# Refinements of Strichartz estimates on tori and applications

Robert Schippa  Korea Institute of Advanced Study, Hoegi-ro 85, Dongdaemun-gu,
02455 Seoul, Republic of Korea<EMAIL_ADDRESS>
###### Abstract.

We show trilinear Strichartz estimates in one and two dimensions on frequency-
dependent time intervals. These improve on the corresponding linear estimates
of periodic solutions to the Schrödinger equation. The proof combines
decoupling iterations with bilinear short-time Strichartz estimates. Secondly,
we use decoupling to show new linear Strichartz estimates on frequency
dependent time intervals. We apply these in case of the Airy propagator to
obtain the sharp Sobolev regularity for the existence of solutions to the
modified Korteweg-de Vries equation.

##  1\. Introduction

Discrete Fourier restriction deals with estimates of the kind

(1) $\|\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{d},\\\ |k|\leq
N\end{subarray}}a_{k}e^{i(x\cdot
k-tk^{2})}\|_{L_{t,x}^{p}([0,1]\times\mathbb{T}^{d})}\lesssim
K_{p}(N)\|a_{k}\|_{\ell_{k}^{2}}.$

After pioneering work by Zygmund [42], Bourgain [1] studied discrete Fourier
restriction to obtain Strichartz estimates for the periodic Schrödinger
propagator:

(2) $\|e^{it\Delta}P_{N}f\|_{L^{p}_{t,x}([0,1]\times\mathbb{T}^{d})}\lesssim
K_{p}(N)\|f\|_{L^{2}},$

where $P_{N}$ denotes frequency projection to frequencies of size $N$. A wide
range of periodic Strichartz estimates with constant $K_{p}(N)$ sharp up to
endpoints were proved in [1, 2]. More recently, Bourgain–Demeter [6] showed
sharp $\ell^{2}$-decoupling inequalities. The $\ell^{2}$-decoupling
inequalities imply the Strichartz estimates on $\mathbb{T}^{d}$ by continuous
approximation; see below.

Let $\mathcal{E}$ denote the Fourier extension operator of the truncated
paraboloid:

$\mathcal{E}f(x,t)=\int_{\\{\xi\in\mathbb{R}^{d},\;|\xi|\leq
1\\}}e^{i(x\cdot\xi-t|\xi|^{2})}f(\xi)d\xi,\quad(x,t)\in\mathbb{R}^{d}\times\mathbb{R}.$

The following estimates were proved in [6]:

(3)
$\|\mathcal{E}f\|_{L^{p}_{t,x}(B_{d+1}(0,R))}\lesssim_{\varepsilon}R^{\varepsilon}\big{(}\sum_{\theta:R^{-\frac{1}{2}}\text{-ball}}\|\mathcal{E}f_{\theta}\|^{2}_{L^{p}_{t,x}(w_{B_{d+1}(0,R)})}\big{)}^{\frac{1}{2}}$

with $R\gg 1$, $2\leq p\leq\frac{2(d+2)}{d}$. $w_{B(0,R)}$ denotes a weight
decaying off $B_{d+1}(0,R)$ polynomially (say with degree $100d$).
$\\{\theta\\}$ denotes an essentially disjoint cover of $B(0,1)$ with
$R^{-\frac{1}{2}}$-balls, and we denote

$\mathcal{E}f_{\theta}=\int_{\theta}e^{i(x\cdot\xi-t|\xi|^{2})}f(\xi)d\xi.$

To solve nonlinear equations, multilinear Strichartz estimates which improve
on the corresponding linear estimate in certain cases turned out to be
crucial.

Consider the multilinear estimates

(4)
$\|\prod_{i=1}^{n}P_{N_{i}}e^{it\Delta}f_{i}\|_{L^{2}_{t,x}([0,T]\times\mathbb{T}^{d})}\lesssim
C(T,N_{1},\ldots,N_{n})\prod_{i=1}^{n}\|f_{i}\|_{L^{2}(\mathbb{T}^{d})}$

for frequencies $N_{1}\gtrsim\ldots\gtrsim N_{n}$. It became clear in
Bourgain’s work [5] that further refinements of (4) for frequency-dependent
times $T=T(N_{1},\ldots,N_{n})$ can be used in the analysis of solutions to
mass-critical Schrödinger equations with infinite energy. Precisely, in [5]
the following qualitative trilinear Strichartz estimate was proved:

######  Theorem 1.1 ([5, Section 4]).

Fix large numbers $N_{j}\in 2^{\mathbb{N}}$ ($1\leq j\leq 3$) and $D$, which
satisfy $N_{1}\geq N_{2}\geq N_{3}$ and for some $1\geq\beta>0$

(5) $N_{1}>N_{3}^{1+\beta},$

and

(6) $D>N_{1}^{\beta}.$

Assume

(7) $\text{supp}(\hat{f}_{j})\subseteq[N_{j},2N_{j}]\qquad(1\leq j\leq 3).$

Then

(8)
$\int_{[0,1/D]\times\mathbb{T}}\prod_{j=1}^{3}|e^{it\Delta}f_{j}|^{2}dxdt\leq
N_{1}^{-\delta^{\prime}}\prod_{j=1}^{3}\|f_{j}\|_{2}^{2},$

where $\delta^{\prime}=\delta^{\prime}(\beta)>0$.

The argument in [5] is qualitative, but relies on induction on scales and
orthogonality considerations similar to decoupling. Also, there is a key step
which can be perceived as partial decoupling. Possibly the proof can be
regarded as a qualitative precursor to efficient congruencing in two
dimensions (cf. [41, 32, 33, 21]). In the following we quantify the derivative
gain in the trilinear estimate, which was left as an open question in [5]:

######  Theorem 1.2.

Let $0<\alpha\leq 1$. Under the assumptions (5) and (7) of Theorem 1.1, we
have

(9)
$\int_{[0,N_{1}^{-\alpha}]\times\mathbb{T}}\prod_{j=1}^{3}|e^{it\Delta}f_{j}|^{2}dxdt\leq
C\prod_{j=1}^{3}\|f_{j}\|_{2}^{2},$

where
$C=C_{\varepsilon,\beta}\log(N_{1})^{12+\varepsilon}N_{1}^{-\frac{\alpha\beta}{8}}$
for $\varepsilon>0$.

For $\alpha=1$, we can take advantage of short-time bilinear Strichartz
estimates, for which we refer to works by Moyua–Vega [35] in one dimension and
Hani [24] in higher dimensions. For $\alpha<1$ we shall lower the effective
size of the Fourier support by orthogonality considerations related to
decoupling. This makes an application of Bernstein’s inequality more
favorable.

The key new ingredient in the proof of Theorem 1.2 is a decoupling iteration
based on efficient congruencing, which Wooley employed for the first time in
[41] to prove Vinogradov’s Mean-Value Theorem in the cubic case. This was
translated to prove decoupling for the paraboloid with an improved decoupling
constant by Li [32, 33]. Since this decoupling iteration allows us to quantify
the decoupling constant fairly precisely, we can show the further logarithmic
refinement:

######  Theorem 1.3.

Let $N_{i}\in 2^{\mathbb{N}}$ be dyadic numbers with $N_{3}\leq
N_{1}\exp\big{(}\frac{-\log N_{1}}{\log\log N_{1}}\big{)}$ and $\alpha>0$.
Suppose that $\text{supp}(\hat{f}_{i})\subseteq[N_{i}/2,N_{i}]$. Then the
following estimate holds:

(10)
$\int_{[0,N_{1}^{-\alpha}]\times\mathbb{T}}|e^{it\Delta}f_{1}|^{2}|e^{it\Delta}f_{2}|^{2}|e^{it\Delta}f_{3}|^{2}dxdt\lesssim_{\alpha}\prod_{i=1}^{3}\|f_{i}\|_{2}^{2}.$

We remark that the currently best linear decoupling constant for the
paraboloid is given by $\log(R)^{c}$ for $R\gg 1$:

$\|\mathcal{E}f\|_{L^{6}_{t,x}(\mathbb{R}^{2})}\lesssim\log(R)^{c}\big{(}\sum_{\theta:R^{-\frac{1}{2}}-\text{ball}}\|\mathcal{E}f_{\theta}\|_{L^{6}_{t,x}(\mathbb{R}^{2})}^{2}\big{)}^{\frac{1}{2}}.$

This was proved by Guth–Maldague–Wang [23] via the ‘High-Low method’; see also
[20] for an improvement in the context of discrete Fourier restriction. This
approach might lead to further improvements of Theorem 1.2.

Next we show a corresponding result in two dimensions using the original
decoupling iteration due to Bourgain–Demeter [6] following the presentation by
Demeter in [15, Chapter 10]. The set-up is as follows:

Let $N_{i}\in 2^{\mathbb{N}}$, $i=1,2,3$, $N_{3}\ll N_{1}$ with
$N_{3}^{1+\beta}=N_{1}$ for some $0<\beta\leq 1$. Let
$f_{i}:\mathbb{T}_{\gamma}^{2}\to\mathbb{C}$ be periodic functions on
$\mathbb{T}_{\gamma}^{2}=\mathbb{R}/(2\pi\mathbb{Z})\times\mathbb{R}/(2\pi\gamma\mathbb{Z})$
with $\gamma\in(\frac{1}{2},1]$. We emphasize that the decoupling arguments do
not distinguish between the square and irrational tori, which means we can
allow for general $\mathbb{T}^{2}_{\gamma}$.

We suppose that for $i=1,2,3$

(11) $\text{supp}(\hat{f}_{i})\subseteq B(\xi_{i}^{*},N_{3})\text{ for some
}\xi_{i}^{*}\in B(0,2N_{1})$

and

(12) $|\xi_{1}^{*}-\xi_{2}^{*}|\gtrsim N_{1}.$

Consider the expression

(13)
$\int_{[0,N_{1}^{-\alpha}]\times\mathbb{T}_{\gamma}^{2}}|e^{it\Delta}f_{1}\,e^{it\Delta}f_{2}\,e^{it\Delta}f_{3}|^{4/3}dxdt.$

This can be regarded as a trilinear version of the Strichartz estimates in two
dimensions on frequency-dependent times at the critical exponent $p=4$.

We impose the following transversality condition: Let
$\xi_{i}\in\text{supp}(\hat{f}_{i})$. Then we require that there is some
$\nu\in(0,1]$ such that

(14) $\xi_{1},\;\xi_{2},\;\xi_{3}\text{ form a triangle of area greater than
}N_{1}^{2}\nu.$

The reason for imposing this condition becomes evident after rescaling: We
have for $\xi^{\prime}_{i}\in\text{supp}(\hat{f}_{i})/N_{1}$ that

$\xi^{\prime}_{1},\;\xi^{\prime}_{2},\;\xi_{3}^{\prime}\text{ form a triangle
of area greater than }\nu,$

which is equivalent to the condition that the normals

$\mathfrak{n}_{i}=\frac{(-2\xi^{\prime}_{i},1)}{|(-2\xi^{\prime}_{i},1)|}$

of the paraboloid $(\xi,-|\xi|^{2})$ with
$N\xi^{\prime}_{i}\in\text{supp}(\hat{f}_{i})$ are quantitatively transverse:

$|\mathfrak{n}_{1}\wedge\mathfrak{n}_{2}\wedge\mathfrak{n}_{3}|\gtrsim\nu.$

######  Theorem 1.4.

Let $\alpha\in(0,1]$. Under the above Fourier support and transversality
assumptions on $f_{i}$, $i=1,2,3$ with $\|f_{i}\|_{2}=1$, the following
estimate holds:

(15)
$\int_{[0,N_{1}^{-\alpha}]\times\mathbb{T}_{\gamma}^{2}}\big{|}e^{it\Delta}f_{1}e^{it\Delta}f_{2}e^{it\Delta}f_{3}\big{|}^{\frac{4}{3}}dxdt\lesssim_{\varepsilon}\nu^{-1+\frac{\beta}{2(1+\beta)}}N_{1}^{-\frac{\alpha\beta}{12}+\varepsilon}.$

######  Remark 1.5.

By applying Hölder’s inequality and the $L^{4}_{t,x}$-Strichartz estimate we
obtain

$\begin{split}\int_{[0,N^{-\alpha}]\times\mathbb{T}^{2}_{\gamma}}\prod_{i=1}^{3}\big{|}e^{it\Delta}f_{i}\big{|}^{\frac{4}{3}}dxdt&\lesssim\big{(}\prod_{i=1}^{3}\|e^{it\Delta}f_{i}\|_{L^{4}_{t,x}([0,N^{-\alpha}]\times\mathbb{T}^{2}_{\gamma})}\big{)}^{\frac{4}{3}}\\\
&\lesssim_{\varepsilon}N^{\varepsilon}\big{(}\prod_{i=1}^{3}\|f_{i}\|_{2}\big{)}^{\frac{4}{3}}.\end{split}$

Hence, if the transversality degenerates too much, then the multilinear
estimates from Theorem 1.4 cannot improve on the linear argument.

We hope that the estimates proved in Theorem 1.2 and Theorem 1.4 can be used
in the analysis of mass-critical nonlinear Schrödinger equations as pioneered
in [5]:

(16) $\left\\{\begin{array}[]{cl}i\partial_{t}u+\Delta
u&=\pm|u|^{\frac{4}{d}}u,\quad(t,x)\in\mathbb{R}\times\mathbb{T}^{d},\\\
u(0)&=u_{0}\in H^{s}(\mathbb{T}^{d}).\end{array}\right.$

In two dimensions, Herr–Kwak [25] recently reported how small mass global
well-posedness for any $s>0$ follows from a refined $L^{4}_{t,x}$-Strichartz
estimate. They show

(17)
$\|P_{S}e^{it\Delta}f\|_{L^{4}([0,(\log\\#S)^{-1}],L^{4}(\mathbb{T}^{2}))}\lesssim\|f\|_{L^{2}(\mathbb{T}^{2})}$

for any bounded set $S\subseteq\mathbb{Z}^{2}$ and that this implies global
well-posedness for $s>0$. Above $P_{S}$ denotes the Fourier projection onto
$S$.

This motivates to study the short-time Strichartz estimate

$\|P_{N}e^{it\Delta}f\|_{L^{p}_{t,x}([0,N^{-\alpha}]\times\mathbb{T}^{d})}\lesssim\|f\|_{L^{2}}$

for $\alpha\in(0,1]$ at the critical exponent $p=\frac{2(d+2)}{d}$. From this
point of view the estimates from Theorems 1.2 and 1.4 give a quantified
version of trilinear Strichartz estimates on frequency-dependent time
intervals. In Section 6 we translate the multilinear refinements to rescaled
tori $\lambda\mathbb{T}^{d}$ on the unit time interval. This is the form, in
which multilinear Strichartz estimates were applied in the context of the
$I$-method; see [12, 17, 14]. Notably, Fan _et al._ [17] proved bilinear
Strichartz estimates on rescaled tori via decoupling, and the question about
trilinear decoupling inequalities was raised. Although the multiscale
inequalities derived in the forthcoming sections do not provide us with
trilinear decoupling, the argument provides us with genuinely trilinear
Strichartz estimates.

The question for global well-posedness of (16) motivates to study Strichartz
estimates without derivative loss on frequency-dependent time intervals:

(18)
$\|P_{N}e^{it\Delta}f\|_{L_{t,x}^{p}([0,N^{-\alpha}]\times\mathbb{T}^{d})}\lesssim\|f\|_{L^{2}(\mathbb{T}^{d})}.$

at the critical exponent $p=\frac{2(d+2)}{d}$.

Recall that for $\alpha=1$ (18) is true on arbitrary compact Riemannian
manifolds (without boundary) due to Burq–Gérard–Tzvetkov [7] (see also
Staffilani–Tataru [39]). On the other hand, Bourgain [1] pointed out that (18)
fails for $\alpha=0$ in one dimension; see also [40] for higher dimensions.
(Since $\ell^{2}$-decoupling implies discrete Fourier restriction, this yields
that the constant $R^{\varepsilon}$ in (3) cannot be removed.) Kishimoto [31]
furthermore pointed out that (16) cannot be analytically well-posed in
$L^{2}(\mathbb{T}^{d})$ for $d=1,2$. We conjecture that, similarly to (17),
the estimate

$\|P_{N}e^{it\Delta}f\|_{L^{6}_{t,x}([0,\log(N)^{-1}]\times\mathbb{T})}\lesssim\|f\|_{L^{2}(\mathbb{T})}$

holds true. This would imply

$\|P_{N}e^{it\Delta}f\|_{L^{6}_{t,x}(\mathbb{T}\times\mathbb{T})}\lesssim\log(N)^{\frac{1}{6}}\|f\|_{L^{2}(\mathbb{T})},$

which matches Bourgain’s counterexample [1].

We analyze (18) via decoupling and obtain smoothing estimates for $2\leq
p<\frac{2(d+2)}{d}$; see Proposition 7.1. For $p=\frac{2(d+2)}{d}$ we find
smoothing as soon we increase the dispersion considering propagators
$e^{it\Delta}\to e^{it(-\Delta)^{a/2}}$ for $a>2$. Note that smoothing is not
expected to hold for the Schrödinger propagator for $\alpha\in(0,2]$, not even
in Euclidean space: For $\alpha=0$ the Knapp example rules out smoothing and
for $\alpha=2$ we can choose highly localized initial data, which exhausts
Sobolev embedding. The smoothing estimates for the Airy propagator
$e^{t\partial_{x}^{3}}$ on $\mathbb{T}$ is recorded in Proposition 7.2.

Finally, in Section 8 we use the Strichartz estimates for the Airy propagator
on frequency-dependent time scales to prove the sharp existence result for the
modified Korteweg-de Vries equation in Sobolev regularity. In [38] a
conditional result on non-existence of the mKdV equation

(19) $\left\\{\begin{array}[]{cl}\partial_{t}u+\partial_{x}^{3}u&=\pm
u^{2}\partial_{x}u,\quad(t,x)\in\mathbb{R}\times\mathbb{T},\\\ u(0)&=u_{0}\in
H^{s}(\mathbb{T})\end{array}\right.$

was proved, which states that existence of solutions to (19) only holds for
the Wick ordered nonlinearity for $s<0$. With the smoothing estimates from
Section 7 at hand, the non- existence result for negative Sobolev regularity
follows. This points out that the established short-time Strichartz estimates
are also relevant in applications to quasilinear equations. We refer to [37]
for further reading.

_Outline of the paper._ In Section 2 we prepare the proof of Theorem 1.2 by
introducing notations and giving a detailed overview of the argument. This is
made rigid in Section 3. In Section 4 we show the logarithmic improvement in
Theorem 1.3. In Section 5 we prove Theorem 1.4. In Section 6 we translate the
refined Strichartz estimates to estimates on tori with large period at finite
times. Next, we show new linear short-time Strichartz estimates with smoothing
effect in Section 7, which are applied to prove the sharp existence result for
the modified Korteweg-de Vries equation in Section 8. In the Appendix we show
how the Fefferman–Córdoba square function estimate implies the bilinear short-
time Strichartz estimate in one dimension.

Basic notations:

  * •

$\mathbb{T}=\mathbb{R}/(2\pi\mathbb{Z})$ denotes the torus in one dimension,

  * •

$(x,t)$ denotes space-time variables and $\xi$ denotes the dual frequency
variable to $x$,

  * •

$\hat{f}$ denotes the Fourier transform of $f:\mathbb{T}^{d}\to\mathbb{C}$:

$\hat{f}(k)=\int_{\mathbb{T}^{d}}e^{-ixk}f(x)dx,\quad k\in\mathbb{Z}^{d}.$

For a function $F:\mathbb{R}\times\mathbb{T}^{d}\to\mathbb{C}$ we denote with
$\hat{F}$ the space-time Fourier transform.

  * •

$H^{s}(\mathbb{T}^{d})$ and $H^{s}(\mathbb{R}^{d})$ denote the $L^{2}$-based
Sobolev spaces with Sobolev regularity $s\in\mathbb{R}$. The Sobolev norms are
given by

$\|f\|_{H^{s}(\mathbb{T}^{d})}^{2}=\sum_{k\in\mathbb{Z}^{d}}\langle
k\rangle^{2s}|\hat{f}(k)|^{2},\quad\|f\|_{H^{s}(\mathbb{R}^{d})}^{2}=\int_{\mathbb{R}^{d}}\langle\xi\rangle^{2s}|\hat{f}(\xi)|^{2}d\xi.$

  * •

We denote with $P_{N}$ smooth Littlewood-Paley projections to frequencies of
modulus in $[N/2,2N]$ for $N\in 2^{\mathbb{N}}$. $P_{1}$ denotes smooth
frequency projection to the ball of radius $2$. Capital numbers usually denote
dyadic numbers $N\in 2^{\mathbb{N}_{0}}$.

  * •

$A\lesssim B$ means there is a harmless constant $C>0$ such that $A\leq CB$.
Further dependencies of the constant $C$ are indicated with subindices, e.g.
$A\lesssim_{\varepsilon}B$ means there is $C_{\varepsilon}=C(\varepsilon)$
such that $A\leq C_{\varepsilon}B$.

##  2\. Preliminaries

In this section we prepare the proof of Theorem 1.2. We collect basic
arguments for future reference, which are presently described in detail.

We shall use the currently best result on linear discrete Fourier restriction
for the paraboloid due to Guo–Li–Yung:

######  Theorem 2.1 ([20, Theorem 1.1]).

For every $\varepsilon>0$, there exists a constant $C_{\varepsilon}>0$ such
that

$\|\sum_{k=1}^{N}f_{n}e^{i(kx-k^{2}t)}\|_{L^{6}(\mathbb{T}^{2})}\leq
C_{\varepsilon}\big{(}\log(N)\big{)}^{2+\varepsilon}\big{(}\sum_{k=1}^{N}|f_{k}|^{2}\big{)}^{\frac{1}{2}}.$

It is conjectured (based on an example due to Bourgain [1]) that the exponent
can be lowered to $1/6$. Note that this would not give an essential
improvement of the present result.

