# Improved decoupling for the moment curve in three dimensions

Robert Schippa<EMAIL_ADDRESS>Department of Mathematics, Karlsruhe
Institute of Technology, Englerstrasse 2, 76131 Karlsruhe, Germany

###### Abstract.

By quantifying a bilinear decoupling iteration for the moment curve in three
dimensions due to Guo–Li–Yung–Zorin-Kranich, we show a logarithmic improvement
of the decoupling constant at critical exponent. Correspondingly, we obtain a
logarithmic improvement of Vinogradov’s mean-value theorem in the cubic case.

###### Key words and phrases:

moment curve in three dimensions, decoupling, Vinogradov’s mean-value theorem

##  1\. Introduction

Let $\delta\in\mathbb{N}^{-1}$ and $\mathcal{P}(\delta)$ be a partition of
$[0,1]$ into intervals of length $\delta$. Let $g(s)=(s,s^{2})$, and let
$(f_{J})_{J\in\mathcal{P}(\delta)}$ be a family of functions
$f_{J}:\mathbb{R}^{2}\to\mathbb{C}$ with
$\text{supp}(\hat{f}_{J})\subseteq\mathcal{V}_{J}$, where $\mathcal{V}_{J}$ is
the parallelepiped of size $\delta\times\delta^{2}$ into directions
$g^{\prime}(c_{J})$ and $g^{\prime\prime}(c_{J})$ with $c_{J}$ denoting the
center of the interval $J$. Let $\mathcal{D}_{2}(\delta)$ denote the smallest
constant such that the decoupling inequality holds:

$\|\sum_{J\in\mathcal{P}(\delta)}f_{J}\|_{L^{6}(\mathbb{R}^{2})}\leq\mathcal{D}_{2}(\delta)\big{(}\sum_{J\in\mathcal{P}(\delta)}\|f_{J}\|^{2}_{L^{6}(\mathbb{R}^{2})}\big{)}^{\frac{1}{2}}.$

Bourgain–Demeter [2] proved that for any $\varepsilon>0$ there is
$C_{\varepsilon}>0$ such that

$\mathcal{D}_{2}(\delta)\leq C_{\varepsilon}\delta^{-\varepsilon}.$

Li [9, Theorem 1.1] observed how the double exponential bound

(1) $\mathcal{D}_{2}(\delta)\leq
A^{A^{\frac{1}{\varepsilon}}}\delta^{-\varepsilon}$

allows one to sharpen the decoupling constant to

$\mathcal{D}_{2}(\delta)\leq\exp(C\frac{\log\delta^{-1}}{\log(\log\delta^{-1})}).$

This recovered the bound proved for discrete restriction by Bourgain [1,
Proposition 2.36] via a divisor counting argument. By a Gauß sum argument,
Bourgain [1, Remark 2, p. 118] showed moreover that

$\mathcal{D}_{2}(\delta)\gtrsim\log(\delta^{-1})^{1/6}.$

Li [9] proved (1) via the bilinear approach. More recently, Guth–Maldague–Wang
[7] improved (1) to

(2) $\mathcal{D}_{2}(\delta)\leq\log(\delta^{-1})^{c}$

for some (possibly large) constant $c$. Subsequently, Guo–Li–Yung [5] improved
the discrete restriction constant to
$C_{\varepsilon}\log(\delta^{-1})^{2+\varepsilon}$ for $\varepsilon>0$. The
approach in [7] differs from [9] as it uses a high-low decomposition. However,
the high-low method becomes more involved in higher dimensions, whereas the
bilinear approach as carried out in higher dimensions by Guo–Li–Yung–Zorin-
Kranich [6] (see also [4]) seems more tractable. Li [9] firstly quantified
decoupling via the bilinear approach for the parabola (see also [8]).

In this note we turn to the moment curve in three dimensions. Let
$\Gamma_{3}(t)=(t,t^{2},t^{3})$ denote the moment curve mapping in three
dimensions. For an interval $J\subseteq[0,1]$ with center $c_{J}$, let
$\mathcal{U}_{J}$ be the parallelepiped of dimensions $4|J|\times
4|J|^{2}\times 4|J|^{3}$, whose center is $\Gamma_{3}(c_{J})$ and sides are
parallel to $\Gamma_{3}^{\prime}(c_{J})$, $\Gamma_{3}^{\prime\prime}(c_{J})$,
$\Gamma_{3}^{(3)}(c_{J})$.

We define the linear decoupling constant for the three-dimensional moment
curve for $\delta\in\mathbb{N}^{-1}$ as smallest constant, which is monotone
decreasing in $\delta$, such that:

$\|\sum_{J\in\mathcal{P}(\delta)}f_{J}\|_{L^{12}(\mathbb{R}^{3})}\leq\mathcal{D}_{3}(\delta)\big{(}\sum_{J\in\mathcal{P}(\delta)}\|f_{J}\|_{L^{12}(\mathbb{R}^{3})}^{2}\big{)}^{1/2}.$

Above $\mathcal{P}(\delta)$ denotes a partition up to points of $I$ into
intervals of length $\delta$, and $f_{J}\in\mathcal{S}(\mathbb{R}^{3})$ with
Fourier support in $\mathcal{U}_{J}$. Bourgain–Demeter–Guth [3] proved that
$\mathcal{D}_{k}(\delta)\leq C_{\varepsilon}\delta^{-\varepsilon}$ for any
$\varepsilon>0$ in any dimension, which yields as corollary the Vinogradov
mean value theorem. The argument in [3] relies on multilinear Kakeya
estimates.

More recently, Guo _et al._ [6] found a shorter proof of decoupling for moment
curves, which relies on bilinear arguments and induction on dimension. We
observe that in one dimension, the decoupling inequality for the moment curve
reduces to Plancherel’s theorem with $\mathcal{D}_{1}(\delta)\leq 10$ and in
two dimensions the logarithmic loss due to Guth–Maldague–Wang [7] is at
disposal. In the present note we quantify the bilinear iteration from [6] for
the moment curve in three dimensions and use improved decoupling inequalities
without $\delta^{-\varepsilon}$-loss in lower dimensions to show the
following:

######  Theorem 1.1 (Improved decoupling for the moment curve in three
dimensions).

There is $0<\delta_{0}<1$ and $C>0$ such that for $0<\delta<\delta_{0}$, we
have the following bound for the decoupling constant of the moment curve in
three dimensions:

(3)
$\mathcal{D}_{3}(\delta)\leq\exp(C\frac{\log\delta^{-1}}{\log(\log(\delta^{-1}))}).$

In the proof we see how losing additional logarithmic factors in the bilinear
approach still allows us to show an estimate like in (1), which implies (3)
after optimizing in $\varepsilon$. The optimization argument was carried out
in [9] without additional logarithms in the context of the parabola.

As mentioned above, the decoupling result for the moment curve yielded
Vinogradov’s mean-value theorem. In the present instance the decoupling result
in Theorem 1.1 yields a logarithmic improvement on the number of simultaneous
solutions to the diophantine equations:

(4) $\left\\{\begin{array}[]{cl}\sum_{i=1}^{6}x_{i}&=\sum_{i=1}^{6}y_{i},\\\
\sum_{i=1}^{6}x_{i}^{2}&=\sum_{i=1}^{6}y_{i}^{2},\\\
\sum_{i=1}^{6}x_{i}^{3}&=\sum_{i=1}^{6}y_{i}^{3}.\end{array}\right.$

For $1\leq x_{i},y_{i}\leq N$ we denote the number of integer solutions to (4)
by $J(N)$. We have the following corollary to Theorem 1.1:

######  Corollary 1.2.

For $N$ sufficiently large, there is $C>0$ such that the following estimate
holds:

$J(N)\leq\exp\big{(}\frac{C\log(N)}{\log(\log(N))}\big{)}N^{6}.$

For an overview of recent work on Vinogradov’s mean-value theorem, we refer to
[10]. Via efficient congruencing Wooley [11, 12] firstly proved

$J(N)\leq C_{\varepsilon}N^{6+\varepsilon}$

for any $\varepsilon>0$. We record the standard argument passing from (3) to
the discrete decoupling and yielding the logarithmic improvement for the sake
of self-containedness.

###### Proof.

As the well-known argument goes, we write with $e(x)=\exp(2\pi ix)$:

$J(N)=\int_{[0,1]^{3}}\big{|}\sum_{j=1}^{N}e(jx_{1}+j^{2}x_{2}+j^{3}x_{3})\big{|}^{12}dx_{1}dx_{2}dx_{3}.$

By change of variables, we find

$J(N)=N^{-6}\int_{[0,N]\times[0,N^{2}]\times[0,N^{3}]}\big{|}\sum_{j=1}^{N}e(x.\Gamma_{3}(j/N))\big{|}^{12}dx_{1}dx_{2}dx_{3}.$

Now we use periodicity in $x_{1}$ with period $N$ and in $x_{2}$ with period
$N^{2}$ to write

$J(N)\lesssim
N^{-9}\int_{[0,N^{3}]^{3}}\big{|}\sum_{j=1}^{N}e(x.\Gamma_{3}(j/N))\big{|}^{12}dx_{1}dx_{2}dx_{3}.$

Let $f_{j}=\exp(x.\Gamma_{3}(j/N))\psi(x)$ with
$\psi\in\mathcal{S}(\mathbb{R}^{3})$ be a Schwartz function such that
$\text{supp}(\hat{\psi})\subseteq N^{-3}$ and $|\psi(x)|\sim 1$ on
$B(0,N^{3})$. By the above we have

$J(N)\lesssim
N^{-9}\int_{[0,N^{3}]^{3}}\big{|}\sum_{j=1}^{N}f_{j}(x)\big{|}^{12}dx_{1}dx_{2}dx_{3}.$

Applying Theorem 1.1 for $N$ large enough gives

$J(N)\lesssim
N^{-9}\exp\big{(}\frac{C\log(N)}{\log(\log(N))}\big{)}\big{(}\sum_{j=1}^{N}\|f_{j}\|^{2}_{L^{12}(w_{B})}\big{)}^{6}.$

Since $|f_{j}(x)|=1$ and
$\|f_{j}\|_{L^{12}(w_{B})}\lesssim(N^{9})^{\frac{1}{12}}$, we find

$J(N)\lesssim\exp\big{(}\frac{C\log(N)}{\log(\log(N))}\big{)}N^{6}.$

The claim follows from choosing $C$ slightly larger. ∎

_Outline of the paper._ In Section 2 we introduce notations, explain the
bilinear reduction, and give an overview of the constants coming up in the
iteration. In Section 3 we recall the improved decoupling results in two
dimensions and show stability. In Section 4 we carry out the decoupling
iteration using asymmetric decoupling constants. Then we can bootstrap the
decoupling constant to prove the claim.

##  2\. Preliminaries

###  2.1. Notations

For $k\in\mathbb{N}$ let $\Gamma_{k}:[0,1]\to\mathbb{R}^{k}$ denote the moment
curve in $\mathbb{R}^{k}$. Let
$\delta\in\mathbb{N}^{-1}=\\{\frac{1}{n}:n\in\mathbb{N}\\}$. For a closed
interval $[a,b]=I\subseteq[0,1]$ with $|I|\delta^{-1}\in\mathbb{N}$ we denote
by $\mathcal{P}(I,\delta)$ the decomposition into closed intervals of length
$\delta$: $I=\bigcup_{j=0}^{N-1}[a+j\delta,a+(j+1)\delta]$ with $N\delta=|I|$.
If $I=[0,1]$, we let $\mathcal{P}(I,\delta)=\mathcal{P}(\delta)$. Above we
defined for an interval $J\subseteq[0,1]$ with center $c_{J}$, the
parallelepiped $\mathcal{U}_{J}$ of dimensions $4|J|\times 4|J|^{2}\times
4|J|^{3}$, whose center is $\Gamma(c_{J})$ and sides are parallel to
$\Gamma_{3}^{\prime}(c_{J})$, $\Gamma_{3}^{\prime\prime}(c_{J})$,
$\Gamma_{3}^{(3)}(c_{J})$. More generally, we define for a curve
$\gamma:[0,1]\to\mathbb{R}^{k}$ the parallelepiped $\mathcal{U}_{J,\gamma}$
with center $c_{J}$ of dimensions $4|J|\times 4|J|^{2}\times\ldots\times
4|J|^{k}$ into directions
$\partial\gamma(c_{J}),\ldots,\partial^{k}\gamma(c_{J})$.

