# Strichartz estimates for quasi-periodic functions and applications

Robert Schippa  UC Berkeley, Department of Mathematics, Evans Hall Berkeley,
CA 94720-3840<EMAIL_ADDRESS>
###### Abstract.

We show Strichartz estimates for quasi-periodic functions with decaying
Fourier coefficients via $\ell^{2}$-decoupling. When we additionally average
in time, further improvements can be obtained. Next, we apply multilinear
refinements to show low regularity local well-posedness for nonlinear
Schrödinger equations. For the cubic nonlinear Schrödinger equation the
approach yields the sharp local well-posedness result.

###### Key words and phrases:

Strichartz estimates, quasi-periodic functions, nonlinear Schrödinger equation

##  1\. Introduction

The local well-posedness of nonlinear dispersive equations with almost
periodic initial data has recently received increased attention. Two prominent
models are the cubic nonlinear Schrödinger equation

(1)
$\left\\{\begin{array}[]{cl}i\partial_{t}u+\partial_{x}^{2}u&=\pm|u|^{2}u,\quad(t,x)\in\mathbb{R}\times\mathbb{R},\\\
u(0)&=u_{0}\end{array}\right.$

and the Korteweg-de Vries equation:

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

Recall that a Bohr-almost periodic function [4] is obtained as a uniform limit
of trigonometric polynomials. The set of almost periodic functions on
$\mathbb{R}$ is denoted by $AP(\mathbb{R})$. For these functions the Fourier
series expansion holds in the $L^{\infty}$-norm:

(3) $f(x)=\sum_{\lambda\in\Lambda}e^{i\lambda
x}a_{\lambda},\quad\lambda\in\Lambda\subseteq\mathbb{R}:\text{ at most
countably infinite}.$

T. Oh [29] proved local well-posedness of (1) in the Banach algebra of
functions with absolutely summable Fourier coefficients. By local well-
posedness we refer to existence, uniqueness, and continuous dependence of the
solutions on the initial data. The result in [29] was recently generalized by
Papenburg [30] who considered more general Banach algebras. Furthermore,
Damanik _et al._ [17, 16] considered power-type nonlinear Schrödinger
equations with a nonlinearity at least quintic and the derivative nonlinear
Schrödinger equation.

Tsugawa [33] applied the Fourier restriction norm method due to Bourgain [5,
6] to show local well-posedness of (2). He showed that his result is sharp in
general in terms of Fourier decay. Presently, we likewise aim to minimize the
decay of the Fourier coefficients, which is required to prove local well-
posedness. This can be viewed as a low regularity well-posedness theory for
nonlinear dispersive equations with quasi-periodic initial data. The presented
arguments generalize the theory in the periodic case, which was initiated in
[5, 6].

On the other hand, Chapouto–Killip–Vişan [10] pointed out that (2) is ill-
posed in $AP(\mathbb{R})$ due to instant loss of continuity. This disproved
the Deift conjecture [18] stating that almost-periodic initial data to (2)
yield solutions almost-periodic in time. Notably, an affirmative answer was
given under more restrictive assumptions on the initial data by Damanik _et
al._ [15, 3, 14]. A possible interpretation to [10] is that continuity is too
strong an assumption for initial data to (nonlinear) dispersive equations.
Instead we consider variants of the following norm:

$\|f\|_{\mathcal{L}_{x}^{2}}^{2}=\lim_{L\to\infty}\frac{1}{2L}\int_{[-L,L]}|f(x)|^{2}dx=\mathcal{M}(|f|^{2}).$

The set of functions $f(x)=\sum_{\lambda\in\Lambda}a_{\lambda}e^{i\lambda x}$
with absolutely summable Fourier coefficients
$(a_{\lambda})_{\lambda\in\Lambda}\in\ell^{1}$ is denoted with
$\mathcal{A}_{\omega}$. For these functions we have

$\|f\|_{\mathcal{L}_{x}^{2}}=\big{(}\sum_{\lambda\in\Lambda}|a_{\lambda}|^{2}\big{)}^{\frac{1}{2}}.$

For $f\in\mathcal{L}^{2}_{x}(\mathbb{R})$ we let
$\sigma(f)=\\{\lambda\in\mathbb{R}^{d}:\mathcal{M}(fe^{-i\lambda\cdot})\neq
0\\}$. We define the space of Besicovitch-almost periodic functions
$\mathcal{B}_{2}(\mathbb{R})$ as closure of trigonometric polynomials under
the $\mathcal{L}^{2}_{x}$-norm. (3) holds in
$\mathcal{L}^{2}_{x}(\mathbb{R})$. The fact that solutions to (1) and (2) for
real-valued initial data satisfy the formal conservation law

$\|u(t)\|_{\mathcal{L}_{x}^{2}}=\|u_{0}\|_{\mathcal{L}_{x}^{2}}$

indicates that an $\mathcal{L}^{2}_{x}$-based space is more appropriate to
solve the quasiperiodic Cauchy problem.

For decaying initial data on the real line or periodic initial data,
dispersive effects are the bedrock to establish a low regularity well-
posedness theory. A manifestation of dispersive effects are Fourier extension
inequalities, which are also known as Strichartz estimates.

For linear Schrödinger equations on the real line these estimates (see [25]
and references therein) read

$\|e^{it\partial_{x}^{2}}u_{0}\|_{L^{p}_{t}(\mathbb{R};L^{q}_{x}(\mathbb{R}))}\lesssim\|u_{0}\|_{L^{2}_{x}(\mathbb{R})},\quad
p,q\geq 2,\quad\frac{2}{p}+\frac{1}{q}=\frac{1}{2}.$

With this at hand, it is straight-forward to solve (1) via the contraction
mapping principle and obtain solutions with analytic dependence on the initial
data in $L^{2}(\mathbb{R})$. Since the mass is conserved, the local well-
posedness theory is readily iterated to prove global well-posedness.

Due to weaker dispersive effects on the circle, less is known about Strichartz
estimates for periodic functions. It still holds the $L^{4}_{t,x}$-Strichartz
estimate without derivative loss (cf. [5, 35]):

(4)
$\|e^{it\partial_{x}^{2}}u_{0}\|_{L_{t}^{4}([0,1],L_{x}^{4}(\mathbb{T}))}\lesssim\|u_{0}\|_{L^{2}(\mathbb{T})}.$

Here we point out how the Córdoba–Fefferman square function estimate [11, 12,
20] and decoupling arguments due to Bourgain–Demeter [9] yield Strichartz
estimates for linear solutions with quasi-periodic initial data. Here it
becomes important to approximate an exponential sum with an oscillatory
integral. The latter is amenable to decoupling. This approach of estimating
exponential sums goes back to Bourgain [8]. We also remark that substitutes of
Strichartz estimates in $L^{p}$-based spaces for $p>2$, again relating to
initial data with less decay compared to $L^{2}$-based spaces, were previously
investigated in [32, 31].

Let $f\in\mathcal{B}_{2}(\mathbb{R})$ be a quasi-periodic function with
$\sigma(f)\subseteq\Lambda$. Here $\nu\in\mathbb{N}$, and suppose that
$\omega\in\mathbb{R}^{\nu}_{>0}$ is _non-resonant_ , i.e.,
$\\{\omega_{1},\ldots,\omega_{\nu}\\}$ are linearly independent over
$\mathbb{Q}$. Let $\Lambda=\omega_{1}\mathbb{Z}+\ldots+\omega_{\nu}\mathbb{Z}$
and denote by $\langle n\rangle_{\omega}=n\cdot\omega$ for
$n\in\mathbb{Z}^{\nu}$. We have the representation in $\mathcal{L}^{2}_{x}$:

$f(x)=\sum_{n\in\mathbb{Z}^{\nu}}e^{i\langle n\rangle_{\omega}\cdot
x}\hat{f}(\langle n\rangle_{\omega}).$

We quantify the Fourier decay in terms of the height $|n|$ of the Fourier
coefficients, which leads us to the following definition of Sobolev-type
spaces:

$\|f\|^{2}_{\mathcal{H}^{s}_{\Lambda}}=\sum_{n\in\mathbb{Z}^{\nu}}\langle
n\rangle^{2s}|\hat{f}(\langle n\rangle_{\omega})|^{2}.$

For $C,N\in 2^{\mathbb{N}_{0}}$ let $R_{C}$ denote the projection to
frequencies of height $C$ and $P_{N}$ denote the projection to frequencies of
size $N$:

$R_{C}f=\sum_{n\in\mathbb{Z}^{\nu}}\chi_{C}(n)e^{i\langle
n\rangle_{\omega}\cdot x}\hat{f}(\langle n\rangle_{\omega}),\quad
P_{N}f=\sum_{n\in\mathbb{Z}^{\nu}}\chi_{N}(\langle
n\rangle_{\omega})e^{i\langle n\rangle_{\omega}\cdot x}\hat{f}(\langle
n\rangle_{\omega}).$

Above $\chi_{A}$ denotes a function, which smoothly localizes to size $A$.

To obtain an $L^{4}$-Strichartz estimate for quasi-periodic functions, we need
to count the number of frequencies $\langle n\rangle_{\omega}$ contained in an
interval $I$ of length $\sim 1$: A counting argument based on the rank of the
lattice gives

$\\#\\{n\in\mathbb{Z}^{\nu}:|n|\sim C,\;\langle n\rangle_{\omega}\in
I\\}\lesssim C^{\nu-1}.$

We define the density parameter as $b=\nu-1$.

The generalization of the estimate (4) reads

(5)
$\|e^{it\partial_{x}^{2}}R_{C}f\|_{L_{t}^{4}([0,1],\mathcal{L}^{4}_{x})}\lesssim_{\varepsilon}C^{\frac{b}{4}+\varepsilon}\|f\|_{\mathcal{L}^{2}_{x}}.$

The above estimate is obtained in two steps:

  * (i)

The decoupling or square function estimate provides us with an almost
orthogonal frequency decomposition of $\langle n\rangle_{\omega}$ into unit
intervals.

  * (ii)

In the second step we use a bound for the height $|n|\leq C$ and a counting
argument to estimate the number of frequencies $\langle n\rangle_{\omega}$
contained in a unit interval.

The first step trivializes the time-evolution on a scale determined by the
frequency $\langle n\rangle_{\omega}$, but we can have many frequencies with a
height $|n|\gg|\langle n\rangle_{\omega}|$. This creates a mismatch between
the almost orthogonal decomposition provided by the square function estimate
and the number of frequencies within the intervals, which is divergent from
the periodic case. Still it appears that the arguments optimally take
advantage of the time-oscillation and the density of frequencies. We give a
simple example, which shows sharpness of (5).

We show the following:

######  Theorem 1.1.

Let $T\in(0,1]$, and
$\sigma(u_{0})\subseteq\Lambda=\omega\cdot\mathbb{Z}^{\nu}$ with
$\omega\in\mathbb{R}^{\nu}_{>0}$ non-resonant, and $b=\nu-1$. Then the
following estimates hold:

(6)
$\displaystyle\|e^{it\partial_{x}^{2}}u_{0}\|_{L_{t}^{4}([0,T],\mathcal{L}^{4}_{x}(\mathbb{R}))}$
$\displaystyle\lesssim
T^{\frac{1}{8}}\|u_{0}\|_{\mathcal{H}^{s}_{\Lambda}}\text{ for
}s>\frac{b}{4},$ (7)
$\displaystyle\|e^{t\partial_{x}^{3}}P_{N}R_{C}u_{0}\|_{L_{t}^{4}([0,T],\mathcal{L}^{4}_{x}(\mathbb{R}))}$
$\displaystyle\lesssim T^{\frac{1}{4}}\langle
T^{-\frac{1}{2}}N^{-\frac{1}{2}}\rangle^{\frac{1}{4}}C^{\frac{b}{4}}\|u_{0}\|_{\mathcal{L}^{2}_{x}}\text{
for }1\leq N\lesssim C.$

The arguments can clearly cover more general dispersion relations. The above
estimates extend the Strichartz estimates for periodic solutions to one-
dimensional Schrödinger and Airy equations due to Bourgain [5, 6]. The higher-
dimensional case, also on irrational tori was settled as a consequence of
sharp $\ell^{2}$-decoupling due to Bourgain–Demeter [9]. We rely on decoupling
to show Strichartz estimates for quasi-periodic functions in higher
dimensions; see Theorem 3.1.

A different type of Strichartz estimate for quasi-periodic functions with
additional time averaging was proved last year by Klaus [26]:

$\|e^{it\partial_{x}^{2}}f\|_{\mathcal{L}_{t,x}^{4}(\mathbb{R}\times\mathbb{R})}\lesssim\|f\|_{\mathcal{L}^{2}_{x}(\mathbb{R})}.$

He considered the special case of quasi-periodic functions
$f\in\mathcal{B}_{2}(\mathbb{R})$ with
$\sigma(f)\subseteq\mathbb{Z}+\sqrt{2}\mathbb{Z}$. The proof relied on a
counting argument similar to the ones from Bourgain [5]. With additional time
averaging, the almost orthogonal decomposition in frequencies $\langle
n\rangle_{\omega}$ can be taken arbitrarily small. We show the following
Strichartz estimates for Besicovitch-almost periodic functions:

######  Theorem 1.2.

Let $u_{0}\in\mathcal{B}_{2}(\mathbb{R})$. The following Strichartz estimates
hold:

(8)
$\displaystyle\|e^{it\partial_{x}^{2}}u_{0}\|_{\mathcal{L}^{4}_{t,x}(\mathbb{R}\times\mathbb{R})}$
$\displaystyle\lesssim\|u_{0}\|_{\mathcal{L}_{x}^{2}(\mathbb{R})},$ (9)
$\displaystyle\|e^{t\partial_{x}^{3}}u_{0}\|_{\mathcal{L}^{4}_{t,x}(\mathbb{R}\times\mathbb{R})}$
$\displaystyle\lesssim\|u_{0}\|_{\mathcal{L}_{x}^{2}(\mathbb{R})}.$

Klaus explained why (8) does not imply local well-posedness of (1) in
$\mathcal{L}_{x}^{2}$. The problem is that applying the $TT^{*}$-argument we
find with

$T:\mathcal{L}_{x}^{2}(\mathbb{R})\to\mathcal{L}^{4}_{t,x}(\mathbb{R}\times\mathbb{R}),\quad
u_{0}\mapsto e^{it\partial_{x}^{2}}u_{0}$

that

$T^{*}:\mathcal{L}^{\frac{4}{3}}_{t,x}(\mathbb{R}\times\mathbb{R})\to\mathcal{L}^{2}_{x}(\mathbb{R}),\quad
F\mapsto\lim_{L\to\infty}\frac{1}{2L}\int_{[-L,L]}e^{-is\partial_{x}^{2}}F(s)ds.$

Consequently,

$TT^{*}:\mathcal{L}^{\frac{4}{3}}_{t,x}(\mathbb{R}\times\mathbb{R})\to\mathcal{L}^{4}_{t,x}(\mathbb{R}\times\mathbb{R}),\quad
F\mapsto\lim_{L\to\infty}\frac{1}{2L}\int_{[-L,L]}e^{i(t-s)\partial_{x}^{2}}F(s)ds.$

This $TT^{*}$-operator is not suitable for estimating the Duhamel formula:

$u(t)=e^{it\partial_{x}^{2}}u_{0}-i\int_{0}^{t}e^{i(t-s)\partial_{x}^{2}}(|u|^{2}u)(s)ds.$

It is the preceding observation, which suggests to study the Strichartz
estimates from Theorem 1.1. We remark that the Strichartz estimates from
Theorem 1.2 extend Strichartz estimates for periodic functions, too: Let $f\in
L^{2}(\mathbb{T})$. Then $e^{it\partial_{x}^{2}}f$ is time-periodic, and we
have

$\|e^{it\partial_{x}^{2}}f\|_{\mathcal{L}^{4}_{t,x}(\mathbb{R}^{2})}=\|e^{it\partial_{x}^{2}}f\|_{L^{4}_{t,x}(\mathbb{T}^{2})}\lesssim\|f\|_{\mathcal{L}^{2}_{x}(\mathbb{R})}=\|f\|_{L^{2}(\mathbb{T})},$

which recovers (4).

Applying $\ell^{2}$-decoupling as a substitute for the square function
estimate, we show Strichartz estimates in higher dimensions with additional
time averaging for quasi-periodic functions in
$\mathcal{H}^{\varepsilon}_{\Lambda}$ in Theorem 4.2 with frequency-vector
satisfying a diophantine condition.