###  2.1. Initial reductions

We can normalize $\|f_{j}\|_{L^{2}}=1$. Next, we apply Hölder’s inequality to
find from Theorem 2.1

(20)
$\begin{split}\int_{[0,N_{1}^{-\alpha}]\times\mathbb{T}}|e^{it\Delta}f_{1}e^{it\Delta}f_{2}e^{it\Delta}f_{3}|^{2}dxdt&\lesssim\big{(}\int_{[0,N_{1}^{-\alpha}]\times\mathbb{T}}|e^{it\Delta}f_{1}|^{2}|e^{it\Delta}f_{3}|^{4}\big{)}^{\frac{1}{2}}\\\
&\quad\times\big{(}\int_{[0,N_{1}^{-\alpha}]\times\mathbb{T}}|e^{it\Delta}f_{2}|^{6}\big{)}^{\frac{1}{3}}\big{(}\int_{[0,N_{1}^{-\alpha}]\times\mathbb{T}}|e^{it\Delta}f_{1}|^{6}\big{)}^{\frac{1}{6}}\\\
&\lesssim_{\varepsilon}\log(N_{1})^{6+\varepsilon}\big{(}\int_{[0,N_{1}^{-\alpha}]\times\mathbb{T}}|e^{it\Delta}f_{1}|^{2}|e^{it\Delta}f_{3}|^{4}\big{)}^{\frac{1}{2}}.\end{split}$

By almost orthogonality we decompose $f_{1}=\sum_{i_{1}}f_{1,i_{1}}$ into
functions with Fourier support contained in intervals of size $N_{3}$ and
assume in the following that $f_{1}$ is Fourier supported in an interval of
size $N_{3}$. It suffices to prove an estimate

(21)
$\int_{[0,N_{1}^{-\alpha}]\times\mathbb{T}}|e^{it\Delta}f_{1}|^{2}|e^{it\Delta}f_{3}|^{4}\lesssim\,C^{\prime}\,\|f_{1}\|_{2}^{2}\|f_{3}\|_{2}^{4}$

with
$C^{\prime}=C_{\varepsilon,\beta}\log(N_{1})^{12+\varepsilon}N_{1}^{-\frac{\alpha\beta}{8}}$
taking into account the initial reduction in (20). To ease notation, redenote
$f_{3}$ with $f_{2}$.

The left hand-side in (21) will be the outset of the asymmetric decoupling due
to Li [32] after continuous approximation (see also follow-up works [33, 21]).

###  2.2. Continuous approximation

We denote $N=N_{1}$ for brevity and

$\hat{f}_{1}(k)=a_{1,k},\quad\hat{f}_{2}(k)=a_{2,k}.$

In the first step we use periodicity and rescaling such that we integrate over
a space-time ball of size $N^{2-\alpha}$. We have

(22)
$\begin{split}&\quad\|e^{it\Delta}f_{1}|e^{it\Delta}f_{2}|^{2}\|_{L_{t,x}^{2}([0,N^{-\alpha}]\times\mathbb{T})}^{2}\\\
&=\big{\|}\sum_{k_{1}\in\mathbb{Z}}e^{i(k_{1}x-tk_{1}^{2})}a_{1,k_{1}}\big{(}\sum_{k_{2}\in\mathbb{Z}}e^{i(k_{2}x-tk_{2}^{2})}a_{2,k_{2}}\big{)}^{2}\big{\|}_{L_{t,x}^{2}([0,N^{-\alpha}]\times\mathbb{T})}^{2}.\end{split}$

We carry out the changes of variables $x\to\frac{x}{N}$, $k_{2}\to Nk_{2}$,
$t\to N^{-2}t$, and denote $b_{i,k_{i}}=a_{i,Nk_{i}}$ accordingly:

$\begin{split}\eqref{eq:BilinearExpression}&=N^{-1}\big{\|}\big{(}\sum_{\begin{subarray}{c}k_{1}\in\mathbb{Z}/N,\\\
k_{1}\in
I_{1}\end{subarray}}e^{i(k_{1}x-N^{2}tk_{1}^{2})}b_{1,k_{1}}\big{)}\big{(}\sum_{\begin{subarray}{c}k_{2}\in\mathbb{Z}/N,\\\
k_{2}\in
I_{2}\end{subarray}}e^{i(k_{2}x-N^{2}tk_{2}^{2})}b_{2,k_{2}}\big{)}^{2}\|_{L_{t,x}^{2}([0,N^{-\alpha}]\times
N\mathbb{T})}^{2}\\\
&=N^{-3}\big{\|}\big{(}\sum_{\begin{subarray}{c}k_{1}\in\mathbb{Z}/N,\\\
k_{1}\in
I_{1}\end{subarray}}e^{i(k_{1}x-N^{2}tk_{1}^{2})}b_{1,k_{1}}\big{)}\big{(}\sum_{\begin{subarray}{c}k_{2}\in\mathbb{Z}/N,\\\
k_{2}\in
I_{2}\end{subarray}}e^{i(k_{2}x-N^{2}tk_{2}^{2})}b_{2,k_{2}}\big{)}^{2}\|_{L_{t,x}^{2}([0,N^{2-\alpha}]\times
N\mathbb{T})}^{2}.\end{split}$

Next, we use periodicity in $x$:

$\eqref{eq:BilinearExpression}=N^{-(4-\alpha)}\big{\|}\big{|}\sum_{\begin{subarray}{c}k_{1}\in\mathbb{Z}/N,\\\
k_{1}\in
I_{1}\end{subarray}}e^{i(k_{1}x-tk_{1}^{2})}b_{1,k_{1}}\big{|}\big{|}\sum_{\begin{subarray}{c}k_{2}\in\mathbb{Z}/N,\\\
k_{2}\in
I_{2}\end{subarray}}e^{i(k_{2}x-tk_{2}^{2})}b_{2,k_{2}}\big{|}^{2}\big{\|}^{2}_{L_{t,x}^{2}([0,N^{2-\alpha}]^{2})}.$

The intervals $I_{i}$ are of length less than $N^{-\frac{\beta}{1+\beta}}$ and
have a distance $\gtrsim 1$. We obtain a factor $N^{-4+\alpha}$ from the
continuous approximation. When we reverse the continuous approximation, the
inverse of this factor will come up. In the following we focus on

$\big{\|}\big{|}\sum_{\begin{subarray}{c}k_{1}\in\mathbb{Z}/N,\\\ k_{1}\in
I_{1}\end{subarray}}e^{i(k_{1}x-tk_{1}^{2})}b_{1,k_{1}}\big{|}\big{|}\sum_{\begin{subarray}{c}k_{2}\in\mathbb{Z}/N,\\\
k_{2}\in
I_{2}\end{subarray}}e^{i(k_{2}x-tk_{2}^{2})}b_{2,k_{2}}\big{|}^{2}\big{\|}^{2}_{L_{t,x}^{2}([0,N^{2-\alpha}]^{2})}.$

Now we approximate the exponential sums by Fourier extension operators:

$|\sum e^{i(k_{1}x-tk_{1}^{2})}b_{1,k_{1}}|\big{|}\sum
e^{i(k_{2}x-tk_{2}^{2})}b_{2,k_{2}}\big{|}^{2}\lesssim|\mathcal{E}\tilde{f}_{1}||\mathcal{E}\tilde{f}_{2}|^{2}\text{
on }B_{N^{2-\alpha}}$

with

$\mathcal{E}\tilde{f}(x,t)=\int_{\\{|\xi|\leq
1\\}}e^{i(x\cdot\xi-t|\xi|^{2})}\tilde{f}(\xi)d\xi.$

The support in $\xi$ for $\tilde{f}_{i}$ is given by $I_{i}$. With $\psi\in
C^{\infty}_{c}(B(0,1))$, $\psi\geq 0$, $\int\psi dx=1$, we have

$\tilde{f}_{i}(\xi)=\sum_{k\in\mathbb{Z}/N}b_{i,k}\lambda\psi(\lambda(\xi-k))$

such that for $\lambda\to\infty$ we obtain in
$\mathcal{S}^{\prime}(\mathbb{R})$

$\tilde{f}_{i}(\xi)=\sum_{k\in\mathbb{Z}/N}b_{i,k}\delta(\xi-k).$

In the following suppose that $\lambda\geq N$ is large enough such that the
above expression becomes

$\begin{split}&\quad\big{\|}\big{|}\sum_{\begin{subarray}{c}k_{1}\in\mathbb{Z}/N,\\\
k_{1}\in
I_{1}\end{subarray}}e^{i(k_{1}x-tk_{1}^{2})}b_{1,k_{1}}\big{|}\big{|}\sum_{\begin{subarray}{c}k_{2}\in\mathbb{Z}/N,\\\
k_{2}\in
I_{2}\end{subarray}}e^{i(k_{2}x-tk_{2}^{2})}b_{2,k_{2}}\big{|}^{2}\big{\|}^{2}_{L_{t,x}^{2}([0,N^{2-\alpha}]\times[0,N^{2-\alpha}])}\\\
&\lesssim\|\mathcal{E}\tilde{f}_{1}|\mathcal{E}\tilde{f}_{2}|^{2}w^{3}_{B_{N^{2-\alpha}}}\|_{L^{2}(\mathbb{R}^{2})}^{2}.\end{split}$

Above we let $w_{B_{N^{2-\alpha}}}$ be a Schwartz weight for
$B_{N^{2-\alpha}}$, which is rapidly decaying off the ball and compactly
Fourier supported. Note that multiplication with $w_{B_{N^{2-\alpha}}}$ is
equivalent to blurring the Fourier support on the scale $N^{-2+\alpha}$.
$\mathcal{E}\tilde{f}_{i}\cdot w_{B_{N^{2-\alpha}}}=g_{i}$ has (space-time)
Fourier support in the $N^{-(2-\alpha)}$-neighbourhood of
$\\{(\xi,-\xi^{2}):\xi\in I_{i}\\}$.

By reversing the continuous approximation we mean taking the limit
$\lambda\to\infty$ such that the Fourier extension operator becomes an
exponential sum. Note that the continuous approximation must be reversed on
the level of extension operators or equivalently, space-time estimates. On the
level of $L_{x}^{2}$-norms it is too costly.

We would like to decouple to the scale $N^{-1}$ (since we rescaled by $N$, the
spacing between the frequencies is now $1/N$). However, the finest spatial
scale we can reach by decoupling respecting the weight function is given by
$N^{-1+\alpha/2}$. At this scale we exit the decoupling iteration.

Let $\pi_{j}:\mathbb{R}^{2}\to\mathbb{R}$, $x\mapsto x_{j}$ denote coordinate
projections and $I_{j}\subseteq[0,1]$ intervals. In the following we aim to
decouple the expression

$\int|g_{I_{1}}|^{4}|g_{I_{2}}|^{2}$

with $\text{supp}(\pi_{1}(\hat{g}_{I_{j}}))\subseteq I_{j}\subseteq[0,1]$ with
$|I_{j}|\lesssim N^{-\frac{\beta}{1+\beta}}$ and $\text{dist}(I_{1},I_{2})\sim
1$. In the following we indicate the spatial frequency support by intervals as
subindices and the weight $w_{B_{N^{2-\alpha}}}$ is absorbed into the
functions.

###  2.3. Overview of the decoupling argument

In the following we carry out a decoupling iteration inspired by efficient
congruencing (cf. [41, 32, 33, 21]). Let $I_{1}$ and $I_{2}$ be intervals of
length $\tilde{\delta}$. By an almost orthogonality argument based on
transversality of the separated regions of the paraboloid, we have morally

$\int|g_{I_{1}}|^{4}|g_{I_{2}}|^{2}\lesssim\sum_{\begin{subarray}{c}I_{2}^{2}\subseteq
I_{2},\\\
|I_{2}^{2}|=\tilde{\delta}^{2}\end{subarray}}\int|g_{I_{1}}|^{4}|g_{I^{2}_{2}}|^{2}$

with intervals $I_{2}^{2}$ partitioning $I_{2}$. Now we exchange the roles of
$g_{I_{1}}$ and $g_{I_{2}}$ by using Cauchy-Schwarz:

(23) $\sum_{\begin{subarray}{c}I_{2}^{2}\subseteq I_{2},\\\
|I_{2}^{2}|=\tilde{\delta}^{2}\end{subarray}}\int|g_{I_{1}}|^{4}|g_{I^{2}_{2}}|^{2}\lesssim\|g_{I_{1}}\|_{L^{6}}^{3}\sum_{\begin{subarray}{c}I_{2}^{2}\subseteq
I_{2},\\\
|I_{2}^{2}|=\tilde{\delta}^{2}\end{subarray}}\big{(}\int|g_{I_{1}}|^{2}|g_{I^{2}_{2}}|^{4}\big{)}^{\frac{1}{2}}.$

The second expression can be decoupled further. The first expression loses a
logarithmic factor, which is acceptable. For the first expression we reverse
the scaling:

$\|\mathcal{E}\tilde{f}_{I_{1}}w_{B_{N^{2-\alpha}}}\|^{6}_{L^{6}_{t,x}(\mathbb{R}^{2})}\lesssim
N^{3}\|\mathcal{E}\tilde{f}_{I_{1}}w_{N^{-\alpha}}(t)w_{N^{1-\alpha}}(x)\|_{L^{6}_{t,x}}^{6}.$

Now we take the limit $\lambda\to\infty$ to recover the exponential sum
(“reversing the continuous approximation”):

$\big{\|}\sum_{k\in\mathbb{Z}}a_{i,k}e^{i(kx-
tk^{2})}w_{N^{-\alpha}}(t)w_{N^{1-\alpha}}(x)\|^{6}_{L^{6}_{t,x}(\mathbb{R}^{2})}.$

Since we are proving $L^{2}$-based estimates, it suffices to show an estimate
for

$N^{3}\big{\|}\sum_{k\in\mathbb{Z}}a_{i,k}e^{i(kx-
tk^{2})}\big{\|}^{6}_{L^{6}_{t}([0,N^{-\alpha}],L^{6}_{x}([0,N^{1-\alpha}]))}.$

By periodicity, we obtain

$\begin{split}&\quad N^{3}\big{\|}\sum_{k\in\mathbb{Z}}a_{i,k}e^{i(kx-
tk^{2})}\big{\|}^{6}_{L^{6}_{t}([0,N^{-\alpha}],L^{6}_{x}([0,N^{1-\alpha}]))}\\\
&=N^{4-\alpha}\big{\|}\sum_{k\in\mathbb{Z}}a_{i,k}e^{i(kx-
tk^{2})}\big{\|}^{6}_{L^{6}_{t}([0,N^{-\alpha}],L^{6}_{x}(\mathbb{T}))}\lesssim
N^{4-\alpha}\log(N)^{12+6\varepsilon}\|a_{i}\|_{\ell^{2}}^{6}.\end{split}$

Roughly speaking, whenever we reverse the continous approximation and switch
back to the periodic setting, we get a factor $N^{4-\alpha}$. This is
compensated by the factor $N^{-4+\alpha}$ from the initial change of variables
to normalize to unit frequencies.

With any iteration we can lower the scale from $\tilde{\delta}$ to
$\tilde{\delta}^{2}$. So, in order to reach the interval of length $N^{-1}$
(actually we can only reach $N^{-1+\frac{\alpha}{2}}$ due to time
localization), $M$ iterations suffice with $M$ given by

$M=\min\\{\ell\in\mathbb{N}:N^{-\frac{\beta}{1+\beta}2^{\ell}}=N^{-1}\\}.$

We choose as upper bound for the number of iterations

$m=\lceil\log_{2}(\beta^{-1}+1)\rceil.$

We can estimate $\|g_{I_{1}}\|_{L^{6}}$ by reversing the approximation steps
for periodic functions, which incurs by Strichartz estimates
$\lesssim_{\varepsilon}\log(N_{1})^{2+\varepsilon}$.

We come back to (23). It suffices to prove an estimate

$\int|g_{I_{1}}|^{2}|g_{I_{2}^{2}}|^{4}\lesssim
CN^{4-\alpha}\|b_{I_{1}}\|_{\ell^{2}}^{2}\|b_{I_{2}^{2}}\|_{\ell^{2}}^{4}.$

because we obtain by almost orthogonality

$\begin{split}\sum_{I^{2}_{2}}\big{(}\int|g_{I_{1}}|^{2}|g_{I_{2}}|^{4}\big{)}^{\frac{1}{2}}&\leq
C^{\frac{1}{2}}N^{4-\alpha}\|b_{I_{1}}\|_{\ell^{2}}\sum_{I^{2}_{2}}\|b_{I_{2}^{2}}\|_{\ell^{2}}^{2}\\\
&\leq
C^{\frac{1}{2}}N^{4-\alpha}\|b_{I_{1}}\|_{\ell^{2}}\|b_{I_{2}}\|^{2}_{\ell^{2}}.\end{split}$

It is important to sum the $\ell^{2}$-norms of the coefficients (after
reversing the continuous approximation on the level of space-time estimates).
We use this decoupling iteration until we reach an expression of the form

$\int|g_{\tilde{I}_{1}}|^{2}|g_{\tilde{I}_{2}}|^{4}$

with $|\tilde{I}_{1}|=|\tilde{I}_{2}|=N^{-1+\frac{\alpha}{2}}$. We again
reverse the continuous approximation to find

$\int|g_{\tilde{I}_{1}}|^{2}|g_{\tilde{I}_{2}}|^{4}\lesssim
N^{4-\alpha}\int_{[0,N^{-\alpha}]\times\mathbb{T}}\big{|}\sum_{k\in\mathbb{Z}}a_{1,k}e^{i(kx-
tk^{2})}\big{|}^{2}\big{|}\sum_{k\in\mathbb{Z}}a_{2,k}e^{i(kx-
tk^{2})}\big{|}^{4}dxdt.$

We use a short-time bilinear Strichartz estimate due to Moyua–Vega [35]. Hani
[24] proved this on general compact manifolds. In the Appendix we show how a
transverse (reverse) square function estimate going back to Fefferman and
Córdoba [18, 13] implies the following by the arguments of the paper:

######  Theorem 2.2 ([24, Theorem 1.2]).

Suppose that $u_{0}$, $v_{0}\in L^{2}(\mathbb{T})$ with $P_{N_{1}}u_{0}=u_{0}$
and $P_{N_{2}}v_{0}=v_{0}$ with $N_{2}\ll N_{1}$. Then the following estimate
holds:

(24)
$\|e^{it\Delta}u_{0}\,e^{it\Delta}v_{0}\|_{L^{2}([0,N_{1}^{-1}],L^{2}(\mathbb{T}))}\lesssim
N_{1}^{-\frac{1}{2}}\|u_{0}\|_{L^{2}(\mathbb{T})}\|v_{0}\|_{L^{2}(\mathbb{T})}.$

By (24) and Bernstein’s inequality to conclude a gain:

$\begin{split}&\quad\int_{[0,N^{-\alpha}]}\int_{\mathbb{T}}\big{|}\sum_{k\in\mathbb{Z}}a_{1,k}e^{i(kx-
tk^{2})}\big{|}^{2}\big{|}\sum_{k\in\mathbb{Z}}a_{2,k}e^{i(kx-
tk^{2})}\big{|}^{4}dxdt\\\
&\lesssim\|e^{it\Delta}f_{1}e^{it\Delta}f_{2}\|^{2}_{L^{2}([0,N^{-\alpha}],L^{2}(\mathbb{T}))}\|e^{it\Delta}f_{2}\|_{L^{\infty}}^{2}\\\
&\lesssim
N_{1}^{-\alpha}\|f_{1}\|_{2}^{2}\|f_{2}\|_{2}^{2}N_{1}^{\frac{\alpha}{2}}\|f_{2}\|_{2}^{2}\\\
&\lesssim
N_{1}^{-\frac{\alpha}{2}}\|f_{1}\|_{2}^{2}\|f_{2}\|_{2}^{4}.\end{split}$

Unwinding the decoupling iteration, this will yield Theorem 1.2. In the next
section we turn to the technical details.

##  3\. Proof of Theorem 1.2

Following the reductions in Section 2, let $I_{j}\subseteq[0,1]$, $j=1,2$ be
two intervals, which satisfy $\text{dist}(I_{1},I_{2})\gtrsim 1$, and
$\tilde{\delta}=|I_{i}|\leq N^{-\frac{\beta}{1+\beta}}$. Let
$g_{i}\in\mathcal{S}(\mathbb{R}^{2})$ with
$\text{supp}(\hat{g}_{i})\subseteq\mathcal{N}_{N^{-2+\alpha}}(\\{(\xi,-\xi^{2}):\xi\in
I_{i}\\})$. We analyze the following expression via asymmetric bilinear
decoupling (cf. [21]):

(25) $\int_{\mathbb{R}^{2}}|g_{I_{1}}|^{2}|g_{I_{2}}|^{4}dx.$

More specifically, we have $g_{I}=\mathcal{E}b_{I}\cdot w_{B_{N^{2-\alpha}}}$.
This means that $g_{I}$ is given by the continuous approximation on the
interval $I$ with Fourier support blurred on the scale $N^{-(2-\alpha)}$. We
recall notations from [21, Section 2].

Let $\Gamma(t)=(t,-t^{2})$. For an interval $I\subseteq[0,1]$ with
$|I|=\tilde{\delta}$, we let

$\mathcal{U}_{I}=\\{(c_{I},c_{I}^{2})+a\cdot\Gamma^{\prime}(c_{I})+b\cdot\Gamma^{\prime\prime}(c_{I})\,:\,|a|\leq\tilde{\delta},\;|b|\leq\tilde{\delta}^{2}\\}.$

This describes roughly the $\tilde{\delta}^{2}$-neighbourhood of the
paraboloid in the interval $I$. By $\mathcal{U}_{I}^{o}$ we denote the polar
set of $\mathcal{U}_{I}$ centered at the origin:

$\mathcal{U}_{I}^{o}=\\{x\in\mathbb{R}^{2}:|\langle
x,\partial^{i}\Gamma(c_{I})\rangle|\leq|I|^{-i},\;1\leq i\leq 2\\}.$

We let

$\phi_{I}(x)=|\mathcal{U}_{I}^{o}|^{-1}\inf\\{t\geq
1\,:\,x/t\in\mathcal{U}_{I}^{o}\\}^{-20}$

such that $\phi_{I}\geq 0$ and $\|\phi_{I}\|_{L^{1}}\lesssim 1$. We have the
following consequence of the uncertainty relation:

######  Lemma 3.1 ([21, Lemma 3.3]).

For $p\in[1,\infty)$ and $J\subseteq[0,1]$, we have

$|g_{J}|^{p}\lesssim_{p,C}|g_{J}|^{p}*\phi_{J}$

for any $g_{J}$ with $\text{supp}(\hat{g}_{J})\subseteq C\mathcal{U}_{J}$.