In the following, for an interval $J$ and curve $\gamma$, let
$\mathcal{U}_{J,\gamma}^{o}$ denote the parallelepiped centered at the origin,
which is dual to $\mathcal{U}_{J,\gamma}$, that is

$\mathcal{U}_{I,\gamma}^{o}=\\{x\in\mathbb{R}^{k}\,:\,\big{|}\langle
x,\partial^{i}\gamma(c_{J})\rangle\big{|}\leq\frac{1}{4}|J|^{-i},\quad 1\leq
i\leq k\\}.$

This is a parallelepiped of size $\sim|J|^{-1}\times|J|^{-2}\times|J|^{-3}$.
We define a bump function adapted to $\mathcal{U}_{I}^{o}$ by

$\phi_{I}(x)=|\mathcal{U}_{I}^{o}|^{-1}\inf\\{t\geq
1\,:x/t\in\mathcal{U}_{I}^{o}\\}^{-10k}.$

This is $L^{1}$-normalized as can be seen from anisotropic dilation:
$\int_{\mathbb{R}^{k}}\phi_{I}(x)dx\leq C_{4,k}$. For $k\in\mathbb{N}$, we
define the critical decoupling exponent for the moment curve $\Gamma_{k}$ as
$p_{k}=k(k+1)$. We define $\mathcal{D}_{k}(\delta)$ as monotone decreasing in
$\delta$ (this means if $\delta^{-1}$ becomes larger, the decoupling constant
is also supposed to become larger) and smallest constant, which satisfies

$\big{\|}\sum_{J\in\mathcal{P}(\delta)}f_{J}\big{\|}_{L^{p_{k}}(\mathbb{R}^{k})}\leq\mathcal{D}_{k}(\delta)\big{(}\sum_{J\in\mathcal{P}(\delta)}\|f_{J}\|_{L^{p_{k}}(\mathbb{R}^{k})}^{2}\big{)}^{\frac{1}{2}}.$

###  2.2. Bilinear reduction, and uncertainty principle

We define the bilinear decoupling constant $\mathcal{B}_{k}(\delta)$ as
smallest constant decreasing in $\delta$ such that

$\begin{split}\big{(}\int_{\mathbb{R}^{k}}|\sum_{J_{1}\in\mathcal{P}(I_{1},\delta)}f_{J_{1}}|^{p_{k}/2}|\sum_{J_{2}\in\mathcal{P}(I_{2},\delta)}f_{J_{2}}|^{p_{k}/2}\big{)}^{1/p_{k}}&\leq\mathcal{B}_{k}(\delta)\big{(}\sum_{J_{1}\in\mathcal{P}(I_{1},\delta)}\|f_{J_{1}}\|^{2}_{L^{p_{k}}}\big{)}^{1/4}\\\
&\quad\times\big{(}\sum_{J_{2}\in\mathcal{P}(I_{2},\delta)}\|f_{J_{2}}\|^{2}_{L^{p_{k}}}\big{)}^{1/4}.\end{split}$

In the above display, we consider intervals $I_{i}\subseteq[0,1]$, $i=1,2$
with $\text{dist}(I_{1},I_{2})\geq\frac{1}{4}$ and
$|I_{i}|\delta\in\mathbb{N}$. We have the following linear-to-bilinear
reduction:

######  Lemma 2.1 (Bilinear reduction, [6, Lemma 2.2]).

If $\delta=2^{-M}$, then there is $C_{1}>0$ such that

$\mathcal{D}_{k}(\delta)\leq
C_{1}\big{(}1+\sum_{n=2}^{M}\mathcal{B}_{k}(2^{-M+n-2})^{2}\big{)}^{1/2}.$

The proof of the above lemma is based on a Whitney decomposition and affine
rescaling, which is already very important for the linear decoupling:

######  Lemma 2.2 (Affine rescaling, [6, Lemma 2.3]).

Let $I\in\mathcal{P}(2^{-n})$ for some integer $n\geq 0$. For any
$\delta\in(0,2^{-n})$ and any tuple of functions
$(f_{J})_{J\in\mathcal{P}(I,\delta)}$ with
$\text{supp}(\hat{f}_{J})\subseteq\mathcal{U}_{J}$ for all $J$, the following
holds:

(5)
$\|f_{I}\|_{L^{p_{k}}(\mathbb{R}^{k})}\leq\mathcal{D}_{k}(2^{n}\delta)\big{(}\sum_{J\in\mathcal{P}(I,\delta)}\|f_{J}\|^{2}_{L^{p_{k}}}\big{)}^{1/2}.$

The affine rescaling yields submultiplicativity:

######  Lemma 2.3.

We have for $\delta$, $\sigma$, $\delta/32\sigma\in\mathbb{N}^{-1}$:

(6)
$\mathcal{D}_{k}(\delta)\leq\mathcal{D}_{k}(\sigma)\mathcal{D}_{k}(\delta/32\sigma).$

###### Proof.

We can suppose that $\sigma\leq\frac{1}{32}$ because for
$\sigma\in[\frac{1}{32},1]$ we have by monotonicity trivially

$\mathcal{D}_{k}(\delta)\leq\mathcal{D}_{k}(\delta/32\sigma).$

We partition $\mathcal{P}(\delta)$ into collections indexed by
$\tilde{J}\in\mathcal{P}(8\sigma)$ such that
$\mathcal{U}_{J}\subseteq\mathcal{U}_{\tilde{J}}$ and write $J\sim\tilde{J}$
(it suffices to take $J$ as a child of $\tilde{J}$ or as a child of a
neighbour), and we write $f_{\tilde{J}}=\sum_{J\sim\tilde{J}}f_{J}$. Then we
can apply decoupling at $8\sigma$ to find

$\begin{split}\big{\|}\sum_{\tilde{J}}f_{\tilde{J}}\big{\|}_{L^{p_{k}}(\mathbb{R}^{k})}&\leq\mathcal{D}_{k}(8\sigma)\big{(}\sum_{\tilde{J}}\|f_{\tilde{J}}\|^{2}_{L^{p_{k}}(\mathbb{R}^{k})}\big{)}^{\frac{1}{2}}\\\
&\leq\mathcal{D}_{k}(8\sigma)\mathcal{D}_{k}(\delta/32\sigma)\big{(}\sum_{J\in\mathcal{P}(\delta)}\|f_{J}\|^{2}_{L^{p_{k}}(\mathbb{R}^{k})}\big{)}^{\frac{1}{2}},\end{split}$

where we used affine rescaling in the second step and that $J$ is either a
child of $\tilde{J}$ or a child of a neighbouring interval. Finally, we use
monotonicity $\mathcal{D}_{k}(8\sigma)\leq\mathcal{D}_{k}(\sigma)$ to conclude
the proof.

∎

In the iteration to estimate $\mathcal{D}_{k}(\delta)$, we use monotonicity of
$\mathcal{B}_{k}(\delta)$ to write

(7) $\mathcal{D}_{k}(\delta)\leq
C_{1}\log(\delta^{-1})\mathcal{B}_{k}(\delta).$

The reason we do not resort to the slightly sharper argument of broad-narrow
reduction, which is used by Li [9], is that the unit distance separation of
the intervals simplifies the forthcoming arguments, and we are losing
logarithmic factors in the iteration anyway.

We also use the following instance of uncertainty prinicple:

######  Lemma 2.4 ([6, Lemma 3.3]).

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

$|g_{J}|^{p}\leq C_{p}(|g_{J}|^{p}*\phi_{J}),$

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

Record the following trivial bound due to Cauchy-Schwarz:

(8) $\mathcal{D}_{k}(\delta)\leq\delta^{-\frac{1}{2}}.$

###  2.3. Overview of constants

In the following we denote by $C_{i}$, $i=1,\ldots,8$, $c$ fixed (possibly
very large) constants, which will be defined in the course of the argument.

  * •

$C_{1}$ is used in the linear-to-bilinear reduction (7),

  * •

$C_{2}$, $C_{3}$ are used to record constants in the arguments involving lower
dimensional decoupling, $c\geq 1$ denotes the exponent in the logarithmic loss
for the $\ell^{2}L^{6}$-decoupling,

  * •

$C_{4}$ depends on the $L^{1}$-norm of an essentially $L^{1}$-normalized
function (see Lemma 4.2),

  * •

$C_{5}$ is a constant, which comes up in the key iteration to lower the scale,

  * •

$C_{6}$ comes from an application of the triangle inequality to lower the
scale once to $\nu$ (see (26)),

  * •

$N=N(\varepsilon,c)$ will later in the proof denote the number of iterations
to lower the scale,

  * •

$C_{7}$, $C_{8}$ are absolute constants used to record intermediate estimates
for $\mathcal{D}_{3}(\delta)$ after carrying out the decoupling iteration (see
Lemmas 4.6, 4.7).

##  3\. Decoupling in one and two dimensions

In this section we argue how the improved decoupling result by
Guth–Maldague–Wang extends to the family of curves presently considered. Some
of the arguments are already contained in [7, Appendix], but we opt to give
the details. Then we shall see how we can use lower dimensional decoupling in
bilinear expressions. Their improved decoupling result is formulated for
normalized curves as follows:

######  Theorem 3.1 ([7, Appendix]).

Let $\gamma:[0,1]\to\mathbb{R}^{2}$, $\gamma(t)=(t,h(t))$ be a curve such that
$h\in C^{2}([-1,2])$ with $h(0)=h^{\prime}(0)=0$ and $\frac{1}{2}\leq
h^{\prime\prime}(t)\leq 2$. Then, there are $C,c>0$ such that for
$(f_{J})_{J\in\mathcal{P}_{\delta}}$ with
$\text{supp}(\hat{f}_{J})\subseteq\mathcal{U}_{J,\gamma}$ we have:

$\big{\|}\sum_{J\in\mathcal{P}_{\delta}}f_{J}\big{\|}_{L^{6}(\mathbb{R}^{2})}\leq
C(\log(\delta^{-1}))^{c}\big{(}\sum_{J}\|f_{J}\|_{L^{6}(\mathbb{R}^{2})}^{2}\big{)}^{\frac{1}{2}}.$

In the following we want to argue that the result extends to more general
curves $\gamma(t)=(\gamma_{1}(t),\gamma_{2}(t))\in C^{5}$ with

(9) $\|\gamma\|_{C^{5}}\leq D_{3}<\infty\text{ and
}0<D_{1}\leq|\gamma^{\prime}(t)\wedge\gamma^{\prime\prime}(t)|\leq
D_{2}<\infty.$

######  Proposition 3.2 (Stability of improved decoupling).

Suppose $\gamma\in C^{5}$ satisfies (9), and let
$(f_{J})_{J\in\mathcal{P}_{\delta}(I)}$ with
$\text{supp}(\hat{f}_{J})\subseteq C^{\prime}\mathcal{U}_{J,\gamma}$. Then,
there is $C(\underline{D},C^{\prime})$ such that

(10) $\big{\|}\sum_{J}f_{J}\big{\|}_{L^{6}(\mathbb{R}^{2})}\leq
C(\log(\delta^{-1}))^{c}\big{(}\sum_{J}\|f_{J}\|_{L^{6}}^{2}\big{)}^{\frac{1}{2}}.$

###### Proof.

In the first step we reduce the curves $\gamma$ to $(t,h(t))$ by finite
decomposition, rotation, and translation, which only depends on
$\underline{D}$: For any point $\gamma(t_{*})$ we can obtain by rotation and
translation that $\gamma(t_{*})=0$, $\dot{\gamma}(t_{*})=(c,0)$ for some
$c>0$, and $\ddot{\gamma}(t_{*})>0$. By the implicit function theorem we
obtain a reparametrization $t=g(s)$ such that $\gamma_{1}(g(s))=s$. The
interval on which the reparametrization exists depends on $c$ and
$\|\gamma\|_{C^{2}}$. $c$ is bounded from above by $\mathcal{D}_{3}$ and from
below by using the torsion:

$\begin{vmatrix}c&\ddot{\gamma}_{1}(t_{*})\\\
0&\ddot{\gamma}_{2}(t_{*})\end{vmatrix}=c|\ddot{\gamma}_{2}(t_{*})|\geq
D_{1}\Rightarrow c\geq\frac{D_{1}}{D_{3}}.$

This means we find finitely many curves $\tilde{\gamma}(s)=(s,h(s))$ with
$h(0)=h^{\prime}(0)=0$ and $0<D_{1}^{\prime}\leq h^{\prime\prime}(s)\leq
D_{2}^{\prime}<\infty$ with $D_{i}^{\prime}=D_{i}^{\prime}(\underline{D})$. We
can compare the rectangles $\tilde{\gamma}$ and $\gamma$ by noting that from
$\tilde{\gamma}(s)=\gamma(g(s))$ follows:

$\dot{\tilde{\gamma}}(s)=\dot{\gamma}(g(s))g^{\prime}(s),\quad\ddot{\tilde{\gamma}}(s)=\ddot{\gamma}(g(s))(g^{\prime}(s))^{2}+\dot{\gamma}(g(s))g^{\prime\prime}(s).$

The bilipschitz comparability of rectangles follows then from
$g^{\prime}(s)\sim_{\underline{D}}1$ and
$|g^{\prime\prime}(s)|\leq\kappa(\underline{D})$. In $s$ parametrization, the
rectangles $C^{\prime}\mathcal{U}_{J,\gamma}$ become centered at
$\gamma(t_{J})=\tilde{\gamma}(s_{J})$ and can be contained in rectangles of
length $C^{\prime\prime}\delta\times C^{\prime\prime}\delta^{2}$ in the
directions $\dot{\tilde{\gamma}}(s_{J})$, $\ddot{\tilde{\gamma}}(s_{J})$. For
this reason we observe
$\text{supp}(\hat{f}_{J})\subseteq\mathcal{C}^{\prime\prime}\mathcal{U}_{J,\tilde{\gamma}}$.