Finally, we apply the Strichartz estimates to show new low regularity well-
posedness of nonlinear dispersive equations with quasi-periodic initial data.
We show the following low regularity local well-posedness result for the
nonlinear Schrödinger equation:

(10)
$\left\\{\begin{array}[]{cl}i\partial_{t}u+\partial_{x}^{2}u&=\pm|u|^{2}u,\quad(t,x)\in\mathbb{R}\times\mathbb{R},\\\
u(0)&=f\in\mathcal{H}^{s}_{\Lambda}(\mathbb{R}).\end{array}\right.$

By local well-posedness we refer to existence, uniqueness, and continuous
dependence of the solution on the initial data.

######  Theorem 1.3 (Local well-posedness of NLS with quasi-periodic initial
data).

Let $\nu\in\mathbb{N}$, and suppose that $\omega\in\mathbb{R}^{\nu}_{>0}$ is
non-resonant. Let $\Lambda=\omega\cdot\mathbb{Z}^{\nu}$ with $b=\nu-1$. Then
(10) is locally well-posed for $s>\frac{b}{2}$.

Previously, Oh [29] considered (10) with
$\Lambda=\mathbb{Z}+\sqrt{2}\mathbb{Z}$ and showed local well-posedness for
$s>1$. More precisely, he showed local well-posedness in
$\mathcal{A}_{\omega}$. Applying the Cauchy-Schwarz inequality yields the
following embedding for $s>1$:

$\mathcal{H}^{s}_{\Lambda}\subseteq\mathcal{A}_{\omega}.$

For $\Lambda=\mathbb{Z}+\sqrt{2}\mathbb{Z}$ Theorem 1.3 yields local well-
posedness for $s>\frac{1}{2}$. We regard this as low regularity local well-
posedness since the space $\mathcal{H}^{s}_{\Lambda}$ is not a Banach algebra
for $s<1$. The improvement of the local well-posedness rests on the fixed-time
estimates (6). The key bilinear estimates are a consequence of almost
orthogonality and Galilean invariance, which read for
$\Lambda=\mathbb{Z}+\sqrt{2}\mathbb{Z}$:

$\|R_{C_{1}}e^{it\partial_{x}^{2}}f_{1}R_{C_{2}}e^{it\partial_{x}^{2}}f_{2}\|_{L_{t}^{2}([0,T],\mathcal{L}^{2}_{x}(\mathbb{R}))}\lesssim_{\varepsilon}T^{\frac{1}{4}}C_{\min}^{\frac{1}{2}+\varepsilon}\|R_{C_{1}}u\|_{\mathcal{L}^{2}_{x}}\|R_{C_{2}}v\|_{\mathcal{L}^{2}_{x}}.$

With the above at hand, it is standard to apply the contraction mapping
principle in adapted function spaces [28, 21, 22]. For this reason we shall be
brief. We remark that, since the solutions are constructed via the contraction
mapping principle, the dependence of solutions on initial data is real
analytic like the nonlinearity. This result is sharp up to endpoints. We show
$C^{3}$-ill-posedness of (1) on $\Lambda=\omega\cdot\mathbb{Z}^{\nu}$ in
$\mathcal{H}^{s}_{\Lambda}$ for $s<\frac{\nu-1}{2}$. Moreover, the arguments
are flexible again and apply to Schrödinger equations with algebraic power
nonlinearity. We record a result for the NLS with algebraic power-
nonlinearity in Section 5: Let
$\Lambda=\omega\cdot\mathbb{Z}^{\nu}\subseteq\mathbb{R}$ be like above and
$m\geq 3$. Then

$\left\\{\begin{array}[]{cl}i\partial_{t}u+\partial_{x}^{2}u&=\pm|u|^{2(m-1)}u,\quad(t,x)\in\mathbb{R}\times\mathbb{R},\\\
u(0)&=u_{0}\in\mathcal{H}^{s}_{\Lambda}\end{array}\right.$

is locally well-posed for
$s>s^{*}=2(\nu-1)\big{(}\frac{1}{2}-\frac{1}{2m}\big{)}+(1-\frac{3}{m})$.
Damanik _et al._ [17] show local well-posedness for coefficients decaying like
$|c(n)|\lesssim(1+|n|)^{-r}$ for $r>4\nu$. Theorem 5.13 implies an admissible
Fourier decay of $r>s^{*}+\frac{\nu}{2}$, which constitutes an improvement.

Finally, we remark that one can understand the ill-posedness result as
limitation of the Harmonic Analysis approach presently taken. Proving global
well-posedness of (1), which would be very natural in
$\mathcal{L}^{2}_{x}(\mathbb{R})$, is beyond the methods of the paper. Global
well-posedness might also depend on the lattice $\Lambda$ and whether the
evolution is focusing or defocusing. The present analysis suggests that a
proof of global well-posedness can only be accomplished employing the complete
integrability. This would parallel the result for decaying data, in the sense
that local well-posedness below $L^{2}(\mathbb{R})$ could only be proved using
the complete integrability [23].

_Outline of the paper._ In Section 2 we elaborate on the properties of almost-
periodic functions and the diophantine conditions. Moreover, we recall the
Córdoba–Fefferman square function estimate and $\ell^{2}$-decoupling. In
Section 3 Strichartz estimates on fixed time scales are proved, in Section 4
we show Strichartz estimates with additional averaging in time. Finally, in
Section 5 we show local well-posedness results for Schrödinger equations with
quasi-periodic initial data. Examples pointing out sharpness of the Strichartz
estimates and of the local well-posedness of the cubic NLS are given in
Section 6. In the Appendix we indicate how the analysis recovers Tsugawa’s
result for the KdV equation.

Basic notations:

  * •

For $A\in\mathbb{C}^{d}$ we denote the Euclidean norm with
$|A|=\sqrt{|A_{1}|^{2}+\ldots+|A_{d}|^{2}}$. We let $\langle A\rangle=1+|A|$.
For $A,B\in\mathbb{R}$ let $A\wedge B=\min(A,B)$, and $A\vee B=\max(A,B)$.

  * •

$A\lesssim B$ denotes the inequality $A\leq CB$ with a harmless constant $C$.
Dependence of $C$ on parameters is indicated with subscripts, e.g.,
$A\lesssim_{\varepsilon}B$ denotes $A\leq C(\varepsilon)B$ with $C$ depending
on $\varepsilon$.

  * •

Dyadic numbers are denoted with capital letters $M,N,\ldots\in
2^{\mathbb{N}_{0}}$.

  * •

For $A\subseteq\mathbb{R}^{d}$ we denote with $\mathcal{N}_{\delta}(A)$ the
$\delta$-neighborhood of $A$.

  * •

For $1\leq p<\infty$, $A\subseteq\mathbb{R}^{d}$, the usual $L^{p}$-norm for
Lebesgue-measurable functions $f:A\to\mathbb{C}$ is given by

$\|f\|_{L^{p}(A)}^{p}=\int_{A}|f(x)|^{p}dx.$

For $p=\infty$ the $L^{\infty}$-norm describes the essential supremum.

  * •

With $\mathcal{L}^{p}_{x}$ we denote the norms on $\mathbb{R}^{d}$ after
averaging, see Section 2.

##  2\. Preliminaries

###  2.1. Almost-periodic functions

In this section we recall key properties of Besicovitch-almost periodic
functions. We refer to [13, 2] for further reading. For $1\leq p<\infty$, and
a measurable function $f:\mathbb{R}^{d}\to\mathbb{C}$ define

$\|f\|^{p}_{\mathcal{L}^{p}_{x}(\mathbb{R}^{d})}=\lim_{L\to\infty}\frac{1}{(2L)^{d}}\int_{[-L,L]^{d}}|f(x)|^{p}dx.$

Moreover, let

$\|f\|_{\mathcal{L}^{\infty}(\mathbb{R}^{d})}:=\|f\|_{L^{\infty}(\mathbb{R}^{d})}.$

We define Besicovitch-almost-periodic functions in higher dimensions as

$\mathcal{B}_{2}(\mathbb{R}^{d})=\overline{\\{f(x)=\sum_{\lambda\in\Lambda_{f}}a_{\lambda}e^{i\lambda
x}:\Lambda_{f}\subseteq\mathbb{R}^{d}\text{ finite
}\\}}^{\|\cdot\|_{\mathcal{L}_{x}^{2}}}.$

The Fourier expansion is valid in $\mathcal{L}^{2}_{x}(\mathbb{R}^{d})$:

$f(x)=\sum_{\lambda\in\Lambda}\hat{f}(\lambda)e^{i\lambda x},\quad
x\in\mathbb{R}^{d}$

with $\Lambda\subseteq\mathbb{R}^{d}$ at most countable infinite.111Precisely,
$\mathcal{B}_{2}$ is comprised of equivalence classes of functions $f\sim g$
with $\mathcal{M}(|f-g|^{2})=0$. By Hölder’s inequality, we obtain the Fourier
coefficients by

$\hat{f}(\lambda)=\lim_{L\to\infty}\frac{1}{2L}\int_{[-L,L]}f(x)e^{-i\lambda
x}dx,$

which satisfies $|\hat{f}(\lambda)|\leq\|f\|_{\mathcal{L}^{2}}$. Define the
mean-value by

$\mathcal{M}(f)=\lim_{L\to\infty}\frac{1}{2L}\int_{[-L,L]}f(x)dx.$

In the following let $\sigma(f)\subseteq\Lambda\subseteq\mathbb{R}^{d}$. We
have the following classification of $f\in\mathcal{B}_{2}(\mathbb{R}^{d})$:

  * •

If $\Lambda$ can be chosen such that
$\Lambda\subseteq\omega^{(1)}\mathbb{Z}\times\ldots\times\omega^{(d)}\mathbb{Z}$
for some $\omega^{(i)}>0$, $i=1,\ldots,d$, then $f$ is a periodic function.

  * •

If $\Lambda$ can be chosen such that
$\Lambda\subseteq\omega^{(1)}\cdot\mathbb{Z}^{\nu_{1}}\times\ldots\times\omega^{(d)}\cdot\mathbb{Z}^{\nu_{d}}$
with $\omega^{(i)}\in\mathbb{R}^{\nu_{i}}_{>0}$ non-resonant, then $f$ is a
quasi-periodic function.

  * •

Else,
$\Lambda\subseteq\sum_{k_{1}=1}^{\infty}\omega^{(1)}_{k}\mathbb{Z}\times\sum_{k_{2}=1}^{\infty}\omega^{(2)}_{k}\mathbb{Z}\ldots\times\sum_{k_{d}=1}^{\infty}\omega^{(d)}_{k}\mathbb{Z}$
for $(\omega^{(i)}_{k})_{k\in\mathbb{N}}\subseteq\mathbb{R}_{>0}$ linearly
independent over $\mathbb{Q}$ for any $i=1,\ldots,d$, then $f$ is an almost
periodic function.

Let $f\in\mathcal{B}_{2}(\mathbb{R})$ be a quasi-periodic function. Then,
there are rationally independent
$\\{\omega_{1},\ldots,\omega_{\nu}\\}\subseteq\mathbb{R}_{>0}$ such that
$\sigma(f)\subseteq\Lambda=\omega\cdot\mathbb{Z}^{\nu}$. For
$n\in\mathbb{Z}^{\nu}$ we let $\langle n\rangle_{\omega}=n\cdot\omega$. For
any frequency $\lambda\in\Lambda$ there is a unique $n\in\mathbb{Z}^{\nu}$
such that $\lambda=\langle n\rangle_{\omega}$. We refer to $|n|$ as the
_height_ of the frequency $\langle n\rangle_{\omega}$. We have the Fourier
series representation with convergence in $\mathcal{L}^{2}_{x}(\mathbb{R})$:

$f(x)=\sum_{n\in\mathbb{Z}^{\nu}}e^{i\langle
n\rangle_{\omega}x}\hat{f}(\langle n\rangle_{\omega}).$

We quantify regularity through the Sobolev-type norms: For $s\geq 0$ define

$\mathcal{H}^{s}_{\Lambda}(\mathbb{R})=\\{f\in\mathcal{B}_{2}(\mathbb{R}):\|f\|_{\mathcal{H}^{s}_{\Lambda}}<\infty\\},\quad\|f\|^{2}_{\mathcal{H}^{s}_{\Lambda}}=\sum_{n\in\mathbb{Z}^{\nu}}\langle
n\rangle^{2s}|\hat{f}(\langle n\rangle_{\omega})|^{2}.$

The density parameter is defined as $b=\nu-1$.

The generalization of the above considerations to higher-dimensional quasi-
periodic functions is straight-forward. Let $d\in\mathbb{N}$,
$(\nu_{1},\ldots,\nu_{d})\in\mathbb{N}^{d}$,
$\omega^{(i)}\in\mathbb{R}_{>0}^{\nu_{i}}$ for $i=1,\ldots,d$ with
$\omega^{(i)}$ non-resonant, and let

(11)
$\Lambda=\omega^{(1)}\cdot\mathbb{Z}^{\nu_{1}}\times\ldots\times\omega^{(d)}\cdot\mathbb{Z}^{\nu_{d}}.$

Define for $n_{1}\in\mathbb{Z}^{\nu_{1}},\ldots,n_{d}\in\mathbb{Z}^{\nu_{d}}$:

$\langle n\rangle_{\omega}=(\langle n_{1}\rangle_{\omega^{(1)}},\ldots,\langle
n_{d}\rangle_{\omega^{(d)}})\in\mathbb{R}^{d}.$

We have the Fourier series expansion in $\mathcal{L}^{2}(\mathbb{R}^{d})$:

$f=\sum_{n_{1}\in\mathbb{Z}^{\nu_{1}},\ldots,n_{d}\in\mathbb{Z}^{\nu_{d}}}e^{i\langle
n\rangle_{\omega}\cdot x}\hat{f}(\langle n\rangle_{\omega})$

and the Sobolev norm:

$\|f\|^{2}_{\mathcal{H}^{s}_{\Lambda}(\mathbb{R}^{d})}=\sum_{n_{1}\in\mathbb{Z}^{\nu_{1}},\ldots,n_{d}\in\mathbb{Z}^{\nu_{d}}}\langle
n\rangle^{2s}|\hat{f}(\langle n\rangle_{\omega})|^{2}.$

We let $b_{i}=\nu_{i}-1$ and $b=b_{1}+\ldots+b_{d}$.

###  2.2. Córdoba–Fefferman square function estimate

We recall the following square function estimate [20, 11, 12] as one of the
key ingredients for the $L^{4}$-Strichartz estimates in one dimension:

######  Theorem 2.1 (Córdoba–Fefferman square function estimate).

Let $\Gamma=\\{(\xi,h(\xi)):\xi\in[-1,1]\\}$ with

$\frac{1}{2}\leq h^{\prime\prime}(\xi)\leq 2$

and $\text{supp}(\hat{F})\subseteq\mathcal{N}_{\delta}(\Gamma)$ for some
$0<\delta\ll 1$. Let $\Theta_{\delta}$ denote a cover of
$\mathcal{N}_{\delta}(\Gamma)$ with finitely overlapping
$10\delta^{\frac{1}{2}}\times 10\delta$-boxes with long-side pointing into
tangential and short side pointing into normal direction. Then the following
estimate holds:

$\|F\|_{L^{4}(\mathbb{R}^{2})}\lesssim\big{\|}\big{(}\sum_{\theta\in\Theta_{\delta}}|F_{\theta}|^{2}\big{)}^{\frac{1}{2}}\big{\|}_{L^{4}(\mathbb{R}^{2})}.$

with $F_{\theta}$ denoting the Fourier projection of $F$ to $\theta$.

