&\lesssim_{\varepsilon}N^{\frac{1}{4}+2\varepsilon}\big{(}\frac{M}{N}\big{)}^{\frac{1}{6}}\prod_{i=1}^{2}L_{i}^{\frac{1}{2}}\|f^{\prime}_{i,N_{i},L_{i}}\|_{L^{2}}.\end{split}$

Taking into account the frequency-localization we find

$\big{|}\int(f_{1,N_{1},L_{1}}*f_{2,N_{2},L_{2}})f_{3,N_{3},L_{3}}\big{|}\lesssim_{\varepsilon}N^{\frac{1}{4}+2\varepsilon}\big{(}\frac{M}{N}\big{)}\big{(}\frac{M}{N}\big{)}^{\frac{1}{6}}(M^{2}N)^{-\frac{1}{2}}\prod_{i=1}^{3}L_{i}^{\frac{1}{2}}\|f_{i,N_{i},L_{i}}\|_{L^{2}}.$

For $M\leq N^{1+a_{1}}$ we find

(58)
$\big{|}\int(f_{1,N_{1},L_{1}}*f_{2,N_{2},L_{2}})f_{3,N_{3},L_{3}}\big{|}\lesssim_{\varepsilon}N_{1}^{\frac{1}{4}+2\varepsilon}N_{1}^{-\frac{3}{2}}N^{\frac{a_{1}}{6}}\prod_{i=1}^{3}L_{i}^{\frac{1}{2}}\|f_{i,N_{i},L_{i}}\|_{L^{2}}.$

For $M\geq N^{1+a_{1}}$ we use instead a bilinear Strichartz estimate provided
by Proposition 4.2:

(59)
$\big{|}\int(f_{1,N_{1},L_{1}}*f_{2,N_{2},L_{2}})f_{3,N_{3},L_{3}}\big{|}\lesssim(M^{2}N)^{-\frac{1}{2}}N^{\frac{1}{2}}N^{-\frac{\alpha}{2}}\prod_{i=1}^{3}L_{i}^{\frac{1}{2}}\|f_{i,N_{i},L_{i}}\|_{L^{2}}.$

Collecting estimates (56), (57), and (58), (59), the latter estimates being
dominant, we complete the proof. ∎

Now we turn to the slightly more involved case of a trilinear interaction
involving one lower, but not too low frequency:

######  Proposition 7.2.

Let $a_{2}\in(0,1/8)$, $\alpha\in(0,1/8)$, $N_{i}\in 2^{\mathbb{N}_{0}}$,
$N_{1}\sim N_{3}\gg N_{2}$, $N_{2}\gtrsim N_{1}^{1-a_{2}}$, and $L_{i}\in
2^{\mathbb{N}_{0}}$ with $L_{i}\gtrsim N_{i}^{\alpha}$. Let
$f_{i,N_{i},L_{i}}\in L^{2}_{\tau,\xi,\eta}$ with
$\text{supp}(f_{i,N_{i},L_{i}})\subseteq D_{N_{i},L_{i}}$.

  * •

Resonant case: Suppose that $L_{\max}\sim N_{1}^{2}N_{2}$. Then the following
estimate holds:

$\begin{split}&\quad\int(f_{1,N_{1},L_{1}}*f_{2,N_{2},L_{2}})f_{3,N_{3},L_{3}}d\xi
d\eta d\tau\\\
&\lesssim_{\varepsilon}N_{2}^{-1}N_{1}^{-\frac{1}{4}+\frac{11a_{2}}{24}+\varepsilon}\prod_{i=1}^{3}L_{i}^{\frac{1}{2}}(1+L_{i}/N_{i}^{3})^{\frac{1}{4}}\|f_{i,N_{i},L_{i}}\|_{2}.\end{split}$

  * •

Non-resonant case: Suppose that $L_{\max}\sim M^{2}N_{2}$ for $M\gg N_{1}$.
Then the following estimate holds:

$\begin{split}&\quad\int(f_{1,N_{1},L_{1}}*f_{2,N_{2},L_{2}})f_{3,N_{3},L_{3}}d\xi
d\eta d\tau\\\
&\lesssim\big{(}N_{2}^{-1}\frac{M^{\frac{1}{6}}(N_{1}N_{2})^{\frac{1}{24}+\varepsilon}}{N_{2}^{\frac{1}{2}}}\wedge
M^{-1}N_{2}^{-\frac{1}{2}}(1+N_{2}^{\frac{1}{2}}N_{1}^{-\frac{\alpha}{2}}))\prod_{i=1}^{3}L_{i}^{\frac{1}{2}}(1+L_{i}/N_{i}^{3})^{\frac{1}{4}}\|f_{i,N_{i},L_{i}}\|_{2}.\end{split}$

When we distinguish in the non-resonant case between $M\lesssim
N_{1}^{1+a_{4}}$ and $M\gg N_{1}^{1+a_{4}}$ we obtain the following:

######  Corollary 7.3.

With notations like above, in the non-resonant case the following estimate
holds for any $a_{4}\in(0,1/8)$:

$\begin{split}&\quad\int(f_{1,N_{1},L_{1}}*f_{2,N_{2},L_{2}})f_{3,N_{3},L_{3}}d\xi
d\eta d\tau\\\
&\lesssim_{\varepsilon}N_{2}^{-1}(N_{1}^{\frac{a_{4}}{6}-\frac{1}{4}+\frac{11a_{2}}{24}+\varepsilon}\vee
N_{1}^{-a_{4}-\frac{\alpha}{2}+\varepsilon})\prod_{i=1}^{3}L_{i}^{\frac{1}{2}}(1+L_{i}/N_{i}^{3})^{\frac{1}{4}}\|f_{i,N_{i},L_{i}}\|_{2}\end{split}$

###### Proof of Proposition 7.2.

By convolution constraint and the resonance relation we always have
$L_{\max}\gtrsim N_{1}^{2}N_{2}$. The case $L_{\max}\sim L_{\text{med}}\gtrsim
N_{1}^{2}N_{2}$ is readily estimated by bilinear Strichartz estimates from
Proposition 4.3. Indeed, by symmetry we can suppose that $L_{1}\sim L_{\max}$.
Then we find

(60)
$\begin{split}&\quad\|(f_{1,N_{1},L_{1}}*f_{2,N_{2},L_{2}})f_{3,N_{3},L_{3}}\|_{L^{2}_{\tau,\xi,\eta}}\\\
&\leq\|f_{1,N_{1},L_{1}}\|_{L^{2}}\|f_{2,N_{2},L_{2}}*f_{3,N_{3},L_{3}}\|_{L^{2}}\\\
&\lesssim(N_{1}^{2}N_{2})^{-\frac{1}{2}}L_{1}^{\frac{1}{2}}N_{2}^{\frac{3}{4}}(L_{2}L_{3})^{\frac{1}{2}}(N_{1}^{2}N_{2})^{-\frac{1}{4}}\prod_{i=1}^{3}\|f_{i,N_{i},L_{i}}\|_{2}\\\
&\lesssim
N_{1}^{-1}N_{2}^{-\frac{1}{2}}\prod_{i=1}^{3}L_{i}^{\frac{1}{2}}\|f_{i,N_{i},L_{i}}\|_{2}.\end{split}$

So, in the following we suppose that $L_{\max}\gg L_{\text{med}}$.

$\bullet$ _Resonant case:_ $L_{\max}\sim N_{1}^{2}N_{2}$. First, we suppose
that the high frequency carries the high modulation $L_{3}=L_{\max}$. In this
case we obtain

$D\sim\big{|}\frac{\eta_{1}}{\xi_{1}}-\frac{\eta_{2}}{\xi_{2}}\big{|}\lesssim
N_{1}.$

By the usual almost orthogonality and Galilean invariance argument we can
suppose that $|\eta_{i}|\lesssim N_{1}N_{i}$. We aim to obtain a bound via two
$L^{4}$-Strichartz estimates. To $f_{1,N_{1},L_{1}}$ we can apply Theorem 3.3:

$\|\mathcal{F}^{-1}_{t,x,y}[f_{1,N_{1},L_{1}}]\|_{L^{4}_{t,x,y}}\lesssim_{\varepsilon}N_{1}^{\frac{1}{8}+\varepsilon}L_{1}^{\frac{1}{2}}\|f_{1,N_{1},L_{1}}\|_{2}.$

For $f_{2,N_{2},L_{2}}$ we aim to apply Proposition 3.8. To this end, we
require an additional decomposition of the support in the $\eta$-frequency
into intervals of length $N_{2}^{2}$ and of the $\xi$-support into intervals
of length $N_{2}^{2}/N_{1}$, which incurs a factor of $(N_{1}/N_{2})$ by two
applications of the Cauchy-Schwarz inequality. This will allow us to apply
Proposition 3.8 with $k\sim N_{1}/N_{2}$. We denote the decomposition by
$f_{2,N_{2},L_{2}}=\sum_{J}f^{J}_{2,N_{2},L_{2}}$. Proposition 3.8 yields

$\|\mathcal{F}^{-1}_{t,x,y}[f^{J}_{2,N_{2},L_{2}}]\|_{L^{4}_{t,x,y}}\lesssim
L_{2}^{\frac{1}{2}}N_{2}^{\frac{1}{8}+\varepsilon}\big{(}\frac{N_{1}}{N_{2}}\big{)}^{\frac{1}{12}}\|f_{2,N_{2},L_{2}}\|_{2}.$

Consequently, by applying Hölder’s inequality and Cauchy-Schwarz inequality
the above yields

(61)
$\begin{split}&\quad\int(f_{1,N_{1},L_{1}}*f_{2,N_{2},L_{2}})f_{3,N_{3},L_{3}}d\xi
d\eta d\tau\\\
&\leq\|f_{1,N_{1},L_{1}}*f_{2,N_{2},L_{2}}\|_{L^{2}_{\tau,\xi,\eta}}\|f_{3,N_{3},L_{3}}\|_{L^{2}_{\tau,\xi,\eta}}\\\
&\lesssim(N_{1}^{2}N_{2})^{-\frac{1}{2}}N_{1}^{\frac{1}{8}+\varepsilon}\big{(}\frac{N_{1}}{N_{2}}\big{)}N_{2}^{\frac{1}{8}+\varepsilon}\big{(}\frac{N_{1}}{N_{2}}\big{)}^{\frac{1}{12}}\prod_{i=1}^{3}L_{i}^{\frac{1}{2}}\|f_{i,N_{i},L_{i}}\|_{2}\\\
&\lesssim
N_{2}^{-1}N_{1}^{-\frac{1}{4}+\frac{11a_{2}}{24}}\prod_{i=1}^{3}L_{i}^{\frac{1}{2}}\|f_{i,N_{i},L_{i}}\|_{2}.\end{split}$

The case $L_{2}=L_{\max}\sim N_{1}^{2}N_{2}$ gives a more favorable bound: In
this case we have the transversality bound

$\big{|}\frac{\eta_{1}}{\xi_{1}}-\frac{\eta_{3}}{\xi_{3}}\big{|}\lesssim
N_{2}.$

Hence, we can suppose by almost orthogonality and Galilean invariance that
$|\eta_{i}|\lesssim N_{1}N_{2}\ll N_{1}^{2}$ for $i=1,3$.

We find from two $L^{4}_{t,x,y}$-Strichartz estimates provided by Theorem 3.6

$\begin{split}&\quad\int(f_{1,N_{1},L_{1}}*f_{2,N_{2},L_{2}})f_{3,N_{3},L_{3}}d\xi
d\eta d\tau\\\
&\leq\|f_{2,N_{2},L_{2}}\|_{L^{2}_{\tau,\xi,\eta}}\prod_{i=1,3}\|\mathcal{F}^{-1}_{t,x,y}[f_{i,N_{i},L_{i}}]\|_{L^{4}_{t,x,y}}\\\
&\lesssim(N_{1}^{\frac{1}{8}-\frac{\alpha}{8}+\varepsilon})^{2}N_{1}^{-\frac{3}{2}}\prod_{i=1}^{3}L_{i}^{\frac{1}{2}}(1+L_{i}/N_{i}^{3})^{\frac{1}{4}}\|f_{i,N_{i},L_{i}}\|_{2}.\end{split}$

This is clearly more favorable than (61). The estimate of the resonant case is
complete.