In the following $\gamma(t)=(t,h(t))$, and we turn to normalization of $h$. We
subdivide $[0,1]$ into intervals $I_{s}$ of length $s$. A Taylor expansion of
$\gamma$ at the center $t_{c}$ gives

$\gamma(t)=\gamma(t_{c})+\gamma^{\prime}(t_{c})(t-t_{c})+\gamma^{\prime\prime}(t_{c})\frac{(t-t_{c})^{2}}{2}+O((t-t_{c})^{3}).$

Since
$|\dot{\gamma}(t_{c})\wedge\ddot{\gamma}(t_{c})|=|h^{\prime\prime}(t_{c})|\neq
0$, there is an anisotropic dilation $\underline{d}=\text{diag}(d_{1},d_{2})$
such that after translation

$\tilde{\gamma}(t)=te_{1}+\frac{t^{2}}{2}e_{2}+G(t)t^{3}e_{2}.$

In the above display $e_{i}$ denote the unit vectors such that
$\gamma(t)=\gamma_{1}(t)e_{1}+\gamma_{2}(t)e_{2}=(\gamma_{1}(t),\gamma_{2}(t))$.
The representation $G(t)t^{3}e_{2}$ with $G\in C^{2}$ for the third order
remainder term in the Taylor expansion (after dilation) follows from the
integral representation of the remainder:

$R_{3}(t)=\int_{0}^{t}\frac{\gamma^{(3)}(s)}{6}(t-s)^{3}ds.$

We obtain

$G(t)=\int_{0}^{t}\frac{\gamma^{(3)}(s)}{6}(1-\frac{s}{t})^{3}ds=t\int_{0}^{1}\frac{\gamma^{(3)}(ts^{\prime})}{6}(1-s^{\prime})^{3}ds^{\prime}$

and for $\gamma\in C^{5}$ we find $G\in C^{2}$ and
$\|G\|_{C^{2}}\leq\kappa(\underline{D})$. Moreover,

$\tilde{\gamma}^{\prime\prime}(t)=e_{2}+(G^{\prime}(t)t^{3}+3G(t)t^{2})^{\prime}e_{2}=e_{2}+(G^{\prime\prime}(t)t^{3}+6t^{2}G^{\prime}(t)+6G(t)t)e_{2}.$

Clearly,
$|G^{\prime\prime}(t)t^{3}+6t^{2}G^{\prime}(t)+6G(t)t|=O_{\underline{D}}(t)$
and choosing $s$ small enough only depending on $\underline{D}$, we finish the
decomposition into curves of the kind $\gamma(t)=(t,h(t))$ with
$h(0)=h^{\prime}(0)=0$ and $\frac{1}{2}\leq h^{\prime\prime}(t)\leq 2$ up to
rigid motions and anisotropic dilations controlled by $\underline{D}$. Now we
consider the decoupling of $(t,h(t))$ with $\text{supp}\hat{f}_{J}\subseteq
C\mathcal{U}_{J,\gamma}$ and shall prove that

$\big{\|}\sum_{J\in\mathcal{P}(\delta)}f_{J}\big{\|}_{L^{6}(\mathbb{R}^{2})}\leq\tilde{C}(C,C^{\prime})(\log(\delta^{-1}))^{c}\big{(}\sum_{J}\|f_{J}\|_{L^{6}(\mathbb{R}^{2})}^{2}\big{)}^{\frac{1}{2}}$

with $C$ like in Theorem 3.1. First, we observe that Theorem 3.1 applies with
$\tilde{C}=C$ for $C^{\prime}\leq 1$. We turn to $C^{\prime}\geq 1$: The minor
technical issue is that the blocks $\mathcal{U}_{J,\gamma}$ are overlapping
more often than in the original collection. We observe that these blocks are
in the $\tilde{\delta}$-neighbourhood for
$\tilde{\delta}=10^{10}(C^{\prime})^{2}\delta$. So we can apply decoupling for
$\tilde{\delta}$, but the decomposition into $\mathcal{U}_{\tilde{J},\gamma}$
for $\tilde{J}\in\mathcal{P}_{\tilde{\delta}}$ is too coarse. For $\tilde{J}$
we choose a collection $\mathcal{J}$ of intervals $J\subseteq\tilde{J}$ such
that $\sum_{\tilde{J}}f_{\tilde{J}}=\sum_{J}f_{J}$ and find:

$\big{\|}\sum_{\tilde{J}\in\mathcal{P}(\tilde{\delta})}f_{\tilde{J}}\big{\|}_{L^{6}(\mathbb{R}^{2})}\leq
C\log((10^{10}(C^{\prime})^{2}\delta)^{-1})^{c}\big{(}\sum_{\tilde{J}\in\mathcal{P}(\tilde{\delta})}\|f_{\tilde{J}}\|_{L^{6}(\mathbb{R}^{2})}^{2}\big{)}^{\frac{1}{2}}.$

Since $\\#\\{J\subseteq\tilde{J}\\}=O((C^{\prime})^{2})$, an application of
Cauchy-Schwarz finishes the proof:

$\big{\|}\sum_{J\in\mathcal{U}_{J}}f_{J}\big{\|}_{L^{6}(\mathbb{R}^{2})}\leq\tilde{C}(C,C^{\prime})(\log(\delta^{-1}))^{c}\big{(}\sum_{J}\|f_{J}\|^{2}_{L^{6}(\mathbb{R}^{2})}\big{)}^{\frac{1}{2}}.$

∎

We summarize uniform decoupling inequalities for families of curves: Suppose
$\ell\in\\{1,2\\}$ and $\gamma:[0,1]\to\mathbb{R}^{\ell}$ is a curve such that

(11) $\|\gamma\|_{C^{5}}\leq D_{3}\text{ and for any
}t\in[0,1]:\,D_{1}\leq\big{|}\bigwedge_{i=1}^{\ell}\partial^{i}\gamma(t)\big{|}\leq
D_{2}.$

######  Proposition 3.3 (Decoupling for curves with torsion for $d=1,2$).

Suppose that $\ell\in\\{1,2\\}$, and $\gamma:[0,1]\to\mathbb{R}^{\ell}$ is a
curve satisfying (11). Then, for any $C>0$, any $\delta\in(0,1)$, and any
tuple of functions $(f_{J})_{J\in\mathcal{P}(\delta)}$ with
$\text{supp}(\hat{f}_{J})\subseteq C\mathcal{U}_{J,\gamma}$ for any $J$, the
following inequality holds:

(12)
$\displaystyle\big{\|}\sum_{J\in\mathcal{P}(\delta)}f_{J}\big{\|}_{L^{p_{\ell}}(\mathbb{R}^{\ell})}$
$\displaystyle\leq
C_{\ell}^{\prime}(C,\underline{D},\delta)\big{(}\sum_{J\in\mathcal{P}(\delta)}\|f_{J}\|^{2}_{L^{p_{\ell}}(\mathbb{R}^{\ell})}\big{)}^{1/2}$

with

$C_{\ell}^{\prime}(C,\underline{D},\delta)=\begin{cases}C^{\prime}(C,\underline{D}),&\quad\ell=1,\\\
C^{\prime}(C,\underline{D})(\log(\delta^{-1}))^{c},&\quad\ell=2.\end{cases}$

###### Proof.

For $\ell=1$ this is obvious, for $\ell=2$ this is Proposition 3.2. ∎

######  Corollary 3.4.

Under the assumptions of Proposition 3.3, for every ball
$B\subseteq\mathbb{R}^{\ell}$ of radius $\delta^{-\ell}$, we have

(13) $\displaystyle\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{B}\big{|}\sum_{J\in\mathcal{P}(\delta)}f_{J}\big{|}^{p_{\ell}}\leq
C_{\ell}^{\prime}(C,\underline{D},\delta)\big{(}\sum_{J\in\mathcal{P}(\delta)}\|f_{J}\|^{2}_{L^{p_{\ell}}(\phi_{B})}\big{)}^{p_{\ell}/2}.$

In the above display
$\phi_{B}(x)=|B|^{-1}(1+\delta^{\ell}\text{dist}(x,B))^{-30}$ denotes an
$L^{1}$-normalized bump function adapted to $B$, and
$\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{B}$ denotes the average integral.

###### Proof.

We apply Proposition 3.3 to functions $f_{J}\psi_{B}$, where $\psi_{B}$ is a
Schwartz function such that $|\psi_{B}|\gtrsim 1$ on $B$ and
$\text{supp}(\hat{\psi}_{B})\subseteq B(0,\delta^{\ell})$. For $B$ centered at
the origin, it suffices to consider
$\psi_{B}(x)=\delta^{-\ell^{\ell}}\int_{\mathbb{R}^{3}}e^{ix.\xi}a(\delta^{-\ell}\xi)d\xi$
with $a\in C^{\infty}_{c}(\mathbb{R}^{3})$ a radially decreasing function
satisfying $a(0)=1$, $a\geq 0$ and having support in $B(0,c)$ for $c$ small
enough. The general case follows from translation. Then

$\begin{split}\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{B}\big{|}\sum_{J}f_{J}\big{|}^{p_{\ell}}&\lesssim\int_{\mathbb{R}^{3}}\big{|}\sum_{J}f_{J}\frac{\psi_{B}}{|B|^{1/p_{\ell}}}\big{|}^{p_{\ell}}\lesssim\int_{\mathbb{R}^{3}}\big{|}\sum_{J}f_{J}\frac{\psi_{B}}{|B|^{1/p_{\ell}}}\big{|}^{p_{\ell}}\\\
&\leq
C^{\prime}(C,\underline{D})C_{\ell}(\delta)\big{(}\sum_{J}\|\frac{f_{J}\psi_{B}}{|B|^{1/p_{\ell}}}\|^{2}_{L^{p_{\ell}}}\big{)}^{p_{\ell}/2}\end{split}$

with

$C_{\ell}(\delta)=\begin{cases}1,&\quad\ell=1,\\\
(\log(\delta^{-1}))^{c},&\quad\ell=2.\end{cases}$

To conclude the proof, we need to argue that

$\sup_{x}\frac{|\psi_{B}(x)|^{p_{\ell}}}{|B|\phi_{B}(x)}\lesssim 1.$

This follows from the rapid decay of $\psi_{B}$ away from $B$ (which is by our
definition of $\psi_{B}$ faster than any polynomial). Let $R=\delta^{-\ell}$
and suppose again $B$ is centered at the origin. For $\text{dist}(x,B)\leq R$
we have $|\psi_{B}(x)|^{p_{\ell}}\leq C$, $|B|\phi_{B}(x)\geq 1$. For
$\text{dist}(x,B)\geq R$, which means $x\geq 2R$, we can estimate

$\psi_{B}(x)=\psi_{1}(R^{-1}x)\leq C_{N}(1+R^{-1}x)^{-N}.$

Therefore, $(\psi_{1}(R^{-1}x))^{p_{\ell}}\leq C_{N}(R^{-1}x)^{-N\cdot
p_{\ell}}$ and $|B|\phi_{B}(x)\sim(R^{-1}x)^{-30}$. Choosing $N$ large enough
yields an acceptable contribution. ∎

######  Lemma 3.5 (Lower degree decoupling (see [6, Lemma 3.5.])).

Let $\ell\in\\{1,2\\}$. Let $\delta\in(0,1)$ and
$(f_{K})_{K\in\mathcal{P}(\delta)}$ be a tuple of functions so that
$\text{supp}\hat{f}_{K}\subseteq\mathcal{U}_{K}$ for every $K$. If $0\leq
a\leq(3-\ell+1)b/\ell$, then for any pair of intervals
$I\in\mathcal{P}(\delta^{a})$, $I^{\prime}\in\mathcal{P}(\delta^{b})$ with
$\text{dist}(I,I^{\prime})\geq 1/4$, we obtain for $\ell=1$:

(14)
$\int_{\mathbb{R}^{3}}(|f_{I}|^{2}*\phi_{I})(|f_{I^{\prime}}|^{10}*\phi_{I^{\prime}})\leq
C_{1}\sum_{J\in\mathcal{P}(I,\delta^{3b})}\int_{\mathbb{R}^{3}}(|f_{J}|^{2}*\phi_{J})(|f_{I^{\prime}}|^{10}*\phi_{I^{\prime}}),$

and for $\ell=2$:

(15)
$\int_{\mathbb{R}^{3}}(|f_{I}|^{6}*\phi_{I})(|f_{I^{\prime}}|^{6}*\phi_{I^{\prime}})\leq
C_{2}(\log(\delta^{-b}))^{c}\big{(}\sum_{J\in\mathcal{P}(I,\delta^{b})}\big{(}\int_{\mathbb{R}^{3}}\big{(}|f_{J}|^{6}*\phi_{J}\big{)}\big{(}|f_{I^{\prime}}|^{6}*\phi_{I^{\prime}}\big{)}\big{)}^{\frac{1}{3}}\big{)}^{3}.$

We need the following transversality observation:

######  Lemma 3.6 ([6, Lemma 3.5]).

Let $\Gamma_{k}(t)=(t,t^{2},\ldots,t^{k}):[0,1]\to\mathbb{R}^{k}$. For any
integers $0\leq\ell\leq k$ and any $\xi_{1},\xi_{2}\in\mathbb{R}$, we have

$|\partial^{1}\Gamma_{k}(\xi_{1})\wedge\ldots\wedge\partial^{\ell}\Gamma_{k}(\xi_{1})\wedge\partial^{1}\Gamma_{k}(\xi_{2})\wedge\ldots\wedge\partial^{k-\ell}\Gamma_{k}(\xi_{2})|\gtrsim_{k,\ell}|\xi_{1}-\xi_{2}|^{\ell(k-\ell)}.$

Now we are ready to prove Lemma 3.5. We repeat the argument from [6] for
convenience to make the involved implicit constants transparent. By
quantifying the decoupling constants, we can improve the
$\delta^{-\varepsilon}$ bound from [6] as claimed.