Coming back to (25), we have

$\int|g_{I_{1}}|^{4}|g_{I_{2}}|^{2}\lesssim\int(|g_{I_{1}}|^{4}*\phi_{I_{1}})(|g_{I_{2}}|^{2}*\phi_{I_{2}}).$

At this point we use the lower-dimensional decoupling (based on Plancherel and
transversality of pieces of the paraboloid on the two separated intervals
$I_{1}$, $I_{2}$). The following estimate is [21, Lemma 3.8] with $l=1$ and
$k=2$:

$\int(|g_{I_{1}}|^{4}*\phi_{I_{1}})(|g_{I_{2}}|^{2}*\phi_{I_{2}})\lesssim\sum_{\begin{subarray}{c}I_{2}^{2}\subseteq
I_{2},\\\
|I_{2}^{2}|=\tilde{\delta}^{2}\end{subarray}}\int\big{(}|g_{I_{1}}|^{4}*\phi_{I_{1}}\big{)}\big{(}|g_{I_{2}^{2}}|^{2}*\phi_{I_{2}^{2}}\big{)}.$

Applying Hölder’s inequality (see [21, Lemma 4.1]) yields

$\sum_{|I_{2}^{2}|=\tilde{\delta}^{2}}\int\big{(}|g_{I_{1}}|^{4}*\phi_{I_{1}}\big{)}\big{(}|g_{I_{2}^{2}}|^{2}*\phi_{I_{2}^{2}}\big{)}\lesssim\|g_{I_{1}}\|_{L^{6}}^{3}\sum_{|I_{2}^{2}|=\tilde{\delta}^{2}}\big{(}\int(|g_{I_{1}}|^{2}*\phi_{I_{1}})(|g_{I_{2}^{2}}|^{4}*\phi_{I_{2}^{2}})\big{)}^{\frac{1}{2}}.$

We summarize our findings as

$\int|g_{I_{1}}|^{4}|g_{I_{2}}|^{2}\leq
C_{1}\|g_{I_{1}}\|_{6}^{3}\sum_{|I_{2}^{2}|=\tilde{\delta}^{2}}\big{(}\int\big{(}|g_{I_{1}}|^{2}*\phi_{I_{1}}\big{)}\big{(}|g_{I_{2}^{2}}|^{4}*\phi_{I_{2}^{2}}\big{)}\big{)}^{\frac{1}{2}},$

which is amenable to $m$-fold iteration:

(26) $\begin{split}&\quad\int|g_{I_{1}}|^{4}|g_{I_{2}}|^{2}\\\ &\leq
C_{1}\|g_{I_{1}}\|_{6}^{3}\sum_{|I_{2}^{2}|=\tilde{\delta}^{2}}\big{(}\int\big{(}|g_{I_{1}}|^{2}*\phi_{I_{1}}\big{)}\big{(}|g_{I^{2}_{2}}|^{4}*\phi_{I_{2}}\big{)}\big{)}^{\frac{1}{2}}\\\
&\leq
C_{1}^{\frac{3}{2}}\|g_{I_{1}}\|^{3}_{6}\sum_{|I_{2}^{2}|=\tilde{\delta}^{2}}\big{(}\sum_{|I_{1}^{2}|=\tilde{\delta}^{4}}\int\big{(}|g_{I_{1}^{2}}|^{2}*\phi_{1^{2}}\big{)}\big{(}|g_{I_{2}^{2}}|^{4}*\phi_{I_{2}^{2}}\big{)}\big{)}^{\frac{1}{2}}\\\
&\vdots\\\ &\leq
C_{1}^{2}\|g_{I_{1}}\|^{3}_{L^{6}}\sum_{|I_{2}^{2}|=\tilde{\delta}^{2}}\|g_{I_{2}^{2}}\|_{L^{6}}^{\frac{3}{2}}\\\
&\quad\times\big{(}\sum_{|I_{1}^{2}|=\tilde{\delta}^{4}}\|g_{I_{1}^{2}}\|_{L^{6}}^{\frac{3}{2}}\big{(}\sum_{|I_{2}^{3}|=\tilde{\delta}^{4}}\|g_{I_{2}^{3}}\|_{L^{6}}^{3/2}\ldots\big{(}\int|g_{I_{1}^{m/2}}|^{4}*\phi_{I_{1}^{m/2}}|g_{I_{2}^{m/2}}|^{2}*\phi_{I_{2}^{m/2}}\big{)}^{\frac{1}{2}}\big{)}^{\frac{1}{2}}\ldots\big{)}^{\frac{1}{2}}.\end{split}$

Recall that $m=\lceil\log_{2}(\beta^{-1}+1)\rceil$ (which we suppose to be
even for notational convenience) for
$\tilde{\delta}=N^{-\frac{\beta}{1+\beta}}$ such that
$|I_{j}^{m/2}|=N^{-1+\frac{\alpha}{2}}$. By the argument in Section 2.3 we
have the following:

######  Lemma 3.2.

Let $g_{I_{j}^{k}}=\mathcal{E}b_{I_{j}^{k}}\cdot w_{B_{N^{2-\alpha}}}$ be like
above. Then

$\|g_{I_{j}^{k}}\|_{L^{6}(\mathbb{R}^{2})}\leq N_{1}^{\frac{4-\alpha}{6}}\cdot
C_{\varepsilon}\log(N_{1})^{2+\varepsilon}\|b_{I_{j}^{k}}\|_{\ell^{2}}.$

To estimate the maximally decoupled expression, we use the following:

######  Lemma 3.3.

Let $\tilde{I}_{i}\subseteq[0,1]$ with
$|\tilde{I}_{i}|=N^{-1+\frac{\alpha}{2}}$ and assume the separation property

$\text{dist}(\tilde{I}_{1},\tilde{I}_{2})\sim 1.$

Then the following estimate holds:

(27)
$\int(|g_{\tilde{I}_{1}}|^{2}*\phi_{\tilde{I}_{1}})(|g_{\tilde{I}_{2}}|^{4}*\phi_{\tilde{I}_{2}})\leq
N^{4-\alpha}\cdot
DN^{-\frac{\alpha}{2}}\|b_{\tilde{I}_{1}}\|_{\ell^{2}}^{2}\|b_{\tilde{I}_{2}}\|_{\ell^{2}}^{4}.$

###### Proof.

By Hölder’s inequality it suffices to estimate

$\int\big{(}|g_{\tilde{I}_{1}}|^{2}*\phi_{\tilde{I}_{1}}|g_{\tilde{I}_{2}}|^{2}*\phi_{\tilde{I}_{2}}\big{)}\|g_{I_{2}}\|_{L^{\infty}}^{2}.$

We rewrite the first factor as

$\iint\big{(}\int|g_{\tilde{I}_{1}}|^{2}(x-x_{1},t-t_{1})|g_{\tilde{I}_{2}}|^{2}(x-x_{2},t-t_{2})dxdt\big{)}\phi_{1}(x_{1},t_{1})\phi_{2}(x_{2},t_{2})dx_{1}dx_{2}dt_{1}dt_{2}.$

We shall prove a uniform bound in $x_{1}$,$x_{2}$ and $t_{1}$,$t_{2}$ for the
integral over $x$ and $t$. To this end, we absorb the pure phase factor
$e^{i(x_{1}-x_{2})\xi}e^{i(t_{1}-t_{2})\xi^{2}}$ into $g_{2}$ and dominate

$w_{B_{N^{2-\alpha}}}(x^{\prime},t^{\prime})w_{B_{N^{2-\alpha}}}(x^{\prime}+x_{1}-x_{2},t^{\prime}+t_{1}-t_{2})\lesssim
w_{B_{N^{2-\alpha}}}(x^{\prime},t^{\prime}).$

We reverse the continuous approximation to find

$\begin{split}&\quad\int\big{|}\int
e^{i(x^{\prime}\xi-t^{\prime}\xi^{2})}f_{1}(\xi)d\xi\int
e^{i(x^{\prime}\xi-t\xi^{2})}\tilde{f}_{2}(\xi)d\xi\big{|}^{2}w_{B_{N^{2-\delta}}}(x^{\prime},t^{\prime})dx^{\prime}dt^{\prime}\\\
&\lesssim
N^{4-\alpha}\|e^{it\Delta}f_{\tilde{I}_{1}}e^{it\Delta}f^{\prime}_{\tilde{I}_{2}}\|^{2}_{L_{t,x}^{2}([0,N^{-\alpha}]\times\mathbb{T})}.\end{split}$

By $f^{\prime}$ we indicate that we have absorbed the phase shift into $f$ and
write $\tilde{I}_{j}=NI_{j}$. Note that
$\|f^{\prime}_{\tilde{I}_{2}}\|_{2}=\|f_{\tilde{I}_{2}}\|_{2}$ because the
phase shift leaves the $\ell^{2}$-norm invariant.

We piece together (24) on intervals of size $N^{-1}$ to obtain

(28)
$\|e^{it\Delta}f_{\tilde{I}_{1}}e^{it\Delta}f_{\tilde{I}_{2}}\|^{2}_{L_{t,x}^{2}([0,N^{-\alpha}]\times\mathbb{T})}\leq
D_{1}N^{-\alpha}\|a_{I_{1}}\|_{\ell^{2}}^{2}\|a_{\tilde{I}_{2}}\|_{\ell^{2}}^{2}.$

We estimate $\|f_{\tilde{I}_{2}}\|_{L^{\infty}}$ by Bernstein’s inequality:

(29) $\|f_{\tilde{I}_{2}}\|_{L^{\infty}}^{2}\leq
D_{2}N^{\frac{\alpha}{2}}\|a_{\tilde{I}_{2}}\|_{\ell^{2}}^{2}.$

Taking (28) and (29) together with $\|\phi_{I}\|_{L^{1}}\lesssim 1$, we
conclude the proof of (25). ∎

We are ready to estimate the expression in (26): The expressions
$\|g_{I_{j}^{k}}\|_{L^{6}}$ are estimated by Lemma 3.2, which yields a factor
of $C_{\varepsilon}\log(N_{1})^{2+\varepsilon}$ at every instance. Note that
the exponents in (26) sum up to a number $\leq 6$ by finite geometric sum. The
bilinear expression is estimated by Lemma 3.3, which yields

$\begin{split}\eqref{eq:DecouplingIteration}&\leq
C_{1}^{2}(C_{\varepsilon}\log(N_{1})^{2+\varepsilon})^{6}(DN_{1}^{-\frac{\alpha}{2}})^{\frac{1}{2^{m}}}\|b_{I_{1}}\|^{3}_{\ell^{2}}\sum_{|I_{2}^{2}|=\tilde{\delta}^{2}}\|b_{I_{2}^{2}}\|_{\ell^{2}}^{\frac{3}{2}}\\\
&\quad\times\big{(}\sum_{|I_{1}^{2}|=\tilde{\delta}^{4}}\|b_{I_{1}^{2}}\|_{\ell^{2}}^{\frac{3}{2}}\big{(}\sum_{|I_{2}^{3}|=\tilde{\delta}^{4}}\|b_{I_{2}^{3}}\|_{\ell^{2}}^{3/2}\ldots\big{(}\|b_{I_{1}^{m/2}}\|_{\ell^{2}}^{4}\|b_{I_{2}^{m/2}}\|_{\ell^{2}}^{2}\big{)}^{\frac{1}{2}}\ldots\big{)}^{\frac{1}{2}}\\\
&\leq
C_{1}^{2}(C_{\varepsilon}\log(N_{1})^{2+\varepsilon})^{6}(DN_{1}^{-\frac{\alpha}{2}})^{\frac{1}{2^{m}}}\|b_{I_{1}}\|^{4}_{\ell^{2}}\|b_{I_{2}}\|_{\ell^{2}}^{2}.\end{split}$

In the ultimate estimate we used the disjointness between the intervals to
carry out the sums. Recalling that $2^{m}\leq 4\beta^{-1}$ we find

$\eqref{eq:DecouplingIteration}\leq
C_{1}^{2}(C_{\varepsilon}\log(N_{1})^{2+\varepsilon})^{6}D^{\frac{1}{\beta}}N^{-\frac{\alpha\beta}{8}}\|b_{I_{1}}\|^{4}_{\ell^{2}}\|b_{I_{2}}\|_{\ell^{2}}^{2}.$

We summarize our findings as

$\|e^{it\Delta}f_{1}(e^{it\Delta}f_{2})^{2}\|^{2}_{L^{2}([0,N^{-\alpha}]\times\mathbb{T})}\leq(C_{\varepsilon}\log(N_{1})^{2+\varepsilon})^{6}D_{\beta}N_{1}^{-\frac{\alpha\beta}{8}}\|f_{1}\|_{L^{2}}^{2}\|f_{2}\|_{L^{2}}^{4}.$

Coming back to (21), we obtain from (20) (recall that we suppose
$\|f_{i}\|_{L^{2}}=1$) and the estimate in the previous display:

$\begin{split}\big{\|}\prod_{j=1}^{3}e^{it\Delta}f_{j}\big{\|}^{2}_{L^{2}([0,N^{-\alpha}]\times\mathbb{T})}&\leq
C_{\varepsilon}\log(N_{1})^{12+\varepsilon}D_{\beta}N_{1}^{-\frac{\alpha\beta}{8}}\\\
&\leq
C_{\varepsilon,\beta}\log(N_{1})^{12+\varepsilon}N_{1}^{-\frac{\alpha\beta}{8}}.\end{split}$

In the last step we redefined $\varepsilon$ and $C_{\varepsilon,\beta}$
accordingly. The proof is complete.

$\hfill\Box$

##  4\. Refinement for weakly separated dyadic frequencies

In this section we prove Theorem 1.3 as a variant of the previous argument. We
shall be brief.

###### Proof of Theorem 1.3.

By the reductions of Section 2.1 it suffices to estimate

(30)
$\log(N)^{6+\varepsilon}\big{(}\int_{[0,N^{-\alpha}]\times\mathbb{T}}|e^{it\Delta}f_{1}|^{2}|e^{it\Delta}f_{2}|^{4}\big{)}^{\frac{1}{2}},$

where $\text{supp}(\hat{f}_{i})\subseteq I_{i}$ an interval of length $N_{3}$
and $\text{dist}(I_{1},I_{2})\gtrsim N_{1}=:N$. We rescale $x\to Nx$, $t\to
N^{2}t$ to find

$\begin{split}\eqref{eq:BilinearDecouplingRefined}&=N^{-3}\int_{[0,N^{2-\alpha}]\times
N\mathbb{T}}|e^{it\Delta}\tilde{f}_{1}|^{2}|e^{it\Delta}\tilde{f}_{2}|^{4}\\\
&\lesssim
N^{-3}N^{-(1-\alpha)}\int_{[0,N^{2-\alpha}]\times[0,N^{2-\alpha}]}|e^{it\Delta}\tilde{f}_{1}|^{2}|e^{it\Delta}\tilde{f}_{2}|^{4}.\end{split}$

In the above display we used periodicity which incurs a factor $N^{1-\alpha}$.
We use continuous approximation like above

$\begin{split}&\lesssim
N^{-3}N^{-(1-\alpha)}\int_{B_{N^{2-\alpha}}}|e^{it\Delta}\tilde{f}_{1}|^{2}|e^{it\Delta}\tilde{f}_{2}|^{4}dtdx\\\
&\lesssim
N^{-3}N^{-(1-\alpha)}\int_{B_{N^{2-\alpha}}}|\mathcal{E}\tilde{f}_{1}|^{2}|\mathcal{E}\tilde{f}_{2}|^{4}w_{B_{N^{2-\alpha}}}^{6}dxdt.\end{split}$

Now we use the decoupling iteration to find that the integral is dominated by

(31)
$\lesssim\log^{13}(N)\big{(}\int|\mathcal{E}f_{1,\theta}|^{2}|\mathcal{E}f_{2,\theta}|^{4}w^{6}_{B_{N^{2-\alpha}}}dxdt\big{)}^{1/2^{m}}$

with $f_{i,\theta}$ supported in intervals of size $N^{-\frac{2-\alpha}{2}}$.
$m$ describes the number of decoupling iterations, which is estimated by
$m\leq\log_{2}(10\log\log(N))$. Indeed, the initial size of the intervals is
$\leq\exp(\frac{-\log N}{\log\log N})$ after rescaling. Then it follows

$\exp\big{(}\frac{-\log N}{\log\log N}\big{)}^{2^{m}}\leq N^{-1}.$

Reversing the continuous approximation and the scaling we find

$\int|\mathcal{E}f_{1,\theta}|^{2}|\mathcal{E}f_{2,\theta}|^{4}w_{B_{N^{2-\alpha}}}^{6}dxdt\lesssim\int_{[0,N^{-\alpha}]\times\mathbb{T}}|e^{it\Delta}f_{1,\tilde{\theta}}|^{2}|e^{it\Delta}f_{2,\tilde{\theta}}|^{4}dxdt$

with $\text{supp}(\hat{f}_{i,\tilde{\theta}})\subseteq I_{i,\tilde{\theta}}$,
which are intervals of length $N^{\frac{\alpha}{2}}$. The expression is
finally estimated by short-time bilinear Strichartz estimates and Bernstein’s
inequality:

(32)
$\begin{split}\int_{[0,N^{-\alpha}]\times\mathbb{T}}|e^{it\Delta}f_{1,\tilde{\theta}}|^{2}|e^{it\Delta}f_{2,\tilde{\theta}}|^{4}dxdt&\lesssim\|e^{it\Delta}f_{2,\tilde{\theta}}\|^{2}_{L^{\infty}_{t,x}}\|e^{it\Delta}f_{1,\tilde{\theta}}e^{it\Delta}f_{2,\tilde{\theta}}\|^{2}_{L^{2}_{t,x}([0,N^{-\alpha}]\times\mathbb{T})}\\\
&\lesssim N^{\frac{\alpha}{2}}N^{-\alpha}=N^{-\frac{\alpha}{2}}.\end{split}$

Collecting estimates (31) and (32), we have

$\int_{[0,N^{-\alpha}]\times\mathbb{T}}\prod_{i=1}^{3}|e^{it\Delta}f_{i}|^{2}dxdt\lesssim\log(N)^{13}N^{-\frac{\alpha}{2^{m+2}}}=\log(N)^{20}N^{-\frac{\alpha}{40\log\log
N}}.$

Since $\log(N)^{20}N^{-\frac{\alpha}{40\log\log N}}\to 0$ as $N\to\infty$, the
proof is complete.

∎

######  Remark 4.1.

We emphasize that it is not the separation property of the frequencies which
is required for the above argument to prove a trilinear Strichartz estimate
without derivative loss. This can always be recovered via a Whitney
decomposition and affine rescaling (cf. [21, Section 2]). What is required is
that the initial frequency intervals after rescaling are of size
$O(\exp\big{(}\frac{-\log N}{\log\log N}\big{)})$. This keeps the number of
decoupling iterations small enough for a favorable estimate.

##  5\. Improved trilinear Strichartz estimates via transversality and
decoupling iteration

###  5.1. Set-up

In the following we prove a trilinear analog in two dimensions. We estimate
the expression

$\int_{[0,N_{1}^{-\alpha}]\times\mathbb{T}^{2}_{\gamma}}|e^{it\Delta}f_{1}e^{it\Delta}f_{2}e^{it\Delta}f_{3}|^{\frac{4}{3}}dxdt$

under support and transversality assumptions as stated in Theorem 1.4.

We recall the following quantitative trilinear Kakeya inequality due to Guth
[22] (see also [9]), which plays a crucial role in the argument:

######  Theorem 5.1.

Let $(T_{j}^{i})_{j}$ for $i=1,2,3$ denote three collections of tubes of size
$\delta^{-1}\times\delta^{-1}\times\delta^{-2}$ such that for the long
directions $\mathfrak{n}^{i}_{j_{i}}$ for tubes $T_{j_{i}}^{i}$ we have the
uniform transversality condition:

$|\mathfrak{n}^{1}_{j_{1}}\wedge\mathfrak{n}^{2}_{j_{2}}\wedge\mathfrak{n}^{3}_{j_{3}}|\gtrsim\nu.$

Let $(c_{j}^{i})_{1\leq j\leq M_{i}}\subseteq\mathbb{R}_{>0}$ with $i=1,2,3$
denote three collections of positive real numbers. Then the following holds:

(33)
$\int\big{(}\prod_{i=1}^{3}\sum_{j=1}^{M_{i}}c_{j}^{(i)}\chi_{T_{j}^{i}}\big{)}^{\frac{1}{2}}\leq
C\nu^{-\frac{1}{2}}\delta^{-3}\big{(}\prod_{i=1}^{3}\sum_{j=1}^{M_{i}}c_{j}^{(i)}\big{)}^{\frac{1}{2}}.$

###  5.2. Rescaling and continuous approximation

We return to the expression (13). The rescaling and approximation argument is
analogous to Subsection 2.2. We shall be brief. For the decoupling iteration
we rescale space and time variables $x\to N_{1}x$ and $t\to N_{1}^{2}t$ and
let $b_{i,Nk}=\hat{f}_{i}(k)$ to find:

(34)
$\begin{split}&\quad\int_{[0,N_{1}^{-\alpha}]\times\mathbb{T}_{\gamma}^{2}}\big{|}e^{it\Delta}f_{1}e^{it\Delta}f_{2}e^{it\Delta}f_{3}\big{|}^{p/3}dxdt\\\
&=N_{1}^{-4}\int_{[0,N_{1}^{2-\alpha}]\times
N_{1}\mathbb{T}_{\gamma}}\big{|}\prod_{i=1}^{3}\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{2}/N_{1},\\\
N_{1}k\in\text{supp}(\hat{f}_{i})\end{subarray}}e^{i(k\cdot
x-tk^{2})}b_{i,k}\big{|}^{p/3}dxdt.\end{split}$

Let $\mathcal{E}\tilde{f}_{i}(x,t)=\int
e^{i(x_{1}\xi_{1}+x_{2}\xi_{2}-t|\xi|^{2})}\tilde{f}_{i}(\xi)d\xi$. We use
periodicity in space and continuous approximation to obtain:

$\eqref{eq:Rescaling2d}\lesssim
N_{1}^{-4}N_{1}^{-2(1-\alpha)}\int_{B_{N_{1}^{2-\alpha}}}\big{|}\mathcal{E}\tilde{f}_{1}\mathcal{E}\tilde{f}_{2}\mathcal{E}\tilde{f}_{3}\big{|}^{p/3}dxdt.$

Since this will simplify the argument, we consider the averaged expression for
a decoupling iteration:

(35) $\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{B_{N_{1}^{2-\alpha}}}|\mathcal{E}\tilde{f}_{1}\mathcal{E}\tilde{f}_{2}\mathcal{E}\tilde{f}_{3}\big{|}^{p/3}dxdt.$

We introduce the space-time weight $w_{B_{N_{1}^{2-\alpha}}}^{p/3}$ into the
integral. This will be absorbed into the functions $\mathcal{E}\tilde{f}_{i}$
to find functions $\tilde{F}_{i}$, which are Fourier supported in the
$N_{1}^{-(2-\alpha)}$ neighbourhood of the truncated paraboloid
$\mathbb{P}^{2}=\\{(\xi,-\xi^{2})\,:\xi\in[-2,2]^{2}\\}$, which is denoted by
$\mathcal{N}_{N_{1}^{-(2-\alpha)}}\mathbb{P}^{2}$. Moreover, the Fourier
support of $\tilde{F}_{i}$ is restricted to space-frequency balls of size
$N_{3}/N_{1}\leq N_{1}^{-\frac{\beta}{1+\beta}}$. We have

(36)
$\eqref{eq:AveragedExpressionReductions}\lesssim\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int\big{|}\tilde{F}_{1}\tilde{F}_{2}\tilde{F}_{3}\big{|}^{p/3}dxdt.$

###  5.3. The decoupling iteration

The qualitative argument follows [15, Section 10.3]. The quantification rests
upon two ingredients:

  * •

the endpoint multilinear Kakeya inequality to make the dependence on the
transversality precise,

  * •

keeping track of the number of decoupling iterations to obtain the precise
gain from short-time bilinear Strichartz estimates.