_Non-resonant case:_ $L_{\max}\sim M^{2}N_{2}$ for $M\gg N_{1}$.

Firstly, suppose that $L_{3}=L_{\max}$ and we can suppose that $D\sim M$ which
is a consequence of the resonance relation. By Galilean invariance we can
suppose that $|\eta_{i}|\lesssim MN_{i}$. We aim to obtain a bound via two
$L^{4}$-Strichartz estimates.

In this case $\mathcal{F}^{-1}_{t,x,y}[f_{i,N_{i},L_{i}}]$, $i=1,2$ will both
be estimated via Proposition 3.8. We decompose the $\xi$-support into
intervals of length $N_{2}^{2}/M$ and the $\eta$-support into intervals of
length $N_{2}^{2}$. This incurs a factor of $(M/N_{2})$. Then we can apply
Hölder’s inequality and Proposition 3.8 to find

$\begin{split}&\quad\|f_{1,N_{1},L_{1}}*f_{2,N_{2},L_{2}}\|_{L^{2}_{\tau,\xi,\eta}}\\\
&\leq\|\mathcal{F}^{-1}_{t,x,y}[f_{1,N_{1},L_{1}}]\|_{L^{4}_{t,x,y}}\|\mathcal{F}^{-1}_{t,x,y}[f_{2,N_{2},L_{2}}]\|_{L^{4}_{t,x,y}}\\\
&\lesssim\big{(}\frac{M}{N_{2}}\big{)}N_{1}^{\frac{1}{8}+\varepsilon}\big{(}\frac{M}{N_{1}}\big{)}^{\frac{1}{12}}L_{1}^{\frac{1}{2}}\|f_{1,N_{1},L_{1}}\|_{L^{2}_{\tau,\xi,\eta}}N_{2}^{\frac{1}{8}+\varepsilon}\big{(}\frac{M}{N_{2}}\big{)}^{\frac{1}{12}}L_{2}^{\frac{1}{2}}\|f_{2,N_{2},L_{2}}\|_{L^{2}}.\end{split}$

By an application of Hölder’s inequality and taking into account $L_{3}\sim
M^{2}N_{2}$ we find

$\begin{split}&\quad\int(f_{1,N_{1},L_{1}}*f_{2,N_{2},L_{2}})f_{3,N_{3},L_{3}}d\xi
d\eta d\tau\\\
&\lesssim\frac{1}{N_{2}}\frac{M^{\frac{1}{6}}(N_{1}N_{2})^{\frac{1}{24}+\varepsilon}}{N_{2}^{\frac{1}{2}}}\prod_{i=1}^{3}L_{i}^{\frac{1}{2}}\|f_{i,N_{i},L_{i}}\|_{L^{2}}.\end{split}$

We impose a bound $M\lesssim N_{1}^{1+a_{4}}$ to find (recalling that
$N_{2}\gtrsim N_{1}^{1-a_{2}}$):

(62) $\int(f_{1,N_{1},L_{1}}*f_{2,N_{2},L_{2}})f_{3,N_{3},L_{3}}d\xi d\eta
d\tau\lesssim\frac{1}{N_{2}}N_{1}^{\frac{1+a_{4}}{6}-\frac{10}{24}+\frac{11a_{2}}{24}}\prod_{i=1}^{3}L_{i}^{\frac{1}{2}}\|f_{i,N_{i},L_{i}}\|_{2}.$

If $M\gg N_{1}^{1+a_{4}}$, we can apply a bilinear Strichartz estimate from
Proposition 4.2:

(63)
$\begin{split}\int(f_{1,N_{1},L_{1}}*f_{2,N_{2},L_{2}})f_{3,N_{3},L_{3}}d\xi
d\eta
d\tau&\lesssim(M^{2}N_{2})^{-\frac{1}{2}}N_{2}^{\frac{1}{2}}N_{1}^{-\frac{\alpha}{2}}\prod_{i=1}^{3}L_{i}^{\frac{1}{2}}\|f_{i,N_{i},L_{i}}\|_{2}\\\
&\lesssim
N_{2}^{-1}N_{1}^{-a_{4}-\frac{\alpha}{2}}\prod_{i=1}^{3}L_{i}^{\frac{1}{2}}\|f_{i,N_{i},L_{i}}\|_{2}.\end{split}$

Next, we handle the case $L_{2}=L_{\max}\sim M^{2}N_{2}$. Here we have the
transversality bound

$\big{|}\frac{\eta_{1}}{\xi_{1}}-\frac{\eta_{3}}{\xi_{3}}\big{|}\sim(M/N_{1})N_{2}.$

This implies by the usual Galilean invariance and almost orthogonality
$|\eta_{i}|\lesssim MN_{2}$. For $M\lesssim N_{1}^{2}/N_{2}$ we can apply
Theorem 3.6 directly:

$\begin{split}&\quad\int(f_{1,N_{1},L_{1}}*f_{2,N_{2},L_{2}})f_{3,N_{3},L_{3}}d\xi
d\eta d\tau\\\
&\leq\|f_{2,N_{2},L_{2}}\|_{L^{2}_{\tau,\xi,\eta}}\|f_{1,N_{1},L_{1}}*f_{3,N_{3},L_{3}}\|_{L^{2}_{\tau,\xi,\eta}}\\\
&\lesssim(M^{2}N_{2})^{-\frac{1}{2}}(N_{1}^{\frac{1}{8}-\frac{\alpha}{8}+\frac{\alpha}{12}+\varepsilon})^{2}\prod_{i=1}^{3}L_{i}^{\frac{1}{2}}\|f_{i,N_{i},L_{i}}\|_{2}\\\
&\lesssim
M^{-\frac{3}{2}}(N_{1}^{\frac{1}{8}-\frac{\alpha}{8}+\frac{\alpha}{12}+\varepsilon})^{2}\prod_{i=1}^{3}L_{i}^{\frac{1}{2}}(1+L_{i}/N_{i}^{3})^{\frac{1}{4}}\|f_{i,N_{i},L_{i}}\|_{2}.\end{split}$

For $M\gg N_{1}^{2}/N_{2}$ we need to decompose the $\eta$-frequencies into
intervals of length $N_{1}^{2}$ with the number of sub-intervals given by
$k\sim MN_{2}/N_{1}^{2}$. Correspondingly, we decompose the $\xi$-frequencies
into intervals of length $N_{1}/k\sim\frac{N_{1}^{3}}{MN_{2}}$. Since we can
suppose by convolution constraint that the $\xi$-frequencies are localized to
an interval of length $N_{2}$, the number of subintervals is given by
$\sim\frac{MN_{2}^{2}}{N_{1}^{4}}$. After additional localization of the
frequency support, we can use Proposition 3.8 to find

$\|f_{1,N_{1},L_{1}}^{\prime}*f_{3,N_{3},L_{3}}^{\prime}\|_{L^{2}_{\tau,\xi,\eta}}\lesssim_{\varepsilon}(N_{1}^{\frac{1}{8}+\varepsilon})^{2}\big{(}\frac{MN_{2}}{N_{1}}\big{)}^{\frac{1}{6}}\prod_{i=1,3}\|f^{\prime}_{i,N_{i},L_{i}}\|_{2}.$

Taking into account the additional frequency localization and the modulation
size $L_{2}\sim M^{2}N_{2}$ we obtain

$\begin{split}&\quad\int(f_{1,N_{1},L_{1}}*f_{2,N_{2},L_{2}})f_{3,N_{3},L_{3}}d\xi
d\eta d\tau\\\
&\leq\|f_{2,N_{2},L_{2}}\|_{L^{2}_{\tau,\xi,\eta}}\|f_{1,N_{1},L_{1}}*f_{3,N_{3},L_{3}}\|_{L^{2}_{\tau,\xi,\eta}}\\\
&\lesssim(M^{2}N_{2})^{-\frac{1}{2}}\big{(}\frac{MN_{2}}{N_{1}^{2}}\big{)}^{\frac{1}{2}}\big{(}\frac{MN_{2}^{2}}{N_{1}^{4}}\big{)}^{\frac{1}{2}}(N_{1}^{\frac{1}{8}+\varepsilon})^{2}\big{(}\frac{MN_{2}}{N_{1}}\big{)}^{\frac{1}{6}}\prod_{i=1}^{3}L_{i}^{\frac{1}{2}}\|f_{i,N_{i},L_{i}}\|_{2}\\\
&\lesssim
M^{-\frac{3}{2}}\frac{MN_{2}^{\frac{3}{2}}}{N_{1}^{3}}N_{1}^{\frac{1}{4}+\varepsilon}\big{(}\frac{MN_{2}}{N_{1}}\big{)}^{\frac{1}{6}}\prod_{i=1}^{3}L_{i}^{\frac{1}{2}}(1+L_{i}/N_{i}^{3})^{\frac{1}{4}}\|f_{i,N_{i},L_{i}}\|_{2}\\\
&\lesssim
M^{-\frac{1}{3}}N_{2}^{-1}N_{1}^{-\frac{1}{4}}\big{(}\frac{N_{2}}{N_{1}}\big{)}^{\frac{1}{6}}\prod_{i=1}^{3}L_{i}^{\frac{1}{2}}(1+L_{i}/N_{i}^{3})^{\frac{1}{4}}\|f_{i,N_{i},L_{i}}\|_{2}.\end{split}$

This estimate improves on (62), for which reason we conclude the proof taking
(62) and (63). ∎

###### Proof of Corollary 7.3.

This is a consequence of (62) and (63), since the estimates recorded below the
aforementioned ones are more favorable. ∎

Lastly, we record the estimate for the case where the small frequency is
quantitatively smaller than the highest frequency. In this case the main
estimate is provided by the bilinear Strichartz estimate from Proposition 4.4.

######  Proposition 7.4.

Let $\alpha\in(0,1/8)$, $N_{i}\in 2^{\mathbb{N}_{0}}$, $N_{1}\sim N_{3}\gg
N_{2}$, and $L_{i}\in 2^{\mathbb{N}_{0}}$ with $L_{i}\gtrsim N_{i}^{\alpha}$.
Let $f_{i,N_{i},L_{i}}\in L^{2}_{\tau,\xi,\eta}$ with
$\text{supp}(f_{i,N_{i},L_{i}})\subseteq D_{N_{i},L_{i}}$. Then the following
estimate holds:

$\begin{split}\int(f_{1,N_{1},L_{1}}*f_{2,N_{2},L_{2}})f_{3,N_{3},L_{3}}d\xi
d\eta d\tau&\lesssim
N_{1}^{-1}(N_{2}^{-\frac{1}{4}}N_{1}^{-\frac{\alpha}{4}}+N_{1}^{-\frac{\alpha}{2}})\log(N_{1})\\\
&\quad\prod_{i=1}^{3}L_{i}^{\frac{1}{2}}(1+L_{i}/N_{i}^{3})^{\frac{1}{4}}\|f_{i,N_{i},L_{i}}\|_{2}.\end{split}$

###### Proof.

This is straightforward from the preceding arguments and the bilinear
Strichartz estimate from Proposition 4.4. Suppose that $L_{\max}\gg
L_{\text{med}}$. First, we suppose that $L_{3}=L_{\max}$, which is
interchangeable with $L_{3}=L_{\max}$. In the resonant case, $L_{\max}\sim
N_{1}^{2}N_{2}$ we can apply Proposition 4.4 with $D\lesssim N_{1}$ to find:

$\begin{split}&\quad\int(f_{1,N_{1},L_{1}}*f_{2,N_{2},L_{2}})f_{3,N_{3},L_{3}}d\xi
d\eta d\tau\\\
&\leq\|f_{3,N_{3},L_{3}}\|_{L^{2}_{\tau,\xi,\eta}}\|f_{1,N_{1},L_{1}}*f_{2,N_{2},L_{2}}\|_{L^{2}_{\tau,\xi,\eta}}\\\
&\lesssim(N_{1}^{2}N_{2})^{-\frac{1}{2}}(N_{2}^{\frac{1}{4}}N_{1}^{-\frac{\alpha}{4}}+N_{2}^{\frac{1}{2}}N_{1}^{-\frac{\alpha}{2}})\log(N_{1})\prod_{i=1}^{3}L_{i}^{\frac{1}{2}}(1+L_{i}/N_{i}^{3})^{\frac{1}{4}}\|f_{i,N_{i},L_{i}}\|_{2}.\end{split}$

In case $L_{2}=L_{\max}\sim N_{1}^{2}N_{2}$ we can use the analysis from
Proposition 7.2, which gives a more favorable estimate.