###### Proof of Lemma 3.5.

Denote $b^{\prime}=(3-\ell+1)b/\ell$ and $k=3$. Fix $\xi^{\prime}\in
I^{\prime}$, let
$V^{m}(\xi^{\prime})=\text{span}(\partial^{1}\Gamma_{k}(\xi^{\prime}),\ldots,\partial^{m}\Gamma_{k}(\xi^{\prime}))$
be the tangent space for $m\in\\{1,2,3\\}$, and let
$\hat{H}=\mathbb{R}^{k}/V^{k-\ell}(\xi^{\prime})$ be the quotient space. Let
$P:\mathbb{R}^{k}\to\hat{H}$ be the projection onto $\hat{H}$. For every
$\xi\in I$, we have by Lemma 3.6 that

$|\partial^{1}(P\circ\Gamma_{k})(\xi)\wedge\ldots\wedge\partial^{\ell}(P\circ\Gamma_{k})(\xi)|\gtrsim
1.$

Moreover, $P(\mathcal{U}_{J})\subseteq
C^{\prime}\mathcal{U}_{J,P\circ\Gamma_{k}}$. Let
$H=V^{k-\ell}(\xi^{\prime})^{\perp}$ be the orthogonal complement in
$\mathbb{R}^{k}$ so that $\hat{H}$ is its Pontryagin dual. Since the Fourier
support of the restriction $f_{J}\big{|}_{H+z}$ to almost every translated
copy is contained in $P(\text{supp}(\hat{f}_{J}))$ and
$P(\mathcal{U}_{J})\subseteq C^{\prime}\mathcal{U}_{J,P\circ\Gamma}$, we can
apply lower dimensional decoupling inequalities. We write by Fubini’s theorem

(16)
$\int_{\mathbb{R}^{k}}\big{(}|f_{I}|^{p_{\ell}}*\phi_{I}\big{)}\big{(}|f_{I^{\prime}}|^{p_{k}-p_{\ell}}*\phi_{I^{\prime}}\big{)}=\int_{z\in\mathbb{R}^{k}}\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{B_{H}(z,\delta^{-b^{\prime}\ell})}\big{(}|f_{I}|^{p_{\ell}}*\phi_{I}\big{)}\big{(}|f_{I^{\prime}}|^{p_{k}-p_{\ell}}*\phi_{I^{\prime}}\big{)},$

where $B_{H}(z,\delta^{-b^{\prime}\ell})$ is the $\ell$-dimensional ball with
radius $\delta^{-b^{\prime}\ell}$ centered at $z$ inside the affine subspace
$H+z$. Since
$B_{H}(0,\delta^{-b^{\prime}\ell})=B_{H}(0,\delta^{-(k-\ell+1)b})\subseteq
C^{\prime}\mathcal{U}_{I^{\prime}}^{o}$, we have

(17) $\sup_{x\in
B_{H}(z,\delta^{-b^{\prime}\ell})}\big{(}|f_{I^{\prime}}|^{p_{k}-p_{\ell}}*\phi_{I^{\prime}})(x)\lesssim\big{(}|f_{I^{\prime}}|^{p_{k}-p_{\ell}}*\phi_{I^{\prime}})(z).$

This allows us to continue to write

(18)
$\eqref{eq:Fubini}\lesssim\int_{z\in\mathbb{R}^{k}}\big{(}\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{B_{H}(z,\delta^{-b^{\prime}\ell})}|f_{I}|^{p_{\ell}}*\phi_{I}\big{)}\big{(}|f_{I^{\prime}}|^{p_{k}-p_{\ell}}*\phi_{I^{\prime}}\big{)}(z).$

Above $*_{H}$ denotes convolution along $H$. Now we can use lower dimensional
decoupling with $\delta^{b^{\prime}}$ in place of $\delta$ in Corollary 3.4:

$\leq
C_{\ell}^{\prime}(C,\underline{D},\delta)\int_{z^{\prime}}\phi_{I}(z-z^{\prime})\big{(}\sum_{J\in\mathcal{P}(I,\delta^{b^{\prime}})}\|f_{J}\|^{2}_{L^{p_{\ell}}(\phi_{B_{H}}(z^{\prime},\delta^{-b^{\prime}\ell})}\big{)}^{p_{\ell}/2}.$

Taking the $p_{\ell}$th root we find for (16):

(19) $\begin{split}\eqref{eq:Fubini}^{1/p_{\ell}}&\leq
C_{\ell}^{\prime}(C,\underline{D},\delta^{b^{\prime}})\big{(}\int_{z,z^{\prime}\in\mathbb{R}^{3}}\big{(}|f_{I^{\prime}}|^{p_{k}-p_{\ell}}*\phi_{I^{\prime}}\big{)}(z)\\\
&\qquad\times\phi_{I}(z-z^{\prime})\big{(}\sum_{J\in\mathcal{P}(I,\delta^{b^{\prime}})}\|f_{J}\|^{2}_{L^{p_{\ell}}(z^{\prime}+H,\phi_{B_{H}}(z^{\prime},\delta^{-b^{\prime}\ell}))}\big{)}^{p_{\ell}/2}\big{)}^{1/p_{\ell}}\\\
&\leq
C_{\ell}^{\prime}(C,\underline{D},\delta^{b^{\prime}})\big{(}\sum_{J\in\mathcal{P}(I,\delta^{b^{\prime}})}\big{(}\int_{z,z^{\prime}\in\mathbb{R}^{3}}\big{(}|f_{I^{\prime}}|^{p_{k}-p_{\ell}}*\phi_{I^{\prime}}\big{)}(z)\\\
&\qquad\times\phi_{I}(z-z^{\prime})\|f_{J}\|^{p_{\ell}}_{L^{p_{\ell}}(\phi_{B_{H}(z^{\prime},\delta^{-b^{\prime}\ell})})}\big{)}^{2/p_{\ell}}\big{)}^{\frac{1}{2}}.\end{split}$

The ultimate estimate follows from Minkowski’s inequality since $2\leq
p_{\ell}$. The double integral inside the brackets can be written as

(20)
$\begin{split}&\quad\int_{\mathbb{R}^{k}}\big{(}|f_{I^{\prime}}|^{p_{k}-p_{\ell}}*\phi_{I^{\prime}}\big{)}\big{(}\phi_{I}*|f_{J}|^{p_{\ell}}*_{H}\phi_{B_{H}(0,\delta^{-b^{\prime}\ell})}\big{)}\\\
&=\int_{\mathbb{R}^{k}}(|f_{I^{\prime}}|^{p_{k}-p_{\ell}}*\phi_{I}*_{H}\phi_{B_{H}(0,\delta^{-b^{\prime}\ell})})(|f_{J}|^{p_{\ell}}*\phi_{I})\\\
&\lesssim\int_{\mathbb{R}^{k}}\big{(}|f_{I^{\prime}}|^{p_{k}-p_{\ell}}*\phi_{I^{\prime}}\big{)}\big{(}|f_{J}|^{p_{\ell}}*\phi_{I}\big{)},\end{split}$

which follows again by $B_{H}(0,\delta^{-b^{\prime}\ell})\subseteq
C\mathcal{U}^{o}_{I^{\prime},\gamma}$. Using the uncertainty principle and
$\mathcal{U}_{I}^{o}\subseteq C\mathcal{U}_{J}^{o}$, we find

(21)
$|f_{J}|^{p_{\ell}}*\phi_{I}\lesssim|f_{J}|^{p_{\ell}}*\phi_{J}*\phi_{I}\lesssim|f_{J}|^{p_{\ell}}*\phi_{J}.$

We merge the implicit constants in (17), (18), (20), and (21) with
$C_{\ell}^{\prime}(C,\underline{D})$ in (19) to complete the proof. ∎

##  4\. Proof of Theorem 1.1

###  4.1. Asymmetric decoupling constant

In the following we define asymmetric decoupling constants, which effectively
allow us to lower the scale by using lower-dimensional decoupling stated in
the previous section. We consider two intervals $I$, $I^{\prime}$ of size
$|I|=\delta^{a}$ and $|I^{\prime}|=\delta^{b}$, $a,b\in[0,1]$, which are
separated at unit distance. Following [6], we define bilinear decoupling
constants as smallest constants, which satisfy the following:

$\int_{\mathbb{R}^{3}}(|f_{I}|^{6}*\phi_{I})(|f_{I^{\prime}}|^{6}*\phi_{I^{\prime}})\leq
M^{12}_{6,a,b}(\delta)\big{(}\sum_{J\in\mathcal{P}(I,\delta)}\|f_{J}\|^{2}_{L^{12}}\big{)}^{3}\big{(}\sum_{J^{\prime}\in\mathcal{P}(I^{\prime},\delta)}\|f_{J^{\prime}}\|^{2}_{L^{12}}\big{)}^{3}.$

Secondly, we define

$\int_{\mathbb{R}^{3}}(|f_{I}|^{2}*\phi_{I})(|f_{I^{\prime}}|^{10}*\phi_{I^{\prime}})\leq
M^{12}_{2,a,b}(\delta)\big{(}\sum_{J\in\mathcal{P}(I,\delta)}\|f_{J}\|^{2}_{L^{12}}\big{)}\big{(}\sum_{J^{\prime}\in\mathcal{P}(I^{\prime},\delta)}\|f_{J^{\prime}}\|^{2}_{L^{12}}\big{)}^{5}.$

We have the following as consequence of (14) and (15):

######  Lemma 4.1 (Lower dimensional decoupling).

Let $a,b\in[0,1]$ such that $0\leq a\leq 3b$. Then

(22) $M_{2,a,b}(\delta)\leq C_{2}M_{2,3b,b}(\delta).$

If $0\leq a\leq b$, then the following estimate holds for some
$c\in\mathbb{N}$:

(23) $M_{6,a,b}(\delta)\leq C_{3}(\log(\delta^{-b}))^{c}M_{6,b,b}(\delta).$

The following is straight-forward from Hölder’s inequality and parabolic
rescaling (cf. [6, Lemma 4.1]):

######  Lemma 4.2 (Hölder’s inequality I).

Let $a,b\in[0,1]$. Then

(24) $M_{2,a,b}(\delta)\leq
C_{4}M_{6,a,b}(\delta)^{1/3}D(\delta/\delta^{b})^{2/3}.$

###### Proof.

We apply Hölder’s inequality to find:

$\displaystyle|f_{I}|^{2}*\phi_{I}$ $\displaystyle\leq
C_{4}(|f_{I}|^{6}*\phi_{I})^{\frac{1}{3}},$
$\displaystyle|f_{I^{\prime}}|^{10}*\phi_{I^{\prime}}$
$\displaystyle\leq(|f_{I^{\prime}}|^{6}*\phi_{I^{\prime}})^{1/3}(|f_{I^{\prime}}|^{12}*\phi_{I^{\prime}})^{2/3}.$

The constant in the first estimate does not depend on the scale due to
$L^{1}$-normalization of $\phi_{I}$:

$\begin{split}\int|f_{I}|^{2}(y)\phi_{I}(x-y)dy&=\int|f_{I}|^{2}(y)\phi_{I}(x-y)^{\frac{1}{3}}\phi_{I}(x-y)^{\frac{2}{3}}dy\\\
&\leq\big{(}\int|f_{I}|^{6}\phi_{I}(x-y)dy\big{)}^{\frac{1}{3}}\big{(}\int\phi_{I}(x-y)dy\big{)}^{\frac{2}{3}}\\\
&=C_{4}(|f_{I}|^{6}*\phi_{I})^{\frac{1}{3}}.\end{split}$

By the above and another application of Hölder’s inequality we find:

$\begin{split}\int_{\mathbb{R}^{3}}(|f_{I}|^{2}*\phi_{I})(|f_{I^{\prime}}|^{10}*\phi_{I^{\prime}})&\leq
C_{4}\int_{\mathbb{R}^{3}}(|f_{I}|^{6}*\phi_{I})^{1/3}(|f_{I^{\prime}}|^{6}*\phi_{I^{\prime}})^{1/3}(|f_{I^{\prime}}|^{12}*\phi_{I^{\prime}})^{2/3}\\\
&\leq
C_{4}\big{(}\int_{\mathbb{R}^{3}}(|f_{I}|^{6}*\phi_{I})(|f_{I^{\prime}}|^{6}*\phi_{I^{\prime}})\big{)}^{1/3}\big{(}\int_{\mathbb{R}^{3}}(|f_{I^{\prime}}|^{12}*\phi_{I^{\prime}})\big{)}^{2/3}\\\
&\leq
C_{4}\big{(}\int_{\mathbb{R}^{3}}(|f_{I}|^{6}*\phi_{I})(|f_{I^{\prime}}|^{6}*\phi_{I^{\prime}})\big{)}^{1/3}\big{(}\int_{\mathbb{R}^{3}}(|f_{I^{\prime}}|^{12}*\phi_{I^{\prime}})\big{)}^{2/3}\end{split}$

