\xi\rangle}\partial_{\xi_{j}}\widetilde{g_{{\epsilon}_{2}}}\big{)}\big{\|}_{L^{2}}+\big{\|}T_{3}^{S}[m_{0}^{j}]\big{(}e^{{\epsilon}_{1}is\langle\xi\rangle}\partial_{s}\widetilde{g_{{\epsilon}_{1}}},e^{{\epsilon}_{2}is\langle\xi\rangle}\partial_{\xi_{j}}\widetilde{g_{{\epsilon}_{2}}}\big{)}\big{\|}_{L^{2}}\\\
&+\big{\|}T_{1}^{S,1}[b_{1}]\big{(}e^{{\epsilon}_{1}is\langle\xi\rangle}\partial_{s}\widetilde{g_{{\epsilon}_{1}}},e^{{\epsilon}_{2}is\langle\xi\rangle}\partial_{\xi}\widetilde{g_{{\epsilon}_{2}}}\big{)}\big{\|}_{L^{2}}+\big{\|}T_{1}^{S,1}[m_{1}^{j}]\big{(}e^{{\epsilon}_{2}is\langle\xi\rangle}\partial_{\xi_{j}}\widetilde{g_{{\epsilon}_{2}}},e^{{\epsilon}_{1}is\langle\xi\rangle}\partial_{s}\widetilde{g_{{\epsilon}_{1}}}\big{)}\big{\|}_{L^{2}}\Big{]}\\\
&+2^{m}\sup_{s\approx 2^{m}}\sum_{k_{1}<5,k_{2}\approx
1}\big{\|}T_{1}^{S,2}[m_{2}^{j}]\big{(}e^{{\epsilon}_{1}is\langle\xi\rangle}\partial_{s}\widetilde{g_{{\epsilon}_{1}}},e^{{\epsilon}_{2}is\langle\xi\rangle}\partial_{\xi_{j}}\widetilde{g_{{\epsilon}_{2}}}\big{)}\big{\|}_{L^{2}}.\end{split}$
(6.38)

Once again it suffices to look at the $T_{1}^{S,1}$ term above, since the
others are identical; the bilinear estimate (5.40b) and the boundedness of
wave operators give

$\displaystyle\begin{split}2^{m}{\|T_{1}^{S,1}[b_{1}]\big{(}e^{{\epsilon}_{1}is\langle\xi\rangle}\partial_{s}\widetilde{g_{{\epsilon}_{1}}},e^{{\epsilon}_{2}is\langle\xi\rangle}\partial_{\xi}\widetilde{g_{{\epsilon}_{2}}}\big{)}\|}_{L^{2}}&\lesssim
2^{m}{\big{\|}e^{isL}\partial_{s}g\big{\|}}_{L^{2-}}{\|\widehat{\mathcal{F}}^{-1}e^{{\epsilon}_{2}is\langle\xi\rangle}(\partial_{\xi}\widetilde{g})\|}_{L^{\infty}}\cdot
2^{C_{0}M_{0}}\\\
&+2^{m}2^{-D}\cdot\mathcal{D}\big{(}e^{{\epsilon}_{1}is\langle\xi\rangle}\partial_{s}\widetilde{g_{{\epsilon}_{1}}},e^{{\epsilon}_{2}is\langle\xi\rangle}\partial_{\xi}\widetilde{g_{{\epsilon}_{2}}}\big{)}.\end{split}$
(6.39)

The first term above can be bounded using (4.21) and (4.27):

$\displaystyle
2^{m}{\big{\|}e^{isL}\partial_{s}g\big{\|}}_{L^{2-}}{\|e^{{\epsilon}_{2}isL}\widehat{\mathcal{F}}^{-1}(\partial_{\xi}\widetilde{g})\|}_{L^{\infty}}\cdot
2^{C_{0}M_{0}}$ $\displaystyle\lesssim
2^{m}\rho(2^{m})\cdot{\|\partial_{\xi}\widetilde{g}\|}_{L^{1}_{\xi}}\cdot
2^{C_{0}\delta_{N}m}$
$\displaystyle\lesssim\big{(}2^{m}\rho(2^{m})\big{)}^{2}2^{(C_{0}+1)\delta_{N}m}\lesssim
2^{(C_{0}+1)\delta_{N}m}\rho(2^{m})^{\beta/2}Z,$

which suffices for our purposes (see (6.9) and Remark 6.3).

Next we estimate the remainder. We can pick $D=5m$ as before and see that the
last term on the right-hand side of (6.39) is fast decaying: using (4.28) and
(4.21) we have

$\displaystyle\begin{split}2^{-D}\mathcal{D}\big{(}e^{{\epsilon}_{1}is\langle\xi\rangle}\partial_{s}\widetilde{g_{{\epsilon}_{1}}},e^{{\epsilon}_{2}is\langle\xi\rangle}\partial_{\xi}\widetilde{g_{{\epsilon}_{2}}}\big{)}&\lesssim
2^{-5m}\cdot{\|\partial_{s}\widetilde{g_{{\epsilon}_{1}}}\|}_{L^{2}}\cdot{\|\partial_{\xi}^{2}\widetilde{g_{{\epsilon}_{1}}}\|}_{L^{2}}\\\
&\lesssim\rho(2^{m})^{2}2^{-2m}\lesssim 2^{-4m}Z.\end{split}$ (6.40)

This completes the estimate (6.31).