In the non-resonant case $L_{\max}\sim M^{2}N_{2}$, $M\gg N_{1}$, we again use
the bilinear estimate in case $L_{3}=L_{\max}$ with $D\sim M$. This yields the
same estimate like above. If $L_{2}=L_{\max}$, we note that we have the
transversality

$\big{|}\frac{\eta_{1}}{\xi_{1}}-\frac{\eta_{3}}{\xi_{3}}\big{|}\sim\frac{MN_{2}}{N_{1}}\gg
N_{2}.$

Then we can use the Hölder’s inequality and the bilinear Strichartz estimate
from Proposition 4.2 to find

$\begin{split}&\quad\int(f_{1,N_{1},L_{1}}*f_{2,N_{2},L_{2}})f_{3,N_{3},L_{3}}d\xi
d\eta d\tau\\\
&\leq\|f_{2,N_{2},L_{2}}\|_{L^{2}_{\tau,\xi,\eta}}\|f_{1,N_{1},L_{1}}*f_{3,N_{3},L_{3}}\|_{L^{2}_{\tau,\xi,\eta}}\\\
&\lesssim(M^{3})^{-\frac{1}{2}}L_{2}^{\frac{1}{2}}(L_{2}/N_{2}^{3})^{\frac{1}{4}}N_{2}^{\frac{1}{2}}L_{\min}^{\frac{1}{2}}\langle
L_{\text{med}}/N_{2}\rangle^{\frac{1}{2}}\log(M)\prod_{i=1}^{3}\|f_{i,N_{i},L_{i}}\|_{2}\\\
&\lesssim
M^{-\frac{3}{2}}\log(M)(1+N_{2}^{\frac{1}{2}}N_{1}^{-\frac{\alpha}{2}})\prod_{i=1}^{3}L_{i}^{\frac{1}{2}}(1+L_{i}/N_{i}^{3})^{\frac{1}{4}}\|f_{i,N_{i},L_{i}}\|_{2}\end{split}$

Summing over $M\gtrsim N$ finishes the proof.

∎

##  8\. Short-time energy estimates

###  8.1. A priori estimates

This section is devoted to prove energy estimates for solutions. We can take
advantage of the conservative derivative nonlinearity and symmetry of the
expression.

######  Proposition 8.1.

Let $T\in(0,1]$, $0<\alpha<\frac{1}{10}$, and $s\geq s^{\prime}\geq
s_{3}(\alpha,\varepsilon)=-\frac{\alpha}{2}+\varepsilon$. Then the following
estimate holds:

(64)
$\|u\|^{2}_{E^{s}(T)}\lesssim\|u_{0}\|_{H^{s,0}}^{2}+T^{\delta}\|u\|^{2}_{F^{s}(T)}\|u\|_{F^{s^{\prime}}(T)}.$

###### Proof.

Since the low frequencies $\|P_{\leq 4}u\|_{E^{s}(T)}=\|P_{\leq
4}u_{0}\|_{H^{s,0}}$ are estimated by the definition of the function spaces,
it suffices to estimate $\|P_{N}u\|_{E^{s}(T)}$ for $N\geq 4$ and carry out
the dyadic square summation. We obtain an estimate for
$\|P_{N}u(t)\|^{2}_{L^{2}}$, $t\in[0,T]$ by the fundamental theorem of
calculus:

$\|P_{N}u(t)\|^{2}_{L^{2}}=\|P_{N}u(0)\|^{2}_{L^{2}}+2\int_{0}^{t}\int_{\mathbb{T}^{2}}P_{N}u(s)\partial_{x}P_{N}(u^{2})(s)dxdyds.$

For technical reasons we suppose that $P_{N}$ denotes in the following a
smooth frequency projection:

$(P_{N}f)\widehat{(}\xi,\eta)=\chi_{1}(\xi/N)\hat{f}(\xi,\eta)$

with $\chi_{1}\in C^{\infty}_{c}$ denoting a suitable bump funciton. Before
decomposing the time integral into intervals of size
$T=T(N_{\max})=N_{\max}^{-\alpha}$, we integrate by parts to let the
derivative always act on the lowest frequency.

We use the standard paraproduct decomposition:

(65) $P_{N}(u^{2})=2P_{N}(uP_{\ll N}u)+P_{N}(P_{\gtrsim N}uP_{\gtrsim N}u).$

Clearly, for the second term it is immediate how the derivative acts on low
frequencies. We turn to the first term. Write for $K\ll N$

(66) $P_{N}(uP_{K}u)=P_{N}uP_{K}u+[P_{N}(uP_{K}u)-P_{N}uP_{K}u].$

For the first term in (66) we obtain

$\begin{split}\int_{\mathbb{T}^{2}}P_{N}u\partial_{x}(P_{N}uP_{K}u)dxdy&=\int_{\mathbb{T}^{2}}(\partial_{x}P_{K}u)(P_{N}u)^{2}dxdy+\int_{\mathbb{T}^{2}}\frac{1}{2}(\partial_{x}(P_{N}u)^{2})P_{K}udxdy\\\
&=\frac{1}{2}\int_{\mathbb{T}^{2}}P_{K}u(P_{N}u)^{2}dxdy.\end{split}$

For the second term we use Parseval’s identity:

$\begin{split}&\quad\int_{\mathbb{T}^{2}}P_{N}u\partial_{x}[P_{N}(uP_{K}u)-P_{N}uP_{K}u]\\\
&=\sum_{\begin{subarray}{c}(\xi_{1},\eta_{1})\in\mathbb{Z}^{2},\\\
(\xi_{2},\eta_{2})\in\mathbb{Z}^{2}\end{subarray}}\widehat{P_{N}u}(-\xi_{1}-\xi_{2},-\eta_{1}-\eta_{2})i(\xi_{1}+\xi_{2})m(\xi_{1},\xi_{2})\hat{u}(\xi_{1},\eta_{1})\hat{u}(\xi_{2},\eta_{2}).\end{split}$

We denote the symbol by

$m(\xi_{1},\xi_{2})=\chi_{N}(\xi_{1}+\xi_{2})\chi_{K}(\xi_{2})-\chi_{N}(\xi_{1})\chi_{K}(\xi_{2})=[\chi_{N}(\xi_{1}+\xi_{2})-\chi_{N}(\xi_{1})]\chi_{K}(\xi_{2}).$

As a consequence of the mean-value theorem, we have the size estimate

$|m(\xi_{1},\xi_{2})|\lesssim\frac{K}{N}\tilde{\chi}_{N}(\xi_{1})\chi_{K}(\xi_{2})=:\tilde{m}(\xi_{1},\xi_{2})$

with $\tilde{\chi}_{N}$ denoting a mildly enlarged version of $\chi_{N}$. With
the regularity estimates

$|\partial_{\xi_{1}}^{\alpha_{1}}\partial_{\xi_{2}}^{\alpha_{2}}m(\xi_{1},\xi_{2})|\lesssim_{\alpha_{1},\alpha_{2}}N^{-\alpha_{1}}K^{-\alpha_{2}}\tilde{m}(\xi_{1},\xi_{2})$

immediate, we can expand $m(\xi_{1},\xi_{2})$ as a Fourier series and
effectively reduce to

$\int_{\mathbb{T}^{2}}P_{N}u\partial_{x}(P_{N}(uP_{K}u)-P_{N}uP_{K}u)dxdy\sim\int_{\mathbb{T}^{2}}P_{N}u\tilde{P}_{N}u(\partial_{x}P_{K}u)dxdy.$

It remains to obtain estimates for

$K\int_{0}^{T}\int_{\mathbb{T}^{2}}\tilde{P}_{N}u\tilde{P}_{N}uP_{K}u\,dxdyds,\quad
K\lesssim N.$

To this end, we consider a smooth partition of unity of $[0,T]$ into intervals
of length $N^{-\alpha}$ to account for the definition of the function spaces:

$1_{[0,T]}(s)\sum_{n\in\mathbb{Z}}\gamma^{3}(N^{\alpha}s-n)=1_{[0,T]}(s).$

Clearly,

$\\#\\{n\in\mathbb{Z}:1_{[0,T]}(s)\gamma^{3}(N^{\alpha}s-n)\not\equiv
0\\}\lesssim TN^{\alpha}.$

Let

$\mathcal{N}=\\{n\in\mathbb{Z}:1_{[0,T]}(s)\gamma^{3}(N^{\alpha}s-n)\not\equiv
0\\}.$

Suppose in the following for $n\in\mathcal{N}$ that

(67) $1_{[0,T]}(t)\gamma^{3}(N^{\alpha}t-n)=\gamma^{3}(N^{\alpha}t-n).$

With the estimates not depending on the location of the time interval $I$,
$|I|\lesssim N^{-\alpha}$, in the following we aim to prove estimates

(68)
$\big{|}\int_{\mathbb{R}\times\mathbb{T}^{2}}\gamma^{3}(N^{\alpha}(t-t_{I}))\tilde{P}_{N}u\tilde{P}_{N}uP_{K}udxdyds\big{|}\lesssim
C(N,K)\|P_{N}u\|^{2}_{F_{N}}\|P_{K}u\|_{F_{K}}$

for $K\lesssim N$.

We introduce notations

$\begin{split}f_{1,N_{1}}&=\mathcal{F}_{t,x,y}[\gamma(N^{\alpha}(t-t_{I}))\tilde{P}_{N}u],\quad
f_{2,N_{2}}=\mathcal{F}_{t,x,y}[\gamma(N^{\alpha}(t-t_{I}))P_{K}u],\\\
f_{3,N_{3}}&=\mathcal{F}_{t,x,y}[\gamma(N^{\alpha}(t-t_{I}))\tilde{P}_{N}u].\end{split}$

Next, we localize in modulation reflecting the time-localization

(69) $\begin{split}f_{i,N_{i},L_{i}}&=1_{D_{N_{i},L_{i}}}f_{i,N_{i}}\;\,\text{
for }L_{i}>N_{1}^{\alpha},\\\ f_{i,N_{i},L_{i}}&=1_{D_{N_{i},\leq
L_{i}}}f_{i,N_{i}}\text{ for }L_{i}=N_{1}^{\alpha}.\end{split}$

We shall prove estimates

(70)
$\begin{split}&\quad\big{|}\int(f_{1,N_{1},L_{1}}*f_{2,N_{2},L_{2}})f_{3,N_{3},L_{3}}d\xi
d\eta d\tau\big{|}\\\ &\lesssim
C(N_{1},N_{2},N_{3})\prod_{i=1}^{3}L_{i}^{\frac{1}{2}}(1+L_{i}/N_{i}^{3})^{\frac{1}{4}}\|f_{i,N_{i},L_{i}}\|_{L^{2}},\end{split}$

from which (32) follows by summation in $L_{i}$ and properties of the function
spaces.