We introduce notations from [15]. Let $Q^{r}$ denote a cube with side length
$2^{r}$ and $\mathbb{I}_{m}$ denote the collection of cubes tiling
$[-2,2]^{2}$ with side length $2^{-m}$. For $I_{i,m}\in\mathbb{I}_{m}$ let
$\mathcal{P}_{I_{i,m}}$ denote the Fourier projection to
$I_{i,m}\times\mathbb{R}$. Let $L^{t}_{\\#}(w_{Q^{r}})$ denote the averaged
Lebesgue norm with a rapidly decreasing weight off $Q^{r}$:

$\|F\|^{t}_{L^{t}_{\\#}(w_{Q^{r}})}=\frac{1}{|Q^{r}|}\int|F|^{t}w_{Q^{r}}dz.$

With this notation, we have

$\text{rhs}\eqref{eq:AveragedExpressionReductionsII}\sim\|\tilde{F}_{1}\tilde{F}_{2}\tilde{F}_{3}\|^{p/3}_{L^{p/3}_{\\#}(w_{Q_{N_{1}^{2-\alpha}}})}.$

Let $\mathcal{Q}_{m,r}(Q^{r})$ denote the partition of $Q^{r}$ with cubes of
side-length $2^{m}$ and $m\leq r$.

Let $\underline{F}=(F_{i})_{i=1,2,3}$ be a family of functions with
$\text{supp}(\hat{F}_{i})\subseteq\mathcal{N}_{\delta^{2}}(\mathbb{P}^{2})$
with $\delta\leq 2^{-m}$ and such that

(37) $|(2\xi_{1},1)\wedge(2\xi_{2},1)\wedge(2\xi_{3},1)|\geq\nu$

for $\xi_{i}\in\pi_{\mathbb{R}^{2}}(\text{supp}(\hat{F}_{i}))$.

In the present context we have

(38) $\delta=N_{1}^{-(1-\frac{\alpha}{2})}.$

We define for $\Delta_{m}\in\mathcal{Q}_{m,r}(Q^{r})$

(39)
$D_{t}(m,\Delta_{m},\underline{F},\delta)=\big{[}\prod_{i=1}^{3}\big{(}\sum_{I_{i,m}\in\mathbb{I}_{m}}\|\mathcal{P}_{I_{i,m}}F_{i}\|^{2}_{L^{t}_{\\#}(w_{\Delta_{m}})}\big{)}^{\frac{1}{2}}\big{]}^{\frac{1}{3}},$

and

(40)
$A_{p}(m,Q^{r},\underline{F},\delta)=\big{(}\frac{1}{|\mathcal{Q}_{m,r}|}\sum_{\Delta_{m}\in\mathcal{Q}_{m,r}(Q^{r})}D_{2}(m,\Delta_{m},\underline{F},\delta)^{p}\big{)}^{\frac{1}{p}}.$

As mentioned on [15, p. 209], using the locally constant property we can
perceive $A_{p}$ as

$|Q^{r}|A_{p}^{p}(m,Q^{r},\underline{F},\delta)\approx\int_{\mathbb{R}^{3}}\prod_{i=1}^{3}\big{(}\sum_{I\in\mathbb{I}_{m}}|\mathcal{P}_{I}F_{i}|^{2}\big{)}^{\frac{p}{3}}w_{Q^{r}}.$

This representation suggests the possibility of exiting multilinearity via
linear and bilinear Strichartz estimates.

Moreover, we have the following representation for $D_{p}$

$\prod_{i=1}^{3}\big{(}\sum_{I_{i,m}\in\mathbb{I}_{m}}\|\mathcal{P}_{I_{i,m}}F_{i}\|^{2}_{L^{p}_{\\#}(w_{Q^{r}}}\big{)}^{\frac{1}{6}}=|Q^{r}|^{-\frac{1}{p}}\prod_{i=1}^{3}\big{(}\sum_{I_{i,m}}\|\mathcal{P}_{I_{i,m}}F_{i}\|^{2}_{L^{p}(w_{Q^{r}})}\big{)}^{\frac{1}{6}}.$

Coming back to (36). We let

$\delta=N_{1}^{-(1-\frac{\alpha}{2})}\text{ and }2^{r}=N_{1}^{2-\alpha}.$

For notational convenience assume $r\in\mathbb{N}$ and $\delta^{-1}\in
2^{\mathbb{N}}$.

We find by Hölder’s inequality

(41) $\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{B_{N_{1}^{2-\alpha}}}\big{|}\tilde{F}_{1}\tilde{F}_{2}\tilde{F}_{3}\big{|}^{4/3}\leq\frac{1}{|\mathcal{Q}_{m,r}|}\sum_{\Delta_{m}\in\mathcal{Q}_{m,r}}\|\tilde{F}_{1}\|^{4/3}_{L^{2}_{\\#}(w_{\Delta})}\|\tilde{F}_{2}\|^{4/3}_{L^{2}_{\\#}(w_{\Delta})}\|\tilde{F}_{3}\|^{4/3}_{L^{2}_{\\#}(w_{\Delta})}.$

The key step is to lower the size of the Fourier support of $\tilde{F}_{i}$ in
(41), which is carried out in two steps. In the first step we show a ball
inflation estimate, which is based on the trilinear Kakeya inequality. In the
second step we use local $L^{2}$-orthogonality after suitable application of
Hölder’s inequality.

######  Proposition 5.2 (Ball inflation).

Let $Q$ be a $\kappa^{-2}$-square, which is tiled with a collection
$\mathcal{Q}$ of squares $\Delta$ with side length $\kappa^{-1}$ and
$\text{supp}(\hat{F}_{i})\subseteq\mathcal{N}_{\kappa^{2}}(\mathbb{P}^{2})$,
which satisfy (37).

For $\tilde{p}\geq 2$, the following estimate holds:

(42)
$\begin{split}&\frac{1}{|\mathcal{Q}|}\sum_{\Delta\in\mathcal{Q}}\prod_{i=1}^{3}\big{(}\sum_{I_{i,m}\in\mathbb{I}_{m}(U_{i})}\|\mathcal{P}_{I_{i,m}}F_{i}\|^{2}_{L^{\tilde{p}}_{\\#}(w_{\Delta})}\big{)}^{\tilde{p}/4}\\\
&\qquad\lesssim\nu^{-\frac{1}{2}}\log(\kappa^{-1})^{c}\prod_{i=1}^{3}\big{(}\sum_{I_{i,m}\in\mathbb{I}_{m}(U_{i})}\|\mathcal{P}_{I_{i,m}}F_{i}\|^{2}_{L^{\tilde{p}}_{\\#}(w_{Q})}\big{)}^{\tilde{p}/4}.\end{split}$

For the qualitative argument we refer to [15, Section 10.3]. By using the
quantitative multilinear Kakeya estimate, we can obtain the dependence of
decoupling on the transversality.

###### Proof.

We use the locally constant properties for each square $I$ of size $\kappa$ to
bring the multilinear Kakeya inequality into play. For
$I\in\mathbb{I}_{m}(U_{i})$ the Fourier support of $\mathcal{P}_{I}F_{i}$ is
inside a rectangle $\mathfrak{r}_{I}$ of size comparable to
$\kappa\times\kappa\times\kappa^{2}$. The long sides are given by $I$. We tile
the cube $4Q$ with translates of the dual rectangle
$T_{I}=\mathfrak{r}_{I}^{o}$, which family is denoted by $\mathcal{T}_{I}$. We
define

$g_{I}^{(i)}(x)=\sup_{y\in
2T_{I}(x)}\|\mathcal{P}_{I}F_{i}\|^{\tilde{p}}_{L^{\tilde{p}}_{\\#}(w_{Q(y,\kappa^{-1})})},$

which are constant on translates of $T_{I}$. It follows for $x\in\Delta$ that

(43)
$\|\mathcal{P}_{I}F_{i}\|^{\tilde{p}}_{L^{\tilde{p}}_{\\#}(w_{\Delta})}\leq
g_{I}^{(i)}(x)$

and (see [15, Eq. (11.15), p. 251])

(44) $\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{4Q}g_{I}^{(j)}(x)dx\lesssim\|\mathcal{P}_{I}F_{j}\|^{\tilde{p}}_{L^{\tilde{p}}_{\\#}(4Q)}.$

Moreover, by the definition of $g_{I}^{(i)}$ it follows

$g_{I}^{(i)}(x)=\sum_{T\in\mathcal{T}_{I}}c_{T}1_{T}.$

We let $g_{i}(x)=\sum_{I\in\mathbb{I}_{m}(U_{i})}g_{I}^{(i)}(x)$.

By dyadic pigeonholing we can suppose that
$\|\mathcal{P}_{I}F_{i}\|_{L^{\tilde{p}}_{\\#}(w_{\Delta})}$ have comparable
size. This incurs a factor $\log(\kappa^{-1})^{c}$. We assume that we have
$M_{i}$ squares $J_{i}$ with
$\|\mathcal{P}_{J_{i}}F_{i}\|_{L^{\tilde{p}}_{\\#}(w_{Q})}$ of comparable
size. By Hölder’s inequality we find

$\begin{split}\text{lhs}\eqref{eq:BallInflation}&\lesssim\log(\kappa^{-1})^{c}\big{(}\prod_{i=1}^{3}M_{i}^{\frac{1}{2}-\frac{1}{\tilde{p}}}\big{)}^{\frac{\tilde{p}}{2}}\frac{1}{|\mathcal{Q}|}\sum_{\Delta\in\mathcal{Q}}\prod_{i=1}^{3}\big{(}\sum_{I\in\mathbb{I}_{m}(U_{i})}\|\mathcal{P}_{I}F_{i}\|^{\tilde{p}}_{L^{\tilde{p}}_{\\#}(w_{\Delta})}\big{)}^{\frac{1}{2}}\\\
&\lesssim\log(\kappa^{-1})^{c}\big{(}\prod_{i=1}^{3}M_{i}^{\frac{1}{2}-\frac{1}{\tilde{p}}}\big{)}^{\frac{\tilde{p}}{2}}\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{Q}\big{(}\prod_{i=1}^{3}g_{i}\big{)}^{\frac{1}{2}}.\end{split}$

The transversality assumption (37) yields the quantitative transversality for
the families of tubes. An application of the endpoint trilinear Kakeya
estimate (33) gives

$\int\big{(}\prod_{i=1}^{3}g_{i}\big{)}^{\frac{1}{3}}\lesssim\nu^{-\frac{1}{2}}\delta^{-3}\big{(}\prod_{j=1}^{3}\sum_{T\in\mathcal{T}_{j}}c_{T}\int\chi_{T}/(\delta^{-4})\big{)}^{\frac{1}{2}}.$

Hence, by (44)

$\frac{1}{|Q|}\int\big{(}\prod_{i=1}^{3}g_{i}\big{)}^{\frac{1}{2}}\lesssim\big{(}\prod_{j=1}^{3}\frac{1}{|Q|}\int\sum_{T\in\mathcal{T}_{j}}c_{T}\chi_{T}\big{)}^{\frac{1}{2}}\lesssim\big{(}\prod_{j=1}^{3}\sum_{J_{j}}\|\mathcal{P}_{J_{j}}F_{j}\|^{\tilde{p}}_{L^{\tilde{p}}_{\\#}(w_{Q})}\big{)}^{\frac{1}{2}}.$

Since we assume that
$\|\mathcal{P}_{I}F_{i}\|_{L^{\tilde{p}}_{\\#}(w_{\Delta})}$ have comparable
size, it follows

$\big{(}\prod_{i=1}^{3}M_{i}^{\frac{1}{2}-\frac{1}{\tilde{p}}}\big{)}^{\frac{\tilde{p}}{2}}\big{(}\prod_{j=1}^{3}\sum_{J_{i}}\|\mathcal{P}_{J_{i}}F_{j}\|^{\tilde{p}}_{L^{\tilde{p}}_{\\#}(w_{Q})}\big{)}^{\frac{1}{2}}\lesssim\big{[}\prod_{j=1}^{3}\big{(}\sum_{J_{i}}\|\mathcal{P}_{J_{i}}F_{j}\|^{2}_{L^{\tilde{p}}_{\\#}(w_{Q})}\big{)}^{\frac{1}{2}}\big{]}^{\frac{\tilde{p}}{2}}.$

The proof is complete.

∎

By an application of Hölder’s inequality and relating $p$ and $\tilde{p}$ we
find the following (see [15, Proposition 10.23]):

######  Proposition 5.3 (Multiscale inequality).

Let $3\leq p\leq 4$ and

$\frac{3}{2p}=\frac{\kappa_{p}}{2}+\frac{1-\kappa_{p}}{p}.$

Then there is $c>0$ such that the following holds:

(45)
$A_{p}(m,Q^{2m},\underline{F},\delta)\lesssim\nu^{-\frac{1}{2p}}\log(\delta^{-1})^{c}A_{p}(2m,Q^{2m},\underline{F},\delta)^{\kappa_{p}}D_{p}(m,Q^{2m},\underline{F},\delta)^{1-\kappa_{p}}.$

###### Proof.

For $p\in[3,4]$ we have that $\tilde{p}=\frac{2p}{3}\geq 2$. This allows us to
use Hölder’s inequality:

$\frac{1}{|\mathcal{Q}|}\sum_{\Delta\in\mathcal{Q}}\prod_{i=1}^{3}\big{(}\sum_{I\in\mathbb{I}_{m}(U_{i})}\|\mathcal{P}_{I}F_{i}\|^{2}_{L^{2}_{\\#}(w_{\Delta})}\big{)}^{\frac{p}{6}}\lesssim\frac{1}{|\mathcal{Q}|}\sum_{\Delta\in\mathcal{Q}}\prod_{i=1}^{3}\big{(}\sum_{I\in\mathbb{I}_{m}(U_{1})}\|\mathcal{P}_{I}F_{i}\|^{2}_{L^{\tilde{p}}_{\\#}(w_{\Delta})}\big{)}^{\frac{\tilde{p}}{4}}.$

Now we can apply Proposition 5.2 to find

$\frac{1}{|\mathcal{Q}|}\sum_{\Delta\in\mathcal{Q}}\prod_{i=1}^{3}\big{(}\sum_{I\in\mathbb{I}_{m}(U_{i})}\|\mathcal{P}_{I}F_{i}\|^{2}_{L^{2}_{\\#}(w_{\Delta})}\big{)}^{\frac{p}{6}}\lesssim\nu^{-\frac{1}{2}}\log(\delta^{-1})^{c}\prod_{i=1}^{3}\big{(}\sum_{I\in\mathbb{I}_{m}(U_{i})}\|\mathcal{P}_{I}F_{i}\|^{2}_{L^{\frac{2p}{3}}_{\\#}(w_{Q})}\big{)}^{\frac{p}{6}}.$

An application of Hölder’s inequality gives

$\begin{split}\prod_{i=1}^{3}\big{(}\sum_{I\in\mathbb{I}_{m}(U_{i})}\|\mathcal{P}_{I}F_{i}\|^{2}_{L^{\frac{2p}{3}}_{\\#}(w_{Q})}\big{)}^{\frac{p}{6}}&\lesssim\prod_{i=1}^{3}\big{(}\sum_{I\in\mathbb{I}_{m}(U_{i})}\|\mathcal{P}_{I}F_{i}\|^{2}_{L^{2}_{\\#}(w_{Q})}\big{)}^{\frac{\kappa_{p}p}{6}}\\\
&\qquad\times\prod_{i=1}^{3}\big{(}\sum_{I\in\mathbb{I}_{m}(U_{i})}\|\mathcal{P}_{I}F_{i}\|^{2}_{L^{p}_{\\#}(w_{Q})}\big{)}^{\frac{(1-\kappa_{p})p}{6}}.\end{split}$

Finally, we can use local $L^{2}$-orthogonality (see [15, Proposition 10.20])
to find

$\sum_{I\in\mathbb{I}_{m}(U_{i})}\|\mathcal{P}_{I}F_{i}\|^{2}_{L^{2}_{\\#}(w_{Q})}\lesssim\sum_{I\in\mathbb{I}_{2m}(U_{i})}\|\mathcal{P}_{I}F_{i}\|^{2}_{L^{2}_{\\#}(w_{Q})}.$

The proof is complete. ∎

We obtain the following from iterating the multiscale inequality: Let

(46) $2^{-\underline{m}}=\frac{N_{3}}{N_{1}}\leq
N_{1}^{-\frac{\beta}{1+\beta}}\text{ and
}2^{\overline{m}}=N_{1}^{2-\alpha}=\delta^{-2}.$

Then by (41) and Proposition 5.3,

(47)
$\begin{split}\frac{1}{|Q^{\overline{m}}|}\int|F_{1}F_{2}F_{3}|^{p/3}dxdt&\lesssim
A_{p}^{p}(\underline{m},Q^{\overline{m}},\underline{F},\delta)\\\
&\lesssim\nu^{-\frac{1}{2}}\log(N_{1})^{c}A_{p}^{p}(2\underline{m},Q^{\overline{m}},\underline{F},\delta)^{\kappa_{p}}D_{p}^{p}(\underline{m},Q^{\overline{m}},\underline{F},\delta)^{1-\kappa_{p}}.\end{split}$

Iterating the multiscale inequality $n$ times, we find

(48)
$\begin{split}A_{p}^{p}(\underline{m},Q^{\overline{m}},\underline{F},\delta)&\lesssim\nu^{-\frac{1}{2}}\log(N_{1})^{c}\big{(}\prod_{\ell=1}^{n-1}\nu^{-\frac{\kappa_{p}^{\ell}}{2}}\log(N_{1})^{c\kappa_{p}^{\ell}}\big{)}\\\
&\quad\times
A_{p}^{p}(2^{n}\underline{m},Q^{\overline{m}},\underline{F},\delta)^{\kappa_{p}^{n}}\prod_{\ell=1}^{n}D_{p}^{p}(2^{\ell-1}\underline{m},Q^{\overline{m}},\underline{F},\delta)^{\kappa_{p}^{\ell-1}(1-\kappa_{p})}.\end{split}$

The number of iterations of the multiscale inequality is at most

(49) $n^{*}=\lceil\log_{2}(\beta^{-1}+1)\rceil$

because

$N_{1}^{-\frac{\beta}{1+\beta}2^{n^{*}}}\leq N_{1}^{-1}.$

###  5.4. Exiting the decoupling iteration with linear and bilinear Strichartz
estimates

We exit the decoupling iteration similar to the one-dimensional case by
abandoning all multilinearity and estimate the resulting expressions via
linear and bilinear Strichartz estimates.

####  5.4.1. Linear and bilinear Strichartz estimates

We recall the following linear Strichartz estimates due to Bourgain–Demeter
[6] combined with Galilean invariance.

######  Theorem 5.4 ([6]).

Let $S\subseteq B(\xi^{*},M)\subseteq\mathbb{R}^{2}$ and $f_{S}=\sum_{k\in
S}e^{ikx}a_{k}:\mathbb{T}_{\gamma}^{2}\to\mathbb{C}$. Then the following
estimate holds:

$\|e^{it\Delta}f_{S}\|_{L_{t}^{4}([0,1],L_{x}^{4}(\mathbb{T}_{\gamma}^{2}))}\lesssim_{\varepsilon}M^{\varepsilon}\|a_{S}\|_{\ell^{2}}.$

The estimate with logarithmically sharpened constant by lattice point counting

$\|e^{it\Delta}f_{S}\|_{L_{t,x}^{4}([0,1]\times\mathbb{T}^{2})}\lesssim\exp\big{(}\frac{c\log
M}{\log\log M}\big{)}\|f_{S}\|_{L^{2}}$

is due to Bourgain [1]. Very recently, Herr–Kwak [25] reported the estimate
with sharp constant $\log(M)^{\frac{1}{4}}$ for $M\geq 2$.