Proof of (6.32). For the second estimate, using the same notation for $b$ as
in (6.34), but without the $\xi/\langle\xi\rangle$ factor, and the notation
(6.18), we can write

$\displaystyle\sup_{t\approx
2^{m}}{\Big{\|}\widehat{\mathcal{F}}^{-1}_{\xi\mapsto
x}e^{it\langle\xi\rangle}\partial_{t}T^{{\epsilon_{1},\epsilon_{2}},(1)}_{m,k_{1},k_{2}}(g,g)\Big{\|}}_{L^{8/7}_{x}}\lesssim\sup_{t\approx
2^{m}}\sup_{k_{1},k_{2}\approx
1}\Big{[}\sum_{i=0,2,3}{\big{\|}T_{i}^{S}[b_{0}]\big{(}e^{{\epsilon}_{1}it\langle\xi\rangle}\partial_{t}\widetilde{g_{{\epsilon}_{1}}},e^{{\epsilon}_{2}it\langle\xi\rangle}\widetilde{g_{{\epsilon}_{2}}}\big{)}\big{\|}}_{L^{8/7}_{x}}$
$\displaystyle+{\big{\|}T_{1}^{S,1}[b_{1}]\big{(}e^{{\epsilon}_{1}it\langle\xi\rangle}\partial_{t}\widetilde{g_{{\epsilon}_{1}}},e^{{\epsilon}_{2}it\langle\xi\rangle}\widetilde{g_{{\epsilon}_{2}}}\big{)}\big{\|}}_{L^{8/7}_{x}}+{\big{\|}T_{1}^{S,1}[b_{1}^{\prime}]\big{(}e^{{\epsilon}_{2}it\langle\xi\rangle}\widetilde{g_{{\epsilon}_{2}}},e^{{\epsilon}_{1}it\langle\xi\rangle}\partial_{t}\widetilde{g_{{\epsilon}_{1}}}\big{)}\big{\|}}_{L^{8/7}_{x}}\Big{]}$
$\displaystyle+\sup_{t\approx 2^{m}}\sum_{k_{1}<5,k_{2}\approx
1}{\big{\|}T_{1}^{S,2}[b_{0}]\big{(}e^{{\epsilon}_{1}it\langle\xi\rangle}\partial_{t}\widetilde{g_{{\epsilon}_{1}}},e^{{\epsilon}_{2}it\langle\xi\rangle}\widetilde{g_{{\epsilon}_{2}}}\big{)}\big{\|}}_{L^{8/7}_{x}}.$

We focus again on the $T_{1}^{S,1}[b_{1}]$ term since the others can be
treated identically. Using (5.40b), and (6.40),

$\displaystyle\sup_{t\approx
2^{m}}{\big{\|}T_{1}^{S,1}[b_{1}]\big{(}e^{{\epsilon}_{1}it\langle\xi\rangle}\partial_{t}\widetilde{g_{{\epsilon}_{1}}},e^{{\epsilon}_{2}it\langle\xi\rangle}\widetilde{g_{{\epsilon}_{2}}}\big{)}\big{\|}}_{L^{8/7}_{x}}$
$\displaystyle\lesssim 2^{C_{0}\delta_{N}m}\sup_{t\approx
2^{m}}{\big{\|}e^{itL}\partial_{t}g\big{\|}}_{L^{24/17-}_{x}}{\big{\|}e^{itL}g\big{\|}}_{L^{6}_{x}}$
$\displaystyle\lesssim 2^{C_{0}\delta_{N}m}\cdot\rho(2^{m})\cdot\rho(2^{m})m,$

having used (4.21) and (4.17) in the last inequality. This is sufficient to
obtain a bound as in (6.32). ∎

###  6.4. Fermi-continuous interaction

The goal of this subsection is to prove the following lemma which deals with
the bilinear (and trilinear) interaction of the good (continuous) and bad
(Fermi) components:

######  Lemma 6.9.

Under the assumptions of Proposition 6.2, for all
${\epsilon_{1},\epsilon_{2}}\in\\{+,-\\}$, we have

$\displaystyle\big{\|}\partial_{\xi}T^{{\epsilon_{1},\epsilon_{2}},(1)}_{m}(g,h)\big{\|}_{L^{2}_{x}}+\big{\|}\partial_{\xi}T^{{\epsilon_{1},\epsilon_{2}},(1)}_{m}(h,g)\big{\|}_{L^{2}_{x}}\lesssim
2^{m}\rho^{1-\beta+\delta^{\prime}}(2^{m}){\varepsilon}^{\beta},$ (6.41)

for some $\delta^{\prime}>0$, and

$\displaystyle{\big{\|}\mathcal{F}^{-1}_{\xi\mapsto
x}e^{it\langle\xi\rangle}\partial_{t}T^{{\epsilon_{1},\epsilon_{2}},(1)}_{m}(g,h)\big{\|}}_{L^{8/7}_{x}}+{\big{\|}\mathcal{F}^{-1}_{\xi\mapsto
x}e^{it\langle\xi\rangle}\partial_{t}T^{{\epsilon_{1},\epsilon_{2}},(1)}_{m}(h,g)\big{\|}}_{L^{8/7}_{x}}\lesssim\langle
t\rangle^{3/4}\rho^{2-3\beta}(t).$ (6.42)

###### Proof.

We adopt similar notation to those in Lemmas 6.6 and 6.8 above.

Proof of (6.41). Applying again the same reduction from the beginning of Lemma
6.6, and using the notation from (6.34), we can focus on estimating the
$L^{2}$-norms of the terms

$\displaystyle\begin{split}C_{1,{\epsilon_{1},\epsilon_{2}}}[g_{{\epsilon}_{1}},h_{{\epsilon}_{2}}],\qquad
C_{1,{\epsilon_{1},\epsilon_{2}}}[h_{{\epsilon}_{1}},g_{{\epsilon}_{2}}],\\\
C_{2,{\epsilon_{1},\epsilon_{2}}}[g_{{\epsilon}_{1}},h_{{\epsilon}_{2}}],\qquad
C_{2,{\epsilon_{1},\epsilon_{2}}}[h_{{\epsilon}_{1}},g_{{\epsilon}_{2}}].\end{split}$
(6.43)

Estimates for the $C_{1}$ terms. We have analogues of the estimate (6.35),
with the definitions (5.34) and (6.21), for all the terms in (6.43). In
particular, we see that the first term is bounded as follows:

$\displaystyle\begin{split}&{\Big{\|}\widehat{\mathcal{F}}^{-1}_{\xi\mapsto
x}C_{1,{\epsilon_{1},\epsilon_{2}}}[g_{{\epsilon}_{1}},h_{{\epsilon}_{2}}](t)\Big{\|}}_{L^{2}_{x}}\lesssim\Big{\|}\int_{0}^{t}s\tau_{m}(s)e^{-isL}\Big{[}\sum_{k_{1},k_{2}\leqslant
M_{0}}T_{0}[b_{0}]\big{(}e^{{\epsilon}_{1}is\langle\xi\rangle}\partial_{s}\widetilde{g_{{\epsilon}_{1}}},e^{{\epsilon}_{2}is\langle\xi\rangle}\varphi_{k_{2}}\widetilde{h_{{\epsilon}_{2}}}\big{)}\\\
&+T_{1}^{S,2}[b_{0}]\big{(}e^{{\epsilon}_{1}is\langle\xi\rangle}\partial_{s}\widetilde{g_{{\epsilon}_{1}}},e^{{\epsilon}_{2}is\langle\xi\rangle}\varphi_{k_{2}}\widetilde{h_{{\epsilon}_{2}}}\big{)}+\sum_{i=2,3}T_{i}^{S}[b_{0}]\big{(}e^{{\epsilon}_{1}is\langle\xi\rangle}\partial_{s}\widetilde{g_{{\epsilon}_{1}}},e^{{\epsilon}_{2}is\langle\xi\rangle}\varphi_{k_{2}}\widetilde{h_{{\epsilon}_{2}}}\big{)}\\\
&+\sum_{|k_{1}-k_{2}|<5,k_{2}\leqslant
M_{0}}\Big{(}T_{1}^{S,1}[b_{1}]\big{(}e^{{\epsilon}_{1}is\langle\xi\rangle}\partial_{s}\widetilde{g_{{\epsilon}_{1}}},e^{{\epsilon}_{2}is\langle\xi\rangle}\varphi_{k_{2}}\widetilde{h_{{\epsilon}_{2}}}\big{)}+T_{1}^{S,1}[b_{1}^{\prime}]\big{(}e^{{\epsilon}_{2}is\langle\xi\rangle}\widetilde{h_{{\epsilon}_{2}}},e^{{\epsilon}_{1}is\langle\xi\rangle}\partial_{s}\widetilde{g_{{\epsilon}_{1}}}\big{)}\Big{)}\Big{]}\,ds\Big{\|}_{L^{2}}.\end{split}$
(6.44)

As before it suffices to treat the contribution from $T_{1}^{S,1}[b_{1}]$
since the others are almost identical. An analogue of Remark 6.7 holds in this
case and therefore we may reduce to sum over $k_{1},k_{2}$ with only $O(m)$
terms. Note also that a direct application of the bilinear bounds of Theorem
5.7, as done before in Lemma 6.8, would not suffice in this case. We first
apply the Strichartz estimate from Lemma 2.6 with the indices
$(\widetilde{q},\widetilde{r},\gamma)=(2+,\infty-,0+),$ and then use (5.40b)
to obtain:

$\displaystyle\begin{split}&{\Big{\|}\int_{0}^{t}s\tau_{m}(s)e^{-isL}T_{1}^{S,1}[b_{1}]\big{(}e^{{\epsilon}_{1}is\langle\xi\rangle}\partial_{s}\widetilde{g_{{\epsilon}_{1}}},e^{{\epsilon}_{2}is\langle\xi\rangle}\varphi_{k_{2}}\widetilde{h_{{\epsilon}_{2}}}\big{)}\,ds\Big{\|}}_{L^{2}_{x}}\\\
&\lesssim 2^{3m/2+}\cdot\sup_{s\approx
2^{m}}{\big{\|}T_{1}^{S,1}[b_{1}]\big{(}e^{{\epsilon}_{1}is\langle\xi\rangle}\partial_{s}\widetilde{g_{{\epsilon}_{1}}},e^{{\epsilon}_{2}is\langle\xi\rangle}\varphi_{k_{2}}\widetilde{h_{{\epsilon}_{2}}}\big{)}\big{\|}}_{L^{1+}_{x}}\\\
&\lesssim 2^{3m/2+}\sup_{s\approx
2^{m}}\Big{[}{\big{\|}e^{isL}\partial_{s}g\big{\|}}_{L^{6/5-}}{\|e^{isL}P_{k_{2}}h\|}_{L^{6}}\cdot
2^{C_{0}M_{0}}+2^{-D}\cdot\mathcal{D}\big{(}e^{{\epsilon}_{1}is\langle\xi\rangle}\partial_{s}\widetilde{g_{{\epsilon}_{1}}},e^{{\epsilon}_{2}is\langle\xi\rangle}\varphi_{k_{2}}\widetilde{h_{{\epsilon}_{2}}}\big{)}\Big{]}.\end{split}$
(6.45)

Using (4.21) and (4.19) we obtain an estimate for the first term on the last
line of (6.45) by

$\displaystyle C2^{3m/2}\cdot
2^{(C_{0}+1)\delta_{N}}\cdot\rho(2^{m})\cdot\rho(2^{m})^{1-\beta}{\varepsilon}^{\beta}$

which is sufficient for (6.41). To estimate the contribution from
$\mathcal{D}$ (recall (5.39)) we can use once again the admissible choice
$D=5m$, and the a priori bound (4.13) on
$\|\partial_{\xi}\widetilde{h}\|_{L^{2}}$ and (4.21).

To treat the term
$C_{1,{\epsilon_{1},\epsilon_{2}}}[h_{{\epsilon}_{1}},g_{{\epsilon}_{2}}]$ we
instead use first the Strichartz estimates with indices
$(\widetilde{q},\widetilde{r},\gamma)=(6,3,2/3)$:

$\displaystyle\begin{split}&{\Big{\|}\int_{0}^{t}s\tau_{m}(s)e^{isL}T_{1}^{S,1}[b_{1}]\big{(}e^{{\epsilon}_{1}is\langle\xi\rangle}\partial_{s}\widetilde{h_{{\epsilon}_{1}}},e^{{\epsilon}_{2}is\langle\xi\rangle}\widetilde{g_{{\epsilon}_{2}}}\big{)}\,ds\Big{\|}}_{L^{2}_{x}}\\\
&\lesssim 2^{11m/6+\delta_{N}m}\cdot\sup_{s\approx
2^{m}}{\big{\|}T_{1}^{S,1}[b_{1}]\big{(}e^{{\epsilon}_{1}is\langle\xi\rangle}\partial_{s}\widetilde{h_{{\epsilon}_{1}}},e^{{\epsilon}_{2}is\langle\xi\rangle}\widetilde{g_{{\epsilon}_{2}}}\big{)}\big{\|}}_{L^{3/2}_{x}}\\\
&\lesssim 2^{11m/6+\delta_{N}m}\cdot\sup_{s\approx
2^{m}}\Big{[}{\|e^{isL}P_{k_{2}}\partial_{s}h\|}_{L^{2+}}{\big{\|}e^{isL}g\big{\|}}_{L^{6-}}\cdot
2^{C_{0}M_{0}}\\\
&\qquad+2^{-D}\cdot\mathcal{D}\big{(}e^{{\epsilon}_{1}is\langle\xi\rangle}\widetilde{g_{{\epsilon}_{1}}},e^{{\epsilon}_{2}is\langle\xi\rangle}\varphi_{k_{2}}\partial_{s}\widetilde{h_{{\epsilon}_{2}}}\big{)}\Big{]}.\end{split}$
(6.46)