This will separate the frequencies for the Schrödinger evolution and
trivialize the time-evolution. Note that for $h(\xi)=\xi^{3}$ the curvature
degenerates at the origin. We show the following variant directly:

######  Theorem 2.2 (Córdoba–Fefferman square function estimate for the
cubic).

Let $\Gamma=\\{(\xi,\xi^{3}):\xi\in[-1,1]\\}$ and
$\text{supp}(\hat{F})\subseteq\mathcal{N}_{\delta}(\Gamma)$ for some
$0<\delta\ll 1$. Let $\Theta_{\delta}$ denote a cover of
$\mathcal{N}_{\delta}(\Gamma)$ with finitely overlapping
$10\delta^{\frac{1}{3}}\times 10\delta$-boxes with long side pointing into
tangential and short side into normal direction. Then the following estimate
holds:

$\|F\|_{L^{4}(\mathbb{R}^{2})}\lesssim\big{\|}\big{(}\sum_{\theta\in\Theta_{\delta}}|F_{\theta}|^{2}\big{)}^{\frac{1}{2}}\big{\|}_{L^{4}(\mathbb{R}^{2})}.$

###### Proof.

We split the Fourier support into positive and negative frequencies and
suppose in the following by symmetry that

$\text{supp}(\hat{F})\subseteq\mathcal{N}_{\delta}(\\{(\xi,\xi^{3}):\,\xi\in[0,1]\\}).$

We decompose $F=\sum_{\theta\in\Theta_{\delta}}F_{\theta}$ and obtain by
Plancherel’s theorem

$\begin{split}\int_{\mathbb{R}^{2}}\big{|}\sum_{\theta\in\Theta_{\delta}}F_{\theta}\big{|}^{4}dx&=\int_{\mathbb{R}^{2}}\big{|}\sum_{\theta_{1}\in\Theta_{\delta}}F_{\theta_{1}}\sum_{\theta_{2}\in\Theta_{\delta}}F_{\theta_{2}}\big{|}^{2}\\\
&=\int_{\mathbb{R}^{2}}\big{|}\sum_{\theta_{1}}\widehat{F_{\theta_{1}}}*\sum_{\theta_{2}}\widehat{F_{\theta_{2}}}\big{|}^{2}\\\
&=\int_{\mathbb{R}^{2}}\big{(}\sum_{\theta_{1}}\widehat{F_{\theta_{1}}}*\sum_{\theta_{2}}\widehat{F_{\theta_{2}}}\big{)}\overline{\big{(}\sum_{\theta_{3}}\widehat{F_{\theta_{3}}}*\sum_{\theta_{4}}\widehat{F_{\theta_{4}}}\big{)}}.\end{split}$

Non-trivial contribution to the integral stems from solutions to the system

$\left\\{\begin{array}[]{cl}\xi_{1}+\xi_{2}&=\xi_{3}+\xi_{4},\\\
\xi_{1}^{3}+\xi_{2}^{3}&=\xi_{3}^{3}+\xi_{4}^{3}+\mathcal{O}(\delta)\end{array}\right.$

for $(\xi_{i},\xi_{i}^{3})\in\theta_{i}$. We shall establish a
biorthogonality: For $\xi_{1}+\xi_{2}\ll\delta^{\frac{1}{3}}$ it is clear that
$\xi_{i}$ are coming from the first $\mathcal{O}(1)$-blocks
$\delta^{\frac{1}{3}}\times\delta$-close to the origin. So we suppose that
$\xi_{1}+\xi_{2}\gtrsim\delta^{\frac{1}{3}}$. We take the cubic power of the
first equation:

$\xi_{1}^{3}+3\xi_{1}^{2}\xi_{2}+3\xi_{1}\xi_{2}^{2}+\xi_{2}^{3}=\xi_{3}^{3}+3\xi_{3}^{2}\xi_{4}+3\xi_{3}\xi_{4}^{2}+\xi_{4}^{3}.$

Subtracting this from the second equation gives

$3\xi_{1}\xi_{2}(\xi_{1}+\xi_{2})=3\xi_{3}\xi_{4}(\xi_{3}+\xi_{4})+\mathcal{O}(\delta).$

This implies

$\xi_{1}\xi_{2}=\xi_{3}\xi_{4}+\mathcal{O}(\delta^{\frac{2}{3}}).$

We square the first equation and subtract the above display to find

$\left\\{\begin{array}[]{cl}\xi_{1}+\xi_{2}&=\xi_{3}+\xi_{4},\\\
\xi_{1}^{2}+\xi_{2}^{2}&=\xi_{3}^{2}+\xi_{4}^{2}+\mathcal{O}(\delta^{\frac{2}{3}}).\end{array}\right.$

Now the claim follows from the biorthogonality underpinning Theorem 2.1. For
the sake of self-containedness we give the details. Write

$(\xi_{1}-\xi_{3})(\xi_{1}+\xi_{3})=(\xi_{4}-\xi_{2})(\xi_{4}+\xi_{2})+\mathcal{O}(\delta^{\frac{2}{3}}).$

If $|\xi_{1}-\xi_{3}|\lesssim\delta^{\frac{1}{3}}$ biorthogonality follows.
So, we suppose that $|\xi_{1}-\xi_{3}|\gg\delta^{\frac{1}{3}}$. In this case
we find

$\left\\{\begin{array}[]{cl}\xi_{1}+\xi_{3}&=\xi_{2}+\xi_{4}+\mathcal{O}(\delta^{\frac{1}{3}}),\\\
\xi_{1}+\xi_{2}&=\xi_{3}+\xi_{4},\end{array}\right.$

which implies $|\xi_{1}-\xi_{4}|\lesssim\delta^{\frac{1}{3}}$, hence
biorthogonality. ∎

###  2.3. $\ell^{2}$-decoupling

Here we recall the $\ell^{2}$-decoupling result due to Bourgain-Demeter [9,
Theorem 1] for future use. Let $h\in C^{2}(B_{d}(0,1),\mathbb{R})$,
$\Gamma=\\{(\xi,h(\xi)):\xi\in B_{d}(0,1)\\}\subseteq\mathbb{R}^{d+1}$ be a
compact elliptic hypersurface with principal curvature $K_{i}\in[C^{-1},C]$
for some $C>0$. Let $p_{d}=\frac{2(d+2)}{d}$, and define

$\alpha(p)=\begin{cases}0,&\quad 2<p<p_{d},\\\
\frac{d}{2}-\frac{d+2}{p},&\quad p_{d}\leq p<\infty.\end{cases}$

######  Theorem 2.3 ($\ell^{2}$-decoupling).

Let $0<\delta\ll 1$, $F\in\mathcal{S}(\mathbb{R}^{d+1})$ with
$\text{supp}(\hat{F})\subseteq\mathcal{N}_{\delta}(\Gamma)$ and $2<p<\infty$.
Then the following estimate holds:

$\|F\|_{L^{p}(B_{\delta^{-1}})}\lesssim_{\varepsilon}\delta^{-(\alpha(p)+\varepsilon)}\big{(}\sum_{\theta\in\Theta_{\delta}}\|F_{\theta}\|^{2}_{L^{p}(w_{B_{\delta^{-1}}})}\big{)}^{\frac{1}{2}}.$

In the above display $\Theta_{\delta}$ contains a finitely overlapping
collection of caps of size comparable to
$\delta^{\frac{1}{2}}\times\ldots\times\delta^{\frac{1}{2}}\times\delta$ with
small side pointing into the normal direction, which covers
$\mathcal{N}_{\delta}(\Gamma)$.

Recall that the above decoupling also implies the global estimate with
$L^{p}$-norm taken over the whole space. We refer to [19] for a textbook
treatment.

##  3\. Strichartz estimates on fixed time intervals

This section is devoted to the proof of Strichartz estimates on fixed time
intervals. We handle the $L^{4}$-estimates in one dimension from Theorem 2.1
first. In the $L^{4}$-estimate we obtain a gain for estimates on small time
scales.

###  3.1. The $L^{4}$-estimates

Recall that the frequencies are contained in the lattice
$\Lambda=\omega_{1}\mathbb{Z}+\ldots+\omega_{\nu}\mathbb{Z}$ with
$\nu\in\mathbb{N}$, $\omega\in\mathbb{R}^{\nu}_{>0}$ non-resonant, and let
$b=\nu-1$.

In the following we prove the estimates

(12)
$\displaystyle\|e^{it\partial_{x}^{2}}u_{0}\|_{L_{t}^{4}([0,T],\mathcal{L}^{4}_{x}(\mathbb{R}))}$
$\displaystyle\lesssim T^{\frac{1}{8}}\|u_{0}\|_{\mathcal{H}^{s}_{x}}\text{
for }s>\frac{b}{4},$ (13)
$\displaystyle\|e^{t\partial_{x}^{3}}P_{N}R_{C}u_{0}\|_{L_{t}^{4}([0,T],\mathcal{L}^{4}_{x}(\mathbb{R}))}$
$\displaystyle\lesssim T^{\frac{1}{4}}C^{\frac{b}{4}}\langle
T^{-\frac{1}{2}}N^{-\frac{1}{2}}\rangle^{\frac{1}{4}}\|u_{0}\|_{\mathcal{L}^{2}_{x}}\text{
for }1\leq N\lesssim C.$

###### Proof of Theorem 1.1.

We begin with the proof of (12). Note that the height bounds the modulus of
frequencies $\langle n\rangle_{\omega}$, which can occur. By the triangle
inequality we find that it suffices to prove for $1\leq N\lesssim C$:

$\|e^{it\partial_{x}^{2}}P_{N}R_{C}f\|_{L_{t}^{4}([0,T],\mathcal{L}_{x}^{4})}\lesssim
T^{\frac{1}{8}}C^{\frac{b}{4}}\|f\|_{\mathcal{L}^{2}_{x}}.$

For $T\geq N^{-2}$ we shall show that

(14)
$\|e^{it\partial_{x}^{2}}P_{N}R_{C}f\|_{L_{t}^{4}([0,T],\mathcal{L}^{4}_{x})}\lesssim\big{(}\sum_{I_{T}:T^{-\frac{1}{2}}-\text{interval}}\|e^{it\partial_{x}^{2}}P_{N}R_{C}P_{I_{T}}f\|^{2}_{L_{t}^{4}(w_{T},\mathcal{L}^{4}_{x})}\big{)}^{\frac{1}{2}}.$

Above $L_{t}^{4}(w_{T})$ denotes the $L^{4}$-norm with a polynomial weight
$w_{T}(t)\lesssim(1+|t|/T)^{-100}$ decaying off $[-T,T]$:

$\|f\|_{L_{t}^{4}(w_{T})}^{4}=\int_{\mathbb{R}}|f(t)|^{4}w_{T}(t)dt.$

Taken the above estimate for granted, it remains to prove an estimate

(15)
$\|e^{it\partial_{x}^{2}}R_{C}P_{I_{T}}f\|_{L_{t}^{4}(w_{T},\mathcal{L}^{4}_{x})}\lesssim
T^{\frac{1}{8}}C^{\frac{b}{4}}\|P_{I_{T}}f\|_{\mathcal{L}^{2}_{x}}$

as the claim will then follow from the almost orthogonality in (14). It
follows from Galilean invariance and lack of oscillation on the time interval
$[0,T]$ for frequencies $\lesssim T^{-\frac{1}{2}}$ that

(16)
$\|e^{it\partial_{x}^{2}}P_{N}R_{C}P_{I}f\|_{L_{t}^{4}([0,T],\mathcal{L}^{4}_{x})}\lesssim
T^{\frac{1}{4}}\|P_{I}R_{C}f\|_{\mathcal{L}^{4}_{x}}.$

It remains to obtain an estimate for the number of lattice points
$n\in\mathbb{Z}^{\nu}$ with $\langle n\rangle\sim C$ and $\langle
n\rangle_{\omega}\in I_{T}$. Firstly, divide $I_{T}$ into intervals $I$ of
unit length. We can bound the number of frequencies contained in an interval
of unit length by $C^{b}=C^{\nu-1}$ by the rank of $\Lambda$. Hence, we obtain
from the Cauchy-Schwarz inequality

$\|P_{I_{T}}R_{C}f\|_{\mathcal{L}^{\infty}_{x}}\lesssim
T^{-\frac{1}{4}}C^{\frac{b}{2}}\|f\|_{\mathcal{L}^{2}_{x}}.$

Interpolation with the trivial $\mathcal{L}^{2}_{x}$-estimate gives

$\|P_{I_{T}}R_{C}f\|_{\mathcal{L}^{4}_{x}}\lesssim
T^{-\frac{1}{8}}C^{\frac{b}{4}}\|f\|_{\mathcal{L}^{2}_{x}},$

which together with (16) implies (15).

We are left with proving the almost orthogonality estimate (14). For
$P_{\lesssim 1}R_{C}e^{it\partial_{x}^{2}}f$ this is immediate from the
Cauchy-Schwarz inequality. We turn to the contribution of
$P_{N}R_{C}e^{it\partial_{x}^{2}}f$ for $1\ll N\lesssim C$. We rescale to unit
frequencies by $t\to N^{2}t$, $x\to Nx$, $\xi\to\xi/N$, and we find

$\|P_{N}R_{C}e^{it\partial_{x}^{2}}f\|^{4}_{L_{t}^{4}([0,T],\mathcal{L}^{4}_{x})}=N^{-2}\|P_{1}R_{C}e^{it\partial_{x}^{2}}f^{\prime}\|^{4}_{L_{t}^{4}([0,N^{2}T],\mathcal{L}^{4}_{x})}.$

We write

$\begin{split}&\quad\|P_{1}R_{C}e^{it\partial_{x}^{2}}f^{\prime}\|^{4}_{L_{t}^{4}([0,N^{2}T],\mathcal{L}^{4}_{x})}\\\
&=\lim_{L\to\infty}\frac{1}{2L}\int_{[0,N^{2}T]\times[-NL,NL]}\big{|}\sum_{\begin{subarray}{c}n\in\mathbb{Z}^{\nu}/N,\\\
|Nn|\sim C\end{subarray}}e^{i(\langle n\rangle_{\omega}x+\langle
n\rangle^{2}_{\omega}t)}\hat{f}(N\langle
n\rangle_{\omega})\big{|}^{4}dxdt.\end{split}$

Choose $L\gg N^{2}T$, and let $w_{A}:\mathbb{R}\to\mathbb{R}$ denote a
function with compactly supported Fourier transform, which satisfies
$|w_{A}(x)|\gtrsim 1$ for $|x|\lesssim A$ and
$\text{supp}(\hat{w}_{A})\subseteq B(0,cA^{-1})$. This allows us to dominate:

$\begin{split}&\quad\int_{[0,N^{2}T]\times[-NL,NL]}\big{|}\sum_{\begin{subarray}{c}n\in\mathbb{Z}^{\nu}/N,\\\
|Nn|\sim C\end{subarray}}e^{i(\langle n\rangle_{\omega}x+\langle
n\rangle^{2}_{\omega}t)}\hat{f}(N\langle n\rangle_{\omega})\big{|}^{4}dxdt\\\
&\lesssim\int_{\mathbb{R}^{2}}\underbrace{w_{N^{2}T}^{4}(t)w_{NL}^{4}(x)\big{|}\sum_{\begin{subarray}{c}n\in\mathbb{Z}^{\nu}/N,\\\
|n|\sim C/N\end{subarray}}e^{i(\langle n\rangle_{\omega}x+\langle
n\rangle_{\omega}^{2}t)}\hat{f}(N\langle
n\rangle_{\omega})\big{|}^{4}}_{|F(x,t)|^{4}}dxdt.\end{split}$

Let $\delta=(N^{2}T)^{-1}$. The support of the space-time Fourier transform of
$F$ is contained in $\mathcal{N}_{\delta}(\\{(\xi,|\xi|^{2}):|\xi|\lesssim
1\\})$. Consequently, applying the Córdoba–Fefferman square function estimate
yields

$\int_{\mathbb{R}^{2}}|F(x,t)|^{4}dxdt\lesssim\int\big{(}\sum_{I_{T}:N^{-1}T^{-\frac{1}{2}}-\text{interval}}|P_{I_{T}}F(x,t)|^{2}\big{)}^{2}dxdt.$

We take the exponent $1/4$, apply Minkowski’s inequality to interchange the
$\ell^{2}$ with the $L^{4}$-norm and rescale to find after taking the limit
$L\to\infty$

$\|P_{N}R_{C}e^{it\partial_{x}^{2}}f\|_{L_{t}^{4}([0,T],\mathcal{L}^{4}_{x})}\lesssim\big{(}\sum_{I_{T}:T^{-\frac{1}{2}}-\text{interval}}\|P_{I_{T}}R_{C}e^{it\partial_{x}^{2}}f\|^{2}_{L_{t}^{4}(w_{T},\mathcal{L}^{4}_{x})}\big{)}^{\frac{1}{2}},$

which is (14).

Next, we consider times $T\leq N^{-2}$. There are no Schrödinger oscillations,
which allows us to estimate

$\|e^{it\partial_{x}^{2}}P_{N}R_{C}f\|_{L_{t}^{4}([0,T],\mathcal{L}^{4}_{x})}\lesssim
T^{\frac{1}{4}}\|P_{N}R_{C}f\|_{\mathcal{L}^{4}_{x}}.$

We divide now the Fourier support $[-2N,-N/2]\cup[N/2,2N]$ into $\sim N$ unit
intervals and for every interval of unit length we have like above at most
$\lesssim C^{b}$ frequencies of height $C$. Consequently, an application of
Bernstein’s inequality yields

$\|P_{N}R_{C}f\|_{\mathcal{L}^{4}_{x}}\lesssim(NC^{b})^{\frac{1}{4}}\|f\|_{\mathcal{L}^{2}_{x}}.$

Since $T\leq N^{-2}$ taking the preceding displays together, we finish the
proof of (12).

