2^{k_{1}}\big{)}\big{(}2^{3k/2+3k_{1}/2}2^{2k_{1}^{+}}+2^{k}2^{-(N/2)k^{+}}\big{)}\cdot
Z.$

Given our assumptions on $k,k_{1}$ this is sufficient to conclude.

Case 2.2: $-5m<k_{1}\leqslant 10/N$. We make use of (8.72), (8.73) with
$\nu^{\prime}=\nu_{i},i=1,2$ to write that

$\displaystyle\||\xi|^{2}II_{m,k,k_{1}}^{(i)}\|_{L^{2}}\lesssim\rho(2^{m})^{1/2}\cdot
2^{3k/2}2^{k_{1}/2}2^{2k_{1}^{+}}\cdot Z.$

For $i=3,$ we use the Strichartz estimate (Lemma 2.6) with parameters
$(\widetilde{q},\widetilde{r},\gamma)=(2,\infty,0)$ and (4.19) to obtain

$\displaystyle\||\xi|^{2}II_{m,k,k_{1}}^{(3)}\|_{L^{2}}\lesssim 2^{k}\cdot
2^{m/2}\rho(2^{m})^{1/2}\cdot
2^{-k_{1}^{+}}\|P_{k_{1}}e^{isL}h\|_{L^{6}_{x}}\lesssim 2^{k}\cdot
2^{-m+10\delta_{N}m}\cdot Z.$

The desired bound follows in all cases.

Case $G=g$. Note that in this case $k_{1}\sim 0,$ therefore we will disregard
the sum over that index. We insert cutoffs
$\varphi_{\ell}^{(\ell_{0})}(\langle\xi\rangle-2\lambda),\ell_{0}:=\lfloor-m+\delta
m\rfloor,$ where $\delta>0$ is chosen suitably small.

Case 1: $k<m/10$. If $\ell>\ell_{0}$ we integrate by parts in $\eta$ in the
formula (8.7). Using (8.72), (8.73), as well as (4.27) and (4.24), we find

$\displaystyle\|II_{m,k}(t,\xi)\|_{L^{2}_{\xi}}$
$\displaystyle\lesssim\rho(2^{m})^{1/2}\cdot
2^{3k/2}\big{(}\|\varphi_{\ell}(\langle\xi\rangle-2\lambda)\nabla_{\eta}\widetilde{g}(s,\eta)\|_{L^{1}_{\eta}}+{\|\widetilde{g}\|}_{L^{1}}\big{)}$
$\displaystyle\lesssim\rho(2^{m})^{1/2}\cdot 2^{3m/20}\cdot
2^{m}\rho(2^{m})m^{2},$

which suffices. A similar estimate holds if $\ell=\ell_{0},$ integrating
directly using (4.24).

Case 2: $k\geqslant m/10$. In this case we use the identity (8.74) and the
same notation in Case 2.1 above. As above we integrate by parts in frequency
if $\ell>\ell_{0}$ and integrate directly otherwise. Using (8.72), (8.73) as
well as (4.27) and (4.24), we find, for $i=1,2$,

$\displaystyle\big{\|}|\xi|^{2}II_{m,k}^{(i)}\big{\|}_{L^{2}_{\xi}}$
$\displaystyle\lesssim\rho(2^{m})^{1/2}\cdot
2^{3k/2}\big{(}\|\varphi_{\ell}(\langle\xi\rangle-2\lambda)\nabla_{\eta}\widetilde{g}(s,\eta)\|_{L^{1}_{\eta}}+{\|\widetilde{g}\|}_{L^{\infty}}\big{)}$
$\displaystyle\lesssim\rho(2^{m})^{1/2}\cdot 2^{3k/2}(2^{m}\rho(2^{m})m^{2}).$

For the last case, we can conclude as we did for the profile $h$ above using
Strichartz estimates as well as (4.17):

$\displaystyle\big{\|}|\xi|^{2}II_{m,k}^{(3)}(t,\xi)\big{\|}_{L^{2}_{\xi}}\lesssim
2^{k}\cdot 2^{m/2}\rho(2^{m})^{1/2}\|e^{itL}g\|_{L^{6}_{x}}\lesssim
2^{k}\cdot\rho(2^{m})m.$

This is sufficient to conclude. ∎

###  8.6. The remainder terms

Here we explain how to estimate the remainder terms of mixed type $RM$ in
(4.9); since these are easier to treat compared to $M_{1}$ and $M_{2}$ above,
we will leave some details to the reader.

#### The term (4.9a)

Using (3.34), the term (4.9a) can be written as

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

The term $RM_{1,1}$ is similar to the term $M_{1}$ treated in subsections
above, where the only difference is the opposite sign of the oscillation of
the amplitude of the internal mode $e^{-is\lambda}$. The arguments above are
clearly insensitive to this change, and we can then apply the same estimates
verbatim. The starting point is the analogue of (8.24) with the phase now
given by $\Phi(\xi,\eta)=\langle\xi\rangle+\lambda-\langle\eta\rangle$. For
example, the contributions corresponding to (8.40) now have phases
$\langle\xi\rangle+\lambda-\epsilon_{1}\langle\sigma\rangle-\epsilon_{2}\langle\rho\rangle$.
This does not impact the estimates of the term (8.42) since (8.45) still holds
by changing the sign of $\lambda$ (we are restricting to
$|\langle\xi\rangle+\lambda-\langle\eta\rangle|\approx 2^{p}$ in this case).
The same goes for (8.50) and (8.54).

The term $RM_{1,2}$ enjoys the same estimates satisfied by $M_{2}$, which are
based on Strichartz estimates and are not affected by the different sign of
the oscillation of the discrete component.

Using $A-B=O(A^{2})$, we see that $RM_{1,3}$ has the form

$\displaystyle\int_{0}^{t}O(A^{2}(s))\int_{{\mathbb{R}}^{3}}e^{-is(\langle\xi\rangle-\langle\eta\rangle-\lambda)}\langle\eta\rangle^{-1}\widetilde{f}(s,\eta)\nu(\xi,\eta)d\eta.$
(8.75)

After taking a $\partial_{\xi}$ derivative at the cost of adding an $s$
factor, we can directly apply the usual Strichartz estimate from Lemma 2.6, as
in the estimate of $M_{2}$, and use the $L^{6}_{x}$ a priori decay estimate
for $e^{itL}f$, see Lemma 4.4, and $O(A^{2}(s))=\langle s\rangle^{-1}$, to get

$\displaystyle\big{\|}\partial_{\xi}RM_{1,3}\big{\|}_{L^{2}}\lesssim\sum_{m}\|sA^{2}(s)\|_{L^{2}_{s\approx
2^{m}}}\sup_{s\approx
2^{m}}\|\phi\,e^{isL}f\|_{L^{1}}\lesssim\sum_{m}2^{m/2}\cdot
2^{(-1+\beta+\delta_{N})m}\varepsilon^{\beta}\lesssim 1.$

#### The term (4.9b)

Writing (4.9b) in terms of $A$ gives expressions of the form

$\displaystyle\int_{0}^{t}A_{\epsilon_{1}}(s)\int_{{\mathbb{R}}^{3}}e^{-is(\langle\xi\rangle-\epsilon_{1}\lambda+\langle\eta\rangle)}\langle\eta\rangle^{-1}\overline{\widetilde{f}}(s,\eta)\nu(\xi,\eta)d\eta
ds,$

and we see that the oscillating phase is lower bounded by
$\max(\langle\eta\rangle,\langle\xi\rangle)$. Then this term is clearly easier
to bound than $M_{1}$ and $M_{2}$ above.

#### The term (4.9c)

This is almost identical to the term $RM_{1,3}$ in (8.75), and can be handled
in the same way described above.

##  9\. Energy estimates

In this section we first prove energy estimates for $h$, along with the
$H^{N}$ scattering statement (1.32) in Theorem 1.2. Then, we prove Lemma 5.4
about control of the weighted norm for high frequencies.

###  9.1. Sobolev norms and scattering

We prove the following:

######  Proposition 9.1.

Under the a priori assumption of Proposition 4.2, there exists $\delta>0$ such
that, for any $1\leqslant t_{1}<t_{2}\leqslant T$ we have

$\displaystyle{\|h(t_{1})-h(t_{2})\|}_{H^{N}}\lesssim{\varepsilon}^{1+\delta}\,t_{1}^{-\delta}$
(9.1)

The above proposition implies the conclusion (4.14) for the $H^{N}$ norm, as
well as scattering in $H^{N}$ for $h(t)$. Recall that $h$ satisfies (4.6). In
the next two subsections, we will then estimate the $H^{N}$ norm of the terms
appearing in (4.6).

####  9.1.1. Source terms

We start by inserting the usual $\tau_{m}$ cutoffs (see (1.74)) in the time
integral defining the source terms (4.7), and denote $S_{i}^{m}$, $i=1,2,3$
the corresponding terms. The following lemma gives an estimate consistent with
(9.1) for these source terms.