Using (4.20), together with Bernstein and the fact that the frequencies of $h$
are bounded by $C2^{M_{0}}$, and (4.17),

$\displaystyle\begin{split}{\big{\|}e^{isL}\partial_{s}h\big{\|}}_{L^{2+}}{\big{\|}e^{isL}g\big{\|}}_{L^{6-}}\lesssim\rho(2^{m})\cdot\rho(2^{m})\cdot
2^{2\delta_{N}m}\lesssim
2^{-2m+2\delta_{N}m}\rho(2^{m})^{\beta/2}Z,\end{split}$

which is sufficient for the desired bound, as per the usual Remark 6.3. Using
also (4.26) we can easily handle the remainder term $\mathcal{D}$ in (6.46) by
making the usual choice $D=5m$.

Estimates for the $C_{2}$ terms. Let us now look at the $C_{2}$ terms from
(6.43); recall the definition from (6.34). We can proceed similarly to the
proof above. First, with the usual notation, we write the analogue of (6.44)
(see also (6.38)) for the first term:

$\displaystyle\begin{split}&{\Big{\|}\widehat{\mathcal{F}}^{-1}_{\xi\mapsto
x}C_{2,{\epsilon_{1},\epsilon_{2}}}[g_{{\epsilon}_{1}},h_{{\epsilon}_{2}}](t)\Big{\|}}_{L^{2}}\lesssim\Big{\|}\int_{0}^{t}\tau_{m}(s)e^{-isL}\Big{[}\sum_{k_{1},k_{2}\leqslant
M_{0}}T_{0}[b_{0}]\big{(}e^{{\epsilon}_{1}is\langle\xi\rangle}\partial_{s}\widetilde{g_{{\epsilon}_{1}}},e^{{\epsilon}_{2}is\langle\xi\rangle}\varphi_{k_{2}}\partial_{\xi}\widetilde{h_{{\epsilon}_{2}}}\big{)}\\\
&+\sum_{i=2,3}T_{i}^{S}[m_{0}^{j}]\big{(}e^{{\epsilon}_{1}is\langle\xi\rangle}\partial_{s}\widetilde{g_{{\epsilon}_{1}}},e^{{\epsilon}_{2}is\langle\xi\rangle}\varphi_{k_{2}}\partial_{\xi_{j}}\widetilde{h_{{\epsilon}_{2}}}\big{)}+T_{1}^{S,2}[m_{2}^{j}]\big{(}e^{{\epsilon}_{1}is\langle\xi\rangle}\partial_{s}\widetilde{g_{{\epsilon}_{1}}},e^{{\epsilon}_{2}is\langle\xi\rangle}\varphi_{k_{2}}\partial_{\xi_{j}}\widetilde{h_{{\epsilon}_{2}}}\big{)}\\\
&+\sum_{|k_{1}-k_{2}|<5,k_{2}\leqslant
M_{0}}\Big{(}T_{1}^{S,1}[b_{1}]\big{(}e^{{\epsilon}_{1}is\langle\xi\rangle}\partial_{s}\widetilde{g_{{\epsilon}_{1}}},e^{{\epsilon}_{2}is\langle\xi\rangle}\varphi_{k_{2}}\partial_{\xi}\widetilde{h_{{\epsilon}_{2}}}\big{)}\\\
&+T_{1}^{S,1}[m_{1}^{j}]\big{(}e^{{\epsilon}_{2}is\langle\xi\rangle}\varphi_{k_{2}}\partial_{\xi_{j}}\widetilde{h_{{\epsilon}_{2}}},e^{{\epsilon}_{1}is\langle\xi\rangle}\partial_{s}\widetilde{g_{{\epsilon}_{1}}}\big{)}\Big{)}\Big{]}\,ds\Big{\|}_{L^{2}}.\end{split}$
(6.47)

Focusing on the usual main contribution, an application of the Strichartz
estimate with $(\widetilde{q},\widetilde{r},\gamma)=(6,3,2/3)$ and (5.40b)
give us

$\displaystyle\begin{split}&{\Big{\|}\int_{0}^{t}\tau_{m}(s)T_{1}^{S,1}[b_{1}]\big{(}e^{{\epsilon}_{1}is\langle\xi\rangle}\partial_{s}\widetilde{g_{{\epsilon}_{1}}},e^{{\epsilon}_{2}is\langle\xi\rangle}\partial_{\xi}\widetilde{h_{{\epsilon}_{2}}}\big{)}\,ds\Big{\|}}_{L^{2}_{x}}\\\
&\lesssim 2^{5m/6}\cdot 2^{M_{0}}\cdot\sup_{s\approx
2^{m}}{\big{\|}T_{1}[b_{1}]\big{(}e^{{\epsilon}_{1}is\langle\xi\rangle}\partial_{s}\widetilde{g_{{\epsilon}_{1}}},e^{{\epsilon}_{2}is\langle\xi\rangle}\varphi_{k_{2}}\partial_{\xi}\widetilde{h_{{\epsilon}_{2}}}\big{)}\big{\|}}_{L^{3/2}_{x}}\\\
&\lesssim 2^{5m/6}\cdot 2^{M_{0}}\cdot\sup_{s\approx
2^{m}}\Big{[}{\big{\|}e^{isL}\partial_{s}g\big{\|}}_{L^{6-}}{\big{\|}\widehat{\mathcal{F}}^{-1}e^{{\epsilon}_{2}is\langle\xi\rangle}(\varphi_{k_{2}}\partial_{\xi}\widetilde{h_{{\epsilon}_{2}}})\big{\|}}_{L^{2+}}\cdot
2^{C_{0}M_{0}}\\\
&\qquad\qquad\qquad\qquad+2^{-D}\cdot\mathcal{D}\big{(}e^{{\epsilon}_{1}is\langle\xi\rangle}\partial_{s}\widetilde{g_{{\epsilon}_{1}}},e^{{\epsilon}_{2}is\langle\xi\rangle}\varphi_{k_{2}}\partial_{\xi}\widetilde{h_{{\epsilon}_{2}}}\big{)}\Big{]}.\end{split}$
(6.48)

Using (4.21) and (4.13) we can bound by an absolute constant the sum over
$k_{2}$ of first term in the last expression above; using (4.21) and the bound

$\displaystyle{\|\partial_{\xi}\partial_{s}\widetilde{g}\|}_{L^{2}}\lesssim
2^{m}\rho(2^{m})$ (6.49)

we can bound the remainder term in (6.48) with the usual choice $D=5m$.