We turn to the proof of (13). For $N\lesssim 1$ we simply use the same
frequency counting argument to bound the number of frequencies with height
$\sim C$ by $C^{b}$, which gives by Hölder’s inequality

$\|P_{N}R_{C}e^{t\partial_{x}^{3}}f\|_{L_{t}^{4}([0,T],\mathcal{L}^{4}_{x})}\lesssim
T^{\frac{1}{4}}C^{\frac{b}{4}}\|P_{N}R_{C}f\|_{\mathcal{L}^{2}_{x}}.$

For $N\gg 1$ and $T\geq N^{-3}$ we shall show the almost orthogonal
decomposition:

(17)
$\|e^{t\partial_{x}^{3}}P_{N}R_{C}f\|_{L_{t}^{4}([0,T],\mathcal{L}^{4}_{x})}\lesssim\big{(}\sum_{I_{T}:T^{-\frac{1}{2}}N^{-\frac{1}{2}}-\text{interval}}\|P_{N}R_{C}e^{t\partial_{x}^{3}}P_{I_{T}}f\|_{L_{t}^{4}(w_{T},\mathcal{L}^{4}_{x})}^{2}\big{)}^{\frac{1}{2}}.$

Then the estimate can be concluded like above by counting the frequencies of
height $C$, which are contained in an interval of length
$N^{-\frac{1}{2}}T^{-\frac{1}{2}}$. To prove (17), we rescale to unit
frequencies to find:

$\begin{split}&\quad\|P_{N}R_{C}e^{t\partial_{x}^{3}}f\|^{4}_{L_{t}^{4}([0,T],\mathcal{L}^{4}_{x})}\\\
&=N^{-4}\lim_{L\to\infty}\frac{1}{2L}\int_{[0,N^{3}T]\times[-NL,NL]}\big{|}\sum_{\begin{subarray}{c}n\in\mathbb{Z}^{\nu}/N,\\\
|\langle n\rangle_{\omega}|\sim 1,\;|n|\sim C\end{subarray}}e^{i(\langle
n\rangle_{\omega}x+t\langle n\rangle_{\omega}^{3})}\hat{f}(N\langle
n\rangle_{\omega})\big{|}^{4}dxdt\\\ &\lesssim
N^{-4}\lim_{L\to\infty}\frac{1}{2L}\int_{\mathbb{R}^{2}}\underbrace{w^{4}_{N^{3}T}(t)w^{4}_{NL}(x)\big{|}\sum_{\begin{subarray}{c}n\in\mathbb{Z}^{\nu}/N,\\\
|\langle n\rangle_{\omega}|\sim 1,\;|n|\sim C\end{subarray}}e^{i(\langle
n\rangle_{\omega}x+t\langle n\rangle_{\omega}^{3})}\hat{f}(N\langle
n\rangle_{\omega})\big{|}^{4}}_{|F(x,t)|^{4}}dxdt.\end{split}$

Let $\delta=(N^{3}T)^{-1}$. Note that
$\text{supp}(\hat{F})\subseteq\mathcal{N}_{\delta}(\\{(\xi,\xi^{3}):\,|\xi|\sim
1\\})$. We apply the square function estimate recalled in Theorem 2.1 to find

$\int_{\mathbb{R}^{2}}|F(x,t)|^{4}dxdt\lesssim\int_{\mathbb{R}^{2}}\big{(}\sum_{\theta\in\Theta_{\delta}}|P_{\theta}F(x,t)|^{2}\big{)}^{2}dxdt.$

Rescaling gives the almost orthogonal decomposition:

$\|P_{N}R_{C}e^{t\partial_{x}^{3}}f\|_{L_{t}^{4}([0,T],\mathcal{L}^{4}_{x})}\lesssim\big{(}\sum_{I_{T}:T^{-\frac{1}{2}}N^{-\frac{1}{2}}-\text{interval}}\|P_{N}R_{C}e^{t\partial_{x}^{3}}P_{I_{T}}f\|^{2}_{L_{t}^{4}(w_{T},\mathcal{L}^{4}_{x}))}\big{)}^{\frac{1}{2}}.$

It remains to count the number of frequencies contained in an
$T^{-\frac{1}{2}}N^{-\frac{1}{2}}$-frequency interval. Dividing
$T^{-\frac{1}{2}}N^{-\frac{1}{2}}$ into unit intervals and using the density
parameter, we find that the number of frequencies contained in an interval
$I_{T}$ is estimated by $\langle T^{-\frac{1}{2}}N^{-\frac{1}{2}}\rangle
C^{b}$. Arguing like above we find

$\|P_{I_{T}}R_{C}e^{t\partial_{x}^{3}}f\|_{L_{t}^{4}(w_{T},\mathcal{L}^{4}_{x})}\lesssim
T^{\frac{1}{4}}\langle
T^{-\frac{1}{2}}N^{-\frac{1}{2}}\rangle^{\frac{1}{4}}C^{\frac{b}{4}}\|P_{I_{T}}R_{C}f\|_{\mathcal{L}^{2}_{x}}.$

For $T\leq N^{-3}$ we again find no significant time oscillations and can
conclude by Hölder’s inequality and counting the frequencies:

$\begin{split}\|e^{t\partial_{x}^{3}}P_{N}R_{C}u_{0}\|_{L_{t}^{4}([0,T],\mathcal{L}^{4}_{x}(\mathbb{R}))}\lesssim
T^{\frac{1}{4}}\|P_{N}R_{C}u_{0}\|_{\mathcal{L}^{4}_{x}(\mathbb{R})}&\lesssim
T^{\frac{1}{4}}(NC^{b})^{\frac{1}{4}}\|u_{0}\|_{\mathcal{L}^{2}_{x}}\\\
&\lesssim
T^{\frac{1}{8}}N^{-\frac{1}{8}}C^{\frac{b}{4}}\|u_{0}\|_{\mathcal{L}^{2}_{x}}.\end{split}$

The proof is complete. ∎

###  3.2. Strichartz estimates from decoupling

In the following we suppose that
$\Lambda=\omega^{(1)}\cdot\mathbb{Z}^{\nu_{1}}\times\ldots\times\omega^{(d)}\cdot\mathbb{Z}^{\nu_{d}}$
for $\omega^{(i)}\in\mathbb{R}^{\nu_{i}}_{>0}$ non-resonant. Let
$b_{i}=\nu_{i}-1$, and $b=b_{1}+\ldots+b_{d}$.

Now we can formulate the Strichartz estimates from $\ell^{2}$-decoupling.

######  Theorem 3.1.

Let $d\in\mathbb{N}$, $\Lambda\subseteq\mathbb{R}^{d}$, and $b$ like above,
$f\in\mathcal{H}^{s}_{\Lambda}$, and $2<p<\infty$. Then the following estimate
holds:

(18)
$\|e^{it\Delta}f\|_{L_{t}^{p}([0,1],\mathcal{L}^{p}_{x}(\mathbb{R}^{d}))}\lesssim\|f\|_{\mathcal{H}^{s}_{\Lambda}}$

for

$s>s^{*}=b\big{(}\frac{1}{2}-\frac{1}{p}\big{)}+\max\big{(}\frac{d}{2}-\frac{d+2}{p},0\big{)}.$

###### Proof.

It suffices to prove for $C\in 2^{\mathbb{N}_{0}}$ with $s$ like above:

$\|R_{C}e^{it\Delta}f\|_{L_{t}^{p}([0,1],\mathcal{L}^{p}_{x}(\mathbb{R}^{d}))}\lesssim
C^{s}\|f\|_{\mathcal{L}^{2}_{x}(\mathbb{R}^{d})}.$

We use Minkowski’s inequality to decompose:

(19)
$\|e^{it\Delta}R_{C}f\|_{L_{t}^{p}([0,1],\mathcal{L}^{p}_{x}(\mathbb{R}^{d}))}\leq\sum_{1\leq
N\lesssim
C}\|e^{it\Delta}P_{N}R_{C}f\|_{L_{t}^{p}([0,1],\mathcal{L}^{p}_{x}(\mathbb{R}^{d}))}.$

We apply Theorem 2.3, using the same scaling and approximation argument like
in the previous proof with $\delta^{-1}=N^{2}$. This gives an almost
orthogonal decomposition

(20)
$\begin{split}&\quad\|e^{it\Delta}P_{N}R_{C}f\|_{L_{t}^{p}([0,1],\mathcal{L}^{p}_{x})}\\\
&\lesssim_{\varepsilon}\begin{cases}&N^{\varepsilon}\big{(}\sum_{I:1-\text{cube}}\|e^{it\Delta}P_{N}R_{C}P_{I}f\|^{2}_{L_{t}^{p}(w_{1},\mathcal{L}^{p}_{x})}\big{)}^{\frac{1}{2}},\quad\quad\quad\quad\;2<p\leq
p_{d},\\\
&N^{\frac{d}{2}-\frac{d+2}{p}+\varepsilon}\big{(}\sum_{I:1-\text{cube}}\|e^{it\Delta}P_{N}R_{C}P_{I}f\|^{2}_{L_{t}^{p}(w_{1},\mathcal{L}^{p}_{x})}\big{)}^{\frac{1}{2}},\quad
p_{d}\leq p<\infty.\end{cases}\end{split}$

After frequency projection to a cube with side-length $1$, the time-evolution
is trivialized and it remains to count the frequencies with height $C$
contained in a unit cube. The projection of the cube to a coordinate axis is
an interval of unit length, which allows us to count the frequencies contained
in the cube by $C^{b}$.

By the Cauchy-Schwarz inequality we obtain the estimate

$\|e^{it\Delta}P_{N}R_{C}P_{I}f\|_{L_{t}^{\infty}([0,1],\mathcal{L}^{\infty}_{x})}\lesssim
C^{\frac{b}{2}}\|R_{C}P_{I}f\|_{\mathcal{L}^{2}_{x}}.$

Interpolation with the trivial $\mathcal{L}^{2}$-estimate gives for
$p\in(2,\infty)$:

(21) $\|R_{C}P_{I}f\|_{L_{t}^{p}([0,1],\mathcal{L}^{p}_{x})}\lesssim
C^{b\big{(}\frac{1}{2}-\frac{1}{p}\big{)}}\|P_{I}f\|_{\mathcal{L}^{2}_{x}}.$

Taking (19)-(21) together, we obtain from dyadic summation

$\|e^{it\Delta}R_{C}f\|_{L_{t}^{p}([0,1],\mathcal{L}^{p}_{x})}\lesssim_{\varepsilon}(C^{\frac{d}{2}-\frac{d+2}{p}}\vee
1)C^{b\big{(}\frac{1}{2}-\frac{1}{p}\big{)}+\varepsilon}\|R_{C}f\|_{\mathcal{L}^{2}_{x}}.$

The proof is complete. ∎

##  4\. Strichartz estimates upon averaging in time

In this section we show Theorem 1.2 and a variant in higher dimensions, see
below. Decoupling and square function estimates are again the key ingredients.
After recording a variant of Littlewood-Paley decomposition, we can suppose a
dyadic frequency localization. Dyadically localized frequencies can be
rescaled to unit frequencies, after which decoupling becomes applicable. Since
we average in time, we can apply decoupling on arbitrarily small scales. This
allows us to perfectly separate the frequencies and show Strichartz estimates
with the only derivative loss coming from decoupling.

###  4.1. Littlewood-Paley decomposition

To prove the Strichartz estimates, we use a variant of Littlewood-Paley
theory. Let $\chi_{1}\in C^{\infty}_{c}(-2,2)$ be a radially decreasing
function with $\chi_{1}(\xi)\equiv 1$ for $|\xi|\leq 1$. Define for $N\in
2^{\mathbb{N}}$, $\chi_{N}:\mathbb{R}^{d}\to\mathbb{R}_{>0}$,
$\chi_{N}(|\xi|)=\chi(|\xi|/N)-\chi(2|\xi|/N)$. This gives a smooth partition
of unity:

$\sum_{N\in 2^{\mathbb{N}_{0}}}\chi_{N}(|\xi|)\equiv 1.$

We define frequency projections via
$(P_{N}f)\widehat{(}\xi)=\chi_{N}(|\xi|)\hat{f}(\xi)$. We have the following:

######  Proposition 4.1 (Littlewood-Paley decomposition).

Let $f\in\mathcal{B}_{2}(\mathbb{R}^{d})$. Then it holds for $p\in[2,\infty)$:

$\|f\|_{\mathcal{L}^{p}_{x}(\mathbb{R}^{d})}\lesssim\big{(}\sum_{N\in
2^{\mathbb{N}_{0}}}\|P_{N}f\|^{2}_{\mathcal{L}^{p}_{x}(\mathbb{R}^{d})}\big{)}^{\frac{1}{2}}.$

###### Proof.

We take suitable Schwartz functions $\varphi_{L}$, which are rapidly decaying
off $[-L,L]^{d}$ to write:

$\|f\|_{\mathcal{L}^{p}_{x}(\mathbb{R}^{d})}^{p}=\lim_{L\to\infty}\frac{1}{(2L)^{d}}\int_{\mathbb{R}^{d}}|\varphi_{L}(x)f(x)|^{p}dx.$

We require that $\text{supp}(\hat{\varphi}_{L})\subseteq B(0,C/L)$ with
$|\varphi_{L}(x)|\gtrsim 1$ for $|x|\leq L$. Now we can apply standard
Littlewood-Paley theory in $L^{p}$-spaces:

$\big{(}\frac{1}{(2L)^{d}}\int_{\mathbb{R}^{d}}|\varphi_{L}(x)f(x)|^{p}dx\big{)}^{\frac{1}{p}}\sim\big{[}\frac{1}{(2L)^{d}}\int_{\mathbb{R}^{d}}\big{(}\sum_{N\in
2^{\mathbb{N}_{0}}}|P_{N}(\varphi_{L}f)|^{2}\big{)}^{\frac{p}{2}}\big{]}^{\frac{1}{p}}.$

We have by Minkowski’s inequality:

(22)
$\big{(}\frac{1}{2L}\int_{\mathbb{R}}|\varphi_{L}(x)f(x)|^{p}dx\big{)}^{\frac{1}{p}}\lesssim\big{[}\sum_{N\in
2^{\mathbb{N}_{0}}}\big{(}\frac{1}{2L}\int_{\mathbb{R}}|P_{N}(\varphi_{L}f)|^{p}\big{)}^{\frac{2}{p}}\big{]}^{\frac{1}{2}}.$

Note that by small Fourier support of $\varphi_{L}$ we have

$P_{N}\varphi_{L}f=P_{N}\varphi_{L}\tilde{P}_{N}f$

with $\tilde{P}_{N}=P_{N/2}+P_{N}+P_{2N}$, setting $P_{1/2}=0$. Then by
uniform boundedness of $P_{N}$ in $L^{p}$ follows from (22):

$\big{(}\frac{1}{(2L)^{d}}\int_{\mathbb{R}^{d}}|\varphi_{L}(x)f(x)|^{p}dx\big{)}^{\frac{1}{p}}\lesssim\big{[}\sum_{N\in
2^{\mathbb{N}_{0}}}\big{(}\frac{1}{(2L)^{d}}\int_{\mathbb{R}^{d}}|\varphi_{L}(x)\tilde{P}_{N}f(x)|^{p}dx\big{)}^{\frac{2}{p}}\big{]}^{\frac{1}{2}}.$

Taking the limit we find by monotone convergence:

$\|f\|_{\mathcal{L}^{p}_{x}}\lesssim\big{(}\sum_{N\in
2^{\mathbb{N}_{0}}}\|\tilde{P}_{N}f\|_{\mathcal{L}^{p}_{x}}^{2}\big{)}^{\frac{1}{2}}\lesssim\big{(}\sum_{N\in
2^{\mathbb{N}_{0}}}\|P_{N}f\|^{2}_{\mathcal{L}^{p}_{x}(\mathbb{R})}\big{)}^{\frac{1}{2}}.$

∎

###  4.2. One-dimensional Strichartz estimates with time-averaging

Next, we prove Theorem 1.2. Invoking Proposition 4.1, it suffices to show
frequency-localized estimates:

(23)
$\displaystyle\|P_{N}e^{it\partial_{x}^{2}}f\|_{\mathcal{L}^{4}_{t,x}(\mathbb{R}\times\mathbb{R})}$
$\displaystyle\lesssim\|f\|_{\mathcal{L}^{2}_{x}(\mathbb{R})},$ (24)
$\displaystyle\|P_{N}e^{t\partial_{x}^{3}}f\|_{\mathcal{L}^{4}_{t,x}(\mathbb{R}\times\mathbb{R})}$
$\displaystyle\lesssim\|f\|_{\mathcal{L}^{2}_{x}(\mathbb{R})}.$

By density and limiting arguments, we can suppose that $f$ is a trigonometric
polynomial:

$f=\sum_{n=1}^{M}e^{i\lambda_{n}x}\hat{f}(\lambda_{n}).$

We turn to the proof of the frequency localized estimates. The estimates for
$N=1$ and $N>1$ are proven slightly different. For $N=1$, we write