Now we comment on the cases, in which the cut-off is not smooth:

$\\#\\{n\in\mathcal{N}:1_{[0,T]}(t)\gamma^{3}(N^{\alpha}t-n)\not\equiv\gamma^{3}(N^{\alpha}t-n)\\}\leq
4.$

For these finitely many cases we can use the following lemma:

######  Lemma 8.2 ([17, p. 291]).

If $I\subseteq\mathbb{R}$ is an interval, $k\in\mathbb{Z}$, $f_{k}\in X_{K}$,
and $f_{k}^{I}=\mathcal{F}(1_{I}(t)\mathcal{F}^{-1}(f_{k}))$, then

$\sup_{L\in
2^{\mathbb{N}}}L^{\frac{1}{2}}(1+L/N^{3})^{\frac{1}{4}}\|\eta_{L}(\tau-\omega(\xi,\eta))f_{k}^{I}\|_{L^{2}}\lesssim\|f_{k}\|_{X_{K}}.$

Then, we can interpolate suitable estimates in (70) with the bilinear
Strichartz estimate provided by Proposition 4.3:

$\big{|}\int(f_{1,N_{1},L_{1}}*f_{2,N_{2},L_{2}})f_{3,N_{3},L_{3}}d\xi d\eta
d\tau\big{|}\lesssim
L_{\min}^{\frac{1}{2}}N_{\min}^{\frac{3}{4}}L_{\text{max}}^{\frac{1}{4}}\prod_{i=1}^{3}\|f_{i,N_{i},L_{i}}\|_{2}.$

This gives the following variant of (70):

$\begin{split}&\quad\big{|}\int(f_{1,N_{1},L_{1}}*f_{2,N_{2},L_{2}})f_{3,N_{3},L_{3}}d\xi
d\eta d\tau\big{|}\\\ &\lesssim
C(N_{1},N_{2},N_{3})N_{\max}^{0+}L_{\max}^{0-}\prod_{i=1}^{3}L_{i}^{\frac{1}{2}}(1+L_{i}/N_{i}^{3})^{\frac{1}{4}}\|f_{i,N_{i},L_{i}}\|_{L^{2}},\end{split}$

which allows us to replace the $\ell^{1}$-summation over modulation with an
$\ell^{\infty}$-norm. In addition we can combine the estimate with Lemma 2.3
to squeeze out the factor $T^{\delta}$ also for the contribution affected by
the sharp time cutoff.

$\bullet$ High$-$High$-$High-estimates: $N_{1}\sim N_{2}\sim N_{3}$. Let
$N=N_{1}$ for brevity. We can apply Proposition 7.1 to find for
$a_{1}\in(0,1/8)$:

$\int(f_{1,N_{1},L_{1}}*f_{2,N_{2},L_{2}})f_{3,N_{3},L_{3}}d\xi d\eta
d\tau\lesssim N^{-1}(N^{-\frac{1}{4}+\frac{a_{1}}{6}+\varepsilon}\vee
N^{-a_{1}-\frac{\alpha}{2}})\prod_{i=1}^{3}L_{i}^{\frac{1}{2}}\|f_{i,N_{i},L_{i}}\|_{2}.$

The first bound in the above estimate yields taking into account powers of $N$
from derivative and time localization, i.e., a factor of $N^{1+\alpha}$:

(71)
$s_{3}(\alpha,\varepsilon)\geq\alpha+\frac{a_{1}}{6}-\frac{1}{4}+\varepsilon.$

The second estimate gives the lower bound:

(72) $s_{3}(\alpha,\varepsilon)\geq\frac{\alpha}{2}-a_{1}+\varepsilon.$

$\bullet$ High$-$Low$-$High$-$interaction: In this case we estimate

$N_{2}\iint\gamma^{3}(N_{1}^{\alpha}(t-t_{I}))P_{N_{1}}uP_{N_{2}}uP_{N_{3}}udxdydt\text{
with }N_{1}\sim N_{3}\gg N_{2}.$

To prove estimates (64), we need to take into account the input regularity
$N_{1}^{2s_{3}}$ or $N_{2}^{2s_{3}}$. In the following we focus on the latter
as the first case is better behaved.

Frequency-dependent time localization incurs a factor of $N_{1}^{\alpha}$.
After introducing notations for the space-time Fourier transform like above,
we are again reduced to show trilinear convolution estimates as handled in
Proposition 7.2:

$\int\big{(}f_{1,N_{1},L_{1}}*f_{2,N_{2},L_{2}}\big{)}f_{3,N_{3},L_{3}}d\xi
d\eta d\tau\lesssim
C(N_{1},N_{2})\prod_{i=1}^{3}L_{i}^{\frac{1}{2}}\|f_{i,N_{i},L_{i}}\|_{L^{2}}.$

First, we consider the case $N_{2}\lesssim N_{1}^{1-a_{2}}$. In this case we
apply Proposition 7.4 to find

$\begin{split}\int\big{(}f_{1,N_{1},L_{1}}*f_{2,N_{2},L_{2}}\big{)}f_{3,N_{3},L_{3}}d\xi
d\eta d\tau&\lesssim
N_{1}^{-1}(N_{2}^{-\frac{1}{4}}N_{1}^{-\frac{\alpha}{4}}+N_{1}^{-\frac{\alpha}{2}})\log(N_{1})\\\
&\quad\times\prod_{i=1}^{3}L_{i}^{\frac{1}{2}}(1+L_{i}/N_{i}^{3})^{\frac{1}{4}}\|f_{i,N_{i},L_{i}}\|_{2}.\end{split}$

Taking into account time localization and derivative loss, we find the
summability condition

$N_{1}^{\alpha-1}N_{2}^{1+s_{3}}(N_{2}^{-\frac{1}{4}}N_{1}^{-\frac{\alpha}{4}}+N_{1}^{-\frac{\alpha}{2}})\log(N_{1})\lesssim
N_{1}^{2s_{3}}.$

This yields

(73)
$s_{3}(\alpha,\varepsilon)\geq\frac{\frac{\alpha}{2}-a_{2}}{1+a_{2}}+\varepsilon.$

For $N_{2}\gtrsim N_{1}^{1-a_{2}}$ we apply Proposition 7.2. In the resonant
case $L_{\max}\sim N_{1}^{2}N_{2}$ we find

$\begin{split}&\quad\int(f_{1,N_{1},L_{1}}*f_{2,N_{2},L_{2}})f_{3,N_{3},L_{3}}d\xi
d\eta d\tau\\\
&\lesssim_{\varepsilon}N_{2}^{-1}N_{1}^{-\frac{1}{4}+\frac{11a_{2}}{24}+\varepsilon}\prod_{i=1}^{3}L_{i}^{\frac{1}{2}}(1+L_{i}/N_{i}^{3})^{\frac{1}{4}}\|f_{i,N_{i},L_{i}}\|_{2}.\end{split}$

Taking into account the regularity of the functions and the factor
$N_{1}^{\alpha}$ from time localization, we obtain the condition

$N_{2}^{s_{3}}N_{1}^{\alpha}N_{1}^{-\frac{1}{4}+\frac{11a_{2}}{24}}\lesssim
N_{1}^{2s_{3}}.$

This gives

(74)
$s_{3}(\alpha,\varepsilon)\geq-(\frac{1}{4}+\alpha+\frac{11a_{2}}{48})/(1+a_{2})+\varepsilon.$

In the non-resonant case we find

$\begin{split}&\quad\int(f_{1,N_{1},L_{1}}*f_{2,N_{2},L_{2}})f_{3,N_{3},L_{3}}d\xi
d\eta d\tau\\\
&\lesssim_{\varepsilon}N_{2}^{-1}(N_{1}^{\frac{a_{4}}{6}-\frac{1}{4}+\frac{11a_{2}}{24}+\varepsilon}\vee
N_{1}^{-a_{4}-\frac{\alpha}{2}+\varepsilon})\prod_{i=1}^{3}L_{i}^{\frac{1}{2}}(1+L_{i}/N_{i}^{3})^{\frac{1}{4}}\|f_{i,N_{i},L_{i}}\|_{2}.\end{split}$

The first estimate gives the condition taking into account time localization
and derivative loss:

$N_{2}^{s_{3}}N_{1}^{\alpha+\frac{a_{4}}{6}-\frac{1}{4}+\frac{11a_{2}}{24}}\lesssim
N_{1}^{2s_{3}}.$

This yields the bound

(75)
$s_{3}(\alpha,\varepsilon)\geq(-\frac{1}{4}+\alpha+\frac{a_{4}}{6}+\frac{11a_{2}}{24})/(1+a_{2})+\varepsilon.$

The second estimate gives the condition

$N_{2}^{s_{3}}N_{1}^{\frac{\alpha}{2}-a_{4}}\lesssim N_{1}^{2s_{3}}.$

This gives the condition

(76)
$s_{3}(\alpha,\varepsilon)\geq\frac{\frac{\alpha}{2}-a_{4}}{1+a_{2}}+\varepsilon.$

We choose $a_{2}=3\alpha/2$ to simplify the conditions (73) and (74):

(77)
$s_{3}(\alpha,\varepsilon)\geq-\frac{\alpha}{1+3\alpha/2}+\varepsilon,\quad
s_{3}(\alpha,\varepsilon)\geq-\frac{3}{8}+\frac{44\alpha}{24}+\varepsilon.$

For $0<a_{1},\ldots,a_{4}<\frac{1}{8}$ parameters, which are chosen like above
to quantify ratios of frequencies, we collect the regularity thresholds:

  * •

High$-$High$-$High-interaction (71):

$s_{3}(\alpha,\varepsilon)\geq\alpha+\frac{a_{1}}{6}-\frac{1}{4}+\varepsilon.$

  * •

High$-$High$-$High-interaction (72):

$s_{3}(\alpha,\varepsilon)\geq\frac{\alpha}{2}-a_{1}+\varepsilon.$

  * •

High$-$Low$-$High-interaction, $N_{2}\lesssim N_{1}^{1-a_{2}}$ (73):

$s_{3}(\alpha,\varepsilon)\geq\frac{\frac{\alpha}{2}-a_{2}}{1+a_{2}}+\varepsilon.$

  * •

High$-$Low$-$High-interaction, $N_{2}\gtrsim N_{1}^{1-a_{2}}$, Resonant case
(74):

$s_{3}(\alpha,\varepsilon)\geq-(\frac{1}{4}+\alpha+\frac{11a_{2}}{48})/(1+a_{2})+\varepsilon.$

  * •

High$-$Low$-$High-interaction, $N_{2}\gtrsim N_{1}^{1-a_{2}}$, Non-resonant
case (75):

$s_{3}(\alpha,\varepsilon)\geq(-\frac{1}{4}+\alpha+\frac{a_{4}}{6}+\frac{11a_{2}}{24})/(1+a_{2})+\varepsilon.$

  * •

High$-$Low$-$High-interaction, Non-resonant case (76):

$s_{3}(\alpha,\varepsilon)\geq\frac{\frac{\alpha}{2}-a_{4}}{1+a_{2}}+\varepsilon.$

We choose $a_{1}=\alpha$, $a_{2}=a_{4}=\frac{3\alpha}{2}$ to find by (72),
(73), (76)555For the latter two we use $\alpha<\frac{1}{8}$.