The bilinear Strichartz estimates due to Hani [24] in two dimensions are given
as follows:

######  Theorem 5.5.

Let $N_{2}\ll N_{1}$ and $S\subseteq B(\xi^{*},M)\subseteq B(0,N_{2})$. Then
it holds

$\|e^{it\Delta}P_{N_{1}}f_{1}e^{it\Delta}P_{N_{2}}P_{S}f_{2}\|_{L^{2}_{t,x}([0,N_{1}^{-1}]\times\mathbb{T}_{\gamma}^{2})}\lesssim\big{(}\frac{M}{N_{1}}\big{)}^{\frac{1}{2}}\|P_{N_{1}}f_{1}\|_{L^{2}}\|P_{N_{2}}f_{2}\|_{L^{2}}.$

We shall use the below estimate, which follows from piecing together intervals
of length $N_{1}^{-1}$ and using Galilean invariance:

######  Corollary 5.6.

Let $0<\alpha\leq 1$. Under the assumptions of Theorem 5.5, the following
estimate holds:

(50)
$\|e^{it\Delta}P_{N_{1}}f_{1}e^{it\Delta}P_{N_{2}}P_{S}f_{2}\|_{L^{2}_{t,x}([0,N_{1}^{-\alpha}]\times\mathbb{T}_{\gamma}^{2})}\lesssim\frac{M^{\frac{1}{2}}}{N_{1}^{\frac{\alpha}{2}}}\|P_{N_{1}}f_{1}\|_{L^{2}}\|P_{N_{2}}P_{S}f_{2}\|_{L^{2}_{x}}.$

####  5.4.2. Linear estimate

When exiting the decoupling iteration, we reverse the scaling and continuous
approximation. Let $I\in\mathbb{I}_{m_{1}}(B(0,1))$ and $\tilde{I}=N_{1}I$. We
indicate the Fourier support by subindices. Then it holds:

$\begin{split}\|\mathcal{P}_{I}F\|^{4}_{L^{4}(w_{N_{1}^{2-\alpha}})}&\lesssim
N_{1}^{4}N_{1}^{2(1-\alpha)}\int_{[0,N_{1}^{-\alpha}]\times\mathbb{T}_{\gamma}^{2}}|e^{it\Delta}a_{\tilde{I}}|^{4}dxdt\\\
&\lesssim_{\varepsilon}N_{1}^{4}N_{1}^{2(1-\alpha)}N_{1}^{\varepsilon}\|a_{\tilde{I}}\|_{\ell^{2}}^{4}.\end{split}$

This provides us with the following estimate for $D_{4}$:

######  Lemma 5.7.

With notations from (46), the following estimate holds:

(51)
$\begin{split}D_{4}^{4}(m,Q,\underline{F},\delta)&=\frac{1}{|Q|}\big{[}\prod_{i=1}^{3}\big{(}\sum_{I_{i,m}\in\mathbb{I}_{m}}\|P_{I_{i,m}}F_{i}\|^{2}_{L^{4}(w_{Q^{r}})}\big{)}^{\frac{1}{2}}\big{]}^{\frac{4}{3}}\\\
&\lesssim_{\varepsilon}\frac{N_{1}^{4+2(1-\alpha)}}{|Q|}N_{1}^{\varepsilon}\big{(}\prod_{i=1}^{3}\|f_{i}\|_{L^{2}}\big{)}^{\frac{4}{3}}.\end{split}$

####  5.4.3. Bilinear estimates

For $A^{4}_{4}$ we expect by the locally constant property

(52)
$\begin{split}&\quad\frac{1}{|\mathcal{Q}_{m,r}|}\sum_{\Delta_{m}\in\mathcal{Q}_{m,r}(Q^{r})}\big{(}\prod_{i=1}^{3}\big{(}\sum_{I_{i,m}}\|\mathcal{P}_{I_{i,m}}F_{i}\|^{2}_{L^{2}_{\\#}(w_{\Delta_{m}})}\big{)}^{\frac{1}{2}}\big{)}^{\frac{p}{3}}\\\
&\approx\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{Q}\prod_{i=1}^{3}\big{(}\sum_{I}|\mathcal{P}_{I}F_{i}|^{2}\big{)}^{\frac{p}{6}}\\\
&\lesssim\frac{1}{|Q|}\int_{Q}\big{(}\sum_{I_{1}}|\mathcal{P}_{I_{1}}F_{1}|^{2}\big{)}^{\frac{2}{3}}\big{(}\sum_{I_{2}}|\mathcal{P}_{I_{2}}F_{2}|^{2}\big{)}^{\frac{2}{3}}\big{(}\sum_{I_{3}}|\mathcal{P}_{I_{3}}F_{3}|^{2}\big{)}^{\frac{2}{3}}.\end{split}$

Next, we use Hölder’s inequality to find

$\begin{split}&\quad\big{(}\int_{Q}\sum_{I_{1}}|\mathcal{P}_{I_{1}}F_{1}|^{2}\sum_{I_{3}}|\mathcal{P}_{I_{3}}F_{3}|^{2}\big{)}^{\frac{2}{3}}\big{(}\int_{Q}\big{(}\sum_{I}|\mathcal{P}_{I}F_{2}|^{2}\big{)}^{2}\big{)}^{\frac{1}{3}}\\\
&=\big{(}\sum_{I_{1},I_{3}}\int_{Q}|\mathcal{P}_{I_{1}}F_{1}\mathcal{P}_{I_{3}}F_{3}|^{2}\big{)}^{\frac{2}{3}}\big{(}\int_{Q}\big{(}\sum_{I}|\mathcal{P}_{I}F_{2}|^{2}\big{)}^{4/2}\big{)}^{\frac{4}{12}}.\end{split}$

After inverting the scaling and reversing the continuous approximation, the
above expression can be estimated by short-time bilinear Strichartz estimate
and linear Strichartz estimates as recorded above.

######  Proposition 5.8.

With above notations, the following estimate holds:

(53)
$A_{4}^{4}(\underline{m},Q^{\overline{m}},\underline{F},\delta)\lesssim_{\varepsilon}\frac{1}{|Q|}N_{1}^{4}N_{1}^{2(1-\alpha)}N_{1}^{-\frac{\alpha}{3}}N_{1}^{\varepsilon}\big{(}\prod_{i=1}^{3}\|f_{i}\|_{2}\big{)}^{\frac{4}{3}}.$

###### Proof.

To make $\approx$ in (52) precise, we use the uncertainty principle. With the
Fourier transform of $\mathcal{P}_{I_{i,m}}F_{i}$ being supported in a
$\delta\times\delta\times\delta$-box, by Lemma 3.1 we find

$|\mathcal{P}_{I_{i,m}}F_{i}|^{2}(x)\lesssim|\mathcal{P}_{I_{i,m}}F_{i}|^{2}*\phi_{I}(x)$

with $\phi_{I}$ rapidly decaying off
$\delta^{-1}\times\delta^{-1}\times\delta^{-1}$ and locally constant at scale
$\delta^{-1}$.

We have for any $x\in Q_{m}$ by locally constant property of $\phi_{I}$

$\begin{split}\|\mathcal{P}_{I_{im}}F_{i}\|^{2}_{L^{2}_{\\#}(w_{Q_{m}})}&=\frac{1}{|Q_{m}|}\int\big{(}|\mathcal{P}_{I_{im}}F_{i}|^{2}*\phi_{I}\big{)}w_{Q_{m}}\\\
&=\sum_{k_{i}\in\mathbb{Z}^{3}}\frac{1}{|Q_{m}|}\int_{Q_{m}+k_{i}\delta^{-1}}\big{(}|\mathcal{P}_{I_{i,m}}F_{i}|^{2}*\phi_{I}\big{)}w_{Q_{m}}\\\
&\lesssim\sum_{k_{i}\in\mathbb{Z}^{3}}(|\mathcal{P}_{I_{i,m}}F_{i}|^{2}*\phi_{I_{i,m}})(x+k_{i}\delta^{-1})\omega(k_{i}).\end{split}$

We have discretized $w_{Q_{m}}$ by $\omega(k)\lesssim(1+|k|)^{-100}$. This
implies the estimate:

$\begin{split}&\quad\frac{1}{|\mathcal{Q}_{m,r}|}\sum_{Q^{m}\in\mathcal{Q}_{m,r}}\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{Q^{m}}\big{(}\prod_{i=1}^{3}\big{(}\sum_{I_{i,m}}\|\mathcal{P}_{I_{i,m}}F_{i}\|^{2}_{L^{2}_{\\#}(w_{Q^{m}})}\big{)}^{\frac{1}{2}}\big{)}^{\frac{p}{3}}\\\
&\lesssim\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{Q}\big{(}\prod_{i=1}^{3}\big{(}\sum_{I_{i,m}}\sum_{k_{i}\in\mathbb{Z}^{3}}\big{(}|\mathcal{P}_{I_{i,m}}F_{i}|^{2}*\phi_{I_{i,m}}\big{)}(x+k_{i}\delta^{-1})\omega(k_{i})\big{)}^{\frac{1}{2}}\big{)}^{\frac{p}{3}}.\end{split}$

We use Hölder’s inequality to obtain

$\begin{split}&\quad\int_{Q}\prod_{j=1}^{3}\big{(}\sum_{I_{j},k_{j}}|\mathcal{P}_{I_{j}}F_{j}|^{2}*\phi_{I}(z+k_{j}\delta^{-1})\omega(k_{j})\big{)}^{\frac{2}{3}}\\\
&\lesssim\big{(}\sum_{\begin{subarray}{c}I_{1},k_{1},\\\
I_{3},k_{3}\end{subarray}}\int_{Q}|\mathcal{P}_{I_{1}}F_{1}|^{2}*\phi_{I_{1}}(z+k_{1}\delta^{-1})\omega(k_{1})\;|\mathcal{P}_{I_{3}}F_{3}|^{2}*\phi_{I_{2}}(z+k_{3}\delta^{-1})\omega(k_{3})\big{)}^{\frac{2}{3}}\\\
&\quad\times\big{(}\int_{Q}\big{(}\sum_{I_{3},k_{3}}|\mathcal{P}_{I_{3}}F_{3}|^{2}*\phi_{I_{3}}(z+k_{3}\delta^{-1})\omega(k_{3})\big{)}^{2}\big{)}^{\frac{1}{3}}.\end{split}$

The proof is concluded with Lemmas 5.9 and 5.10 below. ∎

######  Lemma 5.9.

The following estimate holds:

(54)
$\begin{split}&\quad\int_{Q}|\mathcal{P}_{I_{1}}F_{1}|^{2}*\phi_{I_{1}}(z+k_{1}\delta^{-1})|\mathcal{P}_{I_{2}}F_{2}|^{2}*\phi_{I_{2}}(z+k_{2}\delta^{-1})dz\\\
&\lesssim
N_{1}^{4}N_{1}^{2(1-\alpha)}N_{1}^{-\frac{\alpha}{2}}\|b_{1,I_{1}}\|^{2}_{\ell^{2}}\|b_{2,I_{2}}\|^{2}_{\ell^{2}}.\end{split}$

The proof is similar to the proof of Lemma 3.3. Here we need the separation
condition (12) to use the bilinear Strichartz estimate.

###### Proof.

We write $\mathcal{P}_{I_{j}}F_{j}=\mathcal{E}b_{j,I_{j}}\cdot
w_{B_{N_{1}^{2-\alpha}}}$ and estimate

$\begin{split}&\quad\int_{Q}|\mathcal{P}_{I_{1}}F_{1}|^{2}*\phi_{I_{1}}(x+k_{1}\delta^{-1})|\mathcal{P}_{I_{2}}F_{2}|^{2}*\phi_{I_{2}}(x+k_{2}\delta^{-1})dx\\\
&\lesssim\iint\int_{Q}|\mathcal{E}b_{1,I_{1}}|^{2}(z+k_{1}\delta^{-1}-y_{1})|\mathcal{E}b_{2,I_{2}}|^{2}(z+k_{2}\delta^{-1}-y_{2})dz\;\phi_{I}(y_{1})dy_{1}\;\phi_{I}(y_{2})dy_{2}.\end{split}$

We prove an estimate for the space-time integral over $Q$ which is independent
of the shifts $k_{i}\delta^{-1}-y_{i}$ by absorbing the dilation into
$b_{I_{1}}$:

$\int_{Q}|\mathcal{E}\tilde{f}_{1,I_{1}}|^{2}(z+k_{1}\delta^{-1}-y_{1})|\mathcal{E}\tilde{f}_{2,I_{2}}|^{2}(z+k_{2}\delta^{-1}-y_{2})dz=\int_{Q}|\mathcal{E}f^{\prime}_{1,I_{1}}|^{2}(z)|\mathcal{E}f^{\prime}_{2,I_{2}}|^{2}(z)dz.$

Now we can reverse the change of variables and the continuous approximation to
find

$\lesssim
N_{1}^{4}N_{1}^{2(1-\alpha)}\int_{[0,N_{1}^{-\alpha}]\times\mathbb{T}^{2}_{\gamma}}|e^{it\Delta}f^{\prime}_{1,I_{1}}|^{2}(x,t)|e^{it\Delta}f^{\prime}_{2,I_{2}}|^{2}(x,t)dxdt.$

The functions $f_{i}$ are Fourier supported in balls of size
$N_{1}^{\alpha/2}$ and separated at a distance comparable to $N_{1}$. We can
estimate this by the short-time bilinear Strichartz estimate to find

$\int_{[0,N_{1}^{-\alpha}]\times\mathbb{T}^{2}_{\gamma}}|e^{it\Delta}f^{\prime}_{I_{1}}|^{2}|e^{it\Delta}f^{\prime}_{I_{2}}|^{2}dxdt\lesssim
N_{1}^{-\frac{\alpha}{2}}\|b_{1,I_{1}}\|^{2}_{\ell^{2}}\|b_{2,I_{2}}\|^{2}_{\ell^{2}}.$

∎

We turn to the estimate for the linear part:

######  Lemma 5.10.

The following estimate holds:

$\int_{Q}\big{(}\sum_{I_{2},k_{2}}|\mathcal{P}_{I_{2}}F_{2}|^{2}*\phi_{I_{2}}(x+k_{2}\delta^{-1})\omega(k_{2})\big{)}^{2}\lesssim_{\varepsilon}N_{1}^{4}N_{1}^{2(1-\alpha)}N_{1}^{\varepsilon}\|f_{2}\|_{L^{2}}^{4}.$

###### Proof.

We write the left hand-side as

$\begin{split}&\quad\int_{Q}\sum_{I_{21},k_{21}}|\mathcal{P}_{I_{21}}F_{2}|^{2}*\phi_{I_{21}}(x+k_{21}\delta^{-1})\omega(k_{21})\sum_{I_{22},k_{22}}|\mathcal{P}_{I_{22}}F_{2}|^{2}*\phi_{I_{22}}(x+k_{22}\delta^{-1})\omega(k_{22})\\\
&=\sum_{\begin{subarray}{c}I_{21},k_{21},\\\
I_{22},k_{22}\end{subarray}}\omega(k_{21})\omega(k_{22})\int_{Q}\big{(}|\mathcal{P}_{I_{21}}F_{2}|^{2}*\phi_{21}(x+k_{21}\delta^{-1})\big{)}\big{(}|\mathcal{P}_{I_{22}}F_{2}|^{2}*\phi_{I_{22}}(x+k_{22}\delta^{-1})\big{)}.\end{split}$

We prove the estimate

$\begin{split}&\quad\int_{Q}|\mathcal{P}_{I_{21}}F_{2}|^{2}*\phi_{I_{21}}(x+k_{21}\delta^{-1})|\mathcal{P}_{I_{22}}F_{2}|^{2}*\phi_{I_{22}}(x+k_{22}\delta^{-1})\\\
&\lesssim_{\varepsilon}N_{1}^{4}N_{1}^{2(1-\alpha)}N_{1}^{\varepsilon}\|b_{2,I_{21}}\|_{\ell^{2}}^{2}\|b_{2,I_{22}}\|^{2}_{\ell^{2}},\end{split}$

which is independent of $k_{21}$, $k_{22}$.

The argument follows along the lines of the proof of (54):

$\iint\int_{Q}|\mathcal{P}_{I_{21}}F_{2}|^{2}(x+k_{21}\delta^{-1}-y_{1})|\mathcal{P}_{I_{22}}F_{2}|^{2}(x+k_{22}\delta^{-1}-y_{2})dz\,\phi_{I_{21}}(y_{1})\phi_{I_{22}}(y_{2})dy_{1}dy_{2}$

and turn to an estimate of the integral independent of $k_{21}$, $k_{22}$,
$y_{1}$, $y_{2}$. Write

$\mathcal{P}_{I_{2j}}F_{2}=\mathcal{E}\tilde{f}_{I_{2j}}\cdot
w_{B_{N_{1}^{2-\alpha}}}.$

We can estimate the weight with a positive constant. Like above we can absorb
the shift $k_{2i}\delta^{-1}-y_{i}$ into the argument $\tilde{f}_{I_{i}}$ and
reversing the scaling and the continuous approximation we obtain

$\begin{split}&\quad\int_{Q}|\mathcal{P}_{I_{21}}F_{2}|^{2}(z+k_{21}\delta^{-1}-y_{1})|\mathcal{P}_{I_{22}}F_{2}|^{2}(z+k_{22}\delta^{-1}-y_{2})dz\\\
&\lesssim
N_{1}^{4}N_{1}^{2(1-\alpha)}\int_{[0,N_{1}^{-\alpha}]\times\mathbb{T}^{2}_{\gamma}}|e^{it\Delta}f^{\prime}_{I_{21}}|^{2}|e^{it\Delta}f^{\prime}_{I_{22}}|^{2}dxdt.\end{split}$

This is estimated by Hölder’s inequality and the $L^{4}_{t,x}$-Strichartz
estimates:

$\int_{[0,N_{1}^{-\alpha}]\times\mathbb{T}^{2}_{\gamma}}|e^{it\Delta}\tilde{f}_{I_{21}}|^{2}|e^{it\Delta}\tilde{f}_{I_{22}}|^{2}dtdx\lesssim_{\varepsilon}N_{1}^{\varepsilon}\|b_{I_{21}}\|^{2}_{\ell^{2}}\|b_{I_{22}}\|^{2}_{\ell^{2}}.$

∎

###  5.5. Conclusion

Now we can estimate (47) and (48) in terms of the original functions $f_{i}$.
Recall that the number of iterations is estimated by

$n^{*}=\lceil\log_{2}(\beta^{-1}+1)\rceil.$

For $p=4$ we obtain as consequence of $\kappa_{4}=\frac{1}{2}$:

$\nu^{-\frac{1}{2}}\log(N_{1})^{c}\big{(}\prod_{\ell=1}^{n-1}\nu^{-\frac{\kappa_{p}^{\ell}}{2}}\log(N_{1})^{c\kappa_{p}^{\ell}}\big{)}\leq\nu^{-1+2^{-n}}\log(N_{1})^{2c}.$

We find by Lemma 5.7 and Proposition 5.8 that

(55) $\begin{split}&\quad
A_{p}^{p}(2^{n}\underline{m},Q^{\overline{m}},\underline{F},\delta)^{\kappa_{p}^{n}}\prod_{\ell=1}^{n}D_{p}^{p}(2^{\ell-1}\underline{m},Q^{\overline{m}},\underline{F},\delta)^{\kappa_{p}^{\ell}}\\\
&\lesssim_{\varepsilon}\frac{1}{|Q|}N_{1}^{4}N_{1}^{2(1-\alpha)}N_{1}^{-\frac{\alpha}{3}2^{-n}}N_{1}^{\varepsilon}\big{(}\prod_{i=1}^{3}\|f_{i}\|_{2}\big{)}^{\frac{4}{3}}.\end{split}$

By the bound on the number of iterations, we find

$\nu^{-1+2^{-n}}=\nu^{-1+\frac{\beta}{2(1+\beta)}}.$

Consequently, it holds

$\eqref{eq:MultiScaleEstimateI}\lesssim_{\varepsilon}\frac{1}{|Q|}N_{1}^{4}N_{1}^{2(1-\alpha)}N_{1}^{-\frac{\alpha\beta}{12}}N_{1}^{\varepsilon}\big{(}\prod_{i=1}^{3}\|f_{i}\|_{2}\big{)}^{\frac{4}{3}}.$

Taking the above estimates together we find

$\int_{[0,N_{1}^{-\delta}]\times\mathbb{T}_{\gamma}^{2}}\big{|}e^{it\Delta}f_{1}e^{it\Delta}f_{2}e^{it\Delta}f_{3}\big{|}^{\frac{4}{3}}dxdt\lesssim_{\varepsilon}\nu^{-1+\frac{\beta}{2(1+\beta)}}N_{1}^{-\frac{\alpha\beta}{12}+\varepsilon}\big{(}\prod_{i=1}^{3}\|f_{i}\|_{2}\big{)}^{\frac{4}{3}}.$

The proof of Theorem 1.4 is complete. $\hfill\Box$

###  5.6. Comparison with $L^{3}$-estimates

In this subsection we compare the obtained estimates to an $L^{3}$-estimate
with sharp dependence on transversality.

####  5.6.1. The trilinear restriction estimate

We have the following trilinear $L^{3}_{t,x}$-estimate with sharp dependence
on transversality, which is due to Ramos [36]:

######  Theorem 5.11.

Let $R\gg 1$, $\nu\in(0,1]$, and $f_{i}:\mathbb{R}^{2}\to\mathbb{C}$ with
$\text{supp}(f_{i})\subseteq B_{2}(0,1)\subseteq\mathbb{R}^{2}$ with
$|\mathfrak{n}(\xi_{1})\wedge\mathfrak{n}(\xi_{2})\wedge\mathfrak{n}(\xi_{3})|\gtrsim\nu$
for $\xi_{i}\in\text{supp}(f_{i})$. Then there is $c>0$ such that the
following estimate holds:

$\int_{B_{3}(0,R)}\big{|}\prod_{i=1}^{3}\mathcal{E}f_{i}\big{|}\lesssim\log(R)^{c}\nu^{-\frac{1}{2}}\prod_{i=1}^{3}\|f_{i}\|_{L^{2}(\mathbb{R}^{2})}.$

This yields the following decoupling estimate as observed by Bourgain in [4]
(see [15, Section 9.2] for further context):

######  Corollary 5.12.

Let $f_{i}\in\mathcal{S}(\mathbb{R}^{3})$, $i=1,2,3$, $B^{(i)}\subseteq
B_{2}(0,1)$ with
$\text{supp}(\hat{f}_{i})\subseteq\mathcal{N}_{R^{-1}}(\\{(\xi,-|\xi|^{2}):\xi\in
B^{(i)}\\})$, and it holds for $\xi_{i}\in B^{(i)}$

$|\mathfrak{n}(\xi_{1})\wedge\mathfrak{n}(\xi_{2})\wedge\mathfrak{n}(\xi_{3})|\gtrsim\nu.$

Then the following estimate holds:

(56)
$\int_{B_{3}(0,R)}\big{|}\prod_{i=1}^{3}f_{i}\big{|}\lesssim\log(R)^{c}\nu^{-\frac{1}{2}}\prod_{i=1}^{3}\big{(}\sum_{\begin{subarray}{c}\theta_{i}\subseteq
B^{(i)}:\\\
\theta_{i}\;R^{-\frac{1}{2}}-ball\end{subarray}}\|f_{i,\theta_{i}}\|^{2}_{L^{3}(w_{B_{R}})}\big{)}^{\frac{1}{2}}.$

We obtain the following consequence:

######  Proposition 5.13.

Let $N\in 2^{\mathbb{N}_{0}}$, $\nu\in(0,1]$, $\alpha\in(0,1]$, and
$f_{i}:\mathbb{T}^{2}_{\gamma}\to\mathbb{C}$ with
$\text{supp}(\hat{f}_{i})\subseteq B^{(i)}\subseteq B(0,N)$. We suppose for
the area of the triangle $\Delta(\xi_{1},\xi_{2},\xi_{3})$ spanned by
$\xi_{i}\in B^{(i)}$:

$|\Delta(\xi_{1},\xi_{2},\xi_{3})|\gtrsim N^{2}\nu.$

Then the following estimate holds:

$\int_{[0,N^{-\alpha}]\times\mathbb{T}^{2}_{\gamma}}\prod_{i=1}^{3}\big{|}e^{it\Delta}f_{i}\big{|}dxdt\lesssim\log(N)^{c}N^{-\frac{\alpha}{2}}\nu^{-\frac{1}{2}}\prod_{i=1}^{3}\|f_{i}\|_{L^{2}(\mathbb{T}^{2}_{\gamma})}.$

###### Proof.

The argument relies on continuous approximation and the decoupling estimate
with sharp dependence on transversality from Corollary 5.12. By the rescaling
argument and spatial periodicity we obtain like above

$\begin{split}\int_{[0,N^{-\alpha}]\times\mathbb{T}^{2}_{\gamma})}\prod_{i=1}^{3}\big{|}e^{it\Delta}f_{i}\big{|}&=N^{-4}\int_{[0,N^{2-\alpha}]\times
N\mathbb{T}^{2}_{\gamma}}\prod_{i=1}^{3}\big{|}\sum_{k_{i}\in\mathbb{Z}^{2}/N}e^{i(k_{i}x-tk_{i}^{2})}b_{i,k_{i}}\big{|}\\\
&\lesssim
N^{-4}N^{-2(1-\alpha)}\int_{B_{N^{2-\alpha}}}\prod_{i=1}^{3}|\mathcal{E}\tilde{f}_{i}|dxdt.\end{split}$

We include a weight $w_{B_{N^{2-\alpha}}}^{3}$ and let
$\mathcal{E}\tilde{f}_{i}\cdot w_{B_{N^{2-\alpha}}}=g_{i}$ to use Corollary
5.12 with $R=N^{2-\alpha}$ and $\tilde{\theta}_{i}$ of size
$N^{-(1-\frac{\alpha}{2})}\times N^{-(1-\frac{\alpha}{2})}\times
N^{-(2-\alpha)}$, which yields

$\int_{B_{N^{2-\alpha}}}\big{|}\prod_{i=1}^{3}g_{i}\big{|}\lesssim\log(N)^{c}\nu^{-\frac{1}{2}}\prod_{i=1}^{3}\big{(}\sum_{\tilde{\theta}_{i}}\|g_{i,\tilde{\theta}_{i}}\|^{2}_{L^{3}(w_{B_{N^{2-\alpha}}})}\big{)}^{\frac{1}{2}}.$

We reverse the continuous approximation like previously to obtain111Here we
gloss over the summation of the weights on the right hand side, which is
handled like in previous sections.