For the last term
$C_{2,{\epsilon_{1},\epsilon_{2}}}[h_{{\epsilon}_{1}},g_{{\epsilon}_{2}}]$
from (6.43) (recall the notation (6.34)) the roles of $h$ and $g$ are reversed
compared to the term just treated above. Starting with the analogue of (6.47)
matters are reduced to bounding

$\displaystyle\begin{split}&{\Big{\|}\int_{0}^{t}\tau_{m}(s)e^{isL}T_{1}^{S,1}[b_{1}]\big{(}e^{{\epsilon}_{1}is\langle\xi\rangle}\varphi_{k_{1}}\partial_{s}\widetilde{h_{{\epsilon}_{1}}},e^{{\epsilon}_{2}is\langle\xi\rangle}\partial_{\xi}\widetilde{g_{{\epsilon}_{2}}}\big{)}\,ds\Big{\|}}_{L^{2}_{x}}.\end{split}$
(6.50)

Using (5.40b) we upperbound the above term by

$\displaystyle\begin{split}C2^{m}\cdot\sup_{s\approx
2^{m}}\Big{[}{\big{\|}e^{isL}P_{k_{1}}\partial_{s}h\big{\|}}_{L^{2}_{x}}{\|\partial_{\xi}\widetilde{g}\|}_{L^{1+}_{\xi}}\cdot
2^{C_{0}M_{0}}+2^{-D}\cdot\mathcal{D}\big{(}e^{{\epsilon}_{1}is\langle\xi\rangle}\varphi_{k_{1}}\partial_{s}\widetilde{h_{{\epsilon}_{1}}},e^{{\epsilon}_{2}is\langle\xi\rangle}\partial_{\xi}\widetilde{g_{{\epsilon}_{2}}}\big{)}\Big{]}.\end{split}$

Then we can use (4.20), and interpolate (4.26) with (4.27) to estimate the sum
over $k_{1}$ of the first term above by

$\displaystyle C2^{m}\cdot 2^{(C_{0}+)m}\cdot\rho(2^{m})\cdot
2^{m}\rho(2^{m})$ $\displaystyle=2^{(C_{0}+)m}\cdot
Z^{2}\rho^{2\beta}(2^{m}){\varepsilon}^{-2\beta},$

which suffices (see also Remark 6.3). Using (4.28) we can take care of the
remainder in the usual way. This concludes the proof of (6.41).

Proof of (6.42). Moving on to the second estimate, we adopt the same notation
above and proceed similarly to the proof of (6.32). For the first term on the
left-hand side of (6.42) we can write up to irrelevant constants

$\displaystyle\widehat{\mathcal{F}}^{-1}_{\xi\mapsto
x}e^{it\langle\xi\rangle}\partial_{t}T^{{\epsilon_{1},\epsilon_{2}},(1)}_{m}(g,h)=\sum_{k_{1},k_{2}\leqslant
M_{0}}\Big{[}T_{0}[b_{0}]\big{(}e^{{\epsilon}_{1}it\langle\xi\rangle}\partial_{t}\widetilde{g_{{\epsilon}_{1}}},e^{{\epsilon}_{2}it\langle\xi\rangle}\varphi_{k_{2}}\widetilde{h_{{\epsilon}_{2}}}\big{)}$
$\displaystyle\sum_{i=2,3}T_{i}^{S}[b_{0}]\big{(}e^{{\epsilon}_{1}it\langle\xi\rangle}\partial_{t}\widetilde{g_{{\epsilon}_{1}}},e^{{\epsilon}_{2}it\langle\xi\rangle}\varphi_{k_{2}}\widetilde{h_{{\epsilon}_{2}}}\big{)}+T_{1}^{S,2}[b_{0}]\big{(}e^{{\epsilon}_{1}it\langle\xi\rangle}\partial_{t}\widetilde{g_{{\epsilon}_{1}}},e^{{\epsilon}_{2}it\langle\xi\rangle}\varphi_{k_{2}}\widetilde{h_{{\epsilon}_{2}}}\big{)}\Big{]}$
$\displaystyle+\sum_{|k_{1}-k_{2}|<5,k_{2}\leqslant
M_{0}}\Big{[}T_{1}^{S,1}[b_{1}]\big{(}e^{{\epsilon}_{1}it\langle\xi\rangle}\partial_{t}\widetilde{g_{{\epsilon}_{1}}},e^{{\epsilon}_{2}it\langle\xi\rangle}\varphi_{k_{2}}\widetilde{h_{{\epsilon}_{2}}}\big{)}$
$\displaystyle+T_{1}^{S,1}[b_{1}^{\prime}]\big{(}e^{{\epsilon}_{2}it\langle\xi\rangle}\varphi_{k_{2}}\widetilde{h_{{\epsilon}_{2}}},e^{{\epsilon}_{1}it\langle\xi\rangle}\partial_{t}\widetilde{g_{{\epsilon}_{1}}}\big{)}\Big{]}.$

We focus again on the $T_{1}^{S,1}[b_{1}]$ term since the others can be
treated identically, using Bernstein’s inequality to ensure convergence of the
sums. Using (5.40b) with the choice of $D=5m$, we have, up to a faster
decaying term,

$\displaystyle\sup_{t\approx
2^{m}}{\Big{\|}\widehat{\mathcal{F}}^{-1}_{\xi\mapsto
x}e^{it\langle\xi\rangle}\partial_{t}T^{{\epsilon_{1},\epsilon_{2}},(1)}_{m}(g,h)\Big{\|}}_{L^{8/7}_{x}}$
$\displaystyle\lesssim 2^{C_{0}\delta_{N}m}\sup_{t\approx
2^{m}}{\big{\|}e^{itL}\partial_{t}g\big{\|}}_{L^{24/17}_{x}}{\big{\|}e^{itL}h\big{\|}}_{L^{6-}_{x}}$
$\displaystyle\lesssim
2^{C_{0}\delta_{N}m}\cdot\rho(2^{m})\cdot\rho(2^{m})^{1-\beta}2^{10\delta_{N}m}\varepsilon^{\beta},$

having used (4.21) and (4.19). This suffices for (6.42).