(78) $s_{3}(\alpha,\varepsilon)\geq-\frac{\alpha}{2}+\varepsilon,$

and from (71), (74), (75):

(79)
$s_{3}(\alpha,\varepsilon)\geq-\frac{1}{4}+\frac{7\alpha}{6}+\varepsilon\vee-(\frac{1}{4}+\alpha+\frac{33\alpha}{96})/(1+\frac{3\alpha}{2})+\varepsilon\vee(-\frac{1}{4}+\alpha+\frac{3\alpha}{12}+\frac{33\alpha}{96})/(1+\frac{3\alpha}{2})+\varepsilon.$

Choosing $0<\alpha<\frac{1}{10}$ we see that (79) does not give an additional
constraint to (78). This finishes the proof.

∎

###  8.2. Estimates for differences of solutions

To prove continuous dependence, it remains to establish energy estimates for
differences of solutions $v=u_{1}-u_{2}$:

(80)
$\|v\|^{2}_{E^{s^{\prime}}(T)}\lesssim\|v_{0}\|_{H^{s^{\prime},0}(\mathbb{T}^{2})}^{2}+T^{\delta}\|v\|^{2}_{F^{s^{\prime}}(T)}(\|u_{1}\|_{F^{s}(T)}+\|u_{2}\|_{F^{s}(T)})$

The estimate will be carried out in lower regularity $s^{\prime}=20s$. The
main difference to the a priori estimates established above is that we can no
longer always assume the derivative to act on the lowest frequency. We will
have to estimate expressions

$\iint_{[0,T]\times\mathbb{T}^{2}}P_{N_{1}}\partial_{x}vP_{N_{2}}vP_{N_{3}}u_{i}dxdydt\text{
with }N_{1}\sim N_{3}\gg N_{2},$

where due to reduced symmetry we cannot integrate by parts to assign the
derivative to $P_{N_{2}}v$. Here the lowered regularity comes to rescue.

We show the following:

######  Proposition 8.3.

Let $T\in(0,1]$, $0<\alpha<\frac{1}{36}$, and $s^{\prime}=20s$, $s\geq
s_{4}(\alpha,\varepsilon)=-\frac{\alpha}{2}+\varepsilon$, and
$s\leq-\frac{5\alpha}{12}$. Then we find (80) to hold for $v=u_{1}-u_{2}$ with
$u_{i}\in C([0,T],L^{2}(\mathbb{T}^{2}))$ solutions to (1).

###### Proof.

Like in the previous section we obtain an estimate for
$\|P_{N}v(t)\|^{2}_{L^{2}}$, $t\in[0,T]$ by the fundamental theorem of
calculus:

$\|P_{N}v(t)\|^{2}_{L^{2}}=\|P_{N}v(0)\|^{2}_{L^{2}}+\int_{0}^{t}\int_{\mathbb{T}^{2}}P_{N}v(s)\partial_{x}P_{N}(v(u_{1}+u_{2}))(s)dxdyds.$

The estimates for

$\int_{0}^{t}\int_{\mathbb{T}^{2}}P_{N}v(s)\partial_{x}P_{N}(v\,u_{i})(s)dxdyds$

will be carried out separately. In the following we denote $u_{i}$ with $u$ to
lighten the notation. We use a paraproduct decomposition like above:

(81) $P_{N}(vu)=P_{N}(vP_{\ll N}u)+P_{N}(P_{\ll N}vu)+P_{N}(P_{\gtrsim
N}uP_{\gtrsim N}v).$

For the first term the high frequency is again on $v$, for which reason we can
still integrate by parts and reduce to estimates for

$\int_{0}^{t}\int_{\mathbb{T}^{2}}\tilde{P}_{N}v(s)\tilde{P}_{N}v(s)\partial_{x}P_{K}udxdyds,\quad
K\ll N.$

$\tilde{P}_{N}$ denotes frequency projection to frequencies comparable to $N$
with a slightly larger projection.

Here we have to take into account time localization and regularity of the
functions, which leads to the same conditions like above. This case gives no
additional constraint on the summability properties. Our starting point based
on Proposition 8.1 is to assume

$s\geq s_{4}(\alpha,\varepsilon)=-\frac{\alpha}{2}+\varepsilon$

and $0<\alpha<\frac{1}{10}$.

We turn to the second case, in which we need to establish estimates

$\begin{split}&\quad\big{|}N_{1}\int_{0}^{t}\int_{\mathbb{T}^{2}}P_{N_{1}}vP_{N_{2}}vP_{N_{3}}u\,dxdyds\big{|}\\\
&\lesssim
C(N_{1},N_{2},N_{3})\|P_{N_{1}}v\|_{F_{N_{1}}}\|P_{N_{2}}v\|_{F_{N_{2}}}\|P_{N_{3}}u\|_{F_{N_{3}}},\quad
N_{1}\sim N_{3}\gtrsim N_{2}.\end{split}$

This had also been essentially carried out in the previous section. What is
left is to check the different summation properties now that the derivative
acts on the high frequency.

To obtain estimates in short-time function spaces, we decompose $[0,t]$ into
intervals of length $N_{1}^{-\alpha}$. So, we reduce to estimates established
above

$\begin{split}&\quad\big{|}\int_{\mathbb{R}\times\mathbb{T}^{2}}\gamma^{3}(N_{1}^{\alpha}(t-t_{I}))P_{N_{1}}vP_{N_{2}}vP_{N_{3}}udxdydt\big{|}\\\
&\lesssim
C(N_{1},N_{2})\|P_{N_{1}}v\|_{F_{N_{1}}}\|P_{N_{2}}v\|_{F_{N_{2}}}\|P_{N_{3}}u\|_{F_{N_{3}}}.\end{split}$

The time localization incurs a factor of $N_{1}^{\alpha}$. We introduce
notations

$\begin{split}f_{1,N_{1}}&=\mathcal{F}_{t,x,y}[\gamma(N_{1}^{\alpha}(t-t_{I}))P_{N_{1}}v],\quad
f_{2,N_{2}}=\mathcal{F}_{t,x,y}[\gamma(N_{1}^{\alpha}(t-t_{I}))P_{N_{2}}v],\\\
f_{3,N_{3}}&=\mathcal{F}_{t,x,y}[\gamma(N_{1}^{\alpha}(t-t_{I}))P_{N_{3}}u],\end{split}$

and localize in modulation like in (69).

With the estimates proved in (70),

$\begin{split}&\quad\big{|}\int(f_{1,N_{1},L_{1}}*f_{2,N_{2},L_{2}})f_{3,N_{3},L_{3}}d\xi
d\eta d\tau\big{|}\\\ &\lesssim
C(N_{1},N_{2},N_{3})\prod_{i=1}^{3}L_{i}^{\frac{1}{2}}(1+L_{i}/N_{i}^{3})^{\frac{1}{4}}\|f_{i,N_{i},L_{i}}\|_{L^{2}},\end{split}$

we check the different summability properties. Clearly, nothing changes for
the High$-$High$-$High-interaction.

$\bullet$ High$\times$Low$\rightarrow$High-interaction: $N_{2}\ll N_{1}\sim
N_{3}$. First, we suppose that $N_{2}\lesssim N_{1}^{1-a_{1}}$. Applying
Proposition 7.4 yields

$\begin{split}\int(f_{1,N_{1},L_{1}}*f_{2,N_{2},L_{2}})f_{3,N_{3},L_{3}}d\xi
d\eta d\tau&\lesssim
N_{1}^{-1}(N_{2}^{-\frac{1}{4}}N_{1}^{-\frac{\alpha}{4}}+N_{1}^{-\frac{\alpha}{2}})\log(N_{1})\\\
&\quad\prod_{i=1}^{3}L_{i}^{\frac{1}{2}}(1+L_{i}/N_{i}^{3})^{\frac{1}{4}}\|f_{i,N_{i},L_{i}}\|_{2}.\end{split}$

Taking into account the regularity and time localization we find the condition

$N_{1}^{1+\alpha}(N_{1}^{2}N_{2})^{-\frac{1}{2}}\langle
N_{2}^{\frac{1}{2}}N_{1}^{-\frac{\alpha}{2}}\rangle N_{1}^{40s}\lesssim
N_{1}^{20s}N_{2}^{20s}N_{1}^{s}\Rightarrow
N_{1}^{\frac{\alpha}{2}}N_{1}^{19s}\lesssim N_{2}^{20s}.$

For $a_{1}=\frac{1}{8}$ we obtain

$N_{1}^{\frac{\alpha}{2}+\frac{3}{2}s}\lesssim 1.$

This gives the condition

(82) $s\leq-\frac{\alpha}{3}.$

This leads us to the choice of $s^{\prime}=20s$.

Next, we suppose that $N_{2}\gtrsim N_{1}^{1-a_{1}}$, $a_{1}=1/8$. First
suppose that we are in the resonant case $L_{\max}\sim N_{1}^{2}N_{2}$. In
this case we apply Proposition 7.2 to find

$\begin{split}\big{|}\int(f_{1,N_{1},L_{1}}*f_{2,N_{2},L_{2}})f_{3,N_{3},L_{3}}d\xi
d\eta d\tau\big{|}&\lesssim
N_{2}^{-1}N_{1}^{-\frac{1}{4}+\frac{11}{192}}\prod_{i=1}^{3}L_{i}^{\frac{1}{2}}(1+L_{i}/N_{i}^{3})^{\frac{1}{4}}\|f_{i,N_{i},L_{i}}\|_{L^{2}}\\\
&\lesssim
N_{1}^{-1-\frac{13}{192}}\prod_{i=1}^{3}L_{i}^{\frac{1}{2}}(1+L_{i}/N_{i}^{3})^{\frac{1}{4}}\|f_{i,N_{i},L_{i}}\|_{L^{2}}.\end{split}$

This gives the condition

$N_{1}^{1+\alpha}N_{1}^{40s}N_{1}^{-1-\frac{13}{192}}\lesssim
N_{2}^{20s}N_{1}^{20s}N_{1}^{s}\Leftarrow
N_{1}^{\alpha-\frac{13}{192}}\lesssim N_{1}^{s}.$

This gives the condition on the regularity:

(83) $s_{4}(\alpha,\varepsilon)\geq\alpha-\frac{13}{192}+\varepsilon.$

$\cdot$ Non-resonant case: $L_{\max}\sim M^{2}N_{2}$ for some $M\gg N_{1}$. We
apply Proposition 7.2 in the non-resonant case

$\big{|}\int(f_{1,N_{1},L_{1}}*f_{2,N_{2},L_{2}})f_{3,N_{3},L_{3}}d\xi d\eta
d\tau\big{|}\lesssim\frac{M^{\frac{1}{6}}(N_{1}N_{2})^{\frac{1}{8}+\varepsilon}}{N_{2}(N_{1}N_{2})^{\frac{1}{12}}}\frac{1}{N_{2}^{\frac{1}{2}}}\prod_{i=1}^{3}L_{i}^{\frac{1}{2}}\|f_{i,N_{i},L_{i}}\|_{L^{2}}.$

We choose $M\leq N_{1}^{1+a_{4}}$ and let $a_{4}=\frac{1}{8}$. Then we compute
for the above constant remembering $N_{2}\gtrsim N_{1}^{1-a_{3}}$,
$a_{3}=\frac{1}{8}$ taking into account derivative loss and time localization

$M^{\frac{1}{6}}(N_{1}N_{2})^{\frac{1}{24}}\frac{1}{N_{2}^{\frac{3}{2}}}N_{1}^{1+\alpha}\leq
N_{1}^{\frac{9}{48}}N_{1}^{\frac{1}{12}}N_{1}^{-\frac{21}{16}}N_{1}^{1+\alpha}=N_{1}^{\frac{13}{48}+\alpha}N_{1}^{-\frac{15}{48}}=N_{1}^{\alpha-\frac{1}{24}}.$