From this estimate and parabolic rescaling, (24) is immediate. ∎

Another application of Hölder’s inequality gives the following (again [6,
Lemma 4.2]):

######  Lemma 4.3 (Hölder’s inequality II).

Let $a,b\in[0,1]$. Then

(25) $M_{6,a,b}(\delta)\leq M_{2,a,b}(\delta)^{1/2}M_{2,b,a}(\delta)^{1/2}.$

###### Proof.

By two applications of the Cauchy-Schwarz inequality we find

$\begin{split}&\quad\int_{\mathbb{R}^{3}}(|f_{I}|^{6}*\phi_{I})(|f_{I^{\prime}}|^{6}*\phi_{I^{\prime}})\\\
&\leq\int_{\mathbb{R}^{3}}(|f_{I}|^{2}*\phi_{I})^{1/2}(|f_{I^{\prime}}|^{10}*\phi_{I^{\prime}})^{1/2}\cdot(|f_{I}|^{10}*\phi_{I})^{1/2}(|f_{I^{\prime}}|^{2}*\phi_{I^{\prime}})^{1/2}\\\
&\leq\big{(}\int_{\mathbb{R}^{3}}(|f_{I}|^{2}*\phi_{I})(|f_{I^{\prime}}|^{10}*\phi_{I^{\prime}})\big{)}^{1/2}\big{(}\int_{\mathbb{R}^{3}}(|f_{I}|^{10}*\phi_{I})(|f_{I^{\prime}}|^{2}*\phi_{I^{\prime}})\big{)}^{1/2}.\end{split}$

From this estimate (25) is immediate. ∎

###  4.2. The decoupling iteration

By Lemma 4.1, 4.2, and 4.3, we have the following key iteration step:

######  Lemma 4.4 (Iteration step for the moment curve).

Let $a,b\in[0,1]$ and $0<a\leq 3b$. We find

$M_{2,a,b}(\delta)\leq
C_{5}M_{2,3b,3b}^{1/3}(\delta)\log(\delta^{-3b})^{c}D(\delta/\delta^{b})^{2/3}.$

###### Proof.

By successive applications of the aforementioned lemmas, we find

$\begin{split}M_{2,a,b}(\delta)&\leq C_{2}M_{2,3b,b}(\delta)\leq
C_{2}C_{4}M_{6,3b,b}(\delta)^{1/3}D(\delta/\delta^{b})^{2/3}\\\ &\leq
C_{2}C_{3}C_{4}M_{6,3b,3b}^{1/3}(\delta)\log(\delta^{-3b})^{c}D(\delta/\delta^{b})^{2/3}\\\
&\leq\underbrace{C_{2}C_{3}C_{4}}_{C_{5}}M_{2,3b,3b}^{1/3}(\delta)\log(\delta^{-3b})^{c}D(\delta/\delta^{b})^{2/3}.\end{split}$

∎

To make the iteration effective, we initially divide the unit size intervals
$I$, $I^{\prime}$ considered in $B(\delta)$ into $\nu^{-1}$ smaller intervals,
and then use the previously established iteration. Let $\nu=\delta^{b}$. We
choose $\nu=\delta^{1/3^{N}}$ such that in $N$ iterations of Lemma 4.4 we
reach the scale $\delta$, where decoupling becomes trivial. We use the
estimate

(26) $M_{6,0,0}(\delta)\leq C_{6}\nu^{-\frac{1}{2}}M_{6,b,b}(\delta)$

due to the Cauchy-Schwarz inequality:

$\begin{split}&\quad\big{(}\int\big{(}\big{|}\sum_{J\in\mathcal{P}(I,\delta^{b})}f_{J}\big{|}^{6}*\phi_{I}\big{)}\big{(}\big{|}\sum_{J^{\prime}\in\mathcal{P}(I^{\prime},\delta^{b})}f_{J^{\prime}}\big{|}^{6}*\phi_{I^{\prime}}\big{)}\big{)}^{\frac{1}{6}}\\\
&\leq\sum_{\begin{subarray}{c}J\in\mathcal{P}(I,\delta^{b}),\\\
J^{\prime}\in\mathcal{P}(I^{\prime},\delta^{b})\end{subarray}}\big{(}\int\big{(}|f_{J}|^{6}*\phi_{I}\big{)}\big{(}|f_{J^{\prime}}|^{6}*\phi_{I^{\prime}}\big{)}\big{)}^{\frac{1}{6}}\\\
&\leq
M_{6,b,b}^{2}(\delta)\sum_{\begin{subarray}{c}J\in\mathcal{P}(I,\delta^{b}),\\\
J^{\prime}\in\mathcal{P}(I^{\prime},\delta^{b})\end{subarray}}\big{(}\sum_{K\in\mathcal{P}(J,\delta)}\|f_{K}\|_{L^{12}(\mathbb{R}^{3})}^{2}\big{)}^{\frac{1}{2}}\big{(}\sum_{K^{\prime}\in\mathcal{P}(J^{\prime},\delta)}\|f_{K^{\prime}}\|_{L^{12}(\mathbb{R}^{3})}^{2}\big{)}^{\frac{1}{2}}\\\
&\leq
M_{6,b,b}^{2}(\delta)C_{6}^{2}\nu^{-1}\big{(}\sum_{K\in\mathcal{P}(I,\delta)}\|f_{K}\|_{L^{12}(\mathbb{R}^{3})}^{2}\big{)}^{\frac{1}{2}}\big{(}\sum_{K^{\prime}\in\mathcal{P}(I^{\prime},\delta)}\|f_{K^{\prime}}\|_{L^{12}(\mathbb{R}^{3})}^{2}\big{)}^{\frac{1}{2}}.\end{split}$

By the decoupling result of Bourgain–Demeter–Guth [3] we have

(27) $\mathcal{D}_{3}(\delta)\leq C_{\varepsilon}\delta^{-\varepsilon},$

which gives:

######  Lemma 4.5.

Let $N\in\mathbb{N}$. Suppose that $\delta\in 2^{\mathbb{Z}}$ and
$\delta^{-\frac{1}{3^{N}}}\in\mathbb{N}$. Then the following estimate holds:

(28) $D(\delta)\leq
C_{7}\delta^{\frac{\varepsilon}{3^{N}}\big{(}1+\frac{2N}{3}-\frac{1}{2\varepsilon}\big{)}}\log(\delta^{-1})^{3c}C_{\varepsilon}^{1-\frac{1}{3^{N}}}\delta^{-\varepsilon}.$

###### Proof.

We find iterating Lemma 4.4 $N$ times:

(29) $M_{2,b,b}(\delta)\leq
C_{5}^{2}M_{6,3^{N}b,3^{N}b}^{1/3^{N}}\log(\delta^{-1})^{2c}\prod_{j=0}^{N-1}D(\delta/\delta^{3^{j}b})^{(2/3)\cdot
1/3^{j}}.$

From the bilinear reduction, we have (here we use $\delta\in 2^{\mathbb{Z}}$)

$\mathcal{D}_{3}(\delta)\leq C_{1}\log(\delta^{-1})M_{6,0,0}(\delta).$

We reduce the scale in $M_{6,0,0}(\delta)$ to $\nu$ by (26) such that

$\mathcal{D}_{3}(\delta)\leq
C_{1}C_{6}\nu^{-\frac{1}{2}}\log(\delta^{-1})M_{6,b,b}(\delta).$

Now we plug in (29) to find the following recursive estimate for the linear
decoupling constant:

$\mathcal{D}_{3}(\delta)\leq\underbrace{C_{1}C_{5}^{2}C_{6}}_{C_{7}}\delta^{-\frac{b}{2}}\log(\delta^{-1})^{3c}\prod_{j=0}^{N-1}D(\delta/\nu^{3^{j}})^{\frac{2}{3}\cdot\frac{1}{3^{j}}}.$

By (27), we find

$\begin{split}D(\delta)&\leq
C_{7}\delta^{-\frac{b}{2}}\log(\delta^{-1})^{3c}\prod_{j=0}^{N-1}(C_{\varepsilon}\delta^{-\varepsilon(1-3^{j-N})})^{\frac{2}{3}\cdot\frac{1}{3^{j}}}\\\
&=C_{7}\delta^{-\frac{b}{2}}\log(\delta^{-1})^{3c}C_{\varepsilon}^{1-\frac{1}{3^{N}}}\delta^{-\varepsilon(1-\frac{1}{3^{N}})}\delta^{\varepsilon\frac{2N}{3\cdot
3^{N}}}\\\
&=C_{7}\delta^{\frac{\varepsilon}{3^{N}}(1+\frac{2N}{3}-\frac{1}{2\varepsilon})}\log(\delta^{-1})^{3c}C_{\varepsilon}^{1-\frac{1}{3^{N}}}\delta^{-\varepsilon}.\end{split}$

∎

In the next step, we choose $N=N(\varepsilon)$, which simplifies the above
expression for $\delta\in 2^{\mathbb{Z}}$ and
$\delta^{-\frac{1}{3^{N}}}\in\mathbb{N}$.

######  Lemma 4.6.

Let $0<\varepsilon<\varepsilon_{0}=\varepsilon_{0}(c)$, and $N\in\mathbb{N}$
such that

(30) $1+\frac{2N}{3}-\frac{1}{2\varepsilon}\in[\frac{2}{3},2].$

For $\delta\in(\delta_{n})_{n=n_{0}}^{\infty}$ with
$\delta_{n}=2^{-n3^{10N}}$, $n_{0}=n_{0}(c)$, we have the following:

$\mathcal{D}_{3}(\delta)\leq
C_{7}C_{\varepsilon}^{1-\frac{1}{3^{N}}}\delta^{-\varepsilon}.$

###### Proof.

With the assumptions of Lemma 4.5 satisfied, we find by (29)

$\mathcal{D}_{3}(\delta)\leq
C_{7}\delta^{\frac{\varepsilon}{3^{N}}\big{(}1+\frac{2N}{3}-\frac{1}{2\varepsilon}\big{)}}\log(\delta^{-1})^{3c}C_{\varepsilon}^{1-\frac{1}{3^{N}}}\delta^{-\varepsilon}.$

By (30) this simplifies to

$\mathcal{D}_{3}(\delta)\leq
C_{7}\log(\delta^{-1})^{3c}\delta^{\frac{2\varepsilon}{3^{N}\cdot
3}}C_{\varepsilon}^{1-\frac{1}{3^{N}}}\delta^{-\varepsilon}.$

Since $\delta=2^{-n\cdot 3^{10N}}$, we show that for $n\geq n_{0}(c)$ and
$0<\varepsilon<\varepsilon_{0}$

$\log(\delta^{-1})^{3c}\delta^{\frac{2\varepsilon}{3^{N}\cdot 3}}\leq 1.$

First we note that

$\delta^{\frac{2\varepsilon}{3\cdot 3^{N}}}\leq\delta^{\frac{1}{3^{2N}}}\leq
2^{-n\cdot 3^{8N}}.$

Here we use $\varepsilon\sim\frac{1}{N}$, and for
$0<\varepsilon<\varepsilon_{0}$, $N$ becomes large enough to argue like in the
above estimate. Moreover,

$\log(\delta^{-1})^{3c}\leq n^{3c}3^{30Nc}\log(2)^{3c}\leq n^{3c}3^{30Nc}.$

First, we see that

$3^{30Nc}\leq 2^{\log(3)30Nc}\leq 2^{\frac{n}{2}3^{8N}}$

by choosing $0<\varepsilon<\varepsilon_{0}(c)$ small enough such that
$30Nc\log(3)\leq 3^{8N}/2$ (since $N$ becomes large enough for $N\sim
1/\varepsilon$ such that the inequality holds). Secondly, we can choose $n\geq
n_{0}(c)$ large enough such that

$3\log_{2}(n)c\leq\frac{n}{2}\Rightarrow 2^{\log_{2}(n)3c}\leq
2^{\frac{n3^{8N}}{2}}.$

Then we arrive at the claim

$\mathcal{D}_{3}(\delta)\leq
C_{7}C_{\varepsilon}^{1-\frac{1}{3^{N}}}\delta^{-\varepsilon}.$

∎

This absorbs the additional $\log(\delta^{-1})^{3c}$-factor, which is absent
in Li’s proof of improved decoupling for the parabola [9]. We give the
concluding arguments from [9] for self-containedness.