A similar estimate can be used to get the same bound when the roles of $g$ and
$h$ are exchanged: using Theorem 5.7 followed by (4.20) and (4.18), we have

$\displaystyle\sup_{t\approx
2^{m}}\sum_{k_{1}}{\Big{\|}\widehat{\mathcal{F}}^{-1}_{\xi\mapsto
x}e^{it\langle\xi\rangle}\partial_{t}T^{{\epsilon_{1},\epsilon_{2}},(1)}_{m,k_{1},k_{2}}(h,g)\Big{\|}}_{L^{8/7}_{x}}$
$\displaystyle\lesssim 2^{C_{0}\delta_{N}m}\sup_{t\approx
2^{m}}\sum_{k_{1}}{\big{\|}\partial_{t}P_{k_{1}}h\big{\|}}_{L^{2+}_{x}}{\big{\|}e^{itL}g\big{\|}}_{L^{8/3-}_{x}}$
$\displaystyle\lesssim
2^{(C_{0}+2)\delta_{N}m}\cdot\rho(2^{m})\cdot\rho(2^{m})2^{m}\cdot
2^{-3m/8+}.$

This suffices to bound the second term on the left-hand side of (6.42) as
desired, and concludes the proof of the lemma. ∎

###  6.5. Continuous interactions

We now deal with terms that are quadratic in $h.$ Note that these are
substantially easier than the ones treated before since $h$ essentially
satisfies better estimates than $g$ does.

######  Lemma 6.10.

For all ${\epsilon_{1},\epsilon_{2}}\in\\{+,-\\}$ we have

$\displaystyle\big{\|}\partial_{\xi}T^{{\epsilon_{1},\epsilon_{2}},(1)}_{m}(h,h)\big{\|}_{L^{2}_{x}}\lesssim{\varepsilon}^{\beta}2^{m}\rho^{1-\beta+\delta^{\prime}}(2^{m}),$
(6.51)

for some $\delta^{\prime}>0,$ and

$\displaystyle{\big{\|}\mathcal{F}^{-1}_{\xi\mapsto
x}e^{it\langle\xi\rangle}\partial_{t}T^{{\epsilon_{1},\epsilon_{2}},(1)}_{m}(h,h)\big{\|}}_{L^{8/7}_{x}}\lesssim\frac{\rho_{0}^{a}}{\langle
t\rangle^{5/4}}$ (6.52)

for some $a>0$.

###### Proof.

The proofs can be done similarly to those in Lemma 6.9, and are in fact
easier. We provide some details for completeness.

Following the same steps in the proof of (6.41) we can write an estimate like
(6.44) and reduce to estimating a term like the one in (6.45) with
$\widetilde{g_{{\epsilon}_{1}}}$ replaced by
$\varphi_{k_{1}}\widetilde{h_{{\epsilon}_{1}}}$; for such a term, using
Strichartz estimates with $(\widetilde{q},\widetilde{r},\gamma)=(6,3,2/3)$,
and Theorem 5.7 we have

$\displaystyle\begin{split}&{\Big{\|}\int_{0}^{t}s\tau_{m}(s)e^{-isL}T_{1}^{S,1}[b_{1}]\big{(}e^{{\epsilon}_{1}is\langle\xi\rangle}\partial_{s}\widetilde{h_{{\epsilon}_{1}}},e^{{\epsilon}_{2}is\langle\xi\rangle}\varphi_{k_{2}}\widetilde{h_{{\epsilon}_{2}}}\big{)}\,ds\Big{\|}}_{L^{2}_{x}}\\\
&\lesssim 2^{11m/6+\delta_{N}m}\cdot\sup_{s\approx
2^{m}}{\big{\|}T_{1}[b_{1}]\big{(}e^{{\epsilon}_{1}is\langle\xi\rangle}\varphi_{k_{1}}\partial_{s}\widetilde{h_{{\epsilon}_{1}}},e^{{\epsilon}_{2}is\langle\xi\rangle}\varphi_{k_{2}}\widetilde{h_{{\epsilon}_{2}}}\big{)}\big{\|}}_{L^{3/2}_{x}}\\\
&\lesssim 2^{11m/6+\delta_{N}m}\sup_{s\approx
2^{m}}\Big{[}{\big{\|}e^{isL}P_{k_{1}}\partial_{s}h\big{\|}}_{L^{2+}}{\|e^{isL}P_{k_{2}}h\|}_{L^{6-}}\cdot
2^{C_{0}M_{0}}\\\
&\qquad\qquad\qquad\qquad+2^{-D}\cdot\mathcal{D}\big{(}e^{{\epsilon}_{1}is\langle\xi\rangle}\varphi_{k_{1}}\partial_{s}\widetilde{h_{{\epsilon}_{1}}},e^{{\epsilon}_{2}is\langle\xi\rangle}\varphi_{k_{2}}\widetilde{h_{{\epsilon}_{2}}}\big{)}\Big{]}.\end{split}$
(6.53)

From this we can deduce the desired bound, using the decay from (4.20) and
(4.19), and handling the remainder in the usual way.

The estimate on the $C_{2}$ type terms follows by estimating

$\displaystyle\begin{split}&\sum_{|k_{1}-k_{2}|<5,k_{2}\leqslant
M_{0}}{\Big{\|}\int_{0}^{t}\tau_{m}(s)e^{isL}T_{1}^{S,1}[b_{1}]\big{(}e^{{\epsilon}_{1}is\langle\xi\rangle}\varphi_{k_{1}}\partial_{s}\widetilde{h_{{\epsilon}_{1}}},e^{{\epsilon}_{2}is\langle\xi\rangle}\partial_{\xi}\widetilde{h_{{\epsilon}_{2}}}\big{)}\,ds\Big{\|}}_{L^{2}_{x}}.\end{split}$
(6.54)

This term is analogous to (6.50), but here the bound follows directly after
using the Strichartz estimate with indices
$(\widetilde{q},\widetilde{r},\gamma)=(6,3,2/3)$, followed by an $L^{6-}\times
L^{2+}$ product estimate via (5.40b), Bernstein, and (4.20) and (4.13).