Taking into account the regularity we find the condition

$N_{1}^{40s}N_{1}^{\alpha-\frac{1}{24}}\lesssim
N_{1}^{20s}N_{2}^{20s}N_{1}^{s},$

which leads to the condition on the regularity

(84) $s_{4}(\alpha,\varepsilon)\geq\alpha-\frac{1}{24}+\varepsilon.$

$\cdot$ $M\geq N_{1}^{1+a_{4}}$ can be estimated by the second bound provided
by Proposition 7.2 in the non-resonant case:

$\big{|}\int(f_{1,N_{1},L_{1}}*f_{2,N_{2},L_{2}})f_{3,N_{3},L_{3}}d\xi d\eta
d\tau\big{|}\lesssim(M^{2}N_{2})^{-\frac{1}{2}}\langle
N_{2}^{\frac{1}{2}}N_{1}^{-\frac{\alpha}{2}}\rangle\prod_{i=1}^{3}L_{i}^{\frac{1}{2}}\|f_{i,N_{i},L_{i}}\|_{2}.$

For $a_{4}=\frac{1}{8}$ we find taking into account derivative loss and time
localization $N_{1}^{1+\alpha}$ the summability condition:

$N_{1}^{40s}N_{1}^{-\frac{1}{8}+\frac{\alpha}{2}}\lesssim
N_{1}^{20s}N_{2}^{20s}N_{1}^{s}.$

This yields the regularity condition

(85) $s_{4}(\alpha,\varepsilon)\geq-\frac{1}{8}+\frac{\alpha}{2}+\varepsilon.$

$\bullet$ High$\times$High$\rightarrow$Low-interaction: In this case we need
to establish estimates

$\big{|}\int_{0}^{t}\int_{\mathbb{T}^{2}}P_{N_{2}}vP_{N_{1}}uP_{N_{3}}vdxdydt\big{|}\lesssim
C(N_{1},N_{2},N_{3})\|P_{N_{2}}v\|_{F_{N_{2}}}\|P_{N_{1}}u\|_{F_{N_{1}}}\|P_{N_{3}}v\|_{F_{N_{3}}}$

for $N_{2}\ll N_{1}\sim N_{3}$, which corresponds to the third case in the
paraproduct decomposition (81). The summability condition taking into account
regularity of the functions, time localization and derivative loss is given by

(86) $N_{2}^{40s}N_{2}N_{1}^{\alpha}C(N_{1},N_{2},N_{3})\lesssim
N_{1}^{40s}N_{2}^{s}.$

In the High$\times$Low$\rightarrow$High-case we have checked the summability
condition

$N_{1}^{40s}N_{1}^{1+\alpha}C(N_{1},N_{2},N_{3})\lesssim
N_{1}^{21s}N_{2}^{20s}.$

We use this to validate (86) writing

$\frac{N_{2}^{40s}N_{2}}{N_{1}^{40s}N_{1}}N_{1}^{40s}N_{1}^{1+\alpha}C(N_{1},N_{2},N_{3})\lesssim\frac{N_{2}^{40s}N_{2}}{N_{1}^{40s}N_{1}}N_{1}^{21s}N_{2}^{20s}.$

To conclude, we need to check that

$\frac{N_{2}^{40s}N_{2}}{N_{1}^{40s}N_{1}}N_{1}^{21s}N_{2}^{20s}\lesssim
N_{1}^{40s}N_{2}^{s}.$

This will be the case if

$N_{2}^{1+59s}\lesssim N_{1}^{1+59s}$

and gives the requirement

(87) $s_{4}(\alpha,\varepsilon)\geq-\frac{1}{59}+\varepsilon.$

We collect the conditions on the regularities obtained above:

  * •

High$\times$High$\rightarrow$High-interaction: The analysis from Proposition
8.1 transpires and for $0<\alpha<\frac{1}{10}$ we have the only requirement:

$s_{4}(\alpha,\varepsilon)\geq-\frac{\alpha}{2}+\varepsilon.$

  * •

High$\times$Low$\rightarrow$High-interaction, $N_{2}\lesssim N_{1}^{1-a_{1}}$
(82):

$s\leq-\frac{\alpha}{3}.$

  * •

High$\times$Low$\rightarrow$High-interaction, $N_{2}\gtrsim
N_{1}^{\frac{7}{8}}$, Resonant case (83):

$s_{4}(\alpha,\varepsilon)\geq\alpha-\frac{13}{192}+\varepsilon.$

  * •

High$\times$Low$\rightarrow$High-interaction, $N_{2}\gtrsim
N_{1}^{\frac{7}{8}}$, Non-resonant case (84), (85):

$s_{4}(\alpha,\varepsilon)\geq\alpha-\frac{1}{24}+\varepsilon\wedge
s_{4}(\alpha,\varepsilon)\geq-\frac{1}{8}+\frac{\alpha}{2}+\varepsilon.$

  * •

High$\times$High$\rightarrow$Low-interaction: To ensure summability in this
case with the estimates from the dual High$\times$Low$\rightarrow$High-
interaction at hand, we need to choose

$s_{4}(\alpha,\varepsilon)\geq-\frac{1}{59}+\varepsilon.$

To fit these conditions on the regularities under the umbrella

$s_{4}(\alpha,\varepsilon)\geq-\frac{\alpha}{2}+\varepsilon\wedge
s\leq-\frac{\alpha}{3},$

we impose a lower bound on the time localization. The conditions obtained in
the non-resonant case with high modulation on the high frequency lead us to
the upper bound $0<\alpha<\frac{1}{36}$. We see that choosing
$s=-\frac{5\alpha}{12}$ satisfies all conditions on the regularity for this
range of $\alpha$. The proof is complete. ∎

##  9\. Improved local well-posedness

In this section we conclude the local well-posedness below $L^{2}$ based on
the short-time analysis from previous sections. Starting point of our analysis
is Bourgain’s global $L^{2}$-well-posedness result for real-valued initial
data with persistence of regularity:

######  Theorem 9.1 (Global well-posedness of KP-II, [4, Theorem 1.4]).

Let $u_{0}\in L^{2}(\mathbb{T}^{2})$ be a real-valued function with
$\int_{\mathbb{T}}u_{0}(x,y)dx=0$. Then (1) is globally well-posed.

We denote the data-to-solution mapping by
$S_{*}^{0}:L^{2}(\mathbb{T}^{2})\rightarrow
C(\mathbb{R};L^{2}(\mathbb{T}^{2}))$, which assigns real-valued initial data
$u_{0}$ with $\int_{\mathbb{T}}u_{0}(x,y)dx=0$ to global solutions $u$ to (1).

We show the following improved local well-posedness result, which has been
summarized in the Introduction:

######  Theorem 9.2 (Improved local well-posedness for KP-II).

Let $0>s>-\frac{1}{90}$, and $R>0$. Then $S^{0}_{*}$ admits a unique
continuous extension

$S_{T}^{s}:H^{s,0}(\mathbb{T}^{2})\to C([0,T],H^{s,0}(\mathbb{T}^{2}))$

with $T=T(s,R)$ provided that $\|u_{0}\|_{H^{s,0}(\mathbb{T}^{2})}\leq R$.

The proof is carried out in three steps:

  * (i)

We establish a priori estimates for solutions to (1) at negative Sobolev
regularities on times $T=T(s,\|u_{0}\|_{H^{s,0}(\mathbb{T}^{2})})$.

  * (ii)

We prove a priori estimates for differences of solutions at lower regularities
$s^{\prime}=20s$ on times
$T=T(s,\|u_{01}\|_{H^{s,0}(\mathbb{T}^{2})},\|u_{02}\|_{H^{s,0}(\mathbb{T}^{2})})$.

  * (iii)

The proof of continuous dependence in the $H^{s,0}(\mathbb{T}^{2})$-topology
is concluded using frequency envelopes.

We begin with a priori estimates:

######  Theorem 9.3 (A priori estimates at Sobolev regularity of negative
order).

Let $s>-\frac{1}{20}$ and $u_{0}\in L^{2}(\mathbb{T}^{2})$ a real-valued
function with vanishing mean $\int_{\mathbb{T}}u_{0}(x,y)dx=0$. Then there is
$T=T(s,\|u_{0}\|_{H^{s,0}})$ such that we have the following a priori
estimates:

(88) $\|u\|_{F^{\bar{s}}(T)}\lesssim\|u_{0}\|_{H^{\bar{s},0}(\mathbb{T}^{2})}$

for any $\bar{s}\in[s,0)$. $T=T(s,\|u_{0}\|_{H^{s,0}})$ can be chosen to be
decreasing in $\|u_{0}\|_{H^{s,0}}$.

###### Proof.

We choose $0<\alpha<\frac{1}{10}$ and $\varepsilon>0$ such that
$s=-\frac{\alpha}{2}+\varepsilon$. Note that by Theorem 9.1 we have global
solutions emanating from $u_{0}\in L^{2}(\mathbb{T}^{2})$.

We invoke Lemma 2.2, Proposition 6.1, and Proposition 8.1 to find $\delta>0$
such that the following set of estimates holds with frequency-dependent time
localization given by $T=T(N)=N^{-\alpha}$:

$\left\\{\begin{array}[]{cl}\|u\|_{F^{s}(T)}&\lesssim\|u\|_{E^{s}(T)}+\|\partial_{x}(u^{2})\|_{\mathcal{N}^{s}(T)},\\\
\|\partial_{x}(u^{2})\|_{\mathcal{N}^{s}(T)}&\lesssim
T^{\delta}\|u\|^{2}_{F^{s}(T)},\\\
\|u\|^{2}_{E^{s}(T)}&\lesssim\|u_{0}\|_{H^{s,0}(\mathbb{T}^{2})}^{2}+T^{\delta}\|u\|^{3}_{F^{s}(T)}.\end{array}\right.$

Taking the estimates together gives

(89)
$\|u\|_{F^{s}(T)}\lesssim\|u_{0}\|_{H^{s,0}(\mathbb{T}^{2})}+T^{\delta}\|u\|_{F^{s}(T)}^{2}+T^{\delta}\|u\|_{F^{s}(T)}^{3/2}.$

By the limiting properties (see [17]) of the short-time function spaces it
holds

$\limsup_{T\downarrow
0}\|u\|_{E^{s}(T)}\lesssim\|u_{0}\|_{H^{s,0}(\mathbb{T}^{2})}\text{ and
}\lim_{T\downarrow 0}\|\partial_{x}(u^{2})\|_{N^{s}(T)}=0,$

therefore we find

$\limsup_{T\downarrow
0}\|u\|_{F^{s}(T)}\lesssim\|u_{0}\|_{H^{s,0}(\mathbb{T}^{2})}.$

With this limiting property at hand, it is standard to argue by a continuity
argument that (89) implies (88) for $\bar{s}=s$. To obtain the persistence of
regularity, we consider the same time localization and invoke Lemma 2.2,
Proposition 6.1, and Proposition 8.1 again to find the set of estimates:

$\left\\{\begin{array}[]{cl}\|u\|_{F^{\bar{s}}(T)}&\lesssim\|u\|_{E^{\bar{s}}(T)}+\|\partial_{x}(u^{2})\|_{\mathcal{N}^{\bar{s}}(T)},\\\
\|\partial_{x}(u^{2})\|_{\mathcal{N}^{\bar{s}}(T)}&\lesssim
T^{\delta}\|u\|_{F^{s}(T)}\|u\|_{F^{\bar{s}}(T)},\\\
\|u\|^{2}_{E^{\bar{s}}(T)}&\lesssim\|u_{0}\|_{H^{\bar{s},0}(\mathbb{T}^{2})}^{2}+T^{\delta}\|u\|^{2}_{F^{\bar{s}}(T)}\|u\|_{F^{s}(T)}.\end{array}\right.$

This can be subsumed as

$\|u\|_{F^{\bar{s}}(T)}\lesssim\|u_{0}\|_{H^{\bar{s},0}(\mathbb{T}^{2})}+T^{\delta}\|u\|_{F^{\bar{s}}(T)}(\|u\|_{F^{s}(T)}+\|u\|_{F^{s}(T)}^{\frac{1}{2}}).$

By the a priori estimates

$\|u\|_{F^{s}(T)}\lesssim\|u_{0}\|_{H^{s,0}(\mathbb{T}^{2})}$

already established, we find upon choosing
$T=T(s,\|u_{0}\|_{H^{s,0}(\mathbb{T}^{2})})$ smaller if necessary

$\|u\|_{F^{\bar{s}(T)}}\lesssim\|u_{0}\|_{H^{\bar{s},0}(\mathbb{T}^{2})}.$

The proof is complete. ∎

In the next step we show Lipschitz continuous dependence in a low regularity
$s^{\prime}=20s$ for some $s<0$, which depends only on the
$H^{s,0}(\mathbb{T}^{2})$-norm of the initial data.