$P_{1}f=\sum_{\begin{subarray}{c}\lambda\in\Lambda,\\\ |\lambda|\leq
1\end{subarray}}e^{i\lambda x}a_{\lambda}.$

We consider

$e^{it\partial_{x}^{2}}P_{1}f=\sum_{\begin{subarray}{c}\lambda\in\Lambda,\\\
|\lambda|\leq 1\end{subarray}}e^{i(\lambda x-t\lambda^{2})}a_{\lambda}.$

Choose $L$ large enough such that the frequencies are separated by at least
$cL^{-\frac{1}{2}}$ (this is clearly possible with the frequencies finite).
Then we can write

$\int_{[-L,L]^{2}}|e^{it\partial_{x}^{2}}P_{1}f|^{4}dxdt\lesssim\int_{\mathbb{R}^{2}}|\varphi_{L}(x,t)\cdot
e^{it\partial_{x}^{2}}P_{1}f|^{4}dxdt$

with $|\varphi_{L}|\gtrsim 1$ on $[-L,L]^{2}$ and Fourier support contained in
$B(0,c^{2}L^{-1})$. The space-time Fourier transform of
$\varphi_{L}e^{it\partial_{x}^{2}}P_{1}f$ is clearly contained in
$\mathcal{N}_{cL^{-1}}(\\{(\xi,\xi^{2}):\xi\in[-1,1]\\})$. Let $\delta=L^{-1}$
and let $\Theta_{\delta}$ be a finitely overlapping cover of
$\mathcal{N}_{cL^{-1}}(\\{(\xi,\xi^{2}):\xi\in[-1,1]\\})$ with rectangles of
size $L^{-\frac{1}{2}}\times L^{-1}$, with long-side pointing into tangential
and short side pointing into normal direction. Consequently, we can apply
Theorem 2.1 to find

$\int_{\mathbb{R}^{2}}|\varphi_{L}(x,t)e^{it\partial_{x}^{2}}P_{1}f|^{4}dxdt\lesssim\int\big{(}\sum_{\theta\in\Theta_{\delta}}|P_{\theta}(\varphi_{L}e^{it\partial_{x}^{2}}P_{1}f)|^{2}\big{)}^{2}dxdt.$

Exponentiating with $1/4$ and applying Minkowski’s inequality we find

$\begin{split}\big{(}\int_{\mathbb{R}^{2}}|\varphi_{L}(x,t)e^{it\partial_{x}^{2}}P_{1}f|^{4}dxdt\big{)}^{\frac{1}{4}}&\lesssim\big{(}\int_{\mathbb{R}^{2}}\big{(}\sum_{\theta\in\Theta_{\delta}}|\varphi_{L}e^{it\partial_{x}^{2}}P_{\tilde{\theta}}f|^{2}\big{)}^{2}dxdt\big{)}^{\frac{1}{4}}\\\
&\lesssim\big{[}\sum_{\theta\in\Theta_{\delta}}\big{(}\int_{\mathbb{R}^{2}}|\varphi_{L}e^{it\partial_{x}^{2}}P_{\tilde{\theta}}f|^{4}dxdt\big{)}^{\frac{2}{4}}\big{]}^{\frac{1}{2}}.\end{split}$

By the compact Fourier support of $\varphi_{L}$ we can write
$P_{\theta}\varphi_{L}f=P_{\theta}\varphi_{L}P_{\tilde{\theta}}f$ with
$\tilde{\theta}$ denoting a mild enlargement of $\theta$.

Still, $P_{\tilde{\theta}}f$ contains at most $\mathcal{O}(1)$ frequencies
$(\lambda,\lambda^{2})$ with $\lambda\in\Lambda_{1}$ by our choice of $L$.
Consequently, the oscillations have been trivialized, and we compute the
integral

$\int_{\mathbb{R}^{2}}|\varphi_{L}e^{it\partial_{x}^{2}}P_{\tilde{\theta}}f|^{4}dxdt\lesssim
L^{2}\sum_{\lambda:(\lambda,\lambda^{2})\in\tilde{\theta}}|a_{\lambda}|^{4}.$

This gives

$\big{(}\int_{\mathbb{R}^{2}}|\varphi_{L}(x,t)e^{it\partial_{x}^{2}}P_{1}f|^{4}dxdt\big{)}^{\frac{1}{4}}\lesssim
L^{\frac{1}{2}}\big{(}\sum_{\theta\in\Theta_{\delta}}\big{(}\sum_{\lambda\in\theta}|a_{\lambda}|^{4}\big{)}^{\frac{1}{2}}\big{)}^{\frac{1}{2}}\lesssim
L^{\frac{1}{2}}\big{(}\sum_{\begin{subarray}{c}\lambda\in\Lambda,\\\
|\lambda|\leq 1\end{subarray}}|a_{\lambda}|^{2}\big{)}^{\frac{1}{2}}$

and concludes the proof of (23) for $N=1$.

The estimate

$\|P_{1}e^{t\partial_{x}^{3}}f\|_{\mathcal{L}^{4}_{t,x}(\mathbb{R}\times\mathbb{R})}\lesssim\|f\|_{\mathcal{L}^{2}_{x}(\mathbb{R})}$

is proved along the same lines. The only difference is that we choose $L$
large enough such that frequencies are separated by at least
$L^{-\frac{1}{3}}$. Applying Theorem 2.2 with $\delta=L^{-1}$ yields

$\int_{\mathbb{R}^{2}}\big{|}\varphi_{L}(x,t)e^{t\partial_{x}^{3}}P_{1}f\big{|}^{4}dxdt\lesssim\int\big{(}\sum_{\theta\in\Theta_{\delta}}\big{|}P_{\theta}\big{(}\varphi_{L}e^{it\partial_{x}^{2}}P_{1}f\big{)}\big{|}^{2}\big{)}^{2}dxdt.$

This again trivializes the time evolution and we find following along the
above lines:

$\big{(}\int_{\mathbb{R}^{2}}|\varphi_{L}(x,t)e^{t\partial_{x}^{3}}P_{1}f|^{4}dxdt\big{)}^{\frac{1}{4}}\lesssim
L^{\frac{1}{2}}\big{(}\sum_{\lambda\in\Lambda_{1}}|a_{\lambda}|^{2}\big{)}^{\frac{1}{2}}.$

This concludes (24) for $N=1$.

We turn to the estimates (23) and (24) for $N>1$. In both cases the proof
hinges on Theorem 2.1. We turn to the proof for the Schrödinger propagation.
Firstly, recall that we assume the frequency set of $P_{N}f$ to be finite. We
denote the minimal separation by $\sigma>0$. By rescaling we find

(25)
$\int_{[-L,L]\times[-L,L]}|e^{it\partial_{x}^{2}}P_{N}f|^{4}dxdt=N^{-3}\int_{[-N^{2}L,N^{2}L]\times[-NL,NL]}|e^{it\partial_{x}^{2}}P_{1}f^{\prime}|^{4}dxdt.$

We dominate with suitable Schwartz functions with compact Fourier support:

$\int_{\mathbb{R}^{2}}w_{N^{2}L}^{4}(t)w^{4}_{NL}(x)|e^{it\partial_{x}^{2}}P_{1}f^{\prime}|^{4}dxdt=\int_{\mathbb{R}^{2}}|\underbrace{w_{N^{2}L}(t)w_{NL}(x)e^{it\partial_{x}^{2}}P_{1}f^{\prime}}_{F(x,t)}|^{4}dxdt.$

We have
$\text{supp}(\hat{F})\subseteq\mathcal{N}_{(NL)^{-1}}(\\{(\xi,\xi^{2}):\frac{1}{2}\leq|\xi|\leq
2\\})$. The frequencies are separated by $N^{-1}\sigma$, with $\sigma$
denoting the original separation of frequencies of $P_{N}f$. We aim to apply
Theorem 2.1 with $\delta=(NL)^{-1}$. Choose $L$ large enough such that
$(NL)^{-\frac{1}{2}}\ll N^{-1}\sigma$. Then applying Theorem 2.1 gives

$\int_{\mathbb{R}^{2}}|F(x,t)|^{4}dxdt\lesssim\int_{\mathbb{R}^{2}}\big{(}\sum_{\theta\in\Theta_{\delta}}|F_{\theta}(x,t)|^{2}\big{)}^{2}dxdt.$

With the frequencies being separated, i.e., only finitely many
$(\lambda,\lambda^{2})$ are contributing to $F_{\theta}$, we can conclude the
argument like in the proof for $N=1$.

For the Airy evolution only the rescaling changes:

$\int_{[-L,L]\times[-L,L]}|P_{N}e^{t\partial_{x}^{3}}f|^{4}dxdt=N^{-4}\int_{[-N^{3}L,N^{3}L]\times[-NL,NL]}|P_{1}e^{t\partial_{x}^{3}}f^{\prime}|^{4}dxdt.$

Dominating like above

$\int_{[-N^{3}L,N^{3}L]\times[-NL,NL]}|P_{1}e^{t\partial_{x}^{3}}f^{\prime}|^{4}dxdt\lesssim\int_{\mathbb{R}^{2}}|\underbrace{w_{N^{3}L}(t)w_{NL}(x)P_{1}e^{t\partial_{x}^{3}}f^{\prime}}_{F}|^{4}dxdt.$

Clearly,
$\text{supp}(\hat{F})\subseteq\mathcal{N}_{(NL)^{-1}}(\\{(\xi,\xi^{3}):|\xi|\sim
1\\})$. Since the curve $\\{(\xi,\xi^{3}):|\xi|\sim 1\\}$ has curvature, we
can again apply Theorem 2.1 and conclude the argument like above.

$\hfill\Box$

###  4.3. Strichartz estimates for quasi-periodic functions in higher
dimensions

We record a higher-dimensional version of Theorem 1.2. Let
$\Lambda=\omega^{(1)}\cdot\mathbb{Z}^{\nu_{1}}\times\ldots\times\omega^{(d)}\cdot\mathbb{Z}^{\nu_{d}}\subseteq\mathbb{R}^{d}$
with $\omega^{(i)}$ non-resonant. Moreover, we suppose the following
diophantine condition:

_There are $\alpha_{i},\beta_{i}>0$ such that for any
$n_{i}\in\mathbb{Z}^{\nu_{i}}\backslash 0$ it holds_

(26) $|\langle
n_{i}\rangle_{\underline{\omega}}|\geq\alpha_{i}|n_{i}|^{-\beta_{i}}.$

Denote $\beta=\beta_{1}+\ldots+\beta_{d}$.

We show the following:

######  Theorem 4.2.

Let $\Lambda$ be like above, $p_{d}=\frac{2(d+2)}{d}$, and $\varepsilon>0$.
For $f\in\mathcal{H}^{\varepsilon}_{\Lambda}$ the following estimate holds:

$\|e^{it\Delta}f\|_{\mathcal{L}^{p_{d}}_{t,x}(\mathbb{R}\times\mathbb{R}^{d})}\lesssim\|f\|_{\mathcal{H}^{\varepsilon}_{\Lambda}}.$

###### Proof.

Let $p=p_{d}$ for brevity. We can use a Littlewood-Paley decomposition
provided by Proposition 4.1 to reduce to

$\|P_{N}e^{it\Delta}f\|_{\mathcal{L}^{p}_{t,x}(\mathbb{R}\times\mathbb{R}^{d})}\lesssim\|f\|_{\mathcal{H}^{\varepsilon}_{\Lambda}}.$

For the above, it suffices to show for $C\gtrsim N$:

(27)
$\|P_{N}R_{C}e^{it\Delta}f\|_{\mathcal{L}^{p}_{t,x}(\mathbb{R}\times\mathbb{R}^{d})}\lesssim_{\varepsilon}(NC)^{\varepsilon}\|f\|_{\mathcal{L}^{2}_{x}}.$

We can estimate the gap between two distinct frequencies
$\underline{n}\neq\underline{n}^{\prime}\in\mathbb{Z}^{\nu}$ of height $C$ by
the diophantine condition:

$|\langle\underline{n}\rangle_{\underline{\omega}}-\langle\underline{n}\rangle_{\underline{\omega}}|\gtrsim
C^{-\beta}.$

We can choose $L$ large enough such that $(NL)^{-\frac{1}{2}}\ll C^{-\beta}$.
Then, it holds

(28)
$\begin{split}&\quad\int_{[-L,L]\times[-L,L]^{d}}\big{|}\sum_{\begin{subarray}{c}\lambda\in\Lambda,\\\
|\lambda|\sim N,\\\ h(\lambda)\sim C\end{subarray}}e^{i(\lambda
x-t\lambda^{2})}\hat{f}(\lambda)\big{|}^{p}dxdt\\\
&=N^{-(d+2)}\int_{[-N^{2}L,N^{2}L]\times[-NL,NL]}\big{|}\sum_{\begin{subarray}{c}\lambda^{\prime}\in\Lambda/N,\\\
|\lambda^{\prime}|\sim 1,\\\ h(N\lambda^{\prime})\sim
C\end{subarray}}e^{i(\lambda^{\prime}x^{\prime}-t^{\prime}(\lambda^{\prime})^{)}}\hat{f}(N\lambda^{\prime})\big{|}^{p}dxdt\\\
&\lesssim
N^{-(d+2)}\int_{\mathbb{R}^{d+1}}\big{|}\underbrace{w_{N^{2}L}(t)w_{NL}(x)\sum_{\begin{subarray}{c}\lambda^{\prime}\in\Lambda/N,\\\
|\lambda^{\prime}|\sim 1,\\\ h(N\lambda^{\prime})\sim
C\end{subarray}}e^{i(\lambda^{\prime}x^{\prime}-t^{\prime}(\lambda^{\prime})^{2})}\hat{f}(N\lambda^{\prime})}_{F(x,t)}\big{|}^{p}dxdt.\end{split}$

We have that
$\text{supp}(\hat{F})\subseteq\mathcal{N}_{\delta}(\\{(\xi,-|\xi|^{2}):|\xi|\leq
1\\})$ for $\delta=(NL)^{-1}$. Decoupling gives a decomposition into caps on
the Fourier side of size $\delta^{\frac{1}{2}}$ in the tangential direction
and $\delta$ in the normal direction:

(29)
$\|F\|_{L^{p}(\mathbb{R}^{d+1})}\lesssim_{\varepsilon}(NC)^{\varepsilon}\big{(}\sum_{\theta\in\Theta_{\delta}}\|P_{\theta}F\|^{2}_{L^{p}(\mathbb{R}^{d+1})}\big{)}^{\frac{1}{2}}.$

Like in the previous arguments, we see that there are only finitely many
frequencies $\lambda\in\Lambda$ contributing to $P_{\theta}F$. Hence, we can
carry out the integration, average, and obtain from (28)-(29) the estimate
(27). The proof is complete.

∎

##  5\. Low regularity local well-posedness for the nonlinear Schrödinger
equation with quasi-periodic initial data

This section is devoted to the proof of Theorem 1.3. We begin with the
introduction of adapted function spaces:

###  5.1. Adapted function spaces

We define adapted function spaces behaving well with sharp-time cutoff. These
constitute a logarithmic refinement of Fourier restriction spaces used by
Bourgain [5, 6] in the analysis of the periodic case. We choose these function
spaces because the key bilinear estimates are readily formulated within.

Functions with bounded variation were investigated by Wiener [34] for the
first time. In the following we shall also need the predual $U^{p}$-spaces. We
refer to the detailed exposition of Hadac–Herr–Koch [21] (and [22]) for
details without which the following account might appear a little bit sketchy.

######  Definition 5.1.

Let $\mathcal{Z}$ be the collection of finite non-decreasing sequences
$\\{t_{k}\\}_{k=0}^{K}$ in $\mathbb{R}$. For $1\leq p<\infty$ we define
$V^{p}$ as the space of all right-continuous functions
$u:\mathbb{R}\to\mathbb{C}$ with $\lim_{t\to-\infty}u(t)=0$ and

$\|u\|_{V^{p}}=\sup_{\\{t_{k}\\}_{k=0}^{K}\in\mathcal{Z}}\big{(}\sum_{k=1}^{K}|u(t_{k})-u(t_{k-1})|^{p}\big{)}^{\frac{1}{p}}<\infty.$

The following $U^{p}_{\Delta}$-/$V^{p}_{\Delta}$-spaces will be useful for
intermediate steps. The function space, in which we formulate the well-
posedness results, are the $Y^{s}_{\pm}$-spaces defined below.