In the following lemma we use submultiplicativity to extend this estimate to
all $\delta\in\mathbb{N}^{-1}$:

######  Lemma 4.7.

Let $0<\varepsilon<\varepsilon_{0}=\varepsilon_{0}(c)$ and $n_{0}=n_{0}(c)$
such that Lemma 4.6 is valid. Then there is some $a>0$ such that we find for
all $\delta\in\mathbb{N}^{-1}$

(31) $\mathcal{D}_{3}(\delta)\leq C_{8}2^{n_{0}\cdot
3^{\frac{a}{\varepsilon}}}C_{\varepsilon}^{1-a/\varepsilon}\delta^{-\varepsilon}.$

###### Proof.

Let $N$ be like in (30) and
$\delta\in(\delta_{n})_{n=n_{0}}^{\infty}=(2^{-n\cdot
3^{10N}})_{n=n_{0}}^{\infty}$. If
$\delta\in(\delta_{n_{0}},1]\in\mathbb{N}^{-1}$, we use the trivial estimate

$\mathcal{D}_{3}(\delta)\leq\delta^{-1/2}\leq 2^{\frac{n_{0}}{2}\cdot
3^{10N}}.$

If $\delta\in(\delta_{n+1},\delta_{n}]$ for $n\geq n_{0}$, then
submultiplicativity and Lemma 4.6 imply

$\begin{split}\mathcal{D}_{3}(\delta)&\leq\mathcal{D}_{3}(\delta_{n+1})\leq\mathcal{D}_{3}(\delta_{n})\mathcal{D}_{3}(\delta_{n+1}/32\delta_{n})\leq(C_{7}C_{\varepsilon}^{1-1/3^{N}}\delta_{n}^{-\varepsilon})(32(\delta_{n}/\delta_{n+1}))^{1/2}\\\
&=32^{1/2}C_{7}C_{0}^{1/2}2^{\frac{1}{2}\cdot
3^{10N}}C_{\varepsilon}^{1-1/3^{N}}\delta^{-\varepsilon}.\end{split}$

Taking the two estimates together gives

$D(\delta)\leq C_{8}2^{n_{0}\cdot
3^{10N}}C_{\varepsilon}^{1-1/3^{N}}\delta^{-\varepsilon}.$

This estimate holds for all $\delta\in\mathbb{N}^{-1}$ with $N$ given by (30).
Now we simplify by monotonicity in $N$. By the choice of $N$, we have
$3^{N}\leq 3^{a/\varepsilon}$ for some $a$ and
$\varepsilon<\varepsilon_{0}(c)$. We obtain

$D(\delta)\leq C_{8}2^{n_{0}\cdot
3^{10a/\varepsilon}}C_{\varepsilon}^{1-\frac{1}{3^{a/\varepsilon}}}\delta^{-\varepsilon}.$

∎

We bootstrap this bound to find the following:

######  Lemma 4.8.

There is $\varepsilon_{0}=\varepsilon_{0}(C_{8},c)$ such that for all
$0<\varepsilon<\varepsilon_{0}(c)$ and $\delta\in\mathbb{N}^{-1}$, we have

$\mathcal{D}_{3}(\delta)\leq 2^{3^{100a/\varepsilon}}\delta^{-\varepsilon}.$

###### Proof.

Let $P(C,\lambda)$ be the statement that $D(\delta)\leq C\delta^{-\lambda}$
for all $\delta\in\mathbb{N}^{-1}$. Lemma 4.7 implies that for
$\varepsilon\in(0,\varepsilon_{0}(c))$ and $n_{0}=n_{0}(c)$:

$P(C_{\varepsilon},\varepsilon)\Rightarrow P(C_{8}\cdot 2^{n_{0}\cdot
3^{10a/\varepsilon}}C_{\varepsilon}^{1-1/3^{a/\varepsilon}},\varepsilon).$

After $M$ iterations of the above implication, we obtain

$P(C_{\varepsilon},\varepsilon)\Rightarrow P((C_{8}\cdot 2^{n_{0}\cdot
3^{10a/\varepsilon}})^{\sum_{j=0}^{M-1}(1-1/3^{a/\varepsilon})^{j}}C_{\varepsilon}^{(1-1/3^{a/\varepsilon})^{M}},\varepsilon).$

We can take limits

$C_{\varepsilon}^{(1-1/3^{a/\varepsilon})^{M}}\rightarrow_{M\to\infty}1,\quad\sum_{j=0}^{M-1}(1-1/3^{a/\varepsilon})^{j}\rightarrow_{M\to\infty}3^{a/\varepsilon}.$

Hence, letting $M\to\infty$, we obtain

$P(C_{8}^{3^{a/\varepsilon}}\cdot 2^{n_{0}\cdot
3^{11a/\varepsilon}},\varepsilon).$

By choosing $0<\varepsilon<\varepsilon_{0}(C_{8},n_{0}(c))$ we find for all
$\delta\in\mathbb{N}^{-1}$

$D(\delta)\leq C_{8}^{3^{a/\varepsilon}}2^{n_{0}\cdot
3^{11a/\varepsilon}}\delta^{-\varepsilon}\leq
2^{3^{100a/\varepsilon}}\delta^{-\varepsilon}.$

This finishes the proof. ∎

In the following we fix $\varepsilon_{0}=\varepsilon_{0}(C_{8},c)$ and $a$
such that Lemma 4.8 is valid.

###  4.3. Proof of Theorem 1.1

We can write for $0<\varepsilon<\varepsilon_{0}$

(32) $D(\delta)\leq A^{A^{1/\varepsilon}}\delta^{-\varepsilon}$

for some $A=A(a)\geq e$. It suffices to prove (3) with exponentials and
logarithms based on $A$.

###### Proof of Theorem 1.1.

We optimize (32) by choosing $\varepsilon=\varepsilon(\delta)$. Let

(33)
$B=\log_{A}(1/\delta)>1,\quad\eta=\log_{A}(B)-\log_{A}\log_{A}(B),\quad\varepsilon=1/\eta.$

This leads to the first constraint

(34) $\delta<A^{-1}.$

The constraint on $\varepsilon_{0}$ translates to

$\varepsilon=\frac{1}{\eta}\leq\varepsilon_{0}\Rightarrow\frac{1}{\varepsilon_{0}}\leq\log_{A}(B/\log_{A}(B))\leq\log_{A}(B)=\log_{A}(\log_{A}(1/\delta)).$

This gives the condition on $\delta$:

(35) $\delta<(A^{A^{1/\varepsilon_{0}}})^{-1}=\delta_{0}.$

It is straight-forward by (33) that

$A^{1/\varepsilon}\leq\varepsilon\log_{A}(1/\delta).$

For this reason we obtain

$A^{A^{1/\varepsilon}}\delta^{-\varepsilon}\leq\exp_{A}(2\varepsilon\log_{A}(1/\delta))\leq\exp_{A}(\frac{4\log_{A}(1/\delta)}{\log_{A}\log_{A}(1/\delta)}).$

In the above display we used that

$\varepsilon=\frac{1}{\log_{A}B-\log_{A}\log_{A}B}\leq\frac{2}{\log_{A}B},$

which is true for $\log_{A}B\leq B^{1/2}$. This is true because $A\geq e$ and
$B\geq 1$. Finally, we find with $a=1/\log(A)\leq 1$

$\begin{split}\exp_{A}(\frac{4\log_{A}(1/\delta)}{\log_{A}\log_{A}(1/\delta)})&=\exp_{A}\big{(}\frac{4\log(x)}{\log(a\log(1/\delta))}\big{)}\\\
&\leq\exp_{A}\big{(}\frac{8\log(x)}{\log(\log(1/\delta))}\big{)}=\exp\big{(}\frac{4\log(A)\log(x)}{\log(a\log(1/\delta))}\big{)}.\end{split}$

In the estimate we used $\log(a\log(1/\delta))\geq\log(\log(1/\delta))/2$,
which amounts to $\delta\leq\exp(-\log(A)^{2})$. This is true by (35), and the
proof is complete.

∎

## Acknowledgement

Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research
Foundation) Project-ID 258734477 – SFB 1173.

## References

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  * [8] Zane Kun Li.  Effective $l^{2}$ decoupling for the parabola.  Mathematika, 66(3):681–712, 2020.  With an appendix by Jean Bourgain and Li.
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Let  $|k|\in I_{\ell_1}$, $|j|\in I_{\ell_2}$, $\tau_\ell\approx 2^s$, $2^s>2\la$. Here $\ell_1,\ell_2\ge [\log_2(10\la)]$. See the definition of $I_{\ell_1}$ before (<ref>). The definition of $I_{\ell_2}$ is the same. Let

\[b_{k,\tau_\ell}=\sum_{|j|\in I_{\ell_2}}\rho_2(|j|/\tau_\ell)\rho_2(|k|/\tau_\ell)\frac{h(|j|)(|j|/\tau_\ell)^{2N}}{(\tau_\ell^2-|j|^2)(\tau_\ell^2-|k|^2)}e_j^0(x)V_{jk}.\]

Note that
\[|h(|j|)|\ls \begin{cases}1,\ \ \ \ \ \ \ \ \ \ \ \ \ \ 2^{\ell_2}\approx10\la\\
2^{-N_1\ell_2},\  \ \ \ \ \ \ 2^{\ell_2}>10\la,\ \forall N_1.
\end{cases}\]

Then we get

\begin{align*}|b_{k,\tau_\ell}|&\ls \sum_{2^m\le 2^s}2^{2N(\ell_2-s)}2^{-4s}\cdot 2^{(-n+2-\eta)m}\min(2^{nm},2^{n\ell_2})\cdot \begin{cases}1,\ \ \ \ \ \ \ \ \ \ \ \ \ \  2^{\ell_2}\approx10\la\\
2^{-N_1\ell_2},\ \ \ \ \ \ \ 2^{\ell_2}>10\la
\end{cases}\\
&\ls 2^{(-2N-4)s}2^{(2N+2-\eta)\ell_2}\cdot \begin{cases}1,\ \ \ \ \ \ \ \ \ \ \ \ \ \  2^{\ell_2}\approx10\la\\
2^{-N_1\ell_2},\ \ \ \ \ \  2^{\ell_2}>10\la.
\end{cases}\end{align*}

And similarly,

\begin{align*}|\partial_{\tau_\ell}b_{k,\tau_\ell}|
&\ls 2^{(-2N-5)s}2^{(2N+2-\eta)\ell_2}\cdot \begin{cases}1,\ \ \ \ \ \ \ \ \ \ \ \ \ \  2^{\ell_2}\approx10\la\\
2^{-N_1\ell_2},\ \ \ \ \ \ \ 2^{\ell_2}>10\la.
\end{cases}\end{align*}

By Sobolev inequality,

\begin{align*}
\|\sum_{|k|\in I_{\ell_1}}& b_{k, 2^s}e_k^0(\cdot)\|_{L^{p_0}(M)}\ls 2^{\ell_1/2}2^{\sigma(p_0)\ell_1}	\|\sum_{|k|\in I_{\ell_1}} b_{k, 2^s}e_k^0(\cdot)\|_{L^{2}(M)} \\
&\ls  2^{(n/2+1-\frac14\eta)\ell_1}\cdot\sup_{k,\tau_\ell}|b_{k,\tau_\ell}|.
\end{align*}

And similarly we may obtain the estimate for  $\partial_\tau b_{k,\tau}$.

By applying Lemma <ref> with $\delta=2^{s}$, we have
\begin{align*}
|\int \sum_{\tau_\ell\approx 2^s}&\sum_{|k|\in I_{\ell_1}} b_{k,\tau_\ell} \overline{e_k^0(z)}V(z)e_{\tau_\ell}(z) \overline{e_{\tau_\ell}(x)}dz |\\
\ls & \|V\|_{L^{\frac{n}{2-\eta}}(M)}2^{ns/2}2^{s/2}2^{\sigma(p_0)s}\cdot 2^{(-2N-4)s}2^{(2N+2-\eta)\ell_2}2^{(\frac n2+1-\frac14\eta)\ell_1}\\
&\,\,\cdot\begin{cases}1,\ \ \ \ \ \ \ \ \ \ \ \ \ \  2^{\ell_2}\approx10\la\\
2^{-N_1\ell_2},\ \ \ \ \ \ \ 2^{\ell_2}>10\la\end{cases}\\
\ls& 2^{(\frac n2-\frac14\eta-2N-3)s}2^{(2N+2-\eta)\ell_2}2^{(\frac n2+1-\frac14\eta)\ell_1}\cdot\begin{cases}1,\ \ \ \ \ \ \ \ \ \ \ \ \ \ 2^{\ell_2}\approx10\la\\
2^{-N_1\ell_2},\ \ \ \ \ \ \ 2^{\ell_2}>10\la\end{cases}
\end{align*}

Fix $N\ge n$ and $N_1\ge2N+2$. Note that $\la\ls 2^{\ell_1},2^{\ell_2}\ls 2^s$. Summing over $\ell_1,\ell_2,s$, we obtain (<ref>).