For (6.52), using Theorem 5.7 we can bound the term on the left-hand side, up
to a faster decaying remainder, by

$\displaystyle C\sum_{k_{1},k_{2}\leqslant M_{0}}\sup_{\approx
2^{m}}{\|\partial_{t}P_{k_{1}}h(t)\|}_{L^{2+}}{\|e^{itL}P_{k_{2}}h(t)\|}_{L^{8/3-}}\cdot
2^{C_{0}\delta_{N}m}$ $\displaystyle\lesssim
2^{(C_{0}+1)\delta_{N}m}\cdot\rho(2^{m})\cdot\rho(2^{m})^{1-\beta}2^{5/8m+10\delta_{N}m}\varepsilon^{\beta},$

having used Bernstein and the bounds (4.20) and (4.19) for the last
inequality. This suffices for (6.52) and concludes the proof of the Lemma. ∎

With the proofs of Lemmas 6.6, 6.8, 6.9 and 6.10 (see also Lemmas 6.4 and
6.5), the proof of Proposition 6.2 is complete. To finish the proof of the
main Propositions 5.1 and 5.2, we are left with estimating the regular terms,
see (5.46). We proceed to do this in the next section.

##  7\. Regular quadratic terms

In this section we estimate the bilinear terms corresponding to the regular
parts of the distribution. Looking at (5.46) we define

$\displaystyle\begin{split}T_{m}^{{\epsilon_{1},\epsilon_{2}},(3)}(G,H)&:=\int_{0}^{t}\tau_{m}(s)\int_{{\mathbb{R}}^{6}}\frac{e^{-is\Phi_{{\epsilon_{1},\epsilon_{2}}}(\xi,\eta,\sigma)}}{\langle\eta\rangle\langle\sigma\rangle}\widetilde{G}(s,\eta)\widetilde{H}(s,\sigma)\\\
&\times\varphi_{\leqslant M_{0}}(\langle\eta\rangle)\varphi_{\leqslant
M_{0}}(\langle\sigma\rangle)\varphi_{\leqslant
M_{0}}(\langle\xi\rangle)\mu^{R}(\xi,\eta,\sigma)\,d\eta d\sigma ds,\\\
T_{m}^{{\epsilon_{1},\epsilon_{2}},(4)}(G,H)&:=\int_{0}^{t}\tau_{m}(s)\int_{{\mathbb{R}}^{6}}\frac{e^{-is\Phi_{{\epsilon_{1},\epsilon_{2}}}(\xi,\eta,\sigma)}}{\langle\eta\rangle\langle\sigma\rangle}\widetilde{G}(s,\eta)\widetilde{H}(s,\sigma)\\\
&\times\varphi_{\leqslant M_{0}}(\langle\eta\rangle)\varphi_{\leqslant
M_{0}}(\langle\sigma\rangle)\varphi_{\leqslant
M_{0}}(\langle\xi\rangle)\mu^{Re}(\xi,\eta,\sigma)\,d\eta d\sigma
ds,\end{split}$ (7.1)

where the distributions are defined in Proposition 5.6 and the phase is as in
(5.7). The goal of this section is to prove the analogue of Proposition 6.2
for the regular terms defined above:

######  Proposition 7.1.

Under the assumptions (4.13), for all $m=0,1,...,L+1,$ for all
${\epsilon_{1},\epsilon_{2}}\in\\{+,-\\},$ we have for some $0<\delta<\beta$,
and $j=3,4$:

$\displaystyle\|\nabla_{\xi}T_{m}^{{\epsilon_{1},\epsilon_{2}},(j)}(G_{\epsilon_{1}},H_{\epsilon_{2}})\|_{L^{2}}\lesssim\rho(2^{m})^{1-\beta+\delta}2^{m}{\varepsilon}^{\beta},\qquad
G,H\in\\{g,h\\}.$ (7.2)

Moreover, there exist $0<a<1$ and $\delta^{\prime}>0$ such that

$\displaystyle\bigg{\|}e^{itL}\widetilde{\mathcal{F}}^{-1}_{\xi\mapsto
x}\big{[}T_{m}^{{\epsilon_{1},\epsilon_{2}},(j)}(G_{\epsilon_{1}},H_{\epsilon_{2}})\big{]}\bigg{\|}_{L^{\infty}_{x}}\lesssim\frac{\rho_{0}^{a}}{\langle
t\rangle^{1+a+\delta^{\prime}}},\qquad G,H\in\\{g,h\\}.$ (7.3)

With Proposition 6.2 and Lemmas 5.4 and 5.5, this will complete the proof of
Propositions 5.1 and 5.2.

###  7.1. Preliminaries

We first reduce the proof of (7.2) and (7.3) to some relatively easier
weighted $L^{2}$ bounds. We start by reducing (7.3) to $L^{p}$ estimates for
some $p>6/5$, through the following analogue of Lemma 6.4:

######  Lemma 7.2.

Assume that there exist $0<a<b<1/10$, such that

$\displaystyle\big{\|}\widehat{\mathcal{F}}^{-1}_{\xi\mapsto
x}I(s)\big{\|}_{L^{6/(5+6b)}_{x}}\lesssim\frac{\rho_{0}^{a}}{\langle
s\rangle^{1+2a}},$ (7.4)

where

$\displaystyle
I(s):=e^{is\langle\xi\rangle}\partial_{s}T^{{\epsilon_{1},\epsilon_{2}},(j)}_{m}(G_{\epsilon_{1}},H_{\epsilon_{2}}),\qquad
j=3,4.$ (7.5)

Then

$\displaystyle{\Big{\|}e^{itL}\widetilde{\mathcal{F}}^{-1}_{\xi\mapsto
x}\big{[}T_{m}^{{\epsilon_{1},\epsilon_{2}},(j)}(G_{\epsilon_{1}},H_{\epsilon_{2}})\big{]}\Big{\|}}_{L^{\infty}_{x}}\lesssim\frac{\rho_{0}^{a}}{\langle
t\rangle^{1+a+\delta^{\prime}}},\qquad j=3,4.$