######  Theorem 9.4 (Lipschitz continuous dependence in rough topology).

Let $s>-\frac{1}{90}$ and $u_{01},u_{02}\in L^{2}(\mathbb{T}^{2})$ real-valued
initial data with vanishing mean $\int_{\mathbb{T}}u_{0i}(x,y)dx=0$. Then
there is $T=T(s,\|u_{0}\|_{H^{s,0}})$ such that we have the following estimate
on the difference of solutions $v=u_{1}-u_{2}$ with $u_{i}$ denoting the
global solution emanating from $u_{0i}$:

(90) $\|v\|_{F^{20s}(T)}\lesssim\|v(0)\|_{H^{20s,0}(\mathbb{T}^{2})}.$

$T=T(s,\|u_{0}\|_{H^{s,0}})$ can be chosen to be decreasing in
$\|u_{0}\|_{H^{s,0}}$.

###### Proof.

Choose $0<\alpha<\frac{1}{45}$ and $\varepsilon>0$ such that there
$s\geq-\frac{\alpha}{2}+\varepsilon$ and $s\leq-\frac{5\alpha}{12}$.

We invoke Lemma 2.2, Proposition 6.1, and Proposition 8.3 to find the
following set of estimates for some $\delta>0$ with the time localization
$T=T(N)=N^{-\alpha}$:

$\left\\{\begin{array}[]{cl}\|v\|_{F^{20s}(T)}&\lesssim\|v\|_{E^{20s}(T)}+\|\partial_{x}(v(u_{1}+u_{2}))\|_{\mathcal{N}^{20s}(T)},\\\
\|\partial_{x}(v(u_{1}+u_{2}))\|_{\mathcal{N}^{20s}(T)}&\lesssim
T^{\delta}\|v\|_{F^{20s}(T)}(\|u_{1}\|_{F^{s}(T)}+\|u_{2}\|_{F^{s}(T)}),\\\
\|v\|^{2}_{E^{20s}(T)}&\lesssim\|v(0)\|_{H^{20s,0}}^{2}+T^{\delta}\|v\|^{2}_{F^{20s}(T)}(\|u_{1}\|_{F^{s}(T)}+\|u_{2}\|_{F^{s}(T)}).\end{array}\right.$

Taking the estimates together we find

$\begin{split}\|v\|_{F^{20s}(T)}&\lesssim\|v_{0}\|_{H^{20s,0}(\mathbb{T}^{2})}+T^{\delta}\|v\|_{F^{20s}(T)}(\|u_{1}\|_{F^{s}(T)}+\|u_{2}\|_{F^{s}(T)})\\\
&\quad+T^{\delta}\|v\|_{F^{20s}(T)}(\|u_{1}\|_{F^{s}(T)}+\|u_{2}\|_{F^{s}(T)})^{\frac{1}{2}}.\end{split}$

Choosing $T=T(s,\|u_{0i}\|_{H^{s,0}(\mathbb{T}^{2})})$ we can invoke Theorem
9.3 to obtain

$\|u_{i}\|_{F^{s}(T)}\lesssim\|u_{i0}\|_{H^{s,0}(\mathbb{T}^{2})}.$

Hence, the previous estimate reduces to

$\begin{split}\|v\|_{F^{20s}(T)}&\lesssim\|v(0)\|_{H^{20s,0}(\mathbb{T}^{2})}+T^{\delta}\|v\|_{F^{20s}(T)}(\|u_{10}\|_{H^{s,0}(\mathbb{T}^{2})}+\|u_{20}\|_{H^{s,0}(\mathbb{T}^{2})})\\\
&\quad+T^{\delta}\|v\|_{F^{20s}(T)}(\|u_{10}\|_{H^{s,0}(\mathbb{T}^{2})}+\|u_{20}\|_{H^{s,0}(\mathbb{T}^{2})})^{\frac{1}{2}}.\end{split}$

Choosing $T=T(\|u_{i0}\|_{H^{s,0}(\mathbb{T}^{2})})$ smaller if necessary, we
obtain the estimate (90). ∎

We turn to the final part in the argument: the conclusion of local well-
posedness via frequency envelopes. We follow the presentation of Ifrim–Tataru
[16] and shall be brief to avoid repetition.

We introduce the notation

$\big{[}\frac{N}{M}\big{]}=\max\big{(}\frac{N}{M},\frac{M}{N}\big{)}.$

######  Definition 9.5 (Frequency envelopes in $H^{s,0}(\mathbb{T}^{2})$).

Let $u_{0}\in H^{s,0}(\mathbb{T}^{2})$. A sequence $(c_{N})_{N\in
2^{\mathbb{N}}}$ is referred to as (sharp) frequency envelope for $u_{0}$, if
it satisfies, for some $\delta>0$

  * (a)

energy estimates $\|P_{N}u_{0}\|_{H^{s}}\leq c_{N}$,

  * (b)

the slowly varying property:
$\frac{c_{N}}{c_{M}}\lesssim\big{[}\frac{N}{M}\big{]}^{\delta}$,

  * (c)

$\|u\|^{2}_{H^{s,0}(\mathbb{T}^{2})}\approx\sum_{N\geq 1}c_{N}^{2}$.

Before we construct sharp frequency envelopes for given initial data, we
derive a frequency envelope bound for solutions to truncated initial data.
Suppose that $u_{0}^{H}=P_{\leq H}u_{0}$ and that $(c_{N})$ is a sharp
frequency envelope of $u_{0}^{H}$. Then, the following bounds hold:

  * (i)

Uniform bounds: $\|P_{N}u_{0}^{H}\|_{H^{s,0}(\mathbb{T}^{2})}\lesssim c_{N}$,

  * (ii)

High frequency bounds: $\|u_{0}^{H}\|_{H^{s+j,0}(\mathbb{T}^{2})}\lesssim
H^{j}c_{H}$,

  * (iii)

Difference bounds:
$\|u_{0}^{2H}-u_{0}^{H}\|_{H^{20s,0}(\mathbb{T}^{2})}\lesssim H^{19s}c_{H}$.

As a consequence of the a priori estimates from Theorem 9.3 and the estimates
for solutions to the difference equation Theorem 9.4, we have the following
estimates for the global solutions $u^{H}$ emanating from $u_{0}^{H}$ on the
time scale $T=T(s,\|u_{0}\|_{H^{s,0}(\mathbb{T}^{2})})$:

  * (i)

High frequency bounds:

(91) $\|u^{H}\|_{C_{T}H^{s+j,0}}\lesssim_{j}H^{j}c_{H},\quad j>0,$

  * (ii)

Difference bounds:

(92) $\|u^{2H}-u^{H}\|_{C_{T}H^{20s,0}}\lesssim H^{19s}c_{H}.$

We obtain as a consequence

(93) $\|P_{K}u^{H}\|_{C_{T}H^{s,0}}\lesssim_{N}c_{K}(K/H\vee 1)^{-N}.$

By using the difference bounds (92) we show first that $u^{H}$ is a Cauchy
sequence in $C([0,T],H^{20s,0}(\mathbb{T}^{2}))$. Indeed, for $H^{\prime}>H$,
we have

$\|u^{H^{\prime}}-u^{H}\|_{C_{T}H^{20s,0}}\lesssim\sum_{H\leq
M<H^{\prime}}\|u^{2M}-u^{M}\|_{C_{T}H^{20s,0}}\lesssim
H^{19s}\big{(}\sum_{M\geq H}c_{M}^{2}\big{)}^{\frac{1}{2}},$

which implies convergence to a limiting solution $u\in
C([0,T],H^{20s,0}(\mathbb{T}^{2}))$ satisfying

$\|u-u^{H}\|_{C_{T}H^{20s,0}}\lesssim H^{19s}\big{(}\sum_{M\geq
H}c_{M}^{2}\big{)}^{\frac{1}{2}}.$

Next, we upgrade this convergence to $C([0,T],H^{s,0}(\mathbb{T}^{2}))$:

$\displaystyle\|u-u^{H}\|_{C_{T}H^{s,0}}\leq\|P_{\leq
H}(u-u^{H})\|_{C_{T}H^{s,0}}+\|P_{>H}(u-u^{H})\|_{C_{T}H^{s,0}}$

Concerning the first term, we simply use the above bound to obtain

$\|P_{\leq H}(u-u^{H})\|_{C_{T}H^{s,0}}\lesssim
H^{-19s}\|u-u^{H}\|_{C_{T}H^{20s,0}}\lesssim\big{(}\sum_{M\geq
H}c_{M}^{2}\big{)}^{\frac{1}{2}}.$

We decompose the second term

$\displaystyle\|P_{>H}(u-u^{H})\|_{C_{T}H^{s,0}}\lesssim\Big{(}\sum_{K>H}\|P_{K}(u-u^{H})\|_{C_{T}H^{s,0}}^{2}\Big{)}^{\frac{1}{2}}.$

Then, $\|P_{K}(u-u^{H})\|_{C_{T}H^{s,0}}\leq I_{\leq K}+I_{>K}$ for

$\displaystyle I_{\leq K}=$ $\displaystyle\sum_{M\leq
K}\|P_{K}(u^{2M}-u^{M})\|_{C_{T}H^{s,0}},$ $\displaystyle I_{>K}=$
$\displaystyle\sum_{M>K}\|P_{K}(u^{2M}-u^{M})\|_{C_{T}H^{s,0}}$

From (93) (with $N=1$) and the slowly varying property we obtain

$I_{\leq K}\lesssim\sum_{M\leq
K}\|P_{K}(u^{M})\|_{C_{T}H^{s,0}}\lesssim\sum_{M\leq
K}\frac{M}{K}c_{M}\lesssim c_{K}.$

From (92) and the slowly varying property we infer that

$\displaystyle I_{>K}\lesssim$
$\displaystyle\sum_{M>K}K^{-19s}\|P_{K}(u^{2M}-u^{M})\|_{C_{T}H^{20s,0}}$
$\displaystyle\lesssim$
$\displaystyle\sum_{M>K}\Big{(}\frac{K}{M}\Big{)}^{-19s}c_{M}\lesssim c_{K}.$

In summary, we obtain

(94) $\|u-u^{H}\|_{C_{T}H^{s,0}}\lesssim\big{(}\sum_{M\geq
H}c_{M}^{2}\big{)}^{\frac{1}{2}},$

which tends to $0$ as $H\to\infty$.

Consequently, we obtain an approximation of $u$ with global solutions $u^{H}$.
This will be the key ingredient in the proof of continuous dependence, for
which we consider next a sequence $u_{j,0}\to u_{0}$ in
$H^{s,0}(\mathbb{T}^{2})$. The emanating solutions are denoted by $u_{j}$,
$u$, respectively. The frequency truncations are denoted by $u_{j,0}^{H}$,
$u_{0}^{H}$, respectively, and the emanating solutions of the frequency
truncations by $u^{H}_{j}$ and $u^{H}$.