§.§ Case 3.2.2

For $\mu=0,1,...,N-1$, we need to show

\begin{multline}\label{h.3.2.2}\Big|\sum_{j,k}\sum_{\tau_\ell> 2\la}\rho_2(|j|/\tau_\ell)\rho_2(|k|/\tau_\ell)\frac{h(|j|)|j|^{2\mu}(|k|/\tau_\ell)^{2N}}{\tau_\ell^{2+2\mu}(\tau_\ell^2-|k|^2)}e_j^0(x)V_{jk}\overline{e_k^0(z)}V(z)e_{\tau_\ell}(z) \overline{e_{\tau_\ell}(x)}dz\Big|\\ \ls\la^{n-\frac32\eta}.
\end{multline}

Let  $|k|\in I_{\ell_1}$, $|j|\in I_{\ell_2}$, $\tau_\ell\approx 2^s$, $2^s>2\la$. As before,  $\ell_1,\ell_2\ge [\log_2(10\la)]$.

\[b_{k,\tau_\ell}=\sum_{|j|\in I_{\ell_2}}\rho_2(|j|/\tau_\ell)\rho_2(|k|/\tau_\ell)\frac{h(|j|)|j|^{2\mu}\tau_\ell^{-2-2\mu}(|k|/\tau_\ell)^{2N}}{\tau_\ell^2-|k|^2}e_j^0(x)V_{jk}.\]

We have

\begin{equation}
\begin{aligned}
|b_{k,\tau_\ell}|\ls & 2^{(-2N-4-2\mu)s}2^{2N\ell_1}2^{(2-\eta+2\mu)\ell_2} \\
&\quad\cdot\begin{cases}1,\ \ \ \ \ \ \ \ \ \ \ \ \ \  2^{\ell_1}\le 80\la\\
2^{-N_1\ell_2},\ \ \ \ \ \ \ \ \ 2^{\ell_1}>80\la\ {\rm and}\ 2^{\ell_2}>2^{\ell_1-2}\\
2^{(n-2+\eta)(\ell_2-\ell_1)},\ \ 2^{\ell_1}>80\la\ {\rm and}\ 2^{\ell_2}\approx 10\la\\
2^{(n-2+\eta)(\ell_2-\ell_1)}2^{-N_1\ell_2},\ \ 2^{\ell_1}>80\la\ {\rm and}\ 10\la< 2^{\ell_2}\le  2^{\ell_1-2}.
\end{cases}
\end{aligned}
\end{equation}

And also by Sobolev inequality

\begin{align*}
\|\sum_{|k|\in I_{\ell_1}}& b_{k, 2^s}e_k^0(\cdot)\|_{L^{p_0}(M)}\ls 2^{\ell_1/2}2^{\sigma(p_0)\ell_1}	\|\sum_{|k|\in I_{\ell_1}} b_{k, 2^s}e_k^0(\cdot)\|_{L^{2}(M)} \\
&\ls  2^{(n/2+1-\frac14\eta)\ell_1}\cdot\sup_{k,\tau_\ell}|b_{k,\tau_\ell}|.
\end{align*}

Similarly, we may obtain the estimates for $\partial_\tau b_{k,\tau}$.
By applying Lemma <ref> with $\delta=2^{s}$, we have
\begin{align*}
|\int& \sum_{\tau_\ell\approx 2^s}\sum_{|k|\in I_{\ell_1}} b_{k,\tau_\ell} \overline{e_k^0(z)}V(z)e_{\tau_\ell}(z) \overline{e_{\tau_\ell}(x)}dz | \\
&\ls \|V\|_{L^{\frac{n}{2-\eta}}(M)}2^{ns/2}2^{s/2}2^{(\frac12-\frac14\eta)s}\cdot2^{(-2N-4-2\mu)s}2^{(2-\eta+2\mu)\ell_2}\\
&\quad \cdot 2^{(2N+\frac n2+1-\frac14\eta)\ell_1} \cdot\begin{cases}1,\ \ \ \ \ \ \ \ \ \ \ \ \ \  2^{\ell_1}\le 80\la\\
2^{-N_1\ell_2},\ \ \ \ \ \ \ 2^{\ell_1}>80\la\ {\rm and}\ 2^{\ell_2}>2^{\ell_1-2}\\
2^{(n-2+\eta)(\ell_2-\ell_1)},\ \ 2^{\ell_1}>80\la\ {\rm and}\ 2^{\ell_2}\approx 10\la\\
2^{(n-2+\eta)(\ell_2-\ell_1)}2^{-N_1\ell_2},\ \ 2^{\ell_1}>80\la\ {\rm and}\ 10\la< 2^{\ell_2}\le  2^{\ell_1-2}.
\end{cases}
\end{align*}

Fix $N\ge n$ and $N_1\ge2N+2n$. Note that $\la\ls 2^{\ell_1},2^{\ell_2}\ls 2^s$. Summing over $\ell_1,\ell_2,s$, we obtain (<ref>).

§.§ Case 3.2.3

For $\mu_1,\mu_2=0,1,...,N-1$, we need to show

\begin{multline}
\Big|\sum_{j,k}\sum_{\tau_\ell> 2\la}\rho_2(|j|/\tau_\ell)\rho_2(|k|/\tau_\ell)\frac{h(|j|)|j|^{2\mu_2}|k|^{2\mu_1}}{\tau_\ell^{2(\mu_1+\mu_2+2)}}e_j^0(x)V_{jk}\overline{e_k^0(z)}V(z)e_{\tau_\ell}(z) \overline{e_{\tau_\ell}(x)}dz\Big| \\ \ls\la^{n-\frac32\eta}.
\end{multline}

For any fixed $k\in I_{\ell_1}$ (see the definition of $I_{\ell_1}$ before (<ref>)), we write

\begin{equation}\label{hde1}
\begin{aligned}
\rho_2(|j|/\tau_\ell)\rho_2(|k|/\tau_\ell) &\tau_\ell^{-2(\mu_1+\mu_2+2)}\\
& =\rho_2(|j|/\tau_\ell)\rho_2(|k|/\tau_\ell) c_0\int_0^\infty t^{\mu_1+\mu_2+1}e^{-t\tau_k^2}dt\\
&=c_0\int_0^{2^{-2\ell_1}} t^{\mu_1+\mu_2+1}e^{-t\tau_\ell^2}dt \\
&\quad-\big(1-\rho_2(|j|/\tau_\ell)\rho_2(|k|/\tau_\ell)\big)c_0\int_0^{2^{-2\ell_1}} t^{\mu_1+\mu_2+1}e^{-t\tau_\ell^2}dt\\ &\quad\quad\quad\quad\quad-\frac{\rho_2(|j|/\tau_\ell)\rho_2(|k|/\tau_\ell)w(\tau_\ell^22^{-2\ell_1})}{2^{2\ell_1(\mu_1+\mu_2+2)}},\\
\end{aligned}
\end{equation}

$c_0$ is a fixed normalizing constant and

\begin{equation}\nonumber
\begin{aligned}
w(x)=c_0\int_1^{\infty} t^{\mu_1+\mu_2+1}e^{-tx}dt
\ls e^{-x/2}.
\end{aligned}
\end{equation}

Now we handle these three terms in Cases 3.2.3a, 3.2.3b, 3.2.3c respectively.

§.§ Case 3.2.3a
We need to show

\begin{equation}\label{h.4.1}
\begin{aligned}
|&\sum_{j,k}\sum_{\tau_\ell> 2\la}\int h(|j|)|j|^{2\mu_2}|k|^{2\mu_1}c^0_{k,\tau_\ell}\,e_j^0(x)V_{jk}\overline{e_k^0(z)}V(z)e_{\tau_\ell}(z) \overline{e_{\tau_\ell}(x)}dz| \\
\end{aligned}
\end{equation}

We shall divide our task into proving

\begin{equation}\label{h.4.2}
\begin{aligned}
|&\sum_{j,k}\sum_{\tau_\ell\le2\la}\int h(|j|)|j|^{2\mu_2}|k|^{2\mu_1}c^0_{k,\tau_\ell}\,e_j^0(x)V_{jk}\overline{e_k^0(z)}V(z)e_{\tau_\ell}(z) \overline{e_{\tau_\ell}(x)}dz| \\
\end{aligned}
\end{equation}

as well as

\begin{equation}\label{h.4.3}
\begin{aligned}
|&\sum_{j,k}\sum_{\tau_\ell}\int h(|j|)|j|^{2\mu_2}|k|^{2\mu_1}c^0_{k,\tau_\ell}\,e_j^0(x)V_{jk}\overline{e_k^0(z)}V(z)e_{\tau_\ell}(z) \overline{e_{\tau_\ell}(x)}dz| \\
\end{aligned}
\end{equation}

\[b_{k}=\sum_{|j|\in I_{\ell_2}}h(|j|)|j|^{2\mu_2}|k|^{2\mu_1}e_j^0(x)V_{jk}.\]

Then for $|k|\in I_{\ell_1}$, we have

\begin{align}\label{h.4.5}|b_k|\ls  2^{2\mu_1\ell_1}2^{(2\mu_2+2-\eta)\ell_2}\cdot\begin{cases}1,\ \ \ \ \ \ \ \ \ \ \ \ \ \  2^{\ell_1}\le 80\la\ {\rm and}\ 2^{\ell_2}\approx 10\la\\
2^{-N_1\ell_2},\ \ \ \ \ \ \ 2^{\ell_1}\le 80\la\ {\rm and}\ 2^{\ell_2}>10\la	\\
2^{-N_1\ell_2},\ \ \ \ \ \ \ 2^{\ell_1}>80\la\ {\rm and}\ 2^{\ell_2}>2^{\ell_1-2}\\
2^{(n-2+\eta)(\ell_2-\ell_1)},\ \ 2^{\ell_1}>80\la\ {\rm and}\ 2^{\ell_2}\approx 10\la\\
2^{(n-2+\eta)(\ell_2-\ell_1)}2^{-N_1\ell_2},\ \ 2^{\ell_1}>80\la\ {\rm and}\ 10\la< 2^{\ell_2}\le  2^{\ell_1-2}.
\end{cases}\end{align}

By using Sobolev inequality,

\begin{align*}
\|&\sum_{|k|\in I_{\ell_1}} b_{k}e_k^0(\cdot)\|_{L^{p_0}(M)}\ls  2^{\ell_1/2}2^{\sigma(p_0)\ell_1} \|\sum_{|k|\in I_{\ell_1}} b_{k}e_k^0(\cdot)\|_{L^{2}(M)} \\ &
\ls 2^{(n/2+1-\frac14\eta)\ell_1}\cdot \sup_{k}|b_k|.
\end{align*}

Then by Hölder's inequality, we have for fixed $0\le t\le 2^{-2\ell_1}$

\begin{align*}
|\int& \sum_{|k|\in I_{\ell_1}}\sum_{\tau_\ell\le2\la} b_{k}t^{\mu_1+\mu_2+1} e^{-t\tau_\ell^2}\overline{e_k^0(z)}V(z)e_{\tau_\ell}(z) \overline{e_{\tau_\ell}(x)}dz |\\
\ls & \|V\|_{L^{\frac{n}{2-\eta}}(M)} \|\sum_{|k|\in I_{\ell_1}} b_{k}e_k^0(\cdot)\|_{L^{p_0}(M)}\cdot2^{-2\ell_1(\mu_1+\mu_2+1)} \la^{1/2}\la^{\sigma(p_0)}( \sum_{\tau_\ell\in  [1, 2\la]}|e_{\tau_\ell}(x)|^2 )^{1/2}  \\
\ls &\la^{\frac n2+1-\frac14\eta}2^{(\frac n2-1-\frac14\eta-2\mu_2)\ell_1}2^{(2\mu_2+2-\eta)\ell_2}\\
&\quad \cdot\begin{cases}1,\ \ \ \ \ \ \ \ \ \ \ \ \ \  2^{\ell_1}\le 80\la\ {\rm and}\ 2^{\ell_2}\approx 10\la\\
2^{-N_1\ell_2},\ \ \ \ \ \ \  2^{\ell_1}\le 80\la\ {\rm and}\ 2^{\ell_2}>10\la	\\
2^{-N_1\ell_2},\ \ \ \ \ \ \  2^{\ell_1}>80\la\ {\rm and}\ 2^{\ell_2}>2^{\ell_1-2}\\
2^{(n-2+\eta)(\ell_2-\ell_1)},\ \ 2^{\ell_1}>80\la\ {\rm and}\ 2^{\ell_2}\approx 10\la\\
2^{(n-2+\eta)(\ell_2-\ell_1)}2^{-N_1\ell_2},\ \ 2^{\ell_1}>80\la\ {\rm and}\ 10\la< 2^{\ell_2}\le  2^{\ell_1-2}.
\end{cases}
\end{align*}

Fix $N_1\ge 2N+2n$. Integrating over $0\le t\le 2^{-2\ell_1}$ and summing over $2^{\ell_1},2^{\ell_2}\gs\la$, we obtain (<ref>).