(57)
$\int_{[0,N^{-\alpha}]\times\mathbb{T}^{2}_{\gamma}}\prod_{i=1}^{3}|e^{it\Delta}f_{i}|\lesssim\log(N)^{c}\nu^{-\frac{1}{2}}\prod_{i=1}^{3}\big{(}\sum_{\theta_{i}:N^{\frac{\alpha}{2}}-\text{ball}}\|e^{it\Delta}f_{i,\theta_{i}}\|^{2}_{L^{3}([0,N^{-\alpha}]\times\mathbb{T}^{2}_{\gamma})}\big{)}^{\frac{1}{2}}.$

We estimate by Hölder’s inequality in time and Bernstein’s inequality for
$\theta_{i}$ an $N^{\frac{\alpha}{2}}$-ball:

(58)
$\|e^{it\Delta}f_{i,\theta_{i}}\|_{L^{3}([0,N^{-\alpha}]\times\mathbb{T}^{2}_{\gamma})}\lesssim
N^{-\frac{\alpha}{3}}N^{\alpha\big{(}\frac{1}{2}-\frac{1}{3}\big{)}}\|f_{i,\theta_{i}}\|_{L^{2}(\mathbb{T}^{2}_{\gamma})}.$

Plugging (58) into (57) yields the claimed estimate. ∎

####  5.6.2. An extrapolation conjecture

In the following let $f_{i}:\mathbb{R}^{2}\to\mathbb{C}$ with
$\text{supp}(f_{i})\subseteq B_{2}(0,1)$ for $i=1,2,3$ and define

$\text{Dec}_{p,R}(f)=\big{(}\sum_{\theta_{i}:R^{-\frac{1}{2}}-\text{ball}}\|\mathcal{E}f_{\theta_{i}}\|^{2}_{L^{p}(w_{B_{R}})}\big{)}^{\frac{1}{2}}.$

We observe the following inequality from Plancherel’s theorem and Hölder’s
inequality:

(59)
$\big{(}\int_{B_{3}(0,R)}\prod_{i=1}^{3}\big{|}\mathcal{E}f_{i}\big{|}^{\frac{2}{3}}\big{)}^{\frac{1}{2}}\lesssim\big{(}\prod_{i=1}^{3}\big{(}\text{Dec}_{2,R}(f_{i})\big{)}^{\frac{1}{3}}.$

In the previous paragraph we have seen that under the additional
transversality assumption

(60)
$|\mathfrak{n}(\xi_{1})\wedge\mathfrak{n}(\xi_{2})\wedge\mathfrak{n}(\xi_{3})|\gtrsim\nu$

for $\xi_{i}\in\text{supp}(f_{i})$, the following $L^{3}$-decoupling estimate
holds:

(61)
$\big{\|}\prod_{i=1}^{3}\big{|}\mathcal{E}f_{i}\big{|}^{\frac{1}{3}}\|_{L^{3}(B(0,R))}\lesssim\log(R)^{c}\nu^{-\frac{1}{6}}\big{(}\prod_{i=1}^{3}\text{Dec}_{3,R}(f_{i})\big{)}^{\frac{1}{3}}.$

This leads us to conjecture that under the transversality assumption (60) it
holds

(62)
$\big{(}\int_{B_{R}}\prod_{i=1}^{3}\big{|}\mathcal{E}f_{i}\big{|}^{\frac{4}{3}}\big{)}^{\frac{1}{4}}\lesssim\log(R)^{c}\nu^{-\frac{1}{4}}\big{(}\prod_{i=1}^{3}\text{Dec}_{4,R}(f_{i})\big{)}^{\frac{1}{3}}.$

##  6\. Refined trilinear Strichartz estimates on rescaled tori

In this section we translate the previously proved Strichartz estimates on the
unit torus for frequency-dependent times to the large $\lambda$-torus on the
unit time scale. The latter is the common setting for the $I$-method.

###  6.1. Preliminaries

We use the following conventions: Let $\alpha_{1}=1$ and
$\alpha_{i}\in(1/2,1]$ for $i=2,\ldots,d$. We endow the (possibly irrational)
torus

$\mathbb{T}^{d}_{\lambda}:=(\lambda\mathbb{T})^{d}=\prod_{i=1}^{d}(\mathbb{R}/(2\pi\lambda\alpha_{i}\mathbb{Z}))$

with Lebesgue measure, and we write

$\|f\|_{L^{p}(\mathbb{T}^{d}_{\lambda})}=\|f\|_{L^{p}_{\lambda}}=\big{(}\int_{\mathbb{T}^{d}_{\lambda}}|f(x)|^{p}dx\big{)}^{\frac{1}{p}}$

for $1\leq p<\infty$ with the usual modification for $p=\infty$. Note that we
do not distinguish between rational and irrational tori to lighten the
notation and since the following estimates are independent of rationality.

Denote $\mathbb{Z}_{\lambda}=\mathbb{Z}/\lambda$. The Fourier coefficients of
$f\in L^{1}(\mathbb{T}^{d}_{\lambda})$ are given by

$\hat{f}(k)=\int_{\mathbb{T}^{d}_{\lambda}}e^{-ikx}f(x)dx,\quad
k\in(\mathbb{Z}_{\lambda}/\alpha_{1})\times(\mathbb{Z}_{\lambda}/\alpha_{2})\ldots\times(\mathbb{Z}_{\lambda}/\alpha_{d})=:\mathbb{Z}_{\lambda}^{d}.$

To be consistent with the notation in physical space, we suppress the
$\alpha$-dependence in frequency space. The Fourier inversion formula is given
by

$f(x)=\lambda^{-d}\sum_{k\in\mathbb{Z}^{d}_{\lambda}}e^{ikx}\hat{f}(k).$

The prefactor normalizes the counting measure such that for $\lambda\to\infty$
the normalized counting measure weakly converges to Lebesgue measure.
Moreover, Plancherel’s theorem holds:

$\|f\|^{2}_{L^{2}_{\lambda}}=\frac{1}{\lambda^{d}}\sum_{k\in\mathbb{Z}_{\lambda}^{d}}|\hat{f}(k)|^{2}.$

We record the following improved linear Strichartz estimates, which follow
from the unit-time Strichartz estimates on the unit torus (cf. [1, 6]) and the
short-time Strichartz estimates without loss (cf. [7]) after rescaling:

######  Proposition 6.1.

Let $\lambda\geq 1$, $d\in\\{1,2\\}$, $N\in 2^{\mathbb{N}}$, and
$p=\frac{2(d+2)}{d}$. The following estimate holds:

$\|P_{N}U_{\lambda}(t)f\|_{L^{p}_{t,x}([0,1]\times\mathbb{T}^{d}_{\lambda})}\lesssim
L_{d}(\lambda,N)\|f\|_{L^{2}_{\lambda}}$

with

(63) $L_{d}(\lambda,N)=1\text{ for }\lambda\geq N\text{, and
}L_{d}(\lambda,N)=\begin{cases}C_{\varepsilon}\log(N)^{2+\varepsilon},&d=1\\\
C_{\varepsilon}N^{\varepsilon},&d=2\end{cases}\text{ for }N\geq\lambda.$

###### Proof.

We write

(64)
$\begin{split}\|P_{N}U_{\lambda}(t)f\|^{p}_{L^{p}_{t,x}([0,1]\times\mathbb{T}^{d}_{\lambda})}&=\big{\|}\big{(}\frac{1}{\lambda}\big{)}^{d}\sum_{\begin{subarray}{c}|k|\sim
N,\\\ k\in\mathbb{Z}^{d}_{\lambda}\end{subarray}}e^{i(kx-
tk^{2})}a_{k}\big{\|}^{p}_{L^{p}_{t,x}([0,1]\times\mathbb{T}^{d}_{\lambda})}\\\
&=\lambda^{-dp}\big{\|}\sum_{\begin{subarray}{c}|k^{\prime}|\sim N\lambda,\\\
k^{\prime}\in\mathbb{Z}\end{subarray}}e^{i\big{(}\frac{k^{\prime}x}{\lambda}-\frac{t(k^{\prime})^{2}}{\lambda^{2}}\big{)}}a_{k^{\prime}/\lambda}\big{\|}^{p}_{L^{p}_{t,x}([0,1]\times\mathbb{T}^{d}_{\lambda})}.\end{split}$

By change of variables $x^{\prime}=x/\lambda$, $t^{\prime}=t/\lambda^{2}$ we
find

$\eqref{eq:RescaledLinearStrichartz1}=\lambda^{-dp}\lambda^{2+d}\big{\|}\sum_{\begin{subarray}{c}|k^{\prime}|\sim
N\lambda,\\\
k^{\prime}\in\mathbb{Z}\end{subarray}}e^{i(k^{\prime}x^{\prime}-t^{\prime}(k^{\prime})^{2})}a_{k^{\prime}/\lambda}\big{\|}^{p}_{L^{p}_{t,x}([0,\lambda^{-2}]\times\mathbb{T}^{d})}.$

If $\lambda\geq N$, we can use the short-time Strichartz estimates due to
Burq–Gérard–Tzvetkov [7] without loss to find

$\lesssim\lambda^{-dp}\lambda^{2+d}\big{(}\sum_{\begin{subarray}{c}|k^{\prime}|\sim
N\lambda,\\\
k^{\prime}\in\mathbb{Z}\end{subarray}}a_{k^{\prime}/\lambda}^{2}\big{)}^{\frac{p}{2}}.$

This implies the claim taking $p$th root.

If $\lambda\leq N$, we can use the finite-time Strichartz estimates on the
torus to obtain in case $d=1$:

$\begin{split}&\quad\lambda^{-6}\lambda^{3}\big{\|}\sum_{\begin{subarray}{c}|k^{\prime}|\sim
N\lambda,\\\
k^{\prime}\in\mathbb{Z}\end{subarray}}e^{i(k^{\prime}x^{\prime}-t^{\prime}(k^{\prime})^{2})}a_{k^{\prime}}\big{\|}^{6}_{L^{6}_{t,x}([0,\lambda^{-2}]\times\mathbb{T}^{d})}\\\
&\lesssim\lambda^{-3}\big{(}\log(N\lambda)^{2+\varepsilon}\big{)}^{6}\big{(}\sum_{\begin{subarray}{c}|k^{\prime}|\sim
N\lambda,\\\
k^{\prime}\in\mathbb{Z}\end{subarray}}|a_{k^{\prime}}|^{2}\big{)}^{3}.\end{split}$

If $\lambda\leq N$ and $d=2$, we find

$\begin{split}&\quad\lambda^{-8}\lambda^{4}\big{\|}\sum_{\begin{subarray}{c}|k^{\prime}|\sim
N\lambda,\\\
k^{\prime}\in\mathbb{Z}\end{subarray}}e^{i(k^{\prime}x^{\prime}-t^{\prime}(k^{\prime})^{2})}a_{k^{\prime}}\big{\|}^{4}_{L^{4}_{t,x}([0,\lambda^{-2}]\times\mathbb{T}^{2})}\\\
&\lesssim(\lambda N)^{\varepsilon}\|f\|_{L^{2}_{\lambda}}^{4}.\end{split}$

∎

In one dimension, we shall also use the following rescaled bilinear Strichartz
estimate, which is [14, Proposition 3.7]:

######  Proposition 6.2.

Let $\eta\in C^{\infty}_{c}([-2,2])$ and $\eta\equiv 1$ on $[-1,1]$. Let
$f_{i}:\mathbb{T}^{d}_{\lambda}\to\mathbb{C}$ be $\lambda$-periodic functions
whose Fourier transforms are supported on $\\{k:|k|\sim N_{1}\\}$ and
$\\{k:|k|\sim N_{2}\\}$ respectively with $N_{1}\gg N_{2}$. Then

(65)
$\|\eta(t)U_{\lambda}(t)f_{1}\,U_{\lambda}(t)f_{2}\|_{L_{t}^{2}L_{x}^{2}}\lesssim
B_{1}(\lambda,N_{1})\|f_{1}\|_{L^{2}}\|f_{2}\|_{L^{2}},$

where

(66) $B_{1}(\lambda,N_{1})=\begin{cases}1,\qquad&\text{if }N_{1}\leq 1,\\\
\big{(}\frac{1}{N_{1}}+\frac{1}{\lambda}\big{)}^{\frac{1}{2}},\qquad&\text{if
}N_{1}>1.\end{cases}$

The constant in (65) reflects that as $\lambda\geq N_{1}$ the periodic
propagation behaves essentially like on Euclidean space, and we recover the
refined bilinear Strichartz estimates due to Bourgain [3].

In two dimensions, we have the corresponding result by Fan _et al._ [17]:

######  Theorem 6.3.

Let $f_{i}\in L^{2}(\mathbb{T}^{2}_{\lambda})$, $i=1,2$ with
$\text{supp}(\hat{f}_{i})\subseteq\\{k:|k|\sim N_{i}\\}$, $i=1,2$, for some
$N_{1}\geq N_{2}\gg 1$. Then it holds:

$\|\eta(t)U_{\lambda}(t)f_{1}\,U_{\lambda}(t)f_{2}\|_{L^{2}_{t,x}}\lesssim_{\varepsilon}N_{2}^{\varepsilon}B_{2}(\lambda,N_{1},N_{2})\|f_{1}\|_{2}\|f_{2}\|_{2}$

with

$B_{2}(\lambda,N_{1},N_{2})=\big{(}\frac{1}{\lambda}+\frac{N_{2}}{N_{1}}\big{)}^{\frac{1}{2}}.$

On the square torus the result was already proved by De Silva _et al._ [14,
Proposition 4.6]; see also Burq–Gérard–Tzvetkov [8]. We record the following
consequence by Galilean invariance:

######  Corollary 6.4.

Under the above assumptions, if additionally $N_{3}\leq N_{2}\ll N_{1}$, and
$\text{supp}(\hat{f}_{2})\subseteq\\{k:|k|\sim N_{2}\\}\cap
B(\xi_{2}^{*},N_{3})$ for some $\xi_{2}^{*}\in B(0,N_{2})$, we have

$\|\eta(t)\,U_{\lambda}(t)f_{1}\,U_{\lambda}(t)f_{2}\|_{L^{2}_{t,x}}\lesssim_{\varepsilon}N_{3}^{\varepsilon}B_{2}(\lambda,N_{1},N_{3})\|f_{1}\|_{2}\|f_{2}\|_{2}.$

###  6.2. The one-dimensional case

With linear and bilinear Strichartz estimates on rescaled tori at hand, we
show the following, from which trilinear refinements follow:

######  Proposition 6.5.

Let $\lambda\geq 1$, $N_{i}\in 2^{\mathbb{N}},\,i=1,2$, $N_{1}\gg N_{2}$ and
$N_{1}\geq N_{2}^{1+\beta}$ for some $\beta\in(0,1]$. Suppose that
$\lambda\leq N_{1}$ and
$\text{supp}(\hat{f}_{i})\subseteq\\{\xi\in\mathbb{R}:|\xi|\sim N_{i}\\}$.
Then we have

(67)
$\int_{[0,1]\times\mathbb{T}_{\lambda}}|U_{\lambda}(t)f_{1}|^{2}|U_{\lambda}(t)f_{2}|^{4}dtdx\lesssim
B_{1}(\lambda,N_{1})^{\frac{\beta}{1+\beta}}L_{1}(\lambda,N_{1})\|f_{1}\|_{2}^{2}\|f_{2}\|_{2}^{4}$

with $B_{1}$ defined in (66) and $L_{1}$ defined in (63).

######  Remark 6.6.

(67) still holds for $\lambda\geq N_{1}$, but is inferior to the estimate
obtained from directly using the bilinear Strichartz estimate together with
Bernstein’s inequality. Let $N_{2}=N_{1}^{\frac{1}{1+\beta}}$ and $\lambda\geq
N_{1}$. We find by Hölder’s inequality

(68)
$\int_{[0,1]\times\mathbb{T}_{\lambda}}|U_{\lambda}(t)f_{1}|^{2}|U_{\lambda}(t)f_{2}|^{4}dxdt\lesssim
B_{1}(\lambda,N_{1})^{2}N_{2}\|f_{1}\|_{2}^{2}\|f_{2}\|_{2}^{4}\lesssim
N_{1}^{-\frac{\beta}{1+\beta}}\|f_{1}\|_{2}^{2}\|f_{2}\|_{2}^{4}.$

This strengthens (67), which yields

$\int_{[0,1]\times\mathbb{T}_{\lambda}}|U_{\lambda}(t)f_{1}|^{2}|U_{\lambda}(t)f_{2}|^{4}dxdt\lesssim
N_{1}^{-\frac{\beta}{2(1+\beta)}}\prod_{i=1}^{3}\|f_{1}\|_{2}^{2}\|f_{2}\|_{2}^{4}.$

Note that this also shows that the trilinear refined estimate does not give an
improvement on Euclidean space.

On the other hand, if $N_{2}=N_{1}^{\frac{1}{1+\beta}}$ and $\lambda\leq
N_{1}$, (68) becomes

(69)
$\int_{[0,1]\times\mathbb{T}_{\lambda}}|U_{\lambda}(t)f_{1}|^{2}|U_{\lambda}(t)f_{2}|^{4}dxdt\lesssim\frac{N_{2}}{\lambda}\|f_{1}\|_{2}^{2}\|f_{2}\|_{2}^{4}.$

Compare this to (67), which yields

(70)
$\int_{[0,1]\times\mathbb{T}_{\lambda}}|U_{\lambda}(t)f_{1}|^{2}|U_{\lambda}(t)f_{2}|^{4}dxdt\lesssim\lambda^{-\frac{\beta}{2(1+\beta)}}\log(N_{1})^{c}\|f_{1}\|_{2}^{2}\|f_{2}\|_{2}^{4}.$

(70) improves on (69) once

$N_{1}\log(N_{1})^{c}\lesssim\lambda^{\frac{1}{2}}.$

###### Proof.

The proof is essentially a reprise of Sections 2 and 3, and we shall be brief.
By almost orthogonality, we can suppose that $f_{1}$ is Fourier supported in
an interval of length $N_{2}$ and $\|f_{j}\|_{2}=1$ for $j=1,2$.

We use the continuous approximation and rescale to $N_{1}^{2}\times\lambda
N_{1}$. This yields a scaling factor of $N_{1}^{-3}$. Via
$\lambda$-periodicity the $x$-domain is extended to $N_{1}^{2}$, and we find:

$\int_{[0,1]\times\mathbb{T}_{\lambda}}|U_{\lambda}(t)f_{1}|^{2}|U_{\lambda}(t)f_{2}|^{4}dxdt\lesssim
N_{1}^{-4}\lambda\int_{B_{N_{1}^{2}}}|\mathcal{E}\tilde{f}_{1}|^{2}|\mathcal{E}\tilde{f}_{2}|^{4}dxdt.$

We dominate $1_{B_{N_{1}^{2}}}\lesssim w_{B_{N_{1}^{2}}}^{6}$ with a suitable
Schwartz function, which is compactly supported in Fourier space to find

$\lesssim
N_{1}^{-4}\lambda\int_{\mathbb{R}^{2}}|g_{I_{1}}|^{2}|g_{I_{2}}|^{4}dxdt$

with notations like above: $I_{j}$ denote $1$-separated intervals in $[0,1]$
with length less than
$\tilde{\beta}=N_{2}/N_{1}=N_{1}^{-\frac{\beta}{1+\beta}}$. Moreover, the
functions $g_{I_{j}}$ have Fourier support in the $N_{1}^{-2}$-neighbourhood
of the paraboloid $\\{(\xi,-\xi^{2}):\xi\in I_{j}\\}$. We invoke the
decoupling iteration (after changing notations) to find:

(71) $\begin{split}&\quad\int|g_{I_{1}}|^{2}|g_{I_{2}}|^{4}\\\ &\leq
C_{1}\|g_{I_{2}}\|_{6}^{3}\sum_{|I_{1}^{2}|=\tilde{\beta}^{2}}\big{(}\int\big{(}|g_{I^{2}_{1}}|^{4}*\phi_{I_{1}}\big{)}\big{(}|g_{I_{2}}|^{2}*\phi_{I_{2}}\big{)}\big{)}^{\frac{1}{2}}\\\
&\leq
C_{1}^{\frac{3}{2}}\|g_{I_{2}}\|^{3}_{6}\sum_{|I_{1}^{2}|=\tilde{\beta}^{2}}\big{(}\sum_{|I_{2}^{2}|=\tilde{\beta}^{4}}\int\big{(}|g_{I_{1}^{2}}|^{4}*\phi_{I_{1}^{2}}\big{)}\big{(}|g_{I_{2}^{2}}|^{2}*\phi_{I_{2}^{2}}\big{)}\big{)}^{\frac{1}{2}}\\\
&\;\vdots\\\ &\leq
C_{1}^{2}\|g_{I_{2}}\|^{3}_{L^{6}}\sum_{|I_{1}^{2}|=\tilde{\beta}^{2}}\|g_{I_{1}^{2}}\|_{L^{6}}^{\frac{3}{2}}\\\
&\quad\times\big{(}\sum_{|I_{2}^{2}|=\tilde{\beta}^{4}}\|g_{I_{2}^{2}}\|_{L^{6}}^{\frac{3}{2}}\big{(}\sum_{|I_{1}^{3}|=\tilde{\beta}^{4}}\|g_{I_{1}^{3}}\|_{L^{6}}^{3/2}\ldots\big{(}\int|g_{I_{1}^{m/2}}|^{2}*\phi_{I_{1}^{m/2}}|g_{I_{2}^{m/2}}|^{4}*\phi_{I_{2}^{m/2}}\big{)}^{\frac{1}{2}}\big{)}^{\frac{1}{2}}\ldots\big{)}^{\frac{1}{2}}.\end{split}$

The number of decoupling iterations is estimated by
$m=\lceil\log_{2}(\beta^{-1}+1)\rceil$ (which we suppose to be even to
simplifiy notations) because

$N_{1}^{-\frac{\beta}{1+\beta}2^{m}}\leq
N_{1}^{-\frac{\beta}{1+\beta}(1+\beta^{-1})}=N_{1}^{-1}.$

We estimate the linear and bilinear contributions like in Section 3 by
reversing the continuous approximation and using the Strichartz estimates on
large tori: For the linear part we have

(72)
$\|g_{I}\|_{L^{6}(\mathbb{R}^{2})}\lesssim\big{(}\frac{N_{1}^{4}}{\lambda}\big{)}^{\frac{1}{6}}\|U_{\lambda}(t)\tilde{f}_{I}\|_{L_{t}^{6}([0,1],L^{6}_{x}(\mathbb{T}_{\lambda}))}\lesssim\big{(}\frac{N_{1}^{4}}{\lambda}\big{)}^{\frac{1}{6}}L_{1}(\lambda,N_{1})\|b_{I}\|_{\ell^{2}_{\lambda}}.$

Like in Section 3, regarding the expression obtained after decoupling we
abandon all multilinearity to find222Here we gloss over the technical issue of
estimating the convolution with $L^{1}$-normalized bump function, which can be
handled like in Section 3.