######  Lemma 9.2.

Under the assumptions of Proposition 4.2, there exists $\delta>0$ such that,
for all $m=0,1,\dots$, we have

$\displaystyle\|S_{i}^{m}\|_{H^{N}}\lesssim 2^{-\delta
m}{\varepsilon}^{1+\delta},$ (9.2)

for $i=1,2,3.$

###### Proof.

We first look at

$\displaystyle
S_{1}^{m}(t,\xi):=\int_{0}^{t}e^{-is(\langle\xi\rangle-2\lambda)}(A^{2}(s)-B^{2}(s))\tau_{m}(s)ds\,\widetilde{\theta}(\xi),$

see (4.7a), and split it as

$\displaystyle
S_{1}^{m}=S_{-}^{m}+S_{+}^{m},\qquad\widetilde{S_{-}^{m}}(t,\xi):=\varphi_{\leqslant-m/10}(\langle\xi\rangle-2\lambda)\widetilde{S_{1}^{m}}(t,\xi).$

We then estimate directly by Hölder and (3.34)

$\displaystyle{\|S_{-}^{m}(t)\|}_{H^{N}_{x}}\lesssim
2^{-m/20}{\|\widetilde{S_{1}^{m}}\|}_{L^{\infty}}\lesssim 2^{-m/20}\cdot
2^{m}\rho^{3/2}(2^{m}),$

which suffices since $\rho\lesssim{\varepsilon}^{2}$. For $S_{+}^{m}$ we
instead integrate by parts in $s$ to obtain

$\displaystyle{\big{\|}S_{+}^{m}(t)\big{\|}}_{H^{N}_{x}}$
$\displaystyle\lesssim{\Big{\|}\frac{\varphi_{>-m/10}(\langle\xi\rangle-2\lambda)}{\langle\xi\rangle-2\lambda}\Big{|}\int_{0}^{t}e^{-is(\langle\xi\rangle-2\lambda)}\frac{d}{ds}\Big{[}(A^{2}(s)-B^{2}(s))\tau_{m}(s)\Big{]}ds\,\langle\xi\rangle^{N}\widetilde{\theta}(\xi)\Big{|}\Big{\|}}_{L^{2}_{\xi}}$
$\displaystyle\lesssim 2^{m/20}2^{m}\rho^{2}(2^{m}),$

which is also sufficient. For $S_{2}^{m}$, see (4.7b), we can also integrate
by parts at no cost since $|\langle\xi\rangle-2\lambda|\gtrsim 1$, and then
use (3.34) and (3.5) to obtain an even stronger bound:

$\displaystyle{\|S_{2}^{m}(t)\|}_{H^{N}_{x}}\lesssim 2^{m}\rho^{2}(2^{m}).$

The same arguments apply to the two terms in (4.7c). Looking at the first one,
the other can be treated identically, we can write

$\displaystyle S_{3}^{m}$
$\displaystyle=\int_{0}^{t}e^{-is\langle\xi\rangle}\big{(}|A|^{2}(s)-|B|^{2}(s)\big{)}\,\tau_{m}(s)ds\widetilde{\theta}(\xi)+\int_{0}^{t}e^{-is\langle\xi\rangle}|B|^{2}(s)\,\tau_{m}(s)ds\widetilde{\theta}(\xi)$

and then proceed as for $S_{1}^{m}$ to deal with the first of the two terms
above (this is in fact even easier since the phase does not vanish), and just
integrate by parts, as we did for $S_{2}^{m}$, to deal with the second. ∎

####  9.1.2. Mixed terms

As in the previous paragraph, we localize the mixed terms (4.8)-(4.9) in time
by inserting time cutoffs. We denote the corresponding terms $M_{i}^{m},i=1,2$
and $RM_{i}^{m},i=1,2,3$ respectively. We prove a similar result as in the
previous subsection:

######  Lemma 9.3.

We have for $m=0,1,\dots$

$\displaystyle\|(M_{1}^{m}-M_{2}^{m})\|_{H^{N}}\lesssim 2^{-\delta
m}{\varepsilon}^{1+\delta},$ $\displaystyle\|RM_{i}^{m}\|_{H^{N}}\lesssim
2^{-\delta m}{\varepsilon}^{1+\delta},\quad i=1,2,3,$

for some $\delta>0.$

###### Proof.

We first show how to bound the $H^{N}$ norm of the (time-localized) mixed term

$\displaystyle(M_{1}^{m}-M_{2}^{m})(t)=\int_{0}^{t}e^{-isL}e^{-is\lambda}B(s)\big{(}\phi(x)\,L^{-1}\Im
w(s)\big{)}\,\tau_{m}(s)ds,\quad m=0,1,2,\dots.$ (9.3)

Recall that $f=h-g$ and $e^{itL}f=w=(\partial_{t}+iL)v$; note also that with
the same argument below we could equally as well estimate separately
$M_{1}^{m}$ and $M_{2}^{m}$. The idea here is to use Strichartz estimates and
the gain of one derivative on $w$ \- due to the ‘strongly semilinear’ nature
of (1.6) - followed by interpolation through the Gagliardo-Nirenberg
inequality for $1\leqslant N_{1}\leqslant N$

$\displaystyle\|w\|_{H^{N_{1}}}$
$\displaystyle\lesssim\|w\|_{H^{N}}^{\frac{N_{1}-1}{N-1}}\|w\|_{L^{6}}^{\frac{N-N_{1}}{N-1}}\lesssim\varepsilon^{\frac{N_{1}-1}{N-1}}\cdot\rho(2^{m})^{(1-\beta)\frac{N-N_{1}}{N-1}}2^{\frac{5}{3}m\delta_{N}\frac{N-N_{1}}{N-1}}\varepsilon^{\beta\frac{N-N_{1}}{N-1}},$
(9.4)

where we used (4.19) for the last part of the inequality.

Relying on the boundedness of wave operators, followed by Lemma 2.6, and using
(9.4), we find

$\displaystyle{\|(M_{1}^{m}-M_{2}^{m})(t)\|}_{H^{N}}$
$\displaystyle\lesssim\sum_{n_{1}+n_{2}\leq
N}{\Big{\|}\int_{0}^{t}B(s)e^{-isL}\big{(}D^{n_{2}}\phi(x)\cdot
L^{-1}D^{n_{1}}\Im w(s)\big{)}\,\tau_{m}(s)ds\Big{\|}}_{L^{2}}$
$\displaystyle\lesssim\sum_{n_{1}+n_{2}\leq
N}{\Big{\|}B(s)\tau_{m}(s)\,\langle D\rangle^{n_{2}}\phi\,\cdot\langle
D\rangle^{n_{1}-1}\Im w(s)\Big{\|}}_{L^{2}_{t}L^{1}_{x}}$
$\displaystyle\lesssim{\|B(s)\|}_{L^{2}_{s\approx 2^{m}}}\cdot\sup_{s\approx
2^{m}}{\|w(s)\|}_{H^{N-1}_{x}}$
$\displaystyle\lesssim{\|B(s)\|}_{L^{2}_{s\approx
2^{m}}}\cdot{\varepsilon}^{\frac{N-2}{N-1}}\cdot\Big{(}\rho(2^{m})^{1-\beta}2^{\frac{5}{3}\delta_{N}m}{\varepsilon}^{\beta}\Big{)}^{\frac{1}{N-1}}$
$\displaystyle\lesssim\big{(}2^{m}\rho(2^{m})\big{)}^{1/2}\cdot{\varepsilon}^{\frac{N-2}{N-1}}\cdot{\varepsilon}^{\frac{1-\beta}{N-1}}2^{-\frac{1-\beta}{2(N-1)}m}2^{\frac{5\delta_{N}}{3(N-1)}m}{\varepsilon}^{\frac{\beta}{N-1}}$
$\displaystyle\lesssim{\varepsilon}\cdot
2^{-\frac{1-\beta}{2(N-1)}m}2^{\frac{5\delta_{N}}{3(N-1)}m},$

where we used $\rho(t)\leqslant{\varepsilon}t^{-1/2}$ for the line before
last. Since $(1-\beta)/2>5\delta_{N}/3$, this concludes the proof of the first
estimate.

The remainder terms can be treated following the same strategy decomposing
$A=A-B+B=O(\rho^{1/2})$ (recall (3.34)), therefore we can skip the details. ∎

####  9.1.3. Field interactions

Finally, we look at the nonlinear interactions that are quadratic in the field
$w$, and show the following:

######  Lemma 9.4.

Under the a priori assumptions of Proposition 4.2, we have for $m=0,1,\dots$

$\displaystyle\|F_{m}\|_{H^{N}}$ $\displaystyle\lesssim 2^{-\delta
m}{\varepsilon}^{1+\delta},\qquad
F_{m}:=\int_{0}^{t}\tau_{m}(s)e^{-isL}\big{(}L^{-1}\Im w(s)\big{)}^{2}ds,$
(9.5)

for some $\delta>0.$

###### Proof.

Using the boundedness of wave operators and distributing the derivatives, it
suffices to show that

$\displaystyle\begin{split}{\Big{\|}\int_{0}^{t}e^{-isL}L^{N}\big{(}L^{-1}\Im
w(s)\big{)}\big{(}L^{-1}\Im
w(s)\big{)}\,\tau_{m}(s)ds\Big{\|}}_{L^{2}}\lesssim 2^{-\delta
m}\varepsilon^{1+\delta},\end{split}$ (9.6)

Using Lemma 2.6 with $(\tilde{q},\tilde{r},\gamma)=(6,3,2/3)$, together with
the boundedness of wave operators and a Sobolev product estimate, we have that
the right-hand side of (9.6) is bounded by

$\displaystyle\begin{split}&C\sum_{k\geqslant
0}2^{-k/3}{\big{\|}P_{k}L\big{[}L^{N}\big{(}L^{-1}\Im w\big{)}\big{(}L^{-1}\Im
w\big{)}\big{]}\big{\|}}_{L^{6/5}_{s\approx 2^{m}}L_{x}^{3/2}}\\\
&\lesssim{\big{\|}w\big{\|}}_{L^{6/5}_{s\approx
2^{m}}L^{6}_{x}}{\big{\|}L^{N}w\big{\|}}_{L^{\infty}_{s\approx
2^{m}}L^{2}_{x}}\\\ &\lesssim
2^{5m/6}\rho(2^{m})^{1-\beta}2^{\frac{5}{3}\delta_{N}m}\cdot{\varepsilon}.\end{split}$
(9.7)

Using that $\rho(2^{m})\lesssim 2^{-\alpha m}{\varepsilon}^{2(1-\alpha)}$ with
$\alpha=5/6+10\beta$, with $\beta$ small enough, gives us the desired
estimate.