######  Definition 5.2.

Let $1\leq p<\infty$. We define

$V^{p}_{\Delta}=\\{u:\mathbb{R}\times\mathbb{R}^{d}\to\mathbb{C}:e^{-it\Delta}u\in
V^{p}\mathcal{L}^{2}(\mathbb{R}^{d})\\}$

with norm given by

$\|u\|_{V^{p}_{\Delta}}=\big{(}\sup_{\\{t_{k}\\}_{k=0}^{K}\in\mathcal{Z}}\sum_{k=1}^{K}\|e^{-it_{k}\Delta}u(t_{k})-e^{-it_{k-1}\Delta}u(t_{k-1})\|^{p}_{\mathcal{L}^{2}_{x}}\big{)}^{\frac{1}{p}}<\infty.$

The predual $U^{p}$-spaces admit an atomic decomposition (see [21]). As a
consequence we obtain the transfer principle: Suppose that for $2\leq
p<\infty$ the estimate

$\|R_{M}e^{it\Delta}f\|_{L_{t}^{p}([0,1],\mathcal{L}^{p}_{x}(\mathbb{R}))}\lesssim
C(M)\|f\|_{\mathcal{L}^{2}_{x}}$

holds. Then we have the following consequence:

$\|R_{M}u\|_{L_{t}^{p}([0,1],\mathcal{L}^{p}_{x}(\mathbb{R}))}\lesssim
C(M)\|u\|_{U^{p}_{\Delta}}.$

In the following fix $\Lambda\subseteq\mathbb{R}^{d}$ like in (11).

######  Definition 5.3.

For $s\in\mathbb{R}$, we define $Y^{s}$ as the space of functions
$u:\mathbb{R}\times\mathbb{R}^{d}\to\mathbb{C}$ with
$\text{supp}(\hat{u}(t))\subseteq\Lambda$ such that $e^{it\langle
n\rangle_{\omega}^{2}}\widehat{u(t)}(\langle n\rangle_{\omega})$ lies in
$V^{2}$ for any
$n\in\mathbb{Z}^{\nu_{1}}\times\ldots\times\mathbb{Z}^{\nu_{d}}=:\mathbb{Z}^{\nu}$
and the norm is defined as

(30)
$\|u\|^{2}_{Y^{s}}=\sum_{n\in\mathbb{Z}^{\nu}}(1+|n|^{2})^{s}\|e^{it\langle
n\rangle_{\omega}^{2}}\widehat{u(t)}(\langle
n\rangle_{\omega})\|^{2}_{V^{2}}<\infty.$

For $T>0$ and $u:[0,T]\times\mathbb{R}^{d}\to\mathbb{C}$ we define

$\|u\|^{2}_{Y^{s}_{T}}=\sum_{n\in\mathbb{Z}^{\nu}}(1+|n|^{2})^{s}\|e^{it\langle
n\rangle_{\omega}^{2}}\widehat{u(t)}(\langle
n\rangle_{\omega})\|^{2}_{V^{2}(0,T)}.$

We have the following embeddings:

$Y^{0}_{T}\hookrightarrow V^{2}_{\Delta}\hookrightarrow U^{p}_{\Delta}\text{
for }p>2.$

The first is immediate from the definition and for the second we refer to
[21].

We summarize further properties to be used freely in the following:

######  Proposition 5.4 ([24, Section 2], [27]).

The following holds:

  * •

Let $A,B\subseteq\mathbb{R}^{d}$ be disjoint. For $s\geq 0$ we have

$\|P_{A\cup
B}u\|^{2}_{Y^{s}_{T}}=\|P_{A}u\|^{2}_{Y^{s}}+\|P_{B}u\|^{2}_{Y^{s}}.$

  * •

For $s\geq 0$, $T>0$, and $f\in L_{T}^{1}\mathcal{H}^{s}_{\Lambda}$, denote

$\mathcal{I}(f)(t)=\int_{0}^{t}e^{i(t-s)\Delta}f(s)ds$

we have with $\chi_{T}=\chi_{[0,T)}$ denoting the indicator function:

$\|\chi_{T}\cdot\mathcal{I}(f)\|_{Y^{s}}\lesssim\sup_{\begin{subarray}{c}v\in
Y^{-s}:\\\ \|v\|_{Y^{-s}}\leq 1\end{subarray}}\big{|}\int_{0}^{T}\langle
f,\bar{v}\rangle_{\mathcal{L}^{2}_{x}}dt\big{|}.$

  * •

For $T>0$ and a function $\phi\in\mathcal{H}^{s}_{\Lambda}$ we have

$\|\chi_{T}e^{it\Delta}\phi\|_{Y^{s}}\lesssim\|\phi\|_{\mathcal{H}^{s}_{\Lambda}}$

and for $u\in Y^{s}_{T}$:

$\|u\|_{L^{\infty}_{T}\mathcal{H}^{s}_{\Lambda}}\lesssim\|u\|_{Y^{s}_{T}}.$

###  5.2. Multilinear Strichartz estimates in adapted function spaces

In the following we denote with $Q^{a}_{C}$ for $a\in\mathbb{Z}^{\nu}$ the
following frequency projection:

$Q^{a}_{C}f=\sum_{\begin{subarray}{c}n\in\mathbb{Z}^{\nu},\\\ |n-a|\leq
C\end{subarray}}e^{i\langle n\rangle_{\omega}x}\hat{f}(\langle
n\rangle_{\omega}).$

We record the following consequence of Galilean invariance:

######  Proposition 5.5 (Galilean invariance).

Let $2\leq p\leq\infty$, $T\in(0,1]$. Suppose that the following estimate
holds:

(31)
$\|e^{it\Delta}R_{C}f\|_{L_{t}^{p}([0,T],\mathcal{L}^{p}_{x}(\mathbb{R}^{d}))}\lesssim
T^{\delta}C^{s}\|R_{C}f\|_{\mathcal{L}^{2}_{x}(\mathbb{R}^{d})}.$

Then it holds with implicit constant independent of $a\in\mathbb{Z}^{\nu}$:

(32)
$\|e^{it\Delta}Q_{C}^{a}f\|_{L_{t}^{p}([0,1],\mathcal{L}^{p}_{x}(\mathbb{R}^{d}))}\lesssim
T^{\delta}C^{s}\|Q_{C}^{a}f\|_{\mathcal{L}^{2}_{x}(\mathbb{R}^{d})}.$

###### Proof.

We write

$\begin{split}e^{it\Delta}Q_{C}^{a}f&=\sum_{\begin{subarray}{c}n\in\mathbb{Z}^{\nu},\\\
|n-a|\leq C\end{subarray}}e^{i(\langle n\rangle_{\omega}x-t\langle
n\rangle_{\omega}^{2})}\hat{f}(\langle n\rangle_{\omega})\\\
&=\sum_{\begin{subarray}{c}n^{\prime}\in\mathbb{Z}^{\nu},\\\ |n^{\prime}|\leq
C\end{subarray}}e^{i((\langle n^{\prime}\rangle_{\omega}+\langle
a\rangle_{\omega})x-t(\langle n^{\prime}\rangle_{\omega}+\langle
a\rangle_{\omega})^{2})}\hat{f}(\langle n^{\prime}\rangle_{\omega}+\langle
a\rangle_{\omega})\\\ &=e^{i\langle a\rangle_{\omega}x-it\langle
a\rangle^{2}_{\omega}}\sum_{\begin{subarray}{c}n^{\prime}\in\mathbb{Z}^{\nu},\\\
|n^{\prime}|\leq C\end{subarray}}e^{i(\langle
n^{\prime}\rangle_{\omega}(x-2t\langle a\rangle_{\omega})-t\langle
n^{\prime}\rangle_{\omega}^{2})}\hat{f}(\langle
n^{\prime}+a\rangle_{\omega}).\end{split}$

Consequently, after the linear change of variables $x\to x-2t\langle
a\rangle_{\omega}$ for fixed $t\in[0,T]$, which leaves the
$\mathcal{L}^{p}_{x}$-norm invariant, (31) becomes applicable. This concludes
the proof of (32). ∎

For sake of illustration, we record the following one-dimensional instance,
which is a consequence of Theorem 1.1, the above, and the transfer principle:

######  Corollary 5.6.

With $\Lambda=\omega\cdot\mathbb{Z}^{\nu}\subseteq\mathbb{R}$ and notations
like above, the following estimate holds:

$\|Q^{a}_{C}e^{it\partial_{x}^{2}}f\|_{L_{t}^{4}([0,T],\mathcal{L}^{4}_{x})}\lesssim_{\varepsilon}T^{\frac{1}{8}}C^{\frac{b}{4}+\varepsilon}\|Q_{C}f\|_{Y_{T}^{0}}.$

We can now formulate the bilinear refinement, which will be crucial for the
local well-posedness result for the cubic NLS on $\mathbb{R}$:

######  Proposition 5.7.

Let $\Lambda=\omega\cdot\mathbb{Z}^{\nu}\subseteq\mathbb{R}$ be like above,
and $C_{1},C_{2}\in 2^{\mathbb{N}_{0}}$ with $C_{1}\leq C_{2}$. Then the
following estimate holds:

$\|R_{C_{1}}u_{1}R_{C_{2}}u_{2}\|_{L_{t}^{2}([0,T],\mathcal{L}^{2}_{x})}\lesssim_{\varepsilon}T^{\frac{1}{4}}C_{1}^{\frac{b}{2}+\varepsilon}\|R_{C_{1}}u_{1}\|_{Y_{T}^{0}}\|R_{C_{2}}u_{2}\|_{Y_{T}^{0}}.$

###### Proof.

We use almost orthogonality induced by the convolution constraint to write

$\|R_{C_{1}}u_{1}R_{C_{2}}u_{2}\|^{2}_{L_{t}^{2}([0,T],\mathcal{L}^{2}_{x})}\lesssim\sum_{a:C_{1}-\text{separated}}\|R_{C_{1}}u_{1}Q^{a}_{C_{1}}R_{C_{2}}u_{2}\|_{L_{t}^{2}([0,T],\mathcal{L}^{2}_{x})}^{2}$

with $\sum_{a:C_{1}-\text{separated}}Q_{C_{1}}^{a}R_{C_{2}}f=R_{C_{2}}f$. Then
we can apply the Strichartz estimate from Corollary 5.6 twice and assemble the
frequency projection by the properties of the function spaces:

$\begin{split}\sum_{a}\|R_{C_{1}}u_{1}Q^{a}_{C_{1}}R_{C_{2}}u_{2}\|^{2}_{L_{t}^{2}([0,T],\mathcal{L}^{2}_{x})}&\leq\|R_{C_{1}}u_{1}\|^{2}_{L_{t}^{4}([0,T],\mathcal{L}^{4}_{x})}\sum_{a}\|Q^{a}_{C_{1}}R_{C_{2}}u_{2}\|^{2}_{L_{t}^{4}([0,T],\mathcal{L}^{4}_{x})}\\\
&\lesssim_{\varepsilon}T^{\frac{1}{2}}C_{1}^{b+\varepsilon}\|R_{C_{1}}u_{1}\|^{2}_{Y_{T}^{0}}\|R_{C_{2}}u_{2}\|^{2}_{Y_{T}^{0}}.\end{split}$

∎

The multilinear generalization reads as follows:

######  Proposition 5.8.

Let $d\geq 1$, and
$\Lambda=\omega^{(1)}\cdot\mathbb{Z}^{\nu_{1}}\times\ldots\times\omega^{(d)}\cdot\mathbb{Z}^{\nu_{d}}\subseteq\mathbb{R}^{d}$,
with $b_{i}=\nu_{i}-1$, and let $b=b_{1}+\ldots+b_{d}$. Let $m\geq 3$ for
$d=1$ and $m\geq 2$ for $d\geq 2$, and $M_{1}\geq M_{2}\geq\ldots\geq M_{m}$.
Let $s>b\big{(}\frac{1}{2}-\frac{1}{2m}\big{)}+\frac{d}{2}-\frac{d+2}{2m}$.
Then the following estimate holds for some $\varepsilon>0$, $\delta>0$:

(33) $\|R_{M_{1}}u_{1}\ldots
R_{M_{m}}u_{m}\|_{L_{t}^{2}([0,T],\mathcal{L}^{2}_{x}(\mathbb{R}^{d}))}\lesssim
T^{\delta}M_{2}^{2s}(M_{3}\ldots
M_{m})^{s}\prod_{i=1}^{m}\|R_{M_{i}}u_{i}\|_{Y_{T}^{0}}.$

###### Proof.

By convolution constraint and almost orthogonality, we can decompose the
frequencies of maximal height $M_{1}$ into frequencies of height $M_{2}$ up to
Galilean invariance. We write

$\begin{split}&\quad\|R_{M_{1}}u_{1}\ldots
R_{M_{m}}u_{m}\|_{L_{t}^{2}([0,T],\mathcal{L}^{2}_{x}(\mathbb{R}^{d}))}^{2}\\\
&\lesssim\sum_{a:M_{2}-\text{separated}}\|Q_{M_{2}}^{a}R_{M_{1}}u_{1}R_{M_{2}}u_{2}\ldots
R_{M_{m}}u_{m}\|^{2}_{L_{t}^{2}([0,T],\mathcal{L}^{2}_{x}(\mathbb{R}^{d}))}.\end{split}$

Now we apply Hölder’s inequality

$\begin{split}&\quad\|Q_{M_{2}}^{a}R_{M_{1}}u_{1}\ldots
R_{M_{m}}u_{m}\|_{L_{t}^{2}([0,T],\mathcal{L}^{2}_{x}(\mathbb{R}^{d}))}\\\
&\leq\|Q_{M_{2}}^{a}R_{M_{1}}u_{1}\|_{L_{t}^{2m}([0,T],\mathcal{L}^{2m}_{x}}\prod_{i=2}^{m}\|R_{M_{i}}u_{i}\|_{L_{t}^{2m}([0,T],\mathcal{L}^{2m}_{x}(\mathbb{R}^{d}))}\end{split}$

and next Theorem 3.1 together with Proposition 5.5 and the transfer principle:
This allows us to estimate the above by

(34)
$\begin{split}&\quad\|Q_{M_{2}}^{a}R_{M_{1}}u_{1}\|_{L_{t}^{2m}([0,T],\mathcal{L}^{2m}_{x}}\prod_{i=2}^{m}\|R_{M_{i}}u_{i}\|_{L_{t}^{2m}([0,T],\mathcal{L}^{2m}_{x}(\mathbb{R}^{d}))}\\\
&\lesssim M_{2}^{2s}(M_{3}\ldots
M_{m})^{s}\|Q_{M_{2}}^{a}R_{M_{1}}u_{1}\|_{Y^{0}_{T}}\prod_{i=2}^{m}\|R_{M_{i}}u_{i}\|_{Y^{0}_{T}}.\end{split}$

We interpolate with the following estimate, which is a consequence of Hölder’s
inequality:

(35) $\begin{split}&\quad\|R_{M_{1}}u_{1}\ldots
R_{M_{m}}u_{m}\|_{L_{t}^{2}([0,T],\mathcal{L}^{2}_{x}(\mathbb{R}^{d}))}\\\
&\leq\|R_{M_{1}}u_{1}\|_{L_{t}^{2}([0,T],\mathcal{L}^{2}_{x}(\mathbb{R}^{d}))}\prod_{i=2}^{d}\|R_{M_{i}}u_{i}\|_{L_{t}^{\infty}([0,T],\mathcal{L}^{\infty}_{x}(\mathbb{R}^{d}))}\\\
&\leq T^{\frac{1}{2}}(M_{2}\ldots
M_{d})^{C}\prod_{i=1}^{m}\|R_{M_{i}}u_{i}\|_{Y_{T}^{0}}.\end{split}$

Above $C=\nu_{1}+\ldots+\nu_{d}$. Interpolation of (34) and (35) yields that
there for any $\varepsilon>0$ there is $\delta>0$ such that

$\|R_{M_{1}}u_{1}\ldots
R_{M_{m}}u_{m}\|_{L_{t}^{2}([0,T],\mathcal{L}^{2}_{x}(\mathbb{R}^{d}))}\lesssim
T^{\delta}M_{2}^{2s+\varepsilon}(M_{3}\ldots
M_{m})^{s+\varepsilon}\prod_{i=1}^{m}\|R_{M_{i}}u_{i}\|_{Y_{T}^{0}}.$

Redefining $s\to s+\varepsilon$ finishes the proof. ∎

######  Remark 5.9.

The trilinear estimate for $d=1$ reads

$\|R_{M_{1}}u_{1}R_{M_{2}}u_{2}R_{M_{3}}u_{3}\|_{L_{t}^{2}([0,T],\mathcal{L}^{2}_{x}(\mathbb{R}^{d}))}\lesssim
T^{\delta}M_{2}^{2s}M_{3}^{s}\prod_{i=1}^{3}\|R_{M_{i}}u_{i}\|_{Y_{T}^{0}}$

for $s>\frac{b}{3}$. We remark that an alternative estimate is provided by

$\|R_{M_{1}}u_{1}R_{M_{2}}u_{2}R_{M_{3}}u_{3}\|_{L_{t}^{2}([0,T],\mathcal{L}^{2}_{x}(\mathbb{R}^{d}))}\lesssim
T^{\frac{1}{4}}M_{2}^{2r}M_{3}^{\frac{\nu}{2}}\prod_{i=1}^{3}\|R_{M_{i}}u_{i}\|_{Y_{T}^{0}}$

for $r>\frac{b}{4}$. If $M_{3}\ll M_{2}$, then this estimate is more favorable
than the one recorded above and will also give an improvement of the local
well-posedness result for the quintic NLS. We omit the details since it does
not appear to provide a sharp estimate.