$\int|g_{I_{1}^{m/2}}|^{2}*\phi_{I_{1}^{m/2}}|g_{I_{2}^{m/2}}|^{4}*\phi_{I_{2}^{m/2}}\lesssim\int|g_{I_{1}^{m/2}}|^{2}|g_{I_{2}^{m/2}}|^{2}dtdx\;\|g_{I_{2}^{m/2}}\|_{L^{\infty}_{t,x}}^{2}$

and now scale back, use periodicity, and the estimate from Proposition 6.2
together with Bernstein’s inequality. Note that Bernstein’s inequality does
not lose derivatives in the present context. We find

(73)
$\int|g_{I_{1}^{m/2}}|^{2}|g_{I_{2}^{m/2}}|^{2}dtdx\;\|g_{I_{2}^{m/2}}\|_{L^{\infty}_{t,x}}^{2}\lesssim\big{(}\frac{N_{1}^{4}}{\lambda}\big{)}B_{1}^{2}(\lambda,N_{1})\|b_{I_{1}^{m/2}}\|_{\ell^{2}_{\lambda}}^{2}\|b_{I_{2}^{m/2}}\|_{\ell^{2}_{\lambda}}^{4}.$

Plugging (72) and (73) into (71), we find

$\int|g_{I_{1}}|^{2}|g_{I_{2}}|^{4}dtdx\leq
C_{1}^{2}\big{(}\frac{N_{1}^{4}}{\lambda}\big{)}B_{1}(\lambda,N_{1})^{1/2^{m-1}}\|b_{I_{1}}\|_{\ell^{2}_{\lambda}}^{2}\|b_{I_{2}}\|^{4}_{\ell^{2}_{\lambda}}.$

Since $B_{1}(\lambda,N_{1})^{1/2^{m-1}}\leq
B_{1}(\lambda,N_{1})^{\frac{\beta}{1+\beta}}$, this proves (67) in case
$N_{1}\geq\lambda$. ∎

###  6.3. The two-dimensional case

In this section we show the following variant of Theorem 1.4:

######  Theorem 6.7.

Let $\lambda\geq 1$, $N_{i}\in 2^{\mathbb{N}}$, $i=1,2,3$, $N_{1}\geq N_{2}$,
$N_{1}\geq N_{3}$, and $N_{1}\geq N_{3}^{1+\beta}$ for some $\beta\in(0,1]$.
Suppose that $\lambda\leq N_{1}$,
$\text{supp}(\hat{f}_{i})\subseteq\\{\xi\in\mathbb{R}:|\xi|\sim N_{1}\\}$, and
$\text{supp}(\hat{f}_{i})\subseteq B(\xi_{i}^{*},N_{3})$ for some
$\xi_{i}^{*}\in\mathbb{R}^{2}$ with $|\xi_{i}^{*}|\leq 2N_{1}$. Suppose that
$\xi_{i}\in\text{supp}(\hat{f}_{i})$ satisfy the transversality assumption
(14).

Then the following estimate holds:

(74)
$\int_{[0,1]\times\mathbb{T}_{\lambda}}\prod_{i=1}^{3}|U_{\lambda}(t)f_{i}|^{\frac{4}{3}}dxdt\lesssim_{\varepsilon}N_{1}^{\varepsilon}\nu^{-1+\frac{\beta}{2(1+\beta)}}B_{2}(\lambda,N_{1},1)^{\frac{2\beta}{3(1+\beta)}}\big{(}\prod_{i=1}^{3}\|f_{i}\|_{L^{2}_{\lambda}}\big{)}^{\frac{4}{3}}$

with $B_{2}$ defined in (66).

###### Proof.

The argument follows Section 5 closely. We use rescaling and continuous
approximation to find

$\begin{split}\int_{[0,1]\times\mathbb{T}^{2}_{\lambda}}\big{|}U_{\lambda}(t)f_{1}U_{\lambda}(t)f_{2}U_{\lambda}(t)f_{3}\big{|}^{\frac{4}{3}}dxdt&\lesssim
N_{1}^{4}\int_{[0,N_{1}^{2}]\times
N_{1}\mathbb{T}^{2}_{\lambda}}\big{|}U_{\lambda}(t)\tilde{f}_{1}U_{\lambda}(t)\tilde{f}_{2}U_{\lambda}(t)\tilde{f}_{3}\big{|}^{\frac{4}{3}}dxdt\\\
&\lesssim
N_{1}^{4}(N_{1}\lambda^{-1})^{2}\int_{\mathbb{R}^{3}}\big{|}\mathcal{E}\tilde{f}_{1}\mathcal{E}\tilde{f}_{2}\mathcal{E}\tilde{f}_{3}\big{|}^{\frac{4}{3}}w_{B_{N_{1}^{2}}}^{\frac{4}{3}}dxdt.\end{split}$

Above $w_{B_{N_{1}^{2}}}$ denotes a weight rapidly decaying off
$B_{N_{1}^{2}}$, which is compactly supported in Fourier space. We turn to
decoupling of

$\frac{1}{|B_{N_{1}}|^{2}}\int_{\mathbb{R}^{3}}|\tilde{F}_{1}\tilde{F}_{2}\tilde{F}_{3}|^{\frac{4}{3}}dxdt,$

where $\tilde{F}_{i}$ denotes functions, which are supported in the
$N_{1}^{-2}$-neighbourhood of the truncated paraboloid, their spatial Fourier
support is confined to balls of radius $N_{1}^{-\frac{\beta}{1+\beta}}$ and
$\xi_{i}\in\pi_{\mathbb{R}^{2}}(\text{supp}\hat{F}_{i})$ satisfy

$|(2\xi_{1},1)\wedge(2\xi_{2},1)\wedge(2\xi_{3},1)|\gtrsim\nu.$

We use notations $A_{p}$ and $D_{p}$ from Section 5.3. Moreover, we have

$2^{-\underline{m}}=\frac{N_{3}}{N_{1}}\leq
N_{1}^{-\frac{\beta}{1+\beta}}\text{ and
}2^{\overline{m}}=N_{1}^{2}=\delta^{-2},$

and the multiscale inequality becomes

(75)
$\begin{split}A_{p}^{p}(\underline{m},Q^{\overline{m}},\underline{F},\delta)&\lesssim\nu^{-\frac{1}{2}}\log(N_{1})^{c}\big{(}\prod_{\ell=1}^{n-1}\nu^{-\frac{\kappa_{p}^{\ell}}{2}}\log(N_{1})^{c\kappa_{p}^{\ell}}\big{)}\\\
&\quad\times
A_{p}^{p}(2^{n}\underline{m},Q^{\overline{m}},\underline{F},\delta)^{\kappa_{p}^{n}}\prod_{\ell=1}^{n}D_{p}^{p}(2^{\ell-1}\underline{m},Q^{\overline{m}},\underline{F},\delta)^{\kappa_{p}^{\ell-1}(1-\kappa_{p})}.\end{split}$

Recall that $\kappa_{4}=\frac{1}{2}$ and like above, the number of iterations
of the multiscale inequality is at most

$m=\lceil\log_{2}(\beta^{-1}+1)\rceil.$

Presently, we use linear and bilinear Strichartz estimates on the
$\lambda$-torus for finite times to exit the decoupling iteration. We record
the analogs of Lemma 5.7 and Proposition 5.8:

######  Lemma 6.8.

The following estimate holds:

$\begin{split}D_{4}^{4}(m,Q,\underline{F},\delta)&=\frac{1}{|Q|}\big{[}\prod_{i=1}^{3}\big{(}\sum_{I_{i,m}\in\mathbb{I}_{m}}\|\mathcal{P}_{I_{i,m}}F_{i}\|^{2}_{L^{4}(w_{Q})}\big{)}^{\frac{1}{2}}\big{]}^{\frac{4}{3}}\\\
&\lesssim\frac{N_{1}^{-6}\lambda^{2}}{|Q|}L_{2}^{4}(\lambda,N_{1})\big{(}\prod_{i=1}^{3}\|f_{i}\|_{2}\big{)}^{\frac{4}{3}}.\end{split}$

For the multilinear expression we find via exiting the decoupling iteration
with bilinear Strichartz estimates:

######  Proposition 6.9.

The following estimate holds:

$A_{4}^{4}(2^{n}\underline{m},Q^{\overline{m}},\underline{F},\delta)\lesssim_{\varepsilon}\frac{1}{|Q|}N_{1}^{-6}\lambda^{2}B_{2}^{\frac{4}{3}}(\lambda,N_{1})L_{2}^{\frac{4}{3}}(\lambda,N_{1})\big{(}\prod_{i=1}^{3}\|f_{i}\|_{2}\big{)}^{\frac{4}{3}}.$

###### Proof.

In the following we use the heuristic argument previously described in (52)
relying on the uncertainty principle. This can be made rigid like in the proof
of Proposition 5.8. We have

$\begin{split}&\quad\frac{1}{|\mathcal{Q}_{m,r}|}\sum_{\Delta_{m}\in\mathcal{Q}_{m,r}(Q^{r})}\big{(}\prod_{i=1}^{3}\big{(}\sum_{I_{i,m}\in\mathbb{I}_{m}}\|\mathcal{P}_{I_{i,m}}F_{i}\|^{2}_{L^{2}_{\\#}(w_{\Delta_{m}})}\big{)}^{\frac{1}{2}}\big{)}^{\frac{4}{3}}\\\
&\lesssim\frac{1}{|Q|}\int_{Q}\prod_{i=1}^{3}\big{(}\sum_{I}|\mathcal{P}_{I}F_{i}|^{2}\big{)}^{\frac{4}{6}}\\\
&\lesssim\frac{1}{|Q|}\big{(}\sum_{I_{1},I_{3}}\int_{Q}|\mathcal{P}_{I_{1}}F_{1}\mathcal{P}_{I_{3}}F_{3}|^{2}\big{)}^{\frac{2}{3}}\big{(}\int_{Q}\big{(}\sum_{I}|\mathcal{P}_{I}F_{2}|^{2}\big{)}^{2}\big{)}^{\frac{1}{3}}.\end{split}$

At this point we can reverse the continuous approximation and use rescaled
Strichartz estimates to find

$\lesssim\frac{1}{|Q|}N_{1}^{6}\lambda^{-2}B_{2}(\lambda,N_{1},1)^{\frac{4}{3}}\big{(}\prod_{i=1}^{3}\|f_{i}\|_{L^{2}_{\lambda}}\big{)}^{\frac{4}{3}}.$

∎

Now we can conclude the proof of Theorem 6.7 by employing Lemma 6.8 and
Proposition 6.9 in (75):

$A_{4}^{4}(\underline{m},Q^{\overline{m}},\underline{F},\delta)\lesssim\nu^{-1}\log(N_{1})^{2c}B_{2}(\lambda,N_{1},1)^{\frac{1}{2^{m}}\cdot\frac{4}{3}}L_{2}(\lambda,N_{1})\big{(}\prod_{i=1}^{3}\|f_{i}\|_{L^{2}_{\lambda}}\big{)}^{\frac{4}{3}}.$

With $m=\lceil\log_{2}(\beta^{-1}+1)\rceil$, we find

$\int_{[0,1]\times\mathbb{T}_{\lambda}}\prod_{i=1}^{3}|U_{\lambda}(t)f_{i}|^{\frac{4}{3}}dxdt\lesssim_{\varepsilon}N_{1}^{\varepsilon}B_{2}(\lambda,N_{1},1)^{\frac{2\beta}{3(\beta+1)}}\big{(}\prod_{i=1}^{3}\|f_{i}\|_{L^{2}_{\lambda}}\big{)}^{\frac{4}{3}}.$

∎

##  7\. Strichartz estimates on frequency dependent time intervals

This section is concerned with linear Strichartz estimates on frequency-
dependent time intervals:

(76)
$\|P_{N}e^{it\Delta}f\|_{L^{p}_{t,x}([0,N^{-\alpha}]\times\mathbb{T}^{d})}\lesssim\|f\|_{L^{2}(\mathbb{T}^{d})}.$

At the critical exponent $p=\frac{2(d+2)}{d}$ in the limiting case
$N^{-\alpha}\to\log(N)^{-1}$ this implies global well-posedness of the mass-
critical NLS.

In this section we point out how decoupling implies smoothing for $\alpha>0$
and $2\leq p<\frac{2(d+2)}{d}$ and for increased dispersion at the critical
exponent.

Although the present argument misses (76), we obtain structural information
about possible counterexamples. We have the following variant of (76):

######  Proposition 7.1.

Let $\alpha>0$, and $2\leq p<\frac{2(d+2)}{d}$. Then the following estimate
holds:

(77)
$\|e^{it\Delta}P_{N}f\|_{L^{p}_{t,x}([0,N^{-\alpha}]\times\mathbb{T}^{d})}\lesssim
N^{-\kappa}\|f\|_{L^{2}(\mathbb{T}^{d})}$

for
$\kappa<\alpha\big{(}\frac{d}{2}\big{(}\frac{1}{2}-\frac{1}{p}\big{)}-\frac{1}{p}\big{)}$.

###### Proof.

We use scaling $t\to N^{2}t$, $x\to Nx$ to rescale to unit frequencies:

$e^{it\Delta}P_{N}f=\sum_{|k|\sim
N}e^{i(kx+tk^{2})}a_{k}=\sum_{|k^{\prime}|\sim 1}e^{i(Nk^{\prime}\cdot
x+tN^{2}(k^{\prime})^{2})}a_{Nk^{\prime}}$

and use periodicity to obtain

$\begin{split}\|e^{it\Delta}P_{N}f\|^{p}_{L^{p}_{t,x}([0,N^{-\alpha}]\times\mathbb{T}^{d})}&\to
N^{-2}N^{-d}\|e^{it\Delta}P_{1}\tilde{f}\|^{p}_{L^{p}_{t,x}([0,N^{2-\alpha}]\times(N\mathbb{T})^{d}}\\\
&\sim
N^{-(2+d+(1-\alpha)d)}\|e^{it\Delta}P_{1}\tilde{f}\|^{p}_{L^{p}_{t,x}(B_{N^{2-\alpha}})}.\end{split}$

We use continuous approximation

$e^{it\Delta}P_{1}\tilde{f}=\sum_{k\in(\mathbb{Z}/N)^{d},|k|\sim
1}e^{i(kx+tk^{2})}a_{k}\approx\mathcal{E}\tilde{f}$

to obtain:

$\|e^{it\Delta}P_{1}\tilde{f}\|_{L^{p}_{t,x}(B_{N^{2-\alpha}})}\lesssim\|\mathcal{E}\tilde{f}\|_{L^{p}_{t,x}(w_{B_{N^{2-\alpha}}})}.$

This is amenable to $\ell^{2}$-decoupling333For $d=1$ the relevant range is
$4<p<6$ because for $p=4$ the Fefferman–Córdoba square function estimate holds
without loss., which gives

$\|\mathcal{E}\tilde{f}\|_{L^{p}_{t,x}(w_{B_{N^{2-\alpha}}})}\lesssim_{\varepsilon}N^{\varepsilon}\big{(}\sum_{\theta:N^{-1+\frac{\alpha}{2}}-\text{ball}}\|\mathcal{E}\tilde{f}_{\theta}\|^{2}_{L^{p}_{t,x}(w_{B_{N^{2-\alpha}}})}\big{)}^{\frac{1}{2}}.$

After applying $\ell^{2}$-decoupling, we can reverse the continuous
approximation, the scaling and use periodicity to find

$\begin{split}&\quad\big{(}\sum_{\theta:N^{-1+\frac{\alpha}{2}}-\text{ball}}\|\mathcal{E}\tilde{f}_{\theta}\|^{2}_{L^{p}_{t,x}(w_{B_{N^{2-\alpha}}})}\big{)}^{\frac{1}{2}}\\\
&\lesssim
N^{\frac{2}{p}}N^{\frac{d}{p}}N^{\frac{(1-\alpha)d}{p}}\big{(}\sum_{\theta:N^{\frac{\alpha}{2}}-\text{ball}}\|e^{it\Delta}P_{\theta}f\|^{2}_{L^{p}_{t,x}([0,N^{-\alpha}]\times\mathbb{T}^{d})}\big{)}^{\frac{1}{2}}.\end{split}$

A note of clarification here: the weight actually requires us to estimate

$\|e^{it\Delta}P_{\theta}f\|_{L^{p}_{t,x}(w([0,N^{-\alpha}]\times\mathbb{T}^{d}))}.$

This can be reduced to the above by periodicity in the spatial variables. In
the time variable we simply apply the following argument to

$\|e^{it\Delta}P_{\theta}f\|_{L^{p}_{t,x}([kN^{-\alpha},(k+1)N^{-\alpha}]\times\mathbb{T}^{d})}$

with $k\in\mathbb{Z}$, which is summable in $k$ by the weight $w$ decaying off
$[0,N^{-\alpha}]$ and unitarity of the Schrödinger propagator in
$L^{2}(\mathbb{T}^{d})$.

Lastly, we use Hölder in time and Bernstein’s inequality to find

$\|e^{it\Delta}P_{\theta}f\|_{L^{p}_{t,x}([0,N^{-\alpha}]\times\mathbb{T}^{d})}\lesssim
N^{-\frac{\alpha}{p}}N^{\frac{\alpha}{2}\cdot
d\big{(}\frac{1}{2}-\frac{1}{p}\big{)}}\|P_{\theta}f\|_{L^{2}(\mathbb{T}^{d})}.$

The proof of (77) is complete by $L^{2}$-orthogonality of
$(P_{\theta}f)_{\theta}$. ∎

Clearly, the argument extends to irrational tori.

In another direction the decoupling argument shows that smoothing estimates
hold for increased dispersion on frequency-dependent time intervals with
_arbitrary_ power $N^{-\alpha}$ at the critical exponent. With a nonlinear
application in mind, we formulate the result for the Airy propagator:

######  Proposition 7.2.

Let $\alpha\in(0,2]$. The following estimate holds:

(78)
$\|P_{N}e^{t\partial_{x}^{3}}f\|_{L_{t}^{6}([0,N^{-\alpha}],L_{x}^{6}(\mathbb{T}))}\lesssim
N^{-\kappa}\|f\|_{L^{2}(\mathbb{T})}$

for

$\kappa<\kappa_{0}=\begin{cases}\frac{\alpha}{6},\quad&\alpha\in(0,1],\\\
\frac{1}{6},\quad&\alpha\in(1,2].\end{cases}$

We consider $\alpha<2$ because for larger $\alpha$ we are within the Euclidean
window $T=T(N)=N^{-2}$ and the estimates from Euclidean space are expected to
hold. Regarding the time localization $T=T(N)=N^{-2}$ we have due to Dinh
[16]:

$\|P_{N}e^{t\partial_{x}^{3}}f\|_{L^{6}_{t}([0,N^{-2}],L^{6}_{x}(\mathbb{T}))}\lesssim
N^{-\frac{1}{6}}\|f\|_{L^{2}(\mathbb{T})}.$

This recovers the estimate from Euclidean space. Note that the smoothing will
further improve for $T(N)=N^{-2+\beta}$, $\beta>0$ as a simple consequence of
Hölder’s and Bernstein’s inequality.

###### Proof of Proposition 7.2.

Suppose that $\alpha\in[1,2)$. From scaling and periodicity we find

$\begin{split}\|P_{N}e^{t\partial_{x}^{3}}f\|^{6}_{L_{t}^{6}([0,N^{-\alpha}],L^{6}(\mathbb{T}))}&\lesssim
N^{-4}\|P_{1}e^{t\partial_{x}^{3}}\tilde{f}\|^{6}_{L_{t}^{6}([0,N^{3-\alpha}],L^{6}_{x}(N\mathbb{T}))}\\\
&\lesssim
N^{-6+\alpha}\|P_{1}e^{t\partial_{x}^{3}}\tilde{f}\|^{6}_{L_{t}^{6}([0,N^{3-\alpha}],L^{6}_{x}(B_{N^{3-\alpha}}))}.\end{split}$

After continuous approximation,

$P_{1}e^{t\partial_{x}^{3}}\tilde{f}\approx\mathcal{E}f,\quad\mathcal{E}f=\int
e^{i(x\xi+t\xi^{3})}f(\xi)d\xi,$

the expression is amenable to decoupling on the scale
$\delta=N^{-\frac{3}{2}+\frac{\alpha}{2}}$ (such that
$\delta^{-2}=N^{3-\alpha}$). The application of decoupling incurs a factor
$\log(N)^{c}$ due to Guth–Maldague–Wang [23] and scaling back and reversing
the continuous approximation we obtain

(79)
$\lesssim\log(N)^{c}\big{(}\sum_{\theta:N^{-\frac{1}{2}+\frac{\alpha}{2}}-\text{interval}}\|P_{\theta}e^{t\partial_{x}^{3}}f\|^{2}_{L_{t}^{6}([0,N^{-\alpha}],L^{6}(\mathbb{T}))}\big{)}^{\frac{1}{2}}.$

Here we sum over $N^{-\frac{1}{2}+\frac{\alpha}{2}}$-intervals $\theta$. It
turns out that after using a KdV-Galilean transform we find with $|A|\sim N$
and $M=N^{\frac{1}{2}+\frac{\alpha}{2}}$:

$\begin{split}\sum_{k=A}^{A+M}e^{i(kx+tk^{3})}a_{k}&=e^{iAx}\sum_{\ell=0}^{M}e^{i(\ell
x+t(A+\ell)^{3})}a_{\ell+A}\\\
&=e^{itA^{3}+iAx}\sum_{\ell=0}^{M}e^{i(\ell(x+3A^{2}t)+t(3A\ell^{2}+\ell^{3}))}a_{\ell+A}.\end{split}$

Hence, it suffices to observe that the dispersion relation
$\omega(\ell)=3A\ell^{2}+\ell^{3}$ does not cause oscillations for
$|t|\lesssim N^{-\alpha}$ and $|\ell|\lesssim M$. We obtain by Bernstein’s
inequality and Hölder’s inequality in time for $\theta$ an
$N^{-\frac{1}{2}+\frac{\alpha}{2}}$-interval:

(80)
$\|P_{\theta}e^{t\partial_{x}^{3}}f\|_{L_{t}^{6}([0,N^{-\alpha}],L^{6}(\mathbb{T}))}\lesssim
N^{-\frac{\alpha}{6}}N^{\big{(}-\frac{1}{2}+\frac{\alpha}{2}\big{)}\big{(}\frac{1}{2}-\frac{1}{6}\big{)}}\|P_{\theta}f\|_{L^{2}}=N^{-\frac{1}{6}}\|P_{\theta}f\|_{L^{2}}.$

Then the claim follows from plugging (80) into (79) and almost orthogonality
of $(P_{\theta}f)_{\theta}$ in $L^{2}$.