We sketch the construction of a sharp frequency envelope $(c_{N})$ for
$u_{0}$. As initial guess we consider
$d_{N}=\|P_{N}u_{0}\|_{H^{s,0}(\mathbb{T}^{2})}$. A frequency envelope is
obtained from mollifying: $c_{N}=\sum_{M\in
2^{\mathbb{N}_{0}}}d_{M}\big{[}\frac{M}{N}\big{]}^{-\delta}$, for some
$0<\delta<19|s|$. By Young’s inequality it holds

$\|(c_{N})\|_{\ell^{2}_{N}}\lesssim\|(d_{N})\|_{\ell^{2}}\sum_{M\in
2^{\mathbb{N}_{0}}}M^{-\delta}\lesssim\|u\|_{H^{s,0}},$

therefore property c) is satisfied. To check the slowly varying property, we
note that

$c_{N}=\sum_{J\in
2^{\mathbb{N}_{0}}}d_{J}\Big{[}\frac{J}{N}\Big{]}^{-\delta}\leq\sum_{J\in
2^{\mathbb{N}_{0}}}d_{J}\Big{[}\frac{J}{M}\Big{]}^{-\delta}\Big{[}\frac{M}{N}\Big{]}^{\delta}\lesssim
e_{M}\big{[}\frac{M}{N}]^{\delta}.$

Similarly, we construct a sharp frequency envelope $(c^{(j)}_{N})$ for
$u_{0,j}$ such that we have $c^{(j)}\to c$ in $\ell^{2}$.

We use the triangle inequality and (94) to write

$\begin{split}\|u_{j}-u\|_{C_{T}H^{s,0}}&\leq\|u_{j}-u_{j}^{H}\|_{C_{T}H^{s,0}}+\|u_{j}^{H}-u^{H}\|_{C_{T}H^{s,0}}\\\
&\quad+\|u^{H}-u\|_{C_{T}H^{s,0}}\\\ &\lesssim\|c^{(j)}_{\geq
H}\|_{\ell^{2}}+\|u_{j}^{H}-u^{H}\|_{C_{T}H^{s,0}}+\|c_{\geq
H}\|_{\ell^{2}}.\end{split}$

By Theorem 9.1, we have as $j\to\infty$

$\|u_{j}^{H}-u^{H}\|_{C_{T}H^{s,0}}\leq\|u_{j}^{H}-u^{H}\|_{C_{T}L^{2}}\lesssim\|u_{j,0}^{H}-u_{0}^{H}\|_{L^{2}}\to
0.$

Consequently,

$\limsup_{j\to\infty}\|u_{j}-u\|_{C_{T}H^{s,0}}\lesssim\|c_{\geq
H}\|_{\ell^{2}}.$

Finally, letting $H\to\infty$, we infer

$\lim_{j\to\infty}\|u_{j}-u\|_{C_{T}H^{s,0}}=0$

and the proof is complete. $\hfill\Box$

##  10\. Sharpness of Bourgain’s bilinear Strichartz estimate

In this section, we prove that Bourgain’s estimate provided in Proposition 4.1
is sharp in general. To this end, we will construct $f_{i,N_{i},L_{i}}$ such
that

$\|f_{1,N_{1},L_{1}}*f_{2,N_{2},L_{2}}\|_{L^{2}_{\tau,\xi,\eta}}\sim
N_{2}^{\frac{1}{4}}(L_{1}\wedge L_{2})^{\frac{1}{2}}(L_{1}\vee
L_{2})^{\frac{1}{4}}\prod_{i=1}^{2}\|f_{i,N_{i},L_{i}}\|_{L^{2}_{\tau,\xi,\eta}}.$

We use the same example as in [11] (which in turn goes back an unpublished
manuscript of the third author). Let $f_{i,N_{i},L_{i}}$ be a characteristic
function, supported on a single $\xi$-node with $\xi_{i}=N_{i}$, $1\leq
N_{2}\leq N_{1}$, and the $\eta$-support be intervals of length
$N_{2}^{\frac{1}{2}}$. Then we can arrange that for
$(\tau_{i},\xi_{i},\eta_{i})\in\text{supp}(f_{i,N_{i},L_{i}})$:

$\big{|}\frac{\eta_{1}}{\xi_{1}}-\frac{\eta_{2}}{\xi_{2}}\big{|}\lesssim
N_{2}^{-\frac{1}{2}}.$

Moreover, we let $\tau_{i}$ range in intervals of length $\sim 1$, such that

$\big{|}\tau_{i}-(\xi_{i}^{3}-\frac{\eta_{i}^{2}}{\xi_{i}})\big{|}\lesssim 1.$

Clearly,

$\|f_{i,N_{i},L_{i}}\|_{L^{2}}\sim N_{2}^{\frac{1}{4}}.$

And moreover, on a $\tau$-interval of length $1$, for $\xi=\xi_{1}+\xi_{2}$
and for $\eta$ in an interval of length $\sim N_{2}^{\frac{1}{2}}$ we have

$f_{1,N_{1},L_{1}}*f_{2,N_{2},L_{2}}(\tau,\xi,\eta)\sim N_{2}^{\frac{1}{2}},$

and the support contains an $\eta$-interval of length $\sim
N_{2}^{\frac{1}{2}}$. Consequently,

$\|f_{1,N_{1},L_{1}}*f_{2,N_{2},L_{2}}\|_{L^{2}}\gtrsim
N_{2}^{\frac{1}{2}}N_{2}^{\frac{1}{4}}.$

We obtain

$\|f_{1,N_{1},L_{1}}*f_{2,N_{2},L_{2}}\|_{L^{2}}\gtrsim
N_{2}^{\frac{1}{4}}(L_{1}\wedge L_{2})^{\frac{1}{2}}(L_{1}\vee
L_{2})^{\frac{1}{4}}\prod_{i=1}^{2}\|f_{i,N_{i},L_{i}}\|_{L^{2}}.$

Based on this example, the same computation as in [11] for the cylinder
$\mathbb{T}\times\mathbb{R}$ can be done in our setting, which shows
$C^{3}$-ill-posedness of the Cauchy problem in $H^{s,0}(\mathbb{T}^{2})$ for
$s<-\frac{1}{4}$.

## Acknowledgements

  1. (1)

Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research
Foundation) – Project-ID 317210226 – SFB 1283.

  2. (2)

S.H. thanks the _Laboratoire de Mathématique d’Orsay – Université Paris-
Saclay_ for its hospitality and funding via _CNRS Poste Rouge_ is gratefully
acknowledged.

  3. (3)

R.S. gratefully acknowledges financial support from the Humboldt foundation
(Feodor-Lynen fellowship) and partial support by the NSF grant DMS-2054975.

## References

  * [1] Mark J. Ablowitz and Harvey Segur.  On the evolution of packets of water waves.  J. Fluid Mech., 92(4):691–715, 1979.
  * [2] J. Bourgain.  Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I. Schrödinger equations.  Geom. Funct. Anal., 3(2):107–156, 1993.
  * [3] J. Bourgain.  Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. II. The KdV-equation.  Geom. Funct. Anal., 3(3):209–262, 1993.
  * [4] J. Bourgain.  On the Cauchy problem for the Kadomtsev-Petviashvili equation.  Geom. Funct. Anal., 3(4):315–341, 1993.
  * [5] N. Burq, P. Gérard, and N. Tzvetkov.  Strichartz inequalities and the nonlinear Schrödinger equation on compact manifolds.  Amer. J. Math., 126(3):569–605, 2004.
  * [6] A. Córdoba.  A note on Bochner-Riesz operators.  Duke Math. J., 46(3):505–511, 1979.
  * [7] Antonio Córdoba.  Geometric Fourier analysis.  Ann. Inst. Fourier (Grenoble), 32(3):vii, 215–226, 1982.
  * [8] Ciprian Demeter.  Fourier restriction, decoupling, and applications, volume 184 of Cambridge Studies in Advanced Mathematics.  Cambridge University Press, Cambridge, 2020.
  * [9] V. S. Dryuma.  Analytic solution of the two-dimensional Korteweg-de Vries (KdV) equation.  Soviet Journal of Experimental and Theoretical Physics Letters, 19:387, June 1974.
  * [10] Charles Fefferman.  A note on spherical summation multipliers.  Israel J. Math., 15:44–52, 1973.
  * [11] Axel Grünrock, Mahendra Panthee, and Jorge Drumond Silva.  On KP-II type equations on cylinders.  Ann. Inst. H. Poincaré C Anal. Non Linéaire, 26(6):2335–2358, 2009.
  * [12] Zihua Guo and Tadahiro Oh.  Non-existence of solutions for the periodic cubic NLS below $L^{2}$.  Int. Math. Res. Not. IMRN, (6):1656–1729, 2018.
  * [13] Larry Guth, Dominique Maldague, and Changkeun Oh.  $l^{2}$ decoupling theorem for surfaces in $\mathbb{R}^{3}$.  arXiv e-prints, page arXiv:2403.18431, March 2024.
  * [14] Martin Hadac, Sebastian Herr, and Herbert Koch.  Well-posedness and scattering for the KP-II equation in a critical space.  Ann. Inst. H. Poincaré C Anal. Non Linéaire, 26(3):917–941, 2009.
  * [15] Benjamin Harrop-Griffiths, Rowan Killip, and Monica Vişan.  Sharp well-posedness for the cubic NLS and mKdV in $H^{s}(\mathbb{R})$.  Forum Math. Pi, 12:Paper No. e6, 2024.
  * [16] Mihaela Ifrim and Daniel Tataru.  Local well-posedness for quasi-linear problems: a primer.  Bull. Amer. Math. Soc. (N.S.), 60(2):167–194, 2023.
  * [17] A. D. Ionescu, C. E. Kenig, and D. Tataru.  Global well-posedness of the KP-I initial-value problem in the energy space.  Invent. Math., 173(2):265–304, 2008.
  * [18] Boris Borisovich Kadomtsev and Vladimir I. Petviashvili.  On the stability of solitary waves in weakly dispersing media.  1970\.
  * [19] Rowan Killip and Monica Vişan.  KdV is well-posed in $H^{-1}$.  Ann. of Math. (2), 190(1):249–305, 2019.
  * [20] Rowan Killip, Monica Vişan, and Xiaoyi Zhang.  Low regularity conservation laws for integrable PDE.  Geom. Funct. Anal., 28(4):1062–1090, 2018.
  * [21] H. Koch and N. Tzvetkov.  On the local well-posedness of the Benjamin-Ono equation in $H^{s}({\mathbb{R}})$.  Int. Math. Res. Not., (26):1449–1464, 2003.
  * [22] Herbert Koch and Daniel Tataru.  Conserved energies for the cubic nonlinear Schrödinger equation in one dimension.  Duke Math. J., 167(17):3207–3313, 2018.
  * [23] L. Molinet, J.-C. Saut, and N. Tzvetkov.  Well-posedness and ill-posedness results for the Kadomtsev-Petviashvili-I equation.  Duke Math. J., 115(2):353–384, 2002.
  * [24] Jean-Claude Saut.  Remarks on the generalized Kadomtsev-Petviashvili equations.  Indiana Univ. Math. J., 42(3):1011–1026, 1993.
  * [25] Robert Schippa.  Refinements of Strichartz estimates on tori and applications.  arXiv e-prints, page arXiv:2312.06333, December 2023.
  * [26] Gigliola Staffilani and Daniel Tataru.  Strichartz estimates for a Schrödinger operator with nonsmooth coefficients.  Comm. Partial Differential Equations, 27(7-8):1337–1372, 2002.
  * [27] Terence Tao.  Global regularity of wave maps. II. Small energy in two dimensions.  Comm. Math. Phys., 224(2):443–544, 2001.