###  5.3. Proof of local well-posedness

Recall that $\Lambda=\omega\cdot\mathbb{Z}^{\nu}\subseteq\mathbb{R}$ with
$\omega_{i}\in\mathbb{R}_{>0}$ linearly independent over $\mathbb{Q}$, and
$b=\nu-1$. We consider the Cauchy problem:

(36)
$\left\\{\begin{array}[]{cl}i\partial_{t}u+\partial_{x}^{2}u&=\pm|u|^{2}u,\quad(t,x)\in\mathbb{R}\times\mathbb{R},\\\
u(0)&=u_{0}\in\mathcal{H}^{s}_{\Lambda}(\mathbb{R}).\end{array}\right.$

We formulate a detailed version of Theorem 1.3:

######  Theorem 5.10 (Local well-posedness of NLS).

Let $s>\frac{b}{2}$. Then there is a real-analytic data-to-solution mapping
for (36)

$S_{T}^{s}:\mathcal{H}^{s}_{\Lambda}\to Y^{s}_{T}\subseteq
C([0,T],\mathcal{H}^{s}_{\Lambda}),\quad u_{0}\mapsto u,$

for some $T=T(s,\|u_{0}\|_{\mathcal{H}^{s}_{\Lambda}})$, which is lower semi-
continuous, decreasing in the second variable, and satisfies $\liminf_{a\to
0}T(s,a)\gtrsim 1$.

###### Proof of Theorem 5.10.

We obtain local-in-time solutions as fixed points of the integral equation for
$s>\frac{b}{2}$ given by

$\Phi_{u_{0}}:Y_{T}^{s}\to Y_{T}^{s},\quad
u(t)=\chi_{T}(t)e^{it\partial_{x}^{2}}u_{0}-i\chi_{T}(t)\int_{0}^{t}e^{i(t-s)\partial_{x}^{2}}(|u|^{2}u)(s)ds.$

$T$ will be chosen as $T=T(\|u_{0}\|_{\mathcal{H}_{\Lambda}^{s}})$. Details
regarding the application of the contraction mapping principle can be found in
the exposition on _quantitative well-posedness_ by Bejenaru–Tao [1].

Note that for $u_{0}\in\mathcal{H}^{s}_{\Lambda}$ we have

$\|e^{it\partial_{x}^{2}}u_{0}\|_{L_{T}^{\infty}\mathcal{H}^{s}_{x}}\lesssim\|u_{0}\|_{\mathcal{H}^{s}_{x}}.$

So it remains to establish a trilinear estimate (cf. [1]), which is the
content of the following proposition:

######  Proposition 5.11.

Let $s>\frac{b}{2}$. Then the following trilinear estimate holds for $s\geq
s^{\prime}>\frac{b}{2}$, and $T\in(0,1]$:

(37)
$\big{\|}\chi_{T}(t)\int_{0}^{t}e^{i(t-s)\partial_{x}^{2}}(u_{1}\overline{u_{2}}u_{3})(s)ds\big{\|}_{Y_{T}^{s}}\lesssim
T^{\frac{1}{4}}\|u_{1}\|_{Y^{s}_{T}}\|u_{2}\|_{Y^{s^{\prime}}_{T}}\|u_{3}\|_{Y^{s^{\prime}}_{T}}.$

###### Proof.

To this end, we use duality to write

$\big{\|}\chi_{T}(t)R_{C}\int_{0}^{t}e^{i(t-s)\partial_{x}^{2}}(u_{1}\overline{u_{2}}u_{3})(s)ds\big{\|}_{Y_{T}^{s}}\lesssim
C^{s}\sup_{\|v\|_{Y_{T}^{0}}=0}\iint\overline{R_{C}v}R_{C}(u_{1}\overline{u_{2}}u_{3}).$

In the following we omit the complex conjugation, as it can be readily checked
that the estimates do not depend on the conjugation. This follows from the
invariance of the $L^{4}$-norm under complex conjugation.

We carry out a height frequency decomposition of $R_{C}(u_{1}u_{2}u_{3})$:
Write

$R_{C}(u_{1}u_{2}u_{3})=\sum_{C_{1},C_{2},C_{3}}R_{C}(R_{C_{1}}u_{1}R_{C_{2}}u_{2}R_{C_{3}}u_{3}),$

and it is easy to see that $C_{\max}=\max(C_{1},C_{2},C_{3})\gtrsim C$ by
otherwise impossible frequency interaction.

We analyze two cases:
$(I)$ $C_{\max}\sim C$ and $(II)$ $C_{\max}\gg C$.

In case $(I)$ we suppose that $C_{1}=C_{\max}$. Applying Proposition 5.7 twice
yields

$\begin{split}\iint
R_{C}vR_{C_{1}}u_{1}R_{C_{2}}u_{2}R_{C_{3}}u_{3}dxdt&\leq\|R_{C}vR_{C_{2}}u_{2}\|_{L_{t}^{2}([0,T],\mathcal{L}^{2}_{x})}\|R_{C_{1}}u_{1}R_{C_{3}}u_{3}\|_{L_{t}^{2}([0,T],\mathcal{L}^{2}_{x})}\\\
&\lesssim_{\varepsilon}T^{\frac{1}{4}}C_{2}^{\frac{1}{2}+\varepsilon}C_{3}^{\frac{1}{2}+\varepsilon}\|R_{C}v\|_{Y_{T}^{0}}\prod_{i=1}^{3}\|R_{C_{i}}u_{i}\|_{Y^{0}_{T}}.\end{split}$

At this point (37) follows from dyadic summation in $C_{2},C_{3}$ and square
summation in $C\sim C_{1}$.

$(II)$ In this case we have
$C_{\text{med}}=\max(\\{C_{1},C_{2},C_{3}\\}\backslash C_{\max})\sim C_{\max}$
by otherwise impossible frequency interaction. Suppose for definiteness that
$C_{1}\sim C_{2}\sim C_{\max}$. Applying two bilinear Strichartz estimates
from Proposition 5.7 gives

$\begin{split}\iint
R_{C}vR_{C_{1}}u_{1}R_{C_{2}}u_{2}R_{C_{3}}u_{3}dxdt&\leq\|R_{C}vR_{C_{1}}u_{1}\|_{L_{t}^{2}([0,T],\mathcal{L}^{2}_{x})}\|R_{C_{2}}u_{2}R_{C_{3}}u_{3}\|_{L_{t}^{2}([0,T],\mathcal{L}^{2}_{x})}\\\
&\lesssim_{\varepsilon}T^{\frac{1}{4}}C^{\frac{1}{2}+\varepsilon}C_{3}^{\frac{1}{2}+\varepsilon}\|R_{C}v\|_{Y_{T}^{0}}\prod_{i=1}^{3}\|R_{C_{i}}u_{i}\|_{Y^{0}_{T}}.\end{split}$

Summing in $C$, $C_{3}$ and square summation in $C_{1}\sim C_{2}$ yields the
claim. ∎

With the trilinear estimate at hand, quantitative well-posedness as described
by Bejenaru–Tao [1] is immediate. The proof of Theorem 5.10 is complete. ∎

######  Remark 5.12 (Persistence of regularity).

It is a consequence of Proposition 5.11 that for $s^{\prime}\geq s$, and
$u_{0}\in\mathcal{H}^{s^{\prime}}_{\Lambda}$ the solution exists up to
$T=T(s,\|u_{0}\|_{\mathcal{H}^{s}_{\Lambda}})$ in
$C([0,T],\mathcal{H}^{s^{\prime}}_{\Lambda})$. We can also extend the
persistence of regularity to exponential weights like in the previous works by
Papenburg [30] and Damanik _et al._ [17, 16]:

$\|u_{0}\|^{2}_{\mathcal{H}^{\kappa,\omega}_{\Lambda}}=\sum_{n\in\mathbb{Z}^{\nu}}e^{2\kappa|n|}|\hat{u}_{0}(\langle
n\rangle_{\omega})|^{2}.$

So, we conclude that for real analytic initial data with
$\|u_{0}\|_{\mathcal{H}^{\kappa,\omega}_{\Lambda}}<\infty$ the solution
persists to be real-analytic with the same exponential Fourier decay.

We formulate a detailed local well-posedness result for the nonlinear
Schrödinger equation with algebraic nonlinearity with quasi-periodic initial
data in $\mathbb{R}^{d}$ with $m\geq 2$:

(38)
$\left\\{\begin{array}[]{cl}i\partial_{t}u+\partial_{x}^{2}u&=\pm|u|^{2(m-1)}u,\quad(t,x)\in\mathbb{R}\times\mathbb{R}^{d},\\\
u(0)&=u_{0}\in\mathcal{H}^{s}_{\Lambda}(\mathbb{R}^{d}).\end{array}\right.$

Let $\Lambda$ be like in (11) with $b_{i}=\nu_{i}-1$ and density parameter
$b=b_{1}+\ldots+b_{d}$.

######  Theorem 5.13.

Let $m\geq 3$ for $d=1$, and $m\geq 2$ for $d\geq 2$. Let
$s^{*}=2b\big{(}\frac{1}{2}-\frac{1}{2m}\big{)}+\big{(}d-\frac{d+2}{m}\big{)}$
and $s>s^{*}$. Then there is a real-analytic data-to-solution mapping for (38)

$S_{T}^{s}:\mathcal{H}^{s}_{\Lambda}(\mathbb{R})\to Y_{T}^{s}\subseteq
C([0,T],\mathcal{H}^{s}_{\Lambda}),\quad u_{0}\mapsto u,$

with $T=T(s,\|u_{0}\|_{\mathcal{H}^{s}_{\Lambda}})$, which is lower semi-
continuous, decreasing, and satisfies
$\liminf_{a\to 0}T(s,a)\gtrsim 1$.

###### Proof.

With the linear estimate following like above, for the proof it suffices to
show the multilinear estimate with $s\geq s^{\prime}>s^{*}$:

$\big{\|}\int_{0}^{t}e^{i(t-s)\partial_{x}^{2}}(u_{1}\overline{u}_{2}\ldots
u_{2m-3}\overline{u}_{2m-2}u_{2m-1})dxds\big{\|}_{Y_{T}^{s}}\lesssim
T^{\delta}\|u_{1}\|_{Y^{s}_{T}}\prod_{i=2}^{2m-1}\|u_{i}\|_{Y^{s^{\prime}}_{T}}.$

Invoking duality, we find

$\begin{split}&\quad\big{\|}R_{C}\int_{0}^{t}e^{i(t-s)\partial_{x}^{2}}(u_{1}\overline{u}_{2}\ldots\overline{u}_{2m-2}u_{2m-1})ds\big{\|}_{Y^{s}_{T}}\\\
&\lesssim C^{s}\sup_{\|v\|_{Y^{0}_{T}}=1}\iint
R_{C}\overline{v}R_{C}(u_{1}\overline{u}_{2}\ldots u_{2m-1})dxds.\end{split}$

We omit the complex conjugation in the following to ease notation as the
estimates do not depend on this. We decompose

$u_{i}=\sum_{C_{i}\in 2^{\mathbb{N}_{0}}}R_{C_{i}}u_{i}$

and respecting the convolution constraint, it suffices to show multilinear
estimates for

$\iint R_{C}vR_{C_{1}}u_{1}\ldots R_{C_{2m-1}}u_{2m-1}dxds.$

Let $C_{1}^{*}\geq C_{2}^{*}\geq\ldots\geq C_{2m-1}^{*}$ denote a decreasing
rearrangement. We need to consider the two cases $C_{1}^{*}\sim C$ $(I)$, and
$C_{1}^{*}\sim C_{2}^{*}\gg C$ $(II)$. We turn to the estimate of $(I)$: By
applying Hölder’s inequality and Proposition 5.8 it follows that

$\begin{split}&\quad\big{|}\iint R_{C}vR_{C_{1}}u_{1}\ldots
R_{C_{2m-1}}u_{2m-1}dxds\big{|}\\\
&\leq\|R_{C}vR_{C_{2}^{*}}u_{2}^{*}R_{C_{3}^{*}}u_{3}^{*}\ldots
R_{C_{m}^{*}}u_{m}^{*}\|_{L_{T}^{2}\mathcal{L}^{2}_{x}}\|R_{C_{1}^{*}}u_{1}^{*}R_{C_{m+1}^{*}}u_{m+1}^{*}\ldots
R_{C_{2m-1}^{*}}u_{2m-1}^{*}\|_{L_{T}^{2}\mathcal{L}^{2}_{x}}\\\ &\lesssim
T^{2\delta}(C_{2}^{*})^{2r}(C_{3}^{*}\ldots
C_{m}^{*})^{r}(C_{m+1}^{*})^{2r}(C_{m+2}^{*}\ldots
C_{2m-1}^{*})^{r}\|R_{C}v\|_{Y^{0}_{T}}\prod_{i=1}^{2m-1}\|R_{C_{i}}u_{i}\|_{Y_{T}^{0}}\end{split}$

for $r>b\big{(}\frac{1}{2}-\frac{1}{2m}\big{)}+\frac{d}{2}-\frac{d+2}{2m}$.
The claim follows then from straight-forward dyadic summation.

We turn to Case $(II)$: In the following denote with $C_{1}^{**}\geq
C_{2}^{**}\geq\ldots C_{2m}^{**}$. We use again Hölder’s inequality to obtain

$\begin{split}&\quad\big{|}\iint R_{C}vR_{C_{1}}u_{1}\ldots
R_{2m-1}u_{2m-1}dxds\big{|}\\\
&\leq\|R_{C_{1}^{*}}u_{1}^{*}R_{C_{3}^{*}}u_{3}^{*}\ldots
R_{C_{m+1}^{*}}u_{m+1}^{*}\|_{L_{T}^{2}\mathcal{L}^{2}_{x}}\|R_{C}vR_{C_{2}^{*}}u_{2}^{*}R_{C_{m+2}^{*}}u_{m+2}^{*}\ldots
R_{C_{2m-1}^{*}}u_{2m-1}^{*}\|_{L_{T}^{2}\mathcal{L}^{2}_{x}}\\\ &\lesssim
T^{2\delta}(C_{3}^{**})^{2r}(C_{4}^{**})^{2r}(C_{5}^{**}\ldots
C_{2m}^{**})^{r}\|R_{C}v\|_{Y^{0}_{T}}\prod_{i=1}^{2m-1}\|R_{C_{i}}u_{i}\|_{Y^{0}_{T}}.\end{split}$

With this estimate at hand, the claim is immediate from dyadic summation. The
proof is complete. ∎

##  6\. Examples

In this section we supplement the Strichartz estimates and the local well-
posedness result for the cubic nonlinear Schrödinger equation on
$\Lambda\subseteq\mathbb{R}^{d}$, $d\in\\{1,2\\}$, with examples, which show
sharpness of the regularity up to endpoints. For simplicity, suppose that the
coefficients are of unit length $\omega^{(j)}_{i}\in(1/2,1]$.

###  6.1. Sharpness of Strichartz estimates on finite times

The examples are variations of a common theme: For almost-periodic functions
at given height $|n|\sim C$, there are many frequencies $|\langle
n\rangle_{\omega}|\lesssim 1$, which are consequently not strongly oscillating
for finite times. We only consider the one-dimensional case
$\Lambda=\omega\cdot\mathbb{Z}^{\nu}$. The generalization of the
$\mathcal{L}^{4}$-estimate to the case $\Lambda\subseteq\mathbb{R}^{2}$ is
straight-forward.

######  Proposition 6.1.

Let $p\in\\{4,6\\}$. The estimate

(39)
$\|e^{it\partial_{x}^{2}}R_{C}f\|_{L_{t}^{p}([0,1],\mathcal{L}^{p}_{x})}\lesssim
C^{s}\|f\|_{\mathcal{L}^{2}_{x}}$

fails for $s<(\nu-1)\big{(}\frac{1}{2}-\frac{1}{p}\big{)}$.