∎

###  9.2. Weighted norm at high frequency: Proof of Lemma 5.4

We focus on proving the hardest estimate (5.9), as (5.10) can be proven with
similar arguments. We split $F_{H}$ as a sum of the terms

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

with

$\displaystyle\begin{split}\mu_{H}^{(i,j,k)}(\xi,\eta,\sigma):=\mu(\xi,\eta,\sigma)\varphi_{i}(\langle\xi\rangle)\varphi_{j}(\langle\eta\rangle)\varphi_{k}(\langle\sigma\rangle),\end{split}$

where $i,j,k\in\\{+,-\\},(i,j,k)\neq(+,+,+),$ and we have defined
$\varphi_{+}:=\varphi_{\leqslant M_{0}}$ (frequencies $\lesssim 2^{M_{0}}$)
and $\varphi_{-}:=\varphi_{>M_{0}}$ (frequencies $\gtrsim 2^{M_{0}}$). Define
also

$\displaystyle\mathcal{B}(f,g):=\int_{\mathbb{R}^{6}}f(\eta)g(\sigma)\mu(\xi,\eta,\sigma)d\eta
d\sigma.$

We rely on the distributional identity in Lemma A.3 (ignoring the easier case
where the derivative falls on the outer localizer). Recall that
$\mathcal{R}^{\alpha}$ denotes the euclidean Riesz transform. In the sequel we
will slightly abuse notations and still write $\mathcal{R}^{\alpha}$ for the
symbol of that operator, i.e.,
$\mathcal{R}^{\alpha}_{\xi}=\xi^{\alpha}/|\xi|$. Using Lemma A.3 we estimate

$\displaystyle\|\partial_{\xi}F_{H}^{(i,j,k)}\|_{L^{2}}$
$\displaystyle\lesssim\sum_{m}\|I_{H}^{(i,j,k)}\|_{L^{2}}+\|II_{H}^{(i,j,k)}\|_{L^{2}}+\|III_{H}^{(i,j,k)}\|_{L^{2}},$
$\displaystyle I_{H}^{(i,j,k)}(\xi)$
$\displaystyle:=\mathcal{R}_{\xi}^{\alpha}\varphi_{i}(\xi)\int_{0}^{t}e^{-is\langle\xi\rangle}\mathcal{B}\bigg{(}\partial_{\eta^{\beta}}\bigg{(}\mathcal{R}_{\eta}^{\beta}\frac{e^{is\epsilon_{1}\langle\eta\rangle}}{\langle\eta\rangle}\varphi_{j}(\eta)\widetilde{f_{\epsilon_{1}}}(s,\eta)\bigg{)},\frac{e^{is\epsilon_{2}\langle\sigma\rangle}}{\langle\sigma\rangle}\varphi_{k}(\sigma)\widetilde{f_{\epsilon_{2}}}(s,\sigma)\bigg{)}\tau_{m}(s)ds,$
$\displaystyle II_{H}^{(i,j,k)}(\xi)$
$\displaystyle:=\mathcal{R}_{\xi}^{\alpha}\varphi_{i}(\xi)\int_{0}^{t}e^{-is\langle\xi\rangle}\mathcal{B}\bigg{(}E_{\eta}\bigg{(}\frac{e^{is\epsilon_{1}\langle\eta\rangle}}{\langle\eta\rangle}\varphi_{j}(\eta)\widetilde{f_{\epsilon_{1}}}(s,\eta)\bigg{)},\frac{e^{is\epsilon_{2}\langle\sigma\rangle}}{\langle\sigma\rangle}\varphi_{k}(\sigma)\widetilde{f_{\epsilon_{2}}}(s,\sigma)\bigg{)}\tau_{m}(s)ds,$
$\displaystyle+\mathcal{R}_{\xi}^{\alpha}E_{\xi}\varphi_{i}(\xi)\int_{0}^{t}e^{-is\langle\xi\rangle}\mathcal{B}\bigg{(}\frac{e^{is\epsilon_{1}\langle\eta\rangle}}{\langle\eta\rangle}\varphi_{j}(\eta)\widetilde{f_{\epsilon_{1}}}(s,\eta),\frac{e^{is\epsilon_{2}\langle\sigma\rangle}}{\langle\sigma\rangle}\varphi_{k}(\sigma)\widetilde{f_{\epsilon_{2}}}(s,\sigma)\bigg{)}\tau_{m}(s)ds,$
$\displaystyle III_{H}^{(i,j,k)}(\xi)$
$\displaystyle:=\mathcal{R}_{\xi}^{\alpha}\varphi_{i}(\xi)\int_{0}^{t}is\frac{\xi^{\alpha}}{\langle\xi\rangle}e^{-is\langle\xi\rangle}\mathcal{B}\bigg{(}\frac{e^{is\epsilon_{1}\langle\eta\rangle}}{\langle\eta\rangle}\varphi_{j}(\eta)\widetilde{f_{\epsilon_{1}}}(s,\eta),\frac{e^{is\epsilon_{2}\langle\sigma\rangle}}{\langle\sigma\rangle}\varphi_{k}(\sigma)\widetilde{f_{\epsilon_{2}}}(s,\sigma)\bigg{)}\tau_{m}(s)ds.$

Note that we can also alternatively differentiate the other profile and write

$\displaystyle\|\partial_{\xi}F_{H}^{(i,j,k)}\|_{L^{2}}$
$\displaystyle\lesssim\sum_{m}\|I_{H}^{(i,j,k)^{\prime}}\|_{L^{2}}+\|II_{H}^{(i,j,k)^{\prime}}\|_{L^{2}}+\|III_{H}^{(i,j,k)}\|_{L^{2}},$
$\displaystyle I_{H}^{(i,j,k)^{\prime}}(\xi)$
$\displaystyle:=\mathcal{R}_{\xi}^{\alpha}\varphi_{i}(\xi)\int_{0}^{t}e^{-is\langle\xi\rangle}\mathcal{B}\bigg{(}\frac{e^{is\epsilon_{1}\langle\eta\rangle}}{\langle\eta\rangle}\varphi_{j}(\eta)\widetilde{f_{\epsilon_{1}}}(s,\eta),\partial_{\sigma^{\beta}}\bigg{(}\mathcal{R}_{\sigma}^{\beta}\frac{e^{is\epsilon_{2}\langle\sigma\rangle}}{\langle\sigma\rangle}\varphi_{k}(\sigma)\widetilde{f_{\epsilon_{2}}}(s,\sigma)\bigg{)}\bigg{)}\tau_{m}(s)ds,$
$\displaystyle II_{H}^{(i,j,k)^{\prime}}(\xi)$
$\displaystyle:=\mathcal{R}_{\xi}^{\alpha}\varphi_{i}(\xi)\int_{0}^{t}e^{-is\langle\xi\rangle}\mathcal{B}\bigg{(}\frac{e^{is\epsilon_{1}\langle\eta\rangle}}{\langle\eta\rangle}\varphi_{j}(\eta)\widetilde{f_{\epsilon_{1}}}(s,\eta),E_{\sigma}\bigg{(}\frac{e^{is\epsilon_{2}\langle\sigma\rangle}}{\langle\sigma\rangle}\varphi_{k}(\sigma)\widetilde{f_{\epsilon_{2}}}(s,\sigma)\bigg{)}\tau_{m}(s)ds,$
$\displaystyle+\mathcal{R}_{\xi}^{\alpha}E_{\xi}\varphi_{i}(\xi)\int_{0}^{t}e^{-is\langle\xi\rangle}\mathcal{B}\bigg{(}\frac{e^{is\epsilon_{1}\langle\eta\rangle}}{\langle\eta\rangle}\varphi_{j}(\eta)\widetilde{f_{\epsilon_{1}}}(s,\eta),\frac{e^{is\epsilon_{2}\langle\sigma\rangle}}{\langle\sigma\rangle}\varphi_{k}(\sigma)\widetilde{f_{\epsilon_{2}}}(s,\sigma)\bigg{)}\tau_{m}(s)ds.$

Estimate of $II_{H}^{(i,j,k)}$. We use either dispersive bounds or energy
bounds depending on whether the input is low or high frequency. More
precisely, we use for $2<p\leqslant 6$ and $s\approx 2^{m}$ that, for $j=+,-$,

$\displaystyle\|\widetilde{\mathcal{F}}^{-1}\varphi_{j}(\eta)e^{is\langle\eta\rangle}\widetilde{f}(s,\eta)\|_{L^{p}}\lesssim
2^{\frac{6-3p}{2p}m+5M_{0}}\cdot Z,$ (9.8)

which is a direct consequence of (4.19), and of Sobolev’s embedding with the
Sobolev a priori bound in (4.13) and (4.25). Then, using Lemma A.1, we obtain
that

$\displaystyle\|II_{H}^{(i,j,k)}\|_{L^{2}}$
$\displaystyle\lesssim\int_{0}^{t}\bigg{\|}\widetilde{\mathcal{F}}^{-1}\big{(}e^{is\epsilon_{1}\langle\eta\rangle}\frac{\varphi_{j}(\eta)}{\langle\eta\rangle}\widetilde{f_{\epsilon_{1}}}(s,\eta)\big{)}\bigg{\|}_{L^{3-}}\bigg{\|}\widetilde{\mathcal{F}}^{-1}\big{(}e^{is\epsilon_{2}\langle\sigma\rangle}\frac{\varphi_{k}(\sigma)}{\langle\sigma\rangle}\widetilde{f_{\epsilon_{2}}}(s,\sigma)\big{)}\bigg{\|}_{L^{6}}\tau_{m}(s)ds$
$\displaystyle\lesssim 2^{m}\cdot 2^{20M_{0}}2^{-3m/2}\cdot Z^{2},$

which is acceptable.