Secondly, we suppose that $\alpha\in(0,1]$. We obtain by rescaling and
periodicity

$\begin{split}\|P_{N}e^{t\partial_{x}^{3}}f\|_{L_{t}^{6}([0,N^{-\alpha}],L^{6}_{x}(\mathbb{T}))}^{6}&\lesssim
N^{-5}\|P_{1}e^{t\partial_{x}^{3}}\tilde{f}\|_{L_{t}^{6}([0,N^{3-\alpha}],L^{6}_{x}(B_{N^{2}}))}^{6}\\\
&=N^{-5}\sum_{I:|I|=N^{2}}\|P_{1}e^{t\partial_{x}^{3}}\tilde{f}\|_{L_{t}^{6}(I,L^{6}_{x}(B_{N^{2}}))}^{6}.\end{split}$

In the last step we decomposed the time interval into intervals of length
$N^{2}$, which yields $N^{1-\alpha}$ intervals. On every interval we can use
the arguments for $\alpha\in[1,2]$, i.e. decoupling, reversing the continuous
approximation, and trivial estimate of the resulting exponential sum. This
yields

$\|P_{1}e^{t\partial_{x}^{3}}\tilde{f}\|_{L_{t}^{6}(I,L^{6}_{x}(B_{N^{2}}))}^{6}\lesssim\log(N)^{c}N^{4}\|f\|^{6}_{L^{2}_{x}(\mathbb{T})}.$

Summing $N^{1-\alpha}$ intervals we obtain

$\|P_{N}e^{t\partial_{x}^{3}}f\|_{L_{t}^{6}([0,N^{-\alpha}],L^{6}_{x}(\mathbb{T}))}\lesssim
N^{-\frac{\alpha}{6}}\log(N)^{c}\|f\|_{L^{2}(\mathbb{T})}.$

The proof is complete. ∎

##  8\. Non-existence of solutions to the mKdV equation

The modified Korteweg-de Vries equation is given by

(81) $\left\\{\begin{array}[]{cl}\partial_{t}u+\partial_{x}^{3}u&=\pm
u^{2}\partial_{x}u,\quad(t,x)\in\mathbb{R}\times\mathbb{T},\\\ u(0)&=u_{0}\in
H^{s}(\mathbb{T})\end{array}\right.$

The equation with $+$-sign on the right hand side is referred to as defocusing
equation (due to coercivity of the energy), whereas the equation with $-$-sign
is referred to as focusing.

The well-posedness theory of the modified Korteweg-de Vries equation is
extensively studied (see e.g. [2, 30, 12, 27, 34, 29, 19]) and the following
brief account is by no means exhaustive. We also refer to the references
within the cited works. Unless explicitly stated otherwise, we only consider
real-valued solutions in the following.

In this section we argue how the short-time Strichartz estimates for the Airy
propagator from the previous section implies non-existence of solutions to the
modified Korteweg-de Vries equation in Sobolev spaces with negative
regularity. We give a brief overview of previous results.

Kappeler–Topalov [27] (see also [26]) showed that the defocusing (real-valued)
modified Korteweg-de Vries equation is globally well-posed in
$L^{2}(\mathbb{T})$. In [27] it was exploited that the Miura map is a global
fold. The Miura map assigns solutions to the defocusing mKdV in
$L^{2}(\mathbb{T})$ to solutions to the Korteweg-de Vries equation in
$H^{-1}(\mathbb{T})$. Kappeler-Topalov proved global well-posedness of the
(arguable even more prominent) Korteweg-de Vries equation in
$H^{-1}(\mathbb{T})$ in their work [28].

In [27] solutions in $L^{2}(\mathbb{T})$ were defined through extension of the
data-to-solution mapping from smooth initial data (see also below). This well-
posedness result was supplemented by Molinet [34], who proved that the
Kappeler-Topalov solutions are weak solutions for $s=0$ and the data-to-
solution mapping fails to be weakly continuous as a map from
$L^{2}(\mathbb{T})\to\mathcal{D}^{\prime}([0,T]\times\mathbb{T})$. He also
showed the first existence result for solutions to the focusing equation in
$L^{2}(\mathbb{T})$.

However, the scaling critical regularity is $s=-\frac{1}{2}$ and one might
surmise that some probabilistic well-posedness result could still hold in
Sobolev spaces of negative regularity. Indeed, Kappeler-Molnar [29] proved
local well-posedness of the defocusing equation in Fourier Lebesgue spaces
$\mathcal{F}L^{p}$ for $2<p<\infty$ after renormalization (see (83)). The
arguments rely on complete integrability. This shows that the unrenormalized
equation is ill-posed in Fourier Lebesgue spaces of negative Sobolev
regularity. Further recent local well-posedness results on the complex-valued
mKdV equation in Fourier Lebesgue spaces are due to Chapouto [10, 11]. We also
mention the recent work of Forlano [19], who unified and simplified previous
proofs of global well-posedness for the modified Korteweg-de Vries equation in
Sobolev spaces $H^{s}(\mathbb{K})$ with $0\leq s<\frac{1}{2}$ and
$\mathbb{K}\in\\{\mathbb{T},\mathbb{R}\\}$.

Conditional upon the conjectured $L^{8}_{t,x}$-Strichartz estimate

(82)
$\|P_{N}e^{t\partial_{x}^{3}}f\|_{L_{t,x}^{8}([0,1]\times\mathbb{T})}\lesssim_{\varepsilon}N^{\varepsilon}\|f\|_{L^{2}(\mathbb{T})},$

I have proved in [38] that there is no data-to-solution mapping to the
unrenormalized equation in the following sense in $H^{s}(\mathbb{T})$ for
$s<0$: A map $S:H^{s}(\mathbb{T})\to C([-T,T],H^{s}(\mathbb{T}))$ where
$T=T(\|u_{0}\|_{H^{s}})>0$ is referred to as data-to-solution mapping to (81)
if it satisfies the following properties:

  * (i)

$S(u_{0})$ satisfies the equation (81) in the distributional sense and
$S(u_{0})(0)=u_{0}$.

  * (ii)

There exists a sequence of smooth global solutions $(u_{n})$ such that
$u_{n}\to S(u_{0})$ in $C([-T,T],H^{s})$ as $n\to\infty$.

The argument in [38] does not rely on complete integrability, but frequency-
dependent time localization, and also covers the focusing case. It is
conceivable that this approach extends with some modifications to the complex
case.

Like in [29], the proof of the non-existence of solutions to (81) relies on
the existence of solutions to the renormalized mKdV equation:

(83)
$\left\\{\begin{array}[]{cl}\partial_{t}u+\partial_{x}^{3}u&=\pm\mathfrak{N}(u),\quad(t,x)\in\mathbb{R}\times\mathbb{T},\\\
u(0)&=u_{0}\in H^{s}(\mathbb{T})\end{array}\right.$

with

$\mathfrak{N}(u)\widehat{(}n)=in|\hat{u}(n)|^{2}\hat{u}(n)+in\sum_{\begin{subarray}{c}n=n_{1}+n_{2}+n_{3},\\\
(n_{1}+n_{2})(n_{1}+n_{3})(n_{2}+n_{3})\neq
0\end{subarray}}\hat{u}(n_{1})\hat{u}(n_{2})\hat{u}(n_{3}).$

A consequence of the conjectured $L^{8}_{t,x}$-Strichartz estimate is the
short-time estimate

$\|P_{N}e^{t\partial_{x}^{3}}u_{0}\|_{L_{t}^{6}([0,N^{-\alpha}],L^{6}(\mathbb{T}))}\lesssim_{\varepsilon}N^{-\frac{\alpha}{18}+\varepsilon}\|u_{0}\|_{L^{2}(\mathbb{T})}.$

This was the key ingredient to show the non-existence for solutions to (81) at
negative Sobolev regularity in [38]. The specific choice for the frequency-
dependent analysis in [38] was actually $\alpha=1+$, which gives a smoothing
of $N^{-\frac{1}{18}}$. Since by Proposition 7.2 we have now the improved
smoothing $N^{-\frac{1}{6}+}$, we obtain Theorem 8.1, which was formulated in
[38] conditional upon (82):

######  Theorem 8.1 (Non-existence of solutions to mKdV).

There is $s^{\prime}<0$ so that for $s^{\prime}<s<0$ there exists
$T=T(\|u_{0}\|_{H^{s}})$ such that there exists a local solution $u\in
C([-T,T],H^{s}(\mathbb{T}))$ to (83), and we find the a priori estimate

$\sup_{t\in[0,T]}\|u(t)\|_{H^{s}}\leq C\|u_{0}\|_{H^{s}}$

to hold. Furthermore, solutions to (81) do not exist for $s^{\prime}<s<0$.

## Appendix: Short-time bilinear Strichartz estimates via the
Fefferman–Córdoba square function estimate

In the following we give another proof of the short-time bilinear Strichartz
estimaet in one dimension. The argument uses the continuous approximation from
previous sections and as key ingredient the bilinear Fefferman–Córdoba square
function estimate (see [18, 13] or [15, p. 40] for a recent treatise). The
argument shows how what is known as small cap decoupling or square function
(in $\ell^{2}$) imply Strichartz estimates on frequency dependent times.

We recall the following square function estimate:

######  Theorem 8.2.

Let $g_{1},g_{2}\in\mathcal{S}(\mathbb{R}^{2})$ with
$\text{supp}(\hat{g}_{i})\subseteq\mathcal{N}_{\delta}(\\{(\xi,|\xi|^{2}):\xi\in
I_{i}\\})$, where $I_{1},\,I_{2}\subseteq[0,1]$ denote intervals with
$\text{dist}(I_{1},I_{2})\sim 1$. Then the following estimate holds:

$\int_{\mathbb{R}^{2}}|g_{1}g_{2}|^{2}\lesssim\sum_{\theta_{i}\subseteq
I_{i}:\\\
\delta-\text{interval}}\int_{\mathbb{R}^{2}}|g_{1,\theta_{1}}|^{2}|g_{2,\theta_{2}}|^{2}.$

In the following we observe how this kind of estimate, which decomposes
functions into blocks smaller than the canonical scale, translates to claims
about exponential sums on frequency-dependent times. We recover the short-time
bilinear Strichartz estimate due to Moyua–Vega [35] and Hani [24]:

######  Theorem 8.3.

Let $f_{i}\in L^{2}(\mathbb{T})$ with $\text{supp}(\hat{f}_{i})\subseteq
I_{i}\subseteq[0,N]$ and $\text{dist}(I_{1},I_{2})\sim N$. Then the following
estimate holds:

$\int_{[0,N^{-1}]\times\mathbb{T}}|e^{it\Delta}f_{1}e^{it\Delta}f_{2}|^{2}dxdt\lesssim
N^{-1}\prod_{i=1}^{2}\|f_{i}\|_{L^{2}(\mathbb{T})}^{2}.$

###### Proof.

The rescaling and continuous approximation like above allows us to write:

$\begin{split}&\quad\int_{[0,N^{-1}]\times\mathbb{T}}|e^{it\Delta}f_{1}e^{it\Delta}f_{2}|^{2}dxdt\\\
&\lesssim N^{3}\int_{[0,N]\times
N\mathbb{T}}\big{|}\sum_{\begin{subarray}{c}k_{1}\in\mathbb{Z}/N,\\\
k_{1}\in\tilde{I}_{1}\end{subarray}}e^{i(k_{1}x-tk_{1}^{2})}b_{1,k_{1}}\sum_{\begin{subarray}{c}k_{2}\in\mathbb{Z}/N_{1},\\\
k_{2}\in\tilde{I}_{2}\end{subarray}}e^{i(k_{2}x-tk_{2}^{2})}b_{2,k_{2}}\big{|}^{2}dxdt\\\
&\lesssim
N^{3}\int_{\mathbb{R}^{2}}|\mathcal{E}\tilde{f}_{1}\mathcal{E}\tilde{f}_{2}|^{2}w_{B_{N}}^{4}dxdt.\end{split}$

We check that $\mathcal{E}\tilde{f}_{i}\cdot w_{B_{N}}=g_{i}$ satisfies the
assumptions of Theorem 8.2 with $\delta=N^{-1}$ to find that

$\int_{\mathbb{R}^{2}}|g_{1}g_{2}|^{2}\lesssim\sum_{\begin{subarray}{c}\theta_{i}\subseteq\tilde{I}_{i}:\\\
\delta-\text{interval}\end{subarray}}\int_{\mathbb{R}^{2}}|g_{1,\theta_{1}}g_{2,\theta_{2}}|^{2}.$

For this expression it is possible to reverse the continuous approximation to
find that the exponential sum has been trivialized:

$\begin{split}N^{3}\sum_{\begin{subarray}{c}\theta_{i}\subseteq\tilde{I}_{i}:\\\
\delta-\text{interval}\end{subarray}}\int_{\mathbb{R}^{2}}|g_{1,\theta_{1}}g_{2,\theta_{2}}|^{2}&\lesssim\sum_{k_{i}\in
I_{i}}\int_{[0,N^{-1}]\times\mathbb{T}}|e^{i(k_{1}x-tk_{1}^{2})}\hat{f}_{1}(k_{1})|^{2}|e^{i(k_{2}x-tk_{2}^{2})}\hat{f}_{2}(k_{2})|^{2}dxdt\\\
&\lesssim
N^{-1}\prod_{i=1}^{2}\sum_{k_{i}}|\hat{f}_{i}(k_{i})|^{2}=N^{-1}\|f_{1}\|_{L^{2}(\mathbb{T})}^{2}\|f_{2}\|_{L^{2}(\mathbb{T})}^{2}.\end{split}$

The proof is complete. ∎

## Acknowledgement

I would like to thank Sebastian Herr, Javier Ramos and Po-Lam Yung for helpful
discussions on this set of problems. Financial support by Korea Institute for
Advanced Study, grant No. MG093901 is gratefully acknowledged.

## References

  * [1] 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.
  * [2] 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.
  * [3] J. Bourgain.  Refinements of Strichartz’ inequality and applications to $2$d-NLS with critical nonlinearity.  Internat. Math. Res. Notices, (5):253–283, 1998.
  * [4] J. Bourgain.  Moment inequalities for trigonometric polynomials with spectrum in curved hypersurfaces.  Israel J. Math., 193(1):441–458, 2013.
  * [5] Jean Bourgain.  A remark on normal forms and the “$I$-method” for periodic NLS.  J. Anal. Math., 94:125–157, 2004.
  * [6] Jean Bourgain and Ciprian Demeter.  The proof of the $l^{2}$ decoupling conjecture.  Ann. of Math. (2), 182(1):351–389, 2015.
  * [7] 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.
  * [8] N. Burq, P. Gérard, and N. Tzvetkov.  Bilinear eigenfunction estimates and the nonlinear Schrödinger equation on surfaces.  Invent. Math., 159(1):187–223, 2005.
  * [9] Anthony Carbery and Stefán Ingi Valdimarsson.  The endpoint multilinear Kakeya theorem via the Borsuk-Ulam theorem.  J. Funct. Anal., 264(7):1643–1663, 2013.
  * [10] Andreia Chapouto.  A remark on the well-posedness of the modified KDV equation in the Fourier-Lebesgue spaces.  Discrete Contin. Dyn. Syst., 41(8):3915–3950, 2021.
  * [11] Andreia Chapouto.  A refined well-posedness result for the modified KdV equation in the Fourier-Lebesgue spaces.  J. Dynam. Differential Equations, 35(3):2537–2578, 2023.
  * [12] J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T. Tao.  Sharp global well-posedness for KdV and modified KdV on $\mathbb{R}$ and $\mathbb{T}$.  J. Amer. Math. Soc., 16(3):705–749, 2003.
  * [13] Antonio Cordoba.  Geometric Fourier analysis.  Ann. Inst. Fourier, 32(3):215–226, 1982.
  * [14] Daniela De Silva, Nataša Pavlović, Gigliola Staffilani, and Nikolaos Tzirakis.  Global well-posedness for a periodic nonlinear Schrödinger equation in 1D and 2D.  Discrete Contin. Dyn. Syst., 19(1):37–65, 2007.
  * [15] Ciprian Demeter.  Fourier restriction, decoupling, and applications, volume 184 of Cambridge Studies in Advanced Mathematics.  Cambridge University Press, Cambridge, 2020.
  * [16] Van Duong Dinh.  Strichartz estimates for the fractional Schrödinger and wave equations on compact manifolds without boundary.  J. Differential Equations, 263(12):8804–8837, 2017.
  * [17] Chenjie Fan, Gigliola Staffilani, Hong Wang, and Bobby Wilson.  On a bilinear Strichartz estimate on irrational tori.  Anal. PDE, 11(4):919–944, 2018.
  * [18] Charles Fefferman.  A note on spherical summation multipliers.  Isr. J. Math., 15:44–52, 1973.
  * [19] Justin Forlano.  A remark on the well-posedness of the modified KdV equation in $L^{2}$.  arXiv e-prints, page arXiv:2205.13110, May 2022.
  * [20] Shaoming Guo, Zane Kun Li, and Po-Lam Yung.  Improved discrete restriction for the parabola.  arXiv e-prints, page arXiv:2103.09795, March 2021.
  * [21] Shaoming Guo, Zane Kun Li, Po-Lam Yung, and Pavel Zorin-Kranich.  A short proof of $\ell^{2}$ decoupling for the moment curve.  Amer. J. Math., 143(6):1983–1998, 2021.
  * [22] Larry Guth.  The endpoint case of the Bennett-Carbery-Tao multilinear Kakeya conjecture.  Acta Math., 205(2):263–286, 2010.
  * [23] Larry Guth, Dominique Maldague, and Hong Wang.  Improved decoupling for the parabola.  arXiv e-prints, page arXiv:2009.07953, September 2020.
  * [24] Zaher Hani.  A bilinear oscillatory integral estimate and bilinear refinements to Strichartz estimates on closed manifolds.  Anal. PDE, 5(2):339–363, 2012.
  * [25] Sebastian Herr and Beomjong Kwak.  Strichartz estimates and global well-posedness of the cubic NLS on $\mathbb{T}^{2}$.  arXiv e-prints, page arXiv:2309.14275, September 2023.
  * [26] T. Kappeler and P. Topalov.  Global fold structure of the Miura map on $L^{2}(\mathbb{T})$.  Int. Math. Res. Not., (39):2039–2068, 2004.
  * [27] T. Kappeler and P. Topalov.  Global well-posedness of mKdV in $L^{2}(\mathbb{T},\mathbb{R})$.  Comm. Partial Differential Equations, 30(1-3):435–449, 2005.
  * [28] T. Kappeler and P. Topalov.  Global wellposedness of KdV in $H^{-1}(\mathbb{T},\mathbb{R})$.  Duke Math. J., 135(2):327–360, 2006.
  * [29] Thomas Kappeler and Jan-Cornelius Molnar.  On the well-posedness of the defocusing mKdV equation below $L^{2}$.  SIAM J. Math. Anal., 49(3):2191–2219, 2017.
  * [30] Carlos E. Kenig, Gustavo Ponce, and Luis Vega.  A bilinear estimate with applications to the KdV equation.  J. Amer. Math. Soc., 9(2):573–603, 1996.
  * [31] Nobu Kishimoto.  Remark on the periodic mass critical nonlinear Schrödinger equation.  Proc. Amer. Math. Soc., 142(8):2649–2660, 2014.
  * [32] Zane Kun Li.  Effective $l^{2}$ decoupling for the parabola.  Mathematika, 66(3):681–712, 2020.  With an appendix by Jean Bourgain and Li.
  * [33] Zane Kun Li.  An $l^{2}$ decoupling interpretation of efficient congruencing: the parabola.  Rev. Mat. Iberoam., 37(5):1761–1802, 2021.
  * [34] Luc Molinet.  Sharp ill-posedness results for the KdV and mKdV equations on the torus.  Adv. Math., 230(4-6):1895–1930, 2012.
  * [35] A. Moyua and L. Vega.  Bounds for the maximal function associated to periodic solutions of one-dimensional dispersive equations.  Bull. Lond. Math. Soc., 40(1):117–128, 2008.
  * [36] Javier Ramos.  A trilinear restriction estimate with sharp dependence on transversality.  Amer. J. Math., 140(6):1443–1469, 2018.
  * [37] Robert Schippa.  Short-time Fourier transform restriction phenomena and applications to nonlinear dispersive equations.  Phd thesis, Bielefeld University, Bielefeld, Germany, September 2019.  Available at https://pub.uni-bielefeld.de/download/2938663/2938664/Dissertation_Schippa.pdf.
  * [38] Robert Schippa.  On the existence of periodic solutions to the modified Korteweg–de Vries equation below $H^{1/2}(\mathbb{T})$.  J. Evol. Equ., 20(3):725–776, 2020.
  * [39] Gigliola Staffilani and Daniel Tataru.  Strichartz estimates for a Schrödinger operator with nonsmooth coefficients.  Comm. Partial Differential Equations, 27(7-8):1337–1372, 2002.
  * [40] H. Takaoka and N. Tzvetkov.  On 2D nonlinear Schrödinger equations with data on ${\mathbb{R}}\times\mathbb{T}$.  J. Funct. Anal., 182(2):427–442, 2001.
  * [41] Trevor D. Wooley.  The cubic case of the main conjecture in Vinogradov’s mean value theorem.  Adv. Math., 294:532–561, 2016.
  * [42] A. Zygmund.  On Fourier coefficients and transforms of functions of two variables.  Studia Math., 50:189–201, 1974.