###### Proof.

The proof is almost identical to the one of Lemma 6.4. In this case we first
use Bernstein to go from $L^{\infty}$ to $L^{q}$ with $q:=(1/6-b)^{-1}$.
Notice that $q^{\prime}=(5/6+b)^{-1}=6/(5+6b)$, consistently with the
assumption (7.4). Using that the linear decay rate for the propagator
$e^{itL}$ in $L^{q}$ is $\langle t\rangle^{-(3/2)(1-2/q)}=\langle
t\rangle^{-1-3b}$ and that $b>a$, we obtain the desired conclusion. ∎

We will also need a bilinear lemma for operators with symbol
$\partial_{\xi}\nu_{1}^{R}$. Recall the definition (5.34b)-(5.34c); in what
follows, whenever needed, we will write explicitly the dependence on the
measures by denoting

$\displaystyle\begin{split}T_{1}^{R,1}[\nu;b](g,h)(x)&:=\widehat{\mathcal{F}}^{-1}_{\xi\rightarrow
x}\iint_{{\mathbb{R}}^{3}\times{\mathbb{R}}^{3}}g(\xi-\eta)h(\sigma)\,b(\xi,\eta,\sigma)\,\nu(\eta,\sigma)\,d\eta
d\sigma,\\\
T_{1}^{R,2}[\nu;b](g,h)(x)&:=\widehat{\mathcal{F}}^{-1}_{\xi\rightarrow
x}\iint_{{\mathbb{R}}^{3}\times{\mathbb{R}}^{3}}g(-\eta-\sigma)h(\sigma)\,b(\xi,\eta,\sigma)\,\overline{\nu(\eta,\xi)}\,d\eta
d\sigma,\\\ T[\mu,b](g,h)(x)&:=\widehat{\mathcal{F}}^{-1}_{\xi\rightarrow
x}\iint_{{\mathbb{R}}^{3}\times{\mathbb{R}}^{3}}g(\eta)h(\sigma)\,b(\xi,\eta,\sigma)\,\mu(\xi,\eta,\sigma)\,d\eta
d\sigma.\end{split}$ (7.6)

We then have the following analogue of Theorem 5.7:

######  Lemma 7.3.

With the definitions (7.6), for $b$ (and $A$) as in the statement of Theorem
5.7, and for all $p,q\geqslant 1,r>1$ with $1/p+1/q>1/r$, we have

$\displaystyle{\big{\|}T_{1}^{R,1}[\partial_{1}\nu^{R}_{1};b](G,H)\big{\|}}_{L^{r}}+{\big{\|}T_{1}^{R,2}[\partial_{2}\nu^{R}_{1};b](G,H)\big{\|}}_{L^{r}}\lesssim
2^{C_{0}A}\|\widehat{G}\|_{L^{p}}\|\widehat{H}\|_{L^{q}},$ (7.7)

for some absolute constant $C_{0}$. Moreover, under the same assumptions, we
have

$\displaystyle{\big{\|}T[\partial_{1}\mu^{Re};b](G,H)\big{\|}}_{L^{r}}\lesssim
2^{C_{0}A}\|\widehat{G}\|_{L^{p}}\|\widehat{H}\|_{L^{q}}.$ (7.8)

###### Proof.

(7.7) follows exactly as in the proof of (5.40c), that is, from the standard
bilinear Lemma 6.3 in [41] and the estimate (5.19) for the derivative of
$\nu^{R}_{1}$. (7.8) follows similarly using instead the estimate (5.24). ∎

Next, let us introduce the operators which will appear when applying
$\nabla_{\xi}$ to (7.1). Let

$\displaystyle\begin{split}D_{1,m,{\epsilon_{1},\epsilon_{2}}}(G,H)&:=\int_{0}^{t}s\,\tau_{m}(s)\int_{{\mathbb{R}}^{6}}e^{-is\Phi_{\epsilon_{1},\epsilon_{2}}(\xi,\eta,\sigma)}\widetilde{G}(t,\eta)\,\widetilde{H}(t,\sigma)\,\,b(\xi,\eta,\sigma)\,\mu^{R}(\xi,\eta,\sigma)d\eta
d\sigma\,ds,\\\
D_{2,m,{\epsilon_{1},\epsilon_{2}}}(G,H)&:=\int_{0}^{t}\tau_{m}(s)\int_{{\mathbb{R}}^{6}}e^{-is\Phi_{\epsilon_{1},\epsilon_{2}}(\xi,\eta,\sigma)}\widetilde{G}(s,\eta)\,\widetilde{H}(s,\sigma)\,\,b^{\prime}(\xi,\eta,\sigma)\,\nabla_{\xi}\mu^{R}(\xi,\eta,\sigma)d\eta
d\sigma\,ds,\end{split}$ (7.9)

where

$\displaystyle b^{\prime}(\xi,\eta,\sigma)$
$\displaystyle:=\frac{\varphi_{\leqslant
M_{0}}(\langle\eta\rangle)\varphi_{\leqslant
M_{0}}(\langle\sigma\rangle)\varphi_{\leqslant
M_{0}}(\langle\xi\rangle)}{\langle\eta\rangle\langle\sigma\rangle},\qquad
b(\xi,\eta,\sigma):=\frac{\xi}{\langle\xi\rangle}b^{\prime}(\xi,\eta,\sigma),$
(7.10)

and, for any $k_{1},k_{2}\leqslant M_{0}$

$\displaystyle\begin{split}E_{1,k_{1},k_{2},m,{\epsilon_{1},\epsilon_{2}}}(G,H)&:=\int_{0}^{t}s\,\tau_{m}(s)\int_{{\mathbb{R}}^{6}}e^{-is\Phi_{\epsilon_{1},\epsilon_{2}}(\xi,\eta,\sigma)}\widetilde{G}(s,\eta)\,\widetilde{H}(s,\sigma)\,\\\
&\qquad\qquad\times\,b_{Re}(\xi,\eta,\sigma)\,\mu^{Re}(\xi,\eta,\sigma)d\eta
d\sigma\,ds,\\\