Estimate of $I_{H}^{(i,j,k)}$. The two main cases to consider are when the
derivative falls on the exponential and on the profile:

$\displaystyle\|I_{H}^{(i,j,k)}\|_{L^{2}}$
$\displaystyle\lesssim\|I_{H,1}^{(i,j,k)}\|_{L^{2}}+\|I_{H,2}^{(i,j,k)}\|_{L^{2}}$
$\displaystyle I_{H,1}^{(i,j,k)}$
$\displaystyle:=\mathcal{R}_{\xi}^{\alpha}\varphi_{i}(\xi)\int_{0}^{t}e^{-is\langle\xi\rangle}\mathcal{B}\bigg{(}\frac{e^{is\epsilon_{1}\langle\eta\rangle}}{\langle\eta\rangle}\varphi_{j}(\eta)\mathcal{R}_{\eta}^{\beta}\partial_{\eta^{\beta}}\widetilde{f_{\epsilon_{1}}}(s,\eta),\frac{e^{is\epsilon_{2}\langle\sigma\rangle}}{\langle\sigma\rangle}\varphi_{k}(\sigma)\widetilde{f_{\epsilon_{2}}}(s,\sigma)\bigg{)}ds,$
$\displaystyle I_{H,2}^{(i,j,k)}$
$\displaystyle:=\mathcal{R}_{\xi}^{\alpha}\varphi_{i}(\xi)\int_{0}^{t}e^{-is\langle\xi\rangle}\mathcal{B}\bigg{(}is\epsilon_{1}\frac{\eta^{\beta}}{\langle\eta\rangle}\frac{e^{is\epsilon_{1}\langle\eta\rangle}}{\langle\eta\rangle}\varphi_{j}(\eta)\mathcal{R}_{\eta}^{\beta}\widetilde{f_{\epsilon_{1}}}(s,\eta),\frac{e^{is\epsilon_{2}\langle\sigma\rangle}}{\langle\sigma\rangle}\varphi_{k}(\sigma)\widetilde{f_{\epsilon_{2}}}(s,\sigma)\bigg{)}ds.$

Assume first that $k=-.$ In this case we use Sobolev embedding as well as the
$H^{N}$ bound in the bootstrap and (4.25) to write that

$\displaystyle\|\widetilde{\mathcal{F}}^{-1}\varphi_{-}(\eta)e^{is\langle\eta\rangle}\widetilde{f}(s,\eta)\|_{L^{p}}\lesssim\|\varphi_{-}(\eta)e^{is\langle\eta\rangle}\widetilde{f}(s,\eta)\|_{H^{2}}\lesssim
2^{-M_{0}(N-2)}\|f\|_{H^{N}}\lesssim 2^{-\frac{5(N-2)}{N}m}\varepsilon^{1-}.$

Therefore we can bound

$\displaystyle\|I_{H,1}^{(i,j,-)}\|_{L^{2}}\lesssim\int_{0}^{t}\big{\|}\nabla_{\eta}\widetilde{f}(s,\eta)\big{\|}_{L^{2}}\|\widetilde{\mathcal{F}}^{-1}e^{is\epsilon_{1}\langle\sigma\rangle}\varphi_{-}(\sigma)\widetilde{f}(s,\sigma)\|_{L^{\infty}}\tau_{m}(s)ds\lesssim
2^{m}\cdot 2^{-\frac{5(N-2)}{N}m}\cdot{\varepsilon}^{1-},$

which yields the desired conclusion.

We can bound $I_{H,2}^{(i,j,-)}$ similarly and obtain

$\displaystyle\|I_{H,2}^{(i,j,-)}\|_{L^{2}}\lesssim 2^{2m}\cdot
2^{-\frac{5(N-1)}{N}m}.$

Note that since we can choose which profile to differentiate, the case $j=-$
is handled identically estimating $I_{H}^{(i,-,k)^{\prime}}$ instead.

If $j,k=+$ then necessarily $i=-$ and in this case we write, using boundedness
of wave operators and distributing derivatives,

$\displaystyle\|I_{H,2}^{(-,j,k)}\|_{L^{2}}$ $\displaystyle\lesssim
2^{-5m}\int_{0}^{t}\bigg{\|}D^{N}\widetilde{\mathcal{F}}^{-1}\mathcal{B}\bigg{(}is\epsilon_{1}\frac{\eta^{\alpha}}{|\eta|}\frac{e^{is\epsilon_{1}\langle\eta\rangle}}{\langle\eta\rangle}\varphi_{j}(\eta)\widetilde{f_{\epsilon_{1}}}(s,\eta),\frac{e^{is\epsilon_{2}\langle\sigma\rangle}}{\langle\sigma\rangle}\varphi_{k}(\sigma)\widetilde{f_{\epsilon_{2}}}(s,\sigma)\bigg{)}\bigg{\|}_{L^{2}}\tau_{m}(s)ds$
$\displaystyle\lesssim 2^{-5m}\sum_{n_{1}+n_{2}\leqslant
N}\int_{0}^{t}s\|D^{n_{1}}L^{-1}P_{\leqslant
M_{0}}w\|_{L^{2}}\|D^{n_{2}}L^{-1}P_{\leqslant
M_{0}}w\|_{L^{\infty}}\tau_{m}(s)ds$ $\displaystyle\lesssim 2^{-5m}\cdot
2^{2m}\cdot 2^{M_{0}},$

where for the last line we used the $H^{N}$ assumption in the bootstrap
(4.13), and (4.25). This yields the desired estimate.

For the remaining term $I_{H,1}^{(-,+,+)}$ we split $f=g+h$. For the $h$ part,
we use Strichartz estimates with parameters $(6,3,2/3).$ For the $g$ part we
use the boundedness of wave operators and distribute derivatives. Overall we
obtain

$\displaystyle\|I_{H,1}^{(-,+,+)}\|_{L^{2}}$
$\displaystyle\lesssim\sum_{\ell\geqslant
M_{0}}2^{-\ell/3}\Bigg{(}\Bigg{\|}\widetilde{\mathcal{F}}^{-1}\mathcal{B}\bigg{(}\frac{e^{is\epsilon_{1}\langle\eta\rangle}}{\langle\eta\rangle}|\eta|\varphi_{+}(\eta)\mathcal{R}_{\eta}^{\alpha}\partial_{\eta^{\alpha}}\widetilde{h_{\epsilon_{1}}}(s,\eta),\frac{e^{is\epsilon_{2}\langle\sigma\rangle}}{\langle\sigma\rangle}\varphi_{+}(\sigma)\widetilde{f_{\epsilon_{2}}}(s,\sigma)\bigg{)}\Bigg{\|}_{L^{6/5}_{s\approx
2^{m}}L^{3/2}_{x}}$
$\displaystyle+\Bigg{\|}\widetilde{\mathcal{F}}^{-1}\mathcal{B}\bigg{(}\frac{e^{is\epsilon_{1}\langle\eta\rangle}}{\langle\eta\rangle}\varphi_{+}(\eta)\mathcal{R}_{\eta}^{\alpha}\partial_{\eta^{\alpha}}\widetilde{h_{\epsilon_{1}}}(s,\eta),\frac{e^{is\epsilon_{2}\langle\sigma\rangle}}{\langle\sigma\rangle}|\sigma|\varphi_{+}(\sigma)\widetilde{f_{\epsilon_{2}}}(s,\sigma)\bigg{)}\Bigg{\|}_{L^{6/5}_{s\approx
2^{m}}L^{3/2}_{x}}\Bigg{)}$ $\displaystyle+2^{-N\ell}\cdot
2^{m}\cdot\sum_{n_{1}+n_{2}\leqslant
N}\big{\|}D^{n_{1}}L^{-1}\varphi_{+}(D)\widehat{\mathcal{F}}^{-1}\partial_{\eta}\widetilde{g}(s,\eta)\big{\|}_{L^{2}}\sup_{s\approx
2^{m}}\|D^{n_{2}}L^{-1}w(s)\|_{L^{2}}.$

The first two terms are bounded by $2^{5m/6}\sup_{s\approx
2^{m}}\|\varphi_{+}(L)e^{isL}f(s)\|_{L^{6}_{x}}\cdot Z,$ which yields the
desired result. Using (4.25) we see that the last term is bounded by
$2^{-NM_{0}+m}\cdot 2^{m/2}{\varepsilon}^{1-}$, which is sufficient given our
choice of parameter $M_{0}.$

Estimate of $III_{H}^{(i,j,k)}$. This term can be handled following the same
strategy for $I_{H,2}^{(i,j,k)}.$ The proof of the proposition is concluded.
$\hfill\Box$

##  Appendix A Basic results about the NSD

We recall several results derived in [16] for the interaction distribution of
three distorted complex exponentials as defined in (1.20). We first define
some useful operators and state bounds that they satisfy:

######  Lemma A.1 (Theorem 3.4 in [16]).

With the definitions (2.5), let

$\displaystyle
R^{\alpha}:=\mathcal{W}\frac{x^{\alpha}}{|x|}\mathcal{W}^{\ast},\ \ \
\mathcal{E}:=\big{[}|x|,\mathcal{W}\big{]}\mathcal{W}^{\ast},$

and

$\displaystyle\mathcal{R}^{\alpha}:=\widetilde{\mathcal{F}}R^{\alpha}\widetilde{\mathcal{F}}^{-1},\
\ \ E:=\widetilde{\mathcal{F}}\mathcal{E}\widetilde{\mathcal{F}}^{-1}.$

Both are bounded from $L^{p}$ to $L^{p+\delta}$ for any $1\leqslant p<\infty$
and $\delta>0$ small enough.

######  Remark A.2.

$\mathcal{R}^{\alpha}$ is the euclidean Riesz transform. It commutes with
differentiation.

The next result is an identity for the distributional derivative of $\mu$.