###### Proof.

We consider

$f=\sum_{\begin{subarray}{c}|n|\sim C,\\\ |\langle n\rangle_{\omega}|\lesssim
1\end{subarray}}e^{i\langle n\rangle_{\omega}x}.$

With $\langle n\rangle_{\omega}=n\cdot\omega$, $\omega_{i}\neq 0$, for any
$n_{1},\ldots,n_{\nu-1}\sim C$, we can choose $n_{\nu}\in\mathbb{Z}$ such that
$|\langle n\rangle_{\omega}|\leq 1$. Note that $|n_{d}|\sim C$. Clearly,
$\|f\|_{\mathcal{L}^{2}_{x}}\sim C^{\frac{\nu-1}{2}}$. We compute for
$g=\sum_{n\in\mathbb{Z}^{\nu}}e^{i\langle n\rangle_{\omega}x}a_{n}$ that

$\|g\|^{4}_{\mathcal{L}^{4}_{x}}=\sum_{n\in\mathbb{Z}^{\nu}}\sum_{\begin{subarray}{c}n_{1},n_{3},\\\
n=n_{1}+n_{2}=n_{3}+n_{4}\end{subarray}}a_{n_{1}}a_{n_{2}}\overline{a}_{n_{3}}\overline{a}_{n_{4}}.$

Consequently,

$\|f\|^{4}_{\mathcal{L}^{4}_{x}}\sim C^{3(\nu-1)}.$

This shows that (39) for $p=4$ can only be valid for $s\geq\frac{\nu-1}{4}$.
This establishes the claim for $p=4$.

Moreover,

$\|g\|^{6}_{\mathcal{L}^{6}_{x}}=\sum_{n\in\mathbb{Z}^{\nu}}\sum_{\begin{subarray}{c}n_{1},n_{2},n_{4},n_{5},\\\
n=n_{1}+n_{2}+n_{3}=n_{4}+n_{5}+n_{6}\end{subarray}}a_{n_{1}}a_{n_{2}}a_{n_{3}}\overline{a}_{n_{4}}\overline{a}_{n_{5}}\overline{a}_{n_{6}}.$

For this reason,

$\|f\|_{\mathcal{L}^{6}_{x}}^{6}\sim C^{5(\nu-1)}.$

This shows that (39) for $p=6$ can only be true for $s\geq\frac{\nu-1}{3}$.
The proof is complete. ∎

###  6.2. Ill-posedness of the nonlinear Schrödinger equation

We consider the cubic nonlinear Schrödinger equation on
$\Lambda\subseteq\mathbb{R}^{d}$, $d\in\\{1,2\\}$. We let
$\Lambda=\omega^{(1)}\cdot\mathbb{Z}^{\nu_{1}}\times\omega^{(2)}\cdot\mathbb{Z}^{\nu_{2}}$
and $b=(\nu_{1}-1)+(\nu_{2}-1)$.

######  Proposition 6.2.

The first Picard iterate

(40)
$\big{\|}\int_{0}^{t}e^{i(t-s)\Delta}\big{(}|e^{is\Delta}f|^{2}e^{is\Delta}f\big{)}ds\big{\|}_{C_{T}\mathcal{H}^{s}_{\Lambda}}\lesssim\|f\|^{3}_{\mathcal{H}^{s}_{\Lambda}}.$

is unbounded for $s<\frac{b}{2}$.

###### Proof.

The choice of data is the same as in the previous section. We have

$\big{(}\big{|}e^{is\Delta}f\big{|}^{2}e^{is\Delta}f\big{)}\widehat{(}n)=\sum_{n=n_{1}-n_{2}+n_{3}}e^{is(\langle
n_{1}\rangle_{\omega}^{2}-\langle n_{2}\rangle_{\omega}^{2}+\langle
n_{3}\rangle_{\omega}^{2})},$

which gives

$\begin{split}&\quad\int_{0}^{t}e^{i(t-s)\Delta}\big{|}e^{is\Delta}f\big{|}^{2}e^{is\Delta}fds\\\
&=\sum_{n}e^{it\langle n\rangle_{\omega}^{2}}e^{ix\langle
n\rangle_{\omega}}\underbrace{\sum_{n=n_{1}-n_{2}+n_{3}}\int_{0}^{t}e^{is(\langle
n_{1}\rangle_{\omega}^{2}-\langle n_{2}\rangle^{2}_{\omega}+\langle
n_{3}\rangle_{\omega}^{2}-\langle
n\rangle_{\omega}^{2})}ds}_{c(t,n)}.\end{split}$

Let $\Omega(n_{1},n_{2},n_{3})=\langle n_{1}\rangle_{\omega}^{2}-\langle
n_{2}\rangle^{2}_{\omega}+\langle n_{3}\rangle_{\omega}^{2}-\langle
n_{1}-n_{2}+n_{3}\rangle_{\omega}^{2}$. We compute the integral to be

$\int_{0}^{t}e^{is\Omega(n_{1},n_{2},n_{3})}ds=\begin{cases}\frac{e^{it\Omega}-1}{i\Omega},&\quad\Omega\neq
0,\\\ t,&\quad\Omega=0.\end{cases}$

Since $|\Omega|\lesssim 1$, we have for $0\leq t\leq c\ll 1$ the uniform
estimate

$|c(t,n)|\gtrsim t\sum_{n=n_{1}-n_{2}+n_{3}}1.$

We have $|n_{i}|\sim C$. So there are $\gtrsim C^{b}$ output frequencies of
height $C$ with coefficients satisfying the bound $|c(t,n)|\gtrsim t\cdot
C^{2b}$. This gives

$\big{\|}\int_{0}^{t}e^{i(t-s)\Delta}\big{(}|e^{is\Delta}f|^{2}e^{is\Delta}f\big{)}ds\big{\|}_{C_{T}\mathcal{H}^{s}_{\Lambda}}\gtrsim
TC^{s}C^{2b}C^{\frac{b}{2}}.$

Secondly, $\|f\|_{\mathcal{H}^{s}_{\Lambda}}\sim C^{\frac{b}{2}+s}$.
Consequently, (40) can only be true for $s\geq\frac{b}{2}$. ∎

We conclude with two remarks:

######  Remark 6.3.

Note that it is actually enough to show sharpness of the local well-posedness
result to infer sharpness of the $L^{4}$-Strichartz estimate. Indeed, as soon
as there is an improved Strichartz estimate, the improvement of the local
well-posedness theory follows readily following along the above arguments.

Secondly, comparing with the periodic case, it is tempting to conjecture that
the local well-posedness result is also sharp in the general quasi-periodic
case $\Lambda\subseteq\mathbb{R}^{d}$ for $d\geq 3$. It remains unclear how to
corroborate this by examples though.

######  Remark 6.4 (Quintic NLS).

We remark that the same computation shows that the quintic nonlinear
Schrödinger equation posed on $\Lambda=\omega\cdot\mathbb{Z}^{\nu}$

$\left\\{\begin{array}[]{cl}i\partial_{t}u+\partial_{x}^{2}u&=|u|^{4}u,\quad(t,x)\in\mathbb{R}\times\mathbb{R},\\\
u(0)&=u_{0}\in\mathcal{H}^{s}_{\Lambda}\end{array}\right.$

can only be $C^{3}$-well-posed for $s>\frac{1}{2}$. The well-posedness result
we showed for $s>\frac{2}{3}$ is actually not sharp: If the frequency
distribution is given by

$A=|\iint
R_{M_{1}}u_{1}R_{M_{2}}u_{2}R_{M_{3}}u_{3}R_{M_{4}}u_{4}R_{M_{5}}u_{5}R_{M_{6}}u_{6}dxdt|$

with $M_{1}\sim M_{2}\gg M_{3}\sim M_{4}\gg M_{5}\sim M_{6}$ the application
of two trilinear estimates gives

$A\lesssim
M_{3}^{2r}M_{4}^{2r}M_{5}^{r}M_{6}^{r}\prod_{i=1}^{6}\|R_{M_{i}}u_{i}\|_{Y_{T}^{0}}.$

for $r>\frac{1}{3}$. We have the alternative estimate from two bilinear
$L^{2}$-Strichartz estimates:

$A\lesssim
M_{3}^{2r}M_{4}^{2r}M_{5}M_{6}\prod_{i=1}^{6}\|R_{M_{i}}u_{i}\|_{Y_{T}^{0}}.$

So, choosing a threshold $M_{5}\lesssim M_{3}^{\alpha}$ will allow us to
improve the local well-posedness. But since we cannot reach $s>\frac{1}{2}$ as
suggested by the example, we omit the details.

## Appendix: Quasiperiodic solutions to the KdV equation

We remark that the above arguments to show local well-posedness for the
nonlinear Schrödinger equation do not take advantage of the resonance. In this
section we consider the KdV with derivative nonlinearity:

(41)
$\left\\{\begin{array}[]{cl}\partial_{t}u+\partial_{x}^{3}u&=u\partial_{x}u,\quad(t,x)\in\mathbb{R}\times\mathbb{R},\\\
u(0)&=u_{0}\in\mathcal{H}^{s}_{\Lambda}.\end{array}\right.$

The systematic study of Strichartz estimates is the first cornerstone to show
local well-posedness. The second ingredient already used by Bourgain [6] is to
use large resonance to ameliorate the derivative loss. The resonance function
quantifies to what extend free solutions can form a free solution again
through the nonlinear interaction and reads for the KdV equation:

$\Omega(\xi_{1},\xi_{2})=(\xi_{1}+\xi_{2})^{3}-\xi_{1}^{3}-\xi_{2}^{3}=3(\xi_{1}+\xi_{2})\xi_{1}\xi_{2}.$

The Fourier restriction norm, or the refined adapted spaces, can recover the
square root of the resonance function. Consider a frequency interaction with
output frequency $N\in 2^{\mathbb{Z}}$ and input frequencies $N_{1}$,
$N_{2}\in 2^{\mathbb{Z}}$. Let $N_{\max}=\max(N,N_{1},N_{2})$, and
$N_{\min}=\min(N,N_{1},N_{2})$. The resonance relation recovered above shows
that for $|\xi_{i}|\sim N_{i}$ and $|\xi_{1}+\xi_{2}|\sim N$ we have

$|\Omega(\xi_{1},\xi_{2})|\sim N_{\max}^{2}N_{\min}.$

The purpose of this section is devoted to explain how Tsugawa’s result is a
consequence of the Strichartz estimates at hand and the resonance analysis due
to Bourgain [6]. This will help for future analysis of more involved models.

We also mention the recent result of Papenburg [30] who showed unconditional
local well-posedness for exponentially decaying Fourier coefficients for more
general dispersive equations with derivative nonlinearity. However, the
solutions were only verified to satisfy a weaker exponential Fourier decay.

###  6.3. Local well-posedness in the periodic case

The resonance relation highlights that interactions with low frequencies are
problematic. By subtracting the mean-value and derivative nonlinearity, we can
suppose that the mean is identically vanishing for all times:
$\hat{u}(t,0)=0$. With the $L^{4}$-Strichartz estimates on the torus

$\|e^{t\partial_{x}^{3}}f\|_{L_{t}^{4}([0,T],L^{4}(\mathbb{T}))}\lesssim
T^{\frac{1}{8}}\|f\|_{L^{2}(\mathbb{T})}$

it is nowadays standard to close the contraction mapping argument.

On the torus we define the function spaces222Here we use a Besov variant in
the frequencies to simplify summability.:

$\|u\|_{Y^{s}_{T}}=\sum_{n\in\mathbb{Z}\backslash 0}\langle
n\rangle^{s}\|e^{it\langle n\rangle_{\omega}^{3}}\hat{u}(t,n)\|_{V^{2}_{T}}.$

For the nonlinear estimate, it suffices to prove after invoking duality for
$N_{2}\lesssim N_{1}\sim N$:

(42)
$\big{|}\iint_{[0,T]\times\mathbb{T}}P_{N}vP_{N_{1}}u_{1}P_{N_{2}}u_{2}dxdt\big{|}\lesssim
T^{\delta}N^{-1}\|P_{N}v\|_{Y^{0}_{T}}\|P_{N_{1}}u_{1}\|_{Y^{0}_{T}}N_{2}^{s}\|P_{N_{2}}u_{2}\|_{Y^{0}_{T}}.$

The high frequency gain will compensate the derivative loss.

To show (42), we use the resonance relation. For one function, say $v$, we can
suppose that the modulation is larger than the resonance:333We gloss over the
technicality of considering finite times. Roughly speaking, for finite times
modulations can be assumed to have a minimum size.

$(\xi,\tau)\in\text{supp}(\hat{v}(\xi,\tau))\Rightarrow|\tau-\xi^{3}|\gtrsim
N^{2}N_{2}.$

By Hölder’s inequality and the $L^{4}$-Strichartz estimate we find

$\begin{split}\iint_{[0,T]\times\mathbb{T}}P_{N}vP_{N_{1}}u_{1}P_{N_{2}}u_{2}dxdt&\leq\|P_{N}v\|_{L^{2}_{t,x}}\|P_{N_{1}}u_{1}\|_{L^{4}_{t,x}}\|P_{N_{2}}u_{2}\|_{L^{4}_{t,x}}\\\
&\lesssim
T^{\frac{1}{4}}(N^{2}N_{2})^{-\frac{1}{2}}\|P_{N}v\|_{Y^{0}_{T}}\prod_{i=1}^{2}\|P_{N_{i}}u_{i}\|_{Y^{0}_{T}},\end{split}$

which is (42) for $s=-\frac{1}{2}$.

###  6.4. Local well-posedness in the quasiperiodic case

Next, we extend the argument to (41) on $\Lambda=\omega\cdot\mathbb{Z}^{\nu}$,
$\omega\in\mathbb{R}^{\nu}_{>0}$ non-resonant. Let $s_{1},s_{2}\in\mathbb{R}$.
We define the norm to estimate solutions by

$\|u\|_{Y_{T}^{s_{1},s_{2}}}=\sum_{n\in\mathbb{Z}^{\nu}\backslash 0}\langle
n\rangle_{\omega}^{s_{1}}\langle n\rangle^{s_{2}}\|e^{it\langle
n\rangle_{\omega}^{3}}\hat{u}(t,n)\|_{V^{2}_{T}}.$

As already observed by Tsugawa, there is a significant difference with low
frequencies compared to the periodic case. Removing the zero frequency in the
periodic case is enough to ensure that the frequencies have at least unit
modulus. This is clearly no longer true in the quasiperiodic case and suggests
to penalize low frequencies by homogeneous Sobolev norms with negative
regularity. Note that in the decaying case on the real line the very low
frequencies can be estimated by their small measure and dispersive effects,
both absent in the quasiperiodic case.

The generalized trilinear estimate in the quasiperiodic case reads

(43)
$\begin{split}&\quad\lim_{L\to\infty}\frac{1}{2L}\iint_{[0,T]\times[-L,L]}P_{N}R_{C}vP_{N_{1}}R_{C_{1}}u_{1}P_{N_{2}}R_{C_{2}}u_{2}dxdt\\\
&\lesssim
C_{\min}^{s_{2}}N_{\max}^{-1}N_{\min}^{s_{1}}\|P_{N}R_{C}v\|_{Y^{0}_{T}}\prod_{i=1}^{2}\|P_{N_{i}}R_{C_{i}}u_{i}\|_{Y^{0}_{T}}.\end{split}$

To effectively lower the height, we can carry out an almost orthogonal
decomposition to localize also the maximal height to cubes of size $C_{\min}$
(compare Proposition 5.7). With the following variant of the $L^{4}$-estimate
from (7), which is a consequence of the transfer principle, we find

$\|P_{N}Q_{C}^{a}u\|_{L_{T}^{4}\mathcal{L}^{4}_{x}}\lesssim
T^{\frac{1}{8}}C^{\frac{b}{4}}\|u\|_{Y^{0}_{T}}.$

Then (43) follows like above after almost orthogonal decomposition into cubes
of low height, applying Hölder’s inequality and the $L^{4}$-Strichartz
estimate recorded in the above display. This essentially recovers Tsugawa’s
result [33, Theorem 1.1]. The $L^{4}$-estimate was implicitly proved in [33,
Lemma 3.1].

###  6.5. Summary

It is well-known since the works of Bourgain [5, 6] how to combine the
resonance relation with Strichartz estimates. The resonance relation remains
the same, and it remains to find substitutes for the Strichartz estimates in
the quasi-periodic case. Above we have demonstrated how Strichartz estimates
depending on the height follow from decoupling and counting arguments. We hope
to apply this approach to further models.

## Acknowledgements

Financial support from the Humboldt foundation (Feodor-Lynen fellowship) and
partial support by the NSF grant DMS-2054975 is gratefully acknowledged. Some
of this work was carried out at Bielefeld University in June 2024, and I would
like to thank Sebastian Herr and his working group for their kind hospitality.

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