######  Lemma A.3 (Identity (3.56) in [16]).

We have, in the distributional sense that

$\displaystyle\partial_{\xi^{\alpha}}\mu$
$\displaystyle=\mathcal{R}_{\xi}^{\alpha}\mathcal{R}_{\eta}^{\beta}\partial_{\eta^{\beta}}\mu+\mathcal{R}_{\xi}^{\alpha}E_{\eta}^{\ast}\mu-\mathcal{R}_{\xi}^{\alpha}E_{\xi}^{\ast}\mu$
$\displaystyle=\mathcal{R}_{\xi}^{\alpha}\mathcal{R}_{\sigma}^{\beta}\partial_{\sigma^{\beta}}\mu+\mathcal{R}_{\xi}^{\alpha}E_{\sigma}^{\ast}\mu-\mathcal{R}_{\xi}^{\alpha}E_{\xi}^{\ast}\mu.$

Therefore, we have the following identity

$\displaystyle\begin{split}\partial_{\xi}^{\alpha}\big{(}\widetilde{fg}\big{)}&=\mathcal{R}_{\xi}^{\alpha}\int_{\mathbb{R}^{6}}\mathcal{R}_{\eta}^{\beta}\partial_{\eta^{\beta}}\widetilde{f}(\eta)\widetilde{g}(\sigma)\mu(\xi,\eta,\sigma)d\eta
d\sigma+\mathcal{R}_{\xi}^{\alpha}\int_{\mathbb{R}^{6}}E\widetilde{f}(\eta)\widetilde{g}(\sigma)\mu(\xi,\eta,\sigma)d\eta
d\sigma\\\
&-E\mathcal{R}_{\xi}^{\alpha}\int_{\mathbb{R}^{6}}\widetilde{f}(\eta)\widetilde{g}(\sigma)\mu(\xi,\eta,\sigma)d\eta
d\sigma\\\
&=\mathcal{R}_{\xi}^{\alpha}\int_{\mathbb{R}^{6}}\widetilde{f}(\eta)\mathcal{R}_{\sigma}^{\beta}\partial_{\sigma^{\beta}}\widetilde{g}(\sigma)\mu(\xi,\eta,\sigma)d\eta
d\sigma+\mathcal{R}_{\xi}^{\alpha}\int_{\mathbb{R}^{6}}\widetilde{f}(\eta)E\widetilde{g}(\sigma)\mu(\xi,\eta,\sigma)d\eta
d\sigma\\\
&-E\mathcal{R}_{\xi}^{\alpha}\int_{\mathbb{R}^{6}}\widetilde{f}(\eta)\widetilde{g}(\sigma)\mu(\xi,\eta,\sigma)d\eta
d\sigma.\end{split}$ (A.1)

######  Remark A.4.

Note that we slightly abused notations above and denoted
$\mathcal{R}^{\alpha}_{\xi}$ the symbol of the euclidean Riesz transform.

######  Remark A.5.

A convenient feature in (A.1) is that one can pick which profile to
differentiate.

As a direct consequence we obtain a proof of Lemma 5.5 and nonlinear estimates
for short times in the next two subsections.

###  A.1. Proof of Lemma 5.5

Note that at least one of the profiles is localized at very low frequency. We
pick one of them and differentiate the other profile, using (A.1) as well as
Bernstein’s inequality:

$\displaystyle\|\partial_{\xi}F_{L,\epsilon_{1},\epsilon_{2}}\|_{L^{2}}$
$\displaystyle\lesssim\int_{0}^{t}\big{(}\|\mathcal{R}_{\eta}^{\alpha}\langle\eta\rangle^{-1}\partial_{\eta^{\alpha}}\widetilde{f}\|_{L^{2}}+\bigg{\|}\mathcal{R}_{\eta}^{\alpha}\frac{s\eta^{\alpha}}{\langle\eta\rangle^{2}}e^{is\langle\eta\rangle}\widetilde{f}\bigg{\|}_{L^{2}}+\|Ef\|_{L^{2}}\bigg{)}\|f_{\leqslant
s^{-5}}\|_{L^{\infty}}ds$
$\displaystyle\lesssim\int_{0}^{t}\big{(}s{\varepsilon}+\langle
s\rangle^{1/2+}+{\varepsilon}\big{)}\langle
s\rangle^{-15/2}ds\,{\varepsilon},$

which is sufficient to conclude. We used the crude bound
$\|\partial_{\eta}\widetilde{f}\|_{L^{2}}\lesssim\langle s\rangle^{1/2+},$
which follows from (4.26) and the bootstrap assumption (4.13). $\hfill\Box$

###  A.2. Local existence

Here we prove that (4.15) implies (4.16). Since
$t\leqslant{\varepsilon}^{-1},$ ($t\geqslant 1$) the proof of the bootstrap
estimates is much simpler than in the general case. Therefore, we only give an
outline of the proof, typically discarding similar and lower order terms.

First, we have from Lemma 2.5 that

$\displaystyle\|e^{itL}f\|_{L^{q}}\lesssim
t^{(-1+10\delta_{N})\frac{3(q-2)}{2q}}{\varepsilon}^{1-\beta}.$ (A.2)

We bound the terms that appear in (4.6). The source terms are treated as in
Lemma 4.7, therefore we skip the details. For the mixed terms, we write that
(treating the worse case where the derivative falls on the phase) using
Strichartz estimates with parameters $(q,r,\gamma)=(2,\infty,0)$

$\displaystyle\|\partial_{\xi}M(t)\|_{L^{2}}$
$\displaystyle\lesssim\bigg{\|}\int_{0}^{t}a(s)s\frac{\xi}{\langle\xi\rangle}e^{is\langle\xi\rangle}\int_{\mathbb{R}^{3}}\frac{e^{-is\langle\eta\rangle}}{\langle\eta\rangle}\widetilde{f}(s,\eta)\nu(\xi,\eta)d\eta
ds\bigg{\|}_{L^{2}}$
$\displaystyle\lesssim{\big{\|}s\|e^{itL}f\|_{L^{6}_{x}}\big{\|}}_{L^{2}_{t}}{\varepsilon}\lesssim
t^{1/2+20\delta_{N}}{\varepsilon}^{2-\beta}\leqslant{\varepsilon}^{3/2-2\beta},$

where we used the dispersive estimate (A.2), and disregarded the simpler
contribution to the integral for $s\in[0,1]$

We come to the quadratic terms. Using Lemma A.3 we find, neglecting the lower
order terms,

$\displaystyle\|\partial_{\xi}\widetilde{F}(t)\|_{L^{2}}$
$\displaystyle\lesssim\bigg{\|}\int_{0}^{t}s\int_{\mathbb{R}^{6}}\frac{\eta^{\alpha}}{\langle\eta\rangle^{2}}e^{is\langle\eta\rangle}\widetilde{f}(\eta)\frac{e^{is\langle\sigma\rangle}}{\langle\sigma\rangle}\widetilde{f}(\sigma)\mu(\xi,\eta,\sigma)d\eta
d\sigma ds\bigg{\|}_{L^{2}}$
$\displaystyle+\bigg{\|}\int_{0}^{t}\int_{\mathbb{R}^{6}}\frac{e^{is\langle\eta\rangle}}{\langle\eta\rangle}\partial_{\eta^{\alpha}}\widetilde{f}(\eta)\frac{e^{is\langle\sigma\rangle}}{\langle\sigma\rangle}\widetilde{f}(\sigma)\mu(\xi,\eta,\sigma)d\eta
d\sigma ds\bigg{\|}_{L^{2}}$
$\displaystyle\lesssim\int_{0}^{t}s\|e^{itL}L^{-1}f\|_{L^{6}}\|e^{itL}L^{-1}f\|_{L^{3}}ds+\int_{0}^{t}\|L^{-1}\mathcal{F}^{-1}\big{(}\partial_{\xi}\widetilde{f}\big{)}\|_{L^{3}}\|e^{itL}L^{-1}f\|_{L^{6}}ds$
$\displaystyle\lesssim\int_{0}^{t}s\cdot\langle
s\rangle^{-1+10\delta_{N}}\cdot\langle
s\rangle^{-1/2+5\delta_{N}}ds\,{\varepsilon}^{2-2\beta}+\int_{0}^{t}\langle
s\rangle^{-1}ds\,{\varepsilon}^{3/2-2\beta}$ $\displaystyle\lesssim\langle
t\rangle^{1/2+15\delta_{N}}{\varepsilon}^{2-2\beta}+{\varepsilon}^{3/2-3\beta},$

where we used the bound, coming from the bootstrap assumption and (4.26)

$\displaystyle\|\partial_{\xi}\widetilde{f}\|_{L^{2}}\lesssim\|\partial_{\xi}\widetilde{h}\|_{L^{2}}+\|\partial_{\xi}\widetilde{g}\|_{L^{2}}\lesssim{\varepsilon}^{1-\beta}+t^{3/2}{\varepsilon}^{2}|\log{\varepsilon}|^{2}\leqslant{\varepsilon}^{1/2-},$

which yields the desired result given that $t\leqslant{\varepsilon}^{-1}$
provided $\beta$ is small enough. $\hfill\Box$

##  Appendix B Structure of the NSD and product estimates

In order to deal with nonlinear problems and understand oscillations in
distorted Fourier space, we need a more refined analysis of (the singularities
of) the NSD. We first recall the decomposition result of [41], under the more
generous assumption $V\in\mathcal{S}$ as in Proposition 5.6; we refer the
reader to [41] for the precise statement under finite regularity and
localization assumptions on the potential $V$.

######  Proposition B.1 ([41]).

The distribution (4.1) can be written as

$\displaystyle\begin{split}(2\pi)^{9/2}\mu(\xi,\eta,\sigma)&=(2\pi)^{3/2}\delta_{0}(\xi-\eta-\sigma)-\frac{1}{4\pi}\mu_{1}(\xi,\eta,\sigma)\\\
&-\frac{1}{(4\pi)^{2}}\mu_{2}(\xi,\eta,\sigma)-\frac{1}{(4\pi)^{3}}\mu_{3}(\xi,\eta,\sigma),\end{split}$
(B.1)

where:

  * •

The distribution $\mu_{1}$ is given by

$\displaystyle\mu_{1}(k,\ell,m)$
$\displaystyle=\nu_{1}(-k+\ell,m)+\nu_{1}(-k+m,\ell)+\overline{\nu_{1}(-\ell-m,k)}$
(B.2)

where

$\displaystyle\nu_{1}(p,q)=\nu_{0}(p,q)+\nu_{L}(p,q)+\nu_{R}(p,q),$ (B.3)

and:

    * (1)

The leading order is

$\displaystyle\nu_{0}(p,q):=\frac{b_{0}(p,q)}{|p|}\Big{[}i\pi\,\delta(|p|-|q|)+\mathrm{p.v.}\frac{1}{|p|-|q|}\Big{]}$
(B.4)

with $b_{0}$ satisfying the bounds

$\displaystyle\big{|}\varphi_{P}(p)\varphi_{Q}(q)\,\nabla_{p}^{\alpha}\nabla_{q}^{\beta}b_{0}(p,q)\big{|}\lesssim
2^{-|\alpha|P}\big{(}2^{|\alpha|Q}+2^{(1-|\beta|)Q_{-}}\big{)}\cdot\mathbf{1}_{\\{|P-Q|<5\\}},$
(B.5)

for all $P,Q\leqslant A$, and $|\alpha|+|\beta|\leqslant N_{2}$, for any
arbitrarily large integer101010The value of $N_{2}$ can be related to the
localization and smoothness of the potential $V$, and it can be chosen to be
approximately $N/2$ when $\langle x\rangle^{N}V\in H^{N}$. Since we assume,
for simplicity, that $V\in\mathcal{S}$, we can let $N_{2}$ be arbitrarily
large. $N_{2}$. Recall our notation $Q^{-}=\min(Q,0)$.

    * (2)

The lower order terms $\nu_{L}(p,q)$ can be written as

$\displaystyle\nu_{L}(p,q)=\frac{1}{|p|}\sum_{a=1}^{N_{2}}\sum_{J\in{\mathbb{Z}}}b_{a,J}(p,q)\cdot
2^{J}K_{a}\big{(}2^{J}(|p|-|q|)\big{)}$ (B.6)

with $K_{a}\in\mathcal{S}$ and $b_{a,J}$ satisfying

$\displaystyle\sum_{J\in{\mathbb{Z}}}\big{|}\varphi_{P}(p)\varphi_{Q}(q)\nabla_{p}^{\alpha}\nabla_{q}^{\beta}b_{a,J}(p,q)\big{|}\lesssim
2^{-|\alpha|P}\big{(}2^{|\alpha|Q}+2^{(1-|\beta|)Q_{-}}\big{)}\cdot\mathbf{1}_{\\{|P-Q|<5\\}},$
(B.7)

for all $P,Q\leqslant A$, $|\alpha|+|\beta|\leqslant N_{2}$.

    * (3)

The remainder term $\nu_{R}$ satisfies the estimates

$\displaystyle\begin{split}\big{|}\varphi_{P}(p)\varphi_{Q}(q)\nabla_{p}^{\alpha}\nabla_{q}^{\beta}\nu_{R}(p,q)\big{|}\lesssim
2^{-2\max(P,Q)}\cdot 2^{-(|\alpha|+|\beta|)\max(P,Q)}\cdot
2^{(|\alpha|+|\beta|+2)5A}\end{split}$ (B.8)

for $|P-Q|<5$, and

$\displaystyle\begin{split}\big{|}\varphi_{P}(p)\varphi_{Q}(q)\nabla_{p}^{\alpha}\nabla_{q}^{\beta}\nu_{R}(p,q)\big{|}\lesssim
2^{-2\max(P,Q)}\cdot 2^{-|\alpha|\max(P,Q)}\max(1,2^{(1-|\beta|)Q_{-}})\cdot
2^{(|\alpha|+|\beta|+2)5A}\end{split}$ (B.9)

for $|P-Q|\geqslant 5$, for all $P,Q\leqslant A$, $|\alpha|+|\beta|\leqslant
N_{2}$.

  * •

The measure $\mu_{2}$ is given by

$\displaystyle\mu_{2}(k,\ell,m)$
$\displaystyle=\nu_{2}^{1}(k,\ell,m)+\nu_{2}^{2}(k,\ell,m)+\nu_{2}^{2}(k,m,\ell)$
(B.10)

with the following properties:

    * –

Let $k,\ell,m\in{\mathbb{R}}^{3}$ with $|k|\approx 2^{K},|\ell|\approx 2^{L}$
and $|m|\approx 2^{M}$, and assume that $K,L,M\leq A$ for some $A>0$. Then we
can write

$\displaystyle\nu_{2}^{1}(k,\ell,m)=\nu_{2,0}^{1}(k,\ell,m)+\nu_{2,R}^{1}(k,\ell,m),$
(B.11)

where:

      * (1)

$\nu_{2,0}^{1}(k,\ell,m)$ can be written as

$\displaystyle\nu_{2,0}^{1}(k,\ell,m)=\frac{1}{|k|}\sum_{i=1}^{N_{2}}\sum_{J\in{\mathbb{Z}}}b_{i,J}(k,\ell,m)\cdot
K_{i}\big{(}2^{J}(|k|-|\ell|-|m|)\big{)}$ (B.12)

with $K_{i}\in\mathcal{S}$ and the symbols $b_{i,J}$ satisfy

$\displaystyle\begin{split}&\big{|}\varphi_{K}(k)\varphi_{L}(\ell)\varphi_{M}(m)\nabla_{k}^{a}\nabla_{\ell}^{\alpha}\nabla_{m}^{\beta}b_{i,J}(k,\ell,m)\big{|}\\\
&\lesssim
2^{-|a|K}\cdot\big{(}2^{|a|\max(L,M)}+2^{(1-|\alpha|)L}2^{(1-|\beta|)M}\big{)}\mathbf{1}_{\\{|K-\max(L,M)|<5\\}},\end{split}$
(B.13)

for all $K,L,M\leqslant A$, and $|a|+|\alpha|+|\beta|\leqslant N_{2}$.

      * (2)

For $M\leq L$, the remainder term $\nu_{2,R}^{1}$ satisfies

$\displaystyle\begin{split}\big{|}\nabla_{k}^{a}\nabla_{\ell}^{\alpha}\nabla_{m}^{\beta}\nu_{2,R}^{1}(k,\ell,m)\big{|}&\lesssim
2^{-2\max(K,L)}\cdot 2^{-|a|\max(K,L)}\\\ &\times
2^{-|\alpha|\max(K,L)}\max(1,2^{-(|\beta|-1)M})\cdot
2^{(|a|+|\alpha|+|\beta|+2)5A}\end{split}$ (B.14)

for all $K,L,M\leqslant A$ and $|a|+|\alpha|+|\beta|\leqslant N_{2}$. A
similar statement holds when $L\leqslant M$ exchanging the roles of $L$ and
$M$ (and $\alpha$ and $\beta$).

    * –

Let $k,\ell,m\in{\mathbb{R}}^{3}$ with $|k|\approx 2^{K},|\ell|\approx 2^{L}$
and $|m|\approx 2^{M}$, and $K,L,M\leqslant A$ for some $A>0$. Then we can
write

$\displaystyle\nu_{2}^{2}(k,\ell,m)=\nu_{2,+}^{2}(k,\ell,m)+\nu_{2,-}^{2}(k,\ell,m)+\nu_{2,R}^{2}(k,\ell,m),$
(B.15)

where:

      * (1)

The leading order is

$\displaystyle\nu_{2,\pm}^{2}(k,\ell,m)=\frac{1}{|\ell|}\sum_{i=1}^{N_{2}}\sum_{J\in{\mathbb{Z}}}b_{i,J}^{\pm}(k,\ell,m)\cdot
K_{i}\big{(}2^{J}(|k|\pm|\ell|-|m|)\big{)}$ (B.16)

with

$\displaystyle\begin{split}&\big{|}\varphi_{K}(k)\varphi_{L}(\ell)\varphi_{M}(m)\nabla_{k}^{a}\nabla_{\ell}^{\alpha}\nabla_{m}^{\beta}b_{i,J}^{\pm}(k,\ell,m)\big{|}\\\
&\lesssim
2^{-|\alpha|L}\cdot\big{(}2^{|\alpha|\max(K,M)}+2^{(1-|a|)K}2^{(1-|\beta|)M}\big{)}\mathbf{1}_{\\{\max(K,L,M)-\mbox{\tiny$\mathrm{med}$}(K,L,M)<5\\}},\end{split}$
(B.17)

for all $K,L,M\leqslant A$, and $|a|+|\alpha|+|\beta|\leqslant N_{2}$.

      * (2)

The remainder term satisfies, for $K\leq M$,

$\displaystyle\begin{split}\big{|}\nabla_{k}^{a}\nabla_{\ell}^{\alpha}\nabla_{m}^{\beta}\nu_{2,R}^{2}(k,\ell,m)\big{|}&\lesssim
2^{-2\max(L,M,K)}\cdot 2^{-(|\alpha|+|\beta|)\max(L,M,K)}\\\
&\times\max(1,2^{-(|a|-1)K})\cdot 2^{(|a|+|\alpha|+|\beta|+2)5A}\end{split}$
(B.18)

for all $K,L,M\leqslant A$ and $|a|+|\alpha|+|\beta|\leqslant N_{2}$. A
similar estimate holds when $M\leqslant K$ by exchanging the roles of $M$ and
$K$ (and $a,\beta$).

  * •

Let $k,\ell,m\in{\mathbb{R}}^{3}$ with $|k|\approx 2^{K},|\ell|\approx 2^{L}$
and $|m|\approx 2^{M}$, and assume that $K,L,M\leqslant A$ for some $A>0$.
Then we can write

$\displaystyle\mu_{3}(k,\ell,m)=\mu_{3,0}(k,\ell,m)+\mu_{3,R}(k,\ell,m),$
(B.19)

where:

    * (1)

The leading order has the form

$\displaystyle\mu_{3,0}(k,\ell,m)=\sum_{i=0}^{N_{2}}\sum_{J\in{\mathbb{Z}}}b_{i,J}(k,\ell,m)\cdot
K_{i}\big{(}2^{J}(|k|-|\ell|-|m|)\big{)}$ (B.20)

with $K_{i}\in\mathcal{S}$, and

$\displaystyle\begin{split}&\big{|}\varphi_{K}(k)\varphi_{L}(\ell)\varphi_{M}(m)\nabla_{k}^{a}\nabla_{\ell}^{\alpha}\nabla_{m}^{\beta}b_{i,J}(k,\ell,m)\big{|}\\\
&\lesssim\max(1,2^{-(|a|-1)K})\max(1,2^{-(|\alpha|-1)L})\max(1,2^{-(|\beta|-1)M})\mathbf{1}_{\\{|K-\max(L,M)|<5\\}}\end{split}$
(B.21)

for all $K,L,M\leqslant A$, and $|a|+|\alpha|+|\beta|\leqslant N_{2}$.

    * (2)

The remainder term satisfies

$\displaystyle\begin{split}\big{|}\nabla_{k}^{a}\nabla_{\ell}^{\alpha}\nabla_{m}^{\beta}\mu_{3,R}(k,\ell,m)\big{|}\lesssim\max(1,2^{-(|a|-1)K})\max(1,2^{-(|\alpha|-1)L})\\\
\cdot\max(1,2^{-(|\beta|-1)M})\cdot 2^{4A(|a|+|\alpha|+|\beta|+1)}\end{split}$
(B.22)

for all $K,L,M\leqslant A$ and $|a|+|\alpha|+|\beta|\leqslant N_{2}$.

Next, we collect all the needed bilinear estimates for the operators
associated to the various pieces appearing in the decomposition of Proposition
B.1. These results are from [41].

Given a symbol $b=b(\xi,\eta,\sigma)$, we define the bilinear operators

$\displaystyle T_{0}[b](g,h)(x)$
$\displaystyle:=\widehat{\mathcal{F}}^{-1}_{\xi\rightarrow
x}\iint_{{\mathbb{R}}^{3}\times{\mathbb{R}}^{3}}g(\eta)h(\sigma)\,b(\xi,\eta,\sigma)\,\delta_{0}(\xi-\eta-\sigma)\,d\eta
d\sigma$ (B.23)

and

$\displaystyle T_{1}^{1}[b](g,h)(x)$
$\displaystyle:=\widehat{\mathcal{F}}^{-1}_{\xi\rightarrow
x}\iint_{{\mathbb{R}}^{3}\times{\mathbb{R}}^{3}}g(\xi-\eta)h(\sigma)\,b(\xi,\eta,\sigma)\,\nu_{1}(\eta,\sigma)\,d\eta
d\sigma,$ (B.24) $\displaystyle T_{1}^{2}[b](g,h)(x)$
$\displaystyle:=\widehat{\mathcal{F}}^{-1}_{\xi\rightarrow
x}\iint_{{\mathbb{R}}^{3}\times{\mathbb{R}}^{3}}g(-\eta-\sigma)h(\sigma)\,b(\xi,\eta,\sigma)\,\overline{\nu_{1}(\eta,\xi)}\,d\eta
d\sigma,$ (B.25) $\displaystyle T_{2}[b](g,h)(x)$
$\displaystyle:=\widehat{\mathcal{F}}^{-1}_{\xi\rightarrow
x}\iint_{{\mathbb{R}}^{3}\times{\mathbb{R}}^{3}}g(\eta)h(\sigma)\,b(\xi,\eta,\sigma)\,\mu_{2}(\xi,\eta,\sigma)\,d\eta
d\sigma,$ (B.26) $\displaystyle T_{3}[b](g,h)(x)$
$\displaystyle:=\widehat{\mathcal{F}}^{-1}_{\xi\rightarrow
x}\iint_{{\mathbb{R}}^{3}\times{\mathbb{R}}^{3}}g(\eta)h(\sigma)\,b(\xi,\eta,\sigma)\,\mu_{3}(\xi,\eta,\sigma)\,d\eta
d\sigma.$ (B.27)

######  Theorem B.2 (Bilinear bounds 1).

Assume that:

  * •

The symbol $b$ is such that

$\displaystyle\textrm{supp}(b)\subseteq\big{\\{}(\xi,\eta,\sigma)\in\mathbb{R}^{9}\,:\,|\xi|+|\eta|+|\sigma|\leq
2^{A},\,|\eta|\approx 2^{k_{1}},\,|\sigma|\approx 2^{k_{2}}\big{\\}},$ (B.28)

for some $A\geq 1$.

  * •

For all $|\xi|\approx 2^{k}$, $|\eta|\approx 2^{k_{1}}$ and $|\sigma|\approx
2^{k_{2}}$

$\displaystyle|\nabla_{\xi}^{a}\nabla^{\alpha}_{\eta}\nabla^{\beta}_{\sigma}b(\xi,\eta,\sigma)|\lesssim
2^{-|a|k}2^{-|\alpha|k_{1}}2^{-|\beta|k_{2}}\cdot
2^{(|a|+|\alpha|+|\beta|)A},\qquad|a|,|\alpha|,|\beta|\leq 5.$ (B.29)

  * •

There is $10A\leq D\leq 2^{A/10}$ such that

$\displaystyle\begin{split}\mathcal{D}(g,h):=\|g\|_{L^{2}}\|h\|_{L^{2}}+\min\big{(}{\|\partial_{k}g\|}_{L^{2}}{\|h\|}_{L^{2}},{\|g\|}_{L^{2}}{\|\partial_{k}h\|}_{L^{2}}\big{)}\leq
2^{D}.\end{split}$ (B.30)

Then, the following estimates hold: For any $p,q\in[1,\infty]$ and $r>1$ with

$\displaystyle\frac{1}{p}+\frac{1}{q}>\frac{1}{r},\qquad$ (B.31)

we have

$\displaystyle\begin{split}{\big{\|}T_{1}^{1}[b]\big{(}g,h\big{)}\big{\|}}_{L^{r}}&\lesssim{\big{\|}\widehat{g}\big{\|}}_{L^{p}}{\big{\|}\widehat{h}\big{\|}}_{L^{q}}\cdot
2^{\max(k_{1},k_{2})}\cdot 2^{C_{0}A}+2^{-D}\mathcal{D}(g,h),\end{split}$
(B.32)

and

$\displaystyle\begin{split}{\big{\|}P_{K}T_{1}^{2}[b](g,h)\big{\|}}_{L^{r}}&\lesssim{\big{\|}\widehat{g}\big{\|}}_{L^{p}}{\big{\|}\widehat{h}\big{\|}}_{L^{q}}\cdot
2^{\max(k_{1},k_{2})}\cdot 2^{C_{0}A}+2^{-D}\mathcal{D}(g,h).\end{split}$
(B.33)

In the above $P_{K}$ denotes the flat/regular Littlewood-Paley projection.
Moreover, we have

$\displaystyle\|T_{0}[b](g,h)\|_{L^{r}},\,\|T_{2}[b](g,h)\|_{L^{r}},\,\|T_{3}[b](g,h)\|_{L^{r}}\lesssim\|\widehat{g}\|_{L^{p}}\|\widehat{h}\|_{L^{q}}\cdot
2^{(C_{0}+1)A}$ (B.34)

where111111This is a convenient value of the absolute constant $C_{0}$ but it
can certainly be improved. In the course of nonlinear estimates for the
evolution equation we impose conditions on the smallness of $C_{0}\delta_{N}$
(or similar quantities); see (1.45). Then, a smaller value of $C_{0}$ would
reduce the total number of derivatives $N$ required for our initial data.
$C_{0}:=65$.

We also recall the following statement from [41], which is used in the proof
of Theorem (B.2):

######  Lemma B.3.

Let $j\geqslant 1,$ and consider the bilinear operator

$\displaystyle B_{j}[b](g,h)(x)=\mathcal{F}^{-1}_{\xi\mapsto
x}\int_{\mathbb{R}^{6}}g(\eta-\xi)h(\sigma)b(\xi,\eta,\sigma)\chi\big{(}2^{j}(|\eta|-|\sigma|)\big{)}\,d\eta
d\sigma,$

where $\chi$ is a Schwartz function. Assume that:

  * •

For some $A\geqslant 1$, $K,L,M\in{\mathbb{Z}}$, with $L\gg-j$, we have

$\displaystyle\textrm{supp}(b)\subseteq\big{\\{}(\xi,\eta,\sigma)\in\mathbb{R}^{9}\,:\,|\xi|+|\eta|+|\sigma|\lesssim
2^{A},\,|\xi|\approx 2^{K},\,|\eta|\approx 2^{L},\,|\sigma|\approx 2^{M}\\};$

  * •

The following estimates hold:

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

Then for $p,q,r\in[1,\infty],$ we have

$\displaystyle\|B_{j}[b](g,h)\|_{L^{r}}\lesssim 2^{-j}\cdot 2^{2L}\cdot
2^{8A}\cdot\|\widehat{g}\|_{L^{p}}\|\widehat{h}\|_{L^{q}},\qquad\frac{1}{p}+\frac{1}{q}=\frac{1}{r}.$

## References

  * [1]
  * [2] X. An and A. Soffer.  Fermi’s golden rule and $H^{1}$ scattering for nonlinear Klein-Gordon equations with metastable states.  DCDS A 40 (2020), no. 1, 331-373.
  * [3] S. Agmon.  Spectral properties of Schrödinger operators and scattering theory.  Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 2 (1975), no. 2, 151-218.
  * [4] P. Alsholm and G. Schmidt.  Spectral and Scattering Theory for Schrödinger Operators.  Arch. Rational Mech. Anal. 40 (1970/71), 281-311.
  * [5] D. Bambusi and S. Cuccagna.  On dispersion of small energy solutions to the nonlinear Klein Gordon equation with a potential.  Amer. J. Math. 133 (2011), no. 5, 1421-1468.
  * [6] M. Beceanu and W. Schlag.  Structure formulas for wave operators under a small scaling invariant condition.  J. Spectr. Theory 9 (2019), no. 3, 967-990.
  * [7] G. Chen.  Long-time dynamics of small solutions to $1$d cubic nonlinear Schrödinger equations with a trapping potential.  Preprint arXiv:2106.10106.
  * [8] G. Chen and F. Pusateri.  The $1$d nonlinear Schrödinger equation with a weighted $L^{1}$ potential.  Preprint. To appear in Analysis & PDE.
  * [9] S. Cuccagna and M. Maeda.  Coordinates at small energy and refined profiles for the Nonlinear Schrödinger Equation.  Preprint arXiv:2004.01366.
  * [10] S. Cuccagna and M. Maeda.  A survey on asymptotic stability of ground states of nonlinear Schrodinger equations II.  Preprint arXiv:2009.00573.
  * [11] S. Cuccagna and M. Maeda.  On selection of standing wave at small energy in the $1$D Cubic Schrödinger Equation with a trapping potential.  Preprint arXiv:2109.08108.
  * [12] J.M. Delort  Modified scattering for odd solutions of cubic nonlinear Schrödinger equations with potential in dimension one.  $<$hal-01396705$>$, 2016.
  * [13] J.M. Delort and N. Masmoudi.  On the stability of kink solutions of the $\Phi^{4}$ model in $1+1$ space time dimensions  $<$hal-02862414$>$, 2020.
  * [14] Y. Deng, A. D. Ionescu and B. Pausader.  The Euler-Maxwell system for electrons: global solutions in 2D.  Arch. Ration. Mech. Anal. 225 (2017), no. 2, 771-871.
  * [15] Y. Deng, A. D. Ionescu, B. Pausader, and F. Pusateri.  Global solutions for the $3D$ gravity-capillary water waves system.  Acta Math. 219 (2017), no. 2, 213-402.
  * [16] P. Germain, Z. Hani and S. Walsh.  Nonlinear resonances with a potential: multilinear estimates and an application to NLS.  Int. Math. Res. Not. IMRN 2015, no. 18, 8484-8544.
  * [17] P. Germain, F. Pusateri and F. Rousset.  The nonlinear Schrödinger equation with a potential.  Ann. Inst. H. Poincaré Anal. Non Linéaire 35 (2018), no. 6, 1477-1530.
  * [18] P. Germain and F. Pusateri.  Quadratic Klein-Gordon equations with a potential in $1$d.  Preprint arXiv:2006.15688.
  * [19] P. Germain, F. Pusateri and Z. Zhang.  On $1$d quadratic Klein-Gordon equations with a potential and with symmetries or genericity.  Preprint.
  * [20] M. Goldberg and W. Schlag.  Dispersive estimates for Schrödinger operators in dimensions one and three.  Comm. Math. Phys. 251 (2004), no. 1, 157-178.
  * [21] S. Gustafson, K. Nakanishi and T-P. Tsai.  Scattering theory for the Gross-Pitaevskii equation in three dimensions.  Commun. Contemp. Math. 11 (2009), no. 4, 657-707.
  * [22] R. A. Hunt.  On $L(p,q)$ spaces.  Enseignement Math. 12 (1966), no. 2, 249–276.
  * [23] T. Ikebe.  Eigenfunction expansions associated with the Schrödinger operators and their applications to Scattering Theory.  Arch. Rational Mech. Anal. 5 (1960), 1-34.
  * [24] J.-L. Journé, A. Soffer and C. Sogge.  Decay estimates for Schrödinger operators.  Comm. Pure Appl. Math. 44 (1991), no. 5, 573-604.
  * [25] R. Killip, J. Murphy and M. Visan.  The initial value problem for the cubic-quintic NLS with non-vanishing boundary conditions,  SIAM J. Math. Anal., 50, (2018), no 3, 2681–2739.
  * [26] R. Killip, B. Stovall and M. Visan.  Blowup behavior for the nonlinear Klein-Gordon equation.  Math. Ann., 358 (2014), no 1-2, 289–350.
  * [27] M. Kowalczyk, Y. Martel and C. Munoz.  Kink dynamics in the $\phi^{4}$ model: Asymptotic stability for odd perturbations in the energy space.  J. Amer. Math. Soc. 30 (2017), no. 3, 769-798.
  * [28] M. Kowalczyk, Y. Martel, C. Munoz and H. Van Den Bosch.  A sufficient condition for asymptotic stability of kinks in general $1+1$ scalar fields.  Ann. PDE 7 (2021), no. 1, Paper No. 10, 98 pp.
  * [29] A. Komech and E. Kopylova.  Dispersion decay and scattering theory.  John Wiley & Sons, Inc., Hoboken, NJ, 2012. xxvi+175 pp.
  * [30] A. I. Komech and E. Kopylova.  On Asymptotic Stability of Kink for Relativistic Ginzburg-Landau Equations.  Arch. Ration. Mech. Anal. 202 (2011), no. 1, 213-245.
  * [31] A. I. Komech and E. Kopylova.  On asymptotic stability of moving kink for relativistic Ginzburg-Landau equation.  Comm. Math. Phys. 302 (2011), no. 1, 225-252.
  * [32] T. Léger.  Global existence and scattering for quadratic NLS with potential in 3D.  to appear in Analysis & PDE.
  * [33] T. Léger.  3D quadratic NLS equation with electromagnetic perturbations.  Adv. Math. 375 (2020), 107407, 70 pp.
  * [34] T. Léger and F. Pusateri  Internal-mode induced growth in $3d$ nonlinear Klein-Gordon equations.  Preprint arXiv:2203.05694.
  * [35] H. Lindblad, J. Lührmann and A. Soffer.  Asymptotics for 1D Klein-Gordon equations with variable coefficient quadratic nonlinearities.  Arch. Ration. Mech. Anal. 241 (2021), no. 3, 1459-1527.
  * [36] H. Lindblad, J. Lührmann, W. Schlag and A. Soffer.  On modified scattering for 1D quadratic Klein-Gordon equations with non-generic potentials  Preprint arXiv:2012.15191.
  * [37] J. Lührmann and W. Schlag.  Asymptotic stability of the sine-Gordon kink under odd perturbations.  Preprint arXiv:2106.09605.
  * [38] I. Naumkin. Sharp asymptotic behavior of solutions for cubic nonlinear Schrödinger equations with a potential. _J.Math. Phys._ 57 (2016), no. 5, 051501, 31 pp.
  * [39] S. Klainerman.  Global existence of small amplitude solutions to nonlinear Klein-Gordon equations in four space-time dimensions.  Comm. Pure Appl. Math. 38 (1985), no. 5, 631-641.
  * [40] F. Pusateri and K. Widmayer.  On the global stability of a beta-plane equation.  Analysis & PDE, 11 (2018), no. 7, 1587-1624.
  * [41] F. Pusateri and A. Soffer.  Bilinear estimates in the presence of a large potential and a critical NLS in 3d.  arXiv:2003.00312.
  * [42] W. Schlag.  Dispersive estimates for Schrödinger operators: a survey.  Mathematical aspects of nonlinear dispersive equations, 255-285, Ann. of Math. Stud. 163, Princeton Univ. Press, Princeton, NJ, 2007.
  * [43] W. Schlag.  On pointwise decay of waves.  Preprint arXiv:2012.14448.
  * [44] J. Shatah.  Normal forms and quadratic nonlinear Klein-Gordon equations.  Comm. Pure Appl. Math. 38 (1985), no. 5, 685-696.
  * [45] I. M. Sigal.  Nonlinear wave and Schrödinger equations I: Instability of periodic and quasiperiodic solutions.  Commun. Math. Phys. 153 (1993), no.2, 297-320.
  * [46] B. Simon.  Spectrum and Continuum Eigenfunctions of Schrödinger Operators.  J. Funct. Analysis 42 (1981), 347-355.
  * [47] A. Soffer.  Soliton dynamics and scattering.  International Congress of Mathematicians. Vol. III, 459-471, Eur. Math. Soc., Zürich, 2006.
  * [48] A. Soffer and M. I. Weinstein.  Resonances, radiation damping and instability in Hamiltonian nonlinear wave equations.  Invent. Math. 136 (1999), no. 1, 9-74.
  * [49] T.-P. Tsai and H.-T. Yau.  Asymptotic dynamics of nonlinear Schrödinger equations: resonance-dominated and dispersion-dominated solutions.  Comm. Pure Appl. Math. 55 (2002), no. 2, 153-216.
  * [50] K. Yajima, The $W^{k,p}-$continuity of wave operators for Schrödinger operators, J. Math. Soc. Japan, Vol. 47, No. 3, 1995
  * [51] M. I. Weinstein.  Localized States and Dynamics in the Nonlinear Schrödinger/Gross-Pitaevskii Equation.  Dynamics of Partial Differential Equations. Frontiers in Applied Dynamical Systems: Reviews and Tutorials, vol 3 (2015). Springer, Cham.