To prove (<ref>), by  the heat kernel bounds in Lemma <ref>, we have

\begin{align}\label{ker03}\nonumber
|\sum_{|k|\in I_{\ell_1}}\sum_{\tau_\ell}\int_0^{2^{-2\ell_1}} &b_{k}\, t^{\mu_1+\mu_2+1}e^{-t\tau_\ell^2}\overline{e_k^0(z)}e_{\tau_\ell}(z) \overline{e_{\tau_\ell}(x)}dt|\\ \nonumber
&\ls  \int_0^{2^{-2\ell_1}}t^{\mu_1+\mu_2+1} t^{-\frac n2}e^{-cd_g(z, x)^2/t}dt\cdot \sum_{|k|\in I_{\ell_1}}|b_k|\\\nonumber
&\ls K(z, x)\cdot 2^{n\ell_1}\cdot \sup_{k}|b_k|\end{align}

$$K(z, x)=\begin{cases}
\log(2+(2^{\ell_1} d_g(z, x))^{-1})(1+2^{\ell_1} d_g(z, x))^{-N},\,\,\,\text{if}\,\,\,2\mu_1+2\mu_2+4=n \\
2^{-(2\mu_1+2\mu_2+4-n)\ell_1}(1+2^{\ell_1} d_g(z, x))^{-N}, \,\,\, \ \ \ \ \  \ \ \text{if}\,\,\, 2\mu_1+2\mu_2+4> n \\
d_g(z, x)^{2\mu_1+2\mu_2+4-n}(1+2^{\ell_1} d_g(z, x))^{-N}, \,\,\, \ \ \ \ \  \text{if}\,\,\, 2\mu_1+2\mu_2+4< n.
\end{cases}

It is straightforward to check that

$$\|K(\cdot, x)\|_{L^{\frac{n}{n-2+\eta}}}\ls 2^{-(2\mu_1+2\mu_2+2+\eta)\ell_1},$$

then by Hölder's inequality and (<ref>), we have

\begin{align*}
|\int \sum_{|k|\in I_{\ell_1}}\sum_{\tau_\ell}&\int_0^{2^{-2\ell_1}}b_{k}\, t^{\mu_1+\mu_2+1}e^{-t\tau_\ell^2}\overline{e_k^0(z)}V(z)e_{\tau_\ell}(z) \overline{e_{\tau_\ell}(x)}dtdz |\\
\ls &\|V\|_{L^{\frac{n}{2-\eta}}(M)}  2^{(n-2\mu_2-2-\eta)\ell_1}2^{(2\mu_2+2-\eta)\ell_2} \\
&\quad\cdot\begin{cases}1,\ \ \ \ \ \ \ \ \ \ \ \ \ \  2^{\ell_1}\le 80\la\ {\rm and}\ 2^{\ell_2}\approx 10\la\\
2^{-N_1\ell_2},\ \ \ \ \ \ 2^{\ell_1}\le 80\la\ {\rm and}\ 2^{\ell_2}>10\la	\\
2^{-N_1\ell_2},\ \ \ \ \ \ 2^{\ell_1}>80\la\ {\rm and}\ 2^{\ell_2}>2^{\ell_1-2}\\
2^{(n-2+\eta)(\ell_2-\ell_1)},\ \ 2^{\ell_1}>80\la\ {\rm and}\ 2^{\ell_2}\approx 10\la\\
2^{(n-2+\eta)(\ell_2-\ell_1)}2^{-N_1\ell_2},\ \ 2^{\ell_1}>80\la\ {\rm and}\ 10\la< 2^{\ell_2}\le  2^{\ell_1-2}.
\end{cases}
\end{align*}

Fix $N_1\ge 2N+2n$. Summing over  $2^{\ell_1},2^{\ell_2}\gs \la$, we obtain (<ref>).

§.§ Case 3.2.3b
We need to show

\begin{equation}\label{h.4.8}
|\sum_{j,k}\sum_{\tau_\ell> 2\la}\int h(|j|)|j|^{2\mu_2}|k|^{2\mu_1}c^1_{k,\tau_\ell}\,e_j^0(x)V_{jk}\overline{e_k^0(z)}V(z)e_{\tau_\ell}(z) \overline{e_{\tau_\ell}(x)}dz| \ls\la^{n-\frac32\eta}.
\end{equation}

In this case, we have $|j|\ge 0.8\tau_\ell$ or $|k|\ge 0.8\tau_\ell$ on the support of $1-\rho_2(|k|/\tau_\ell)\rho_2(|j|/\tau_\ell)$. Let $\tau_\ell\in[2^s,2^{s+1}]$, $2^s>2\la$,
and for $|j|\in I_{\ell_2}$, let

\begin{equation}\label{h.4.9}
b_{k,\tau_\ell}=\sum_{|j|\in I_{\ell_2}} h(|j|)|j|^{2\mu_2}|k|^{2\mu_1}e_j^0(x)V_{jk}(1-\rho_2(|k|/\tau_\ell)\rho_2(|j|/\tau_\ell)).
\end{equation}

Then for $|k|\in I_{\ell_1}$, we have

\begin{align}\label{h.4.10}|b_{k,\tau_\ell}|\ls  2^{2\mu_1\ell_1}2^{(2\mu_2+2-\eta)\ell_2}\cdot\begin{cases}1,\ \ \ \ \ \ \ \ \ \ \ \ \ \  2^{\ell_1}\le 80\la\ {\rm and}\ 2^{\ell_2}\approx 10\la\\
2^{-N_1\ell_2},\ \ \ \ \ \ \ 2^{\ell_1}\le 80\la\ {\rm and}\ 2^{\ell_2}>10\la	\\
2^{-N_1\ell_2},\ \ \ \ \ \ \ 2^{\ell_1}>80\la\ {\rm and}\ 2^{\ell_2}>2^{\ell_1-2}\\
2^{(n-2+\eta)(\ell_2-\ell_1)},\ \ 2^{\ell_1}>80\la\ {\rm and}\ 2^{\ell_2}\approx 10\la\\
2^{(n-2+\eta)(\ell_2-\ell_1)}2^{-N_1\ell_2},\ \ 2^{\ell_1}>80\la\ {\rm and}\ 10\la< 2^{\ell_2}\le  2^{\ell_1-2}.
\end{cases}\end{align}

and then by Sobolev inequality

\begin{align*}
\|\sum_{|k|\in I_{\ell_1}}& b_{k, 2^s}e_k^0(\cdot)\|_{L^{p_0}(M)}\ls 2^{\ell_1/2}2^{\sigma(p_0)\ell_1}	\|\sum_{|k|\in I_{\ell_1}} b_{k, 2^s}e_k^0(\cdot)\|_{L^{2}(M)} \\
&\ls  2^{(n/2+1-\frac14\eta)\ell_1}\cdot\sup_{k,\tau_\ell}|b_{k,\tau_\ell}|.
\end{align*}

And similarly we may obtain the estimates for $\partial_\tau b_{k,\tau}$.
By applying Lemma <ref> with $\delta=2^s$, we have for  fixed $0\le t\le 2^{-2\ell_1}$,

\begin{align*}
|\int \sum_{\tau_\ell\approx 2^s }&\sum_{|k|\in I_{\ell_1}} b_{k,\tau_\ell}t^{\mu_1+\mu_2+1}e^{-t\tau_\ell^2} \overline{e_k^0(z)}V(z)e_{\tau_\ell}(z) \overline{e_{\tau_\ell}(x)}dz |\\
&\ls \|V\|_{L^{\frac{n}{2-\eta}}(M)}2^{ns/2}2^{s/2}2^{(\frac12-\frac14\eta)s}\cdot 2^{(n/2-1-\frac14\eta-2\mu_2)\ell_1}2^{(2\mu_2+2-\eta)\ell_2} \\
&\quad\cdot\begin{cases}1,\ \ \ \ \ \ \ \ \ \ \ \ \ \  2^{\ell_1}\le 80\la\ {\rm and}\ 2^{\ell_2}\approx 10\la\\
2^{-N_1\ell_2},\ \ \ \ \ \ \ 2^{\ell_1}\le 80\la\ {\rm and}\ 2^{\ell_2}>10\la	\\
2^{-N_1\ell_2},\ \ \ \ \ \ \ 2^{\ell_1}>80\la\ {\rm and}\ 2^{\ell_2}>2^{\ell_1-2}\\
2^{(n-2+\eta)(\ell_2-\ell_1)},\ \ 2^{\ell_1}>80\la\ {\rm and}\ 2^{\ell_2}\approx 10\la\\
2^{(n-2+\eta)(\ell_2-\ell_1)}2^{-N_1\ell_2},\ \ 2^{\ell_1}>80\la\ {\rm and}\ 10\la< 2^{\ell_2}\le  2^{\ell_1-2}.
\end{cases}
\end{align*}

Fix $N_1\ge 2N+2n$. Note that $2^s\ls 2^{\ell_1}$ or $2^s\ls 2^{\ell_2}$. Integrating over $0\le t\le 2^{-2\ell_1}$ and summing over $s,\ell_1,\ell_2$, we obtain (<ref>).

§.§ Case 3.2.3c
We need to show

\begin{equation}\label{h.5.1}
|\sum_{j,k}\sum_{\tau_\ell> 2\la}\int h(|j|)|j|^{2\mu_1}|k|^{2\mu_2}c^2_{k,\tau_\ell}\,e_j^0(x)V_{jk}\overline{e_k^0(z)}V(z)e_{\tau_\ell}(z) \overline{e_{\tau_\ell}(x)}dz|\ls\la^{n-\frac32\eta}.
\end{equation}

In this case, we have $|j|\le0.9\tau_\ell$ and $|k|\le 0.9\tau_\ell$ on the support of $\rho_2(|k|/\tau_\ell)\rho_2(|j|/\tau_\ell)$.
Let $\tau_\ell\in[2^s,2^{s+1}]$, $2^s>2\la$,
and for $|j|\in I_{\ell_2}$, let

\begin{equation}\label{h.5.2}
b_{k,\tau_\ell}=\sum_{|j|\in I_{\ell_2}} h(|j|)|j|^{2\mu_1}|k|^{2\mu_2}e_j^0(x)V_{jk}\rho_2(|k|/\tau_\ell)\rho_2(|j|/\tau_\ell).
\end{equation}

Then for $|k|\in I_{\ell_1}$, we have

\begin{align}\label{h.5.3}|b_{k,\tau_\ell}|\ls  2^{2\mu_1\ell_1}2^{(2\mu_2+2-\eta)\ell_2}\cdot\begin{cases}1,\ \ \ \ \ \ \ \ \ \ \ \ \ \  2^{\ell_1}\le 80\la\ {\rm and}\ 2^{\ell_2}\approx 10\la\\
2^{-N_1\ell_2},\ \ \ \ \ \ \ 2^{\ell_1}\le 80\la\ {\rm and}\ 2^{\ell_2}>10\la	\\
2^{-N_1\ell_2},\ \ \ \ \ \ \ 2^{\ell_1}>80\la\ {\rm and}\ 2^{\ell_2}>2^{\ell_1-2}\\
2^{(n-2+\eta)(\ell_2-\ell_1)},\ \ 2^{\ell_1}>80\la\ {\rm and}\ 2^{\ell_2}\approx 10\la\\
2^{(n-2+\eta)(\ell_2-\ell_1)}2^{-N_1\ell_2},\ \ 2^{\ell_1}>80\la\ {\rm and}\ 10\la< 2^{\ell_2}\le  2^{\ell_1-2}
\end{cases}\end{align}

and then by Sobolev inequality

\begin{align*}
\|\sum_{|k|\in I_{\ell_1}}& b_{k, 2^s}e_k^0(\cdot)\|_{L^{p_0}(M)}\ls 2^{\ell_1/2}2^{\sigma(p_0)\ell_1}	\|\sum_{|k|\in I_{\ell_1}} b_{k, 2^s}e_k^0(\cdot)\|_{L^{2}(M)} \\
&\ls  2^{(n/2+1-\frac14\eta)\ell_1}\cdot\sup_{k,\tau_\ell}|b_{k,\tau_\ell}|.
\end{align*}

And similarly we may obtain the estimates for $\partial_\tau b_{k,\tau}$.
Recall that $|w(x)|\ls e^{-x/2}$. So for $\tau_\ell\approx 2^s$ we have

\[|w(\tau_\ell^22^{-2\ell_1})|\ls 2^{-N_2(s-\ell_1)},\ \forall N_2.\]
By applying Lemma <ref> with $\delta=2^s$, we have for  fixed $0\le t\le 2^{-2\ell_1}$,

\begin{align*}
|\int& \sum_{\tau_\ell\approx 2^s }\sum_{|k|\in I_{\ell_1}} b_{k,\tau_\ell}\frac{w(\tau_\ell^22^{-2\ell_1})}{2^{2\ell_1(\mu_1+\mu_2+2)}} \overline{e_k^0(z)}V(z)e_{\tau_\ell}(z) \overline{e_{\tau_\ell}(x)}dz |\\
&\ls  \|V\|_{L^{\frac{n}{2-\eta}}(M)}2^{ns/2}2^{s/2}2^{(\frac12-\frac14\eta)s}2^{-N_2(s-\ell_1)}\cdot2^{(\frac n2-1-\frac14\eta-2\mu_2)\ell_1}2^{(2\mu_2+2-\eta)\ell_2}\\
&\quad\cdot\begin{cases}1,\ \ \ \ \ \ \ \ \ \ \ \ \ \  2^{\ell_1}\le 80\la\ {\rm and}\ 2^{\ell_2}\approx 10\la\\
2^{-N_1\ell_2},\ \ \ \ \ \ \ 2^{\ell_1}\le 80\la\ {\rm and}\ 2^{\ell_2}>10\la	\\
2^{-N_1\ell_2},\ \ \ \ \ \ \ 2^{\ell_1}>80\la\ {\rm and}\ 2^{\ell_2}>2^{\ell_1-2}\\
2^{(n-2+\eta)(\ell_2-\ell_1)},\ \ 2^{\ell_1}>80\la\ {\rm and}\ 2^{\ell_2}\approx10\la\\
2^{(n-2+\eta)(\ell_2-\ell_1)}2^{-N_1\ell_2},\ \ 2^{\ell_1}>80\la\ {\rm and}\ 10\la< 2^{\ell_2}\le  2^{\ell_1-2}.
\end{cases}
\end{align*}

Fix $N_2\ge n$ and $N_1\ge N_2+2N+2n$. Note that $\la\ls 2^{\ell_1},2^{\ell_2}\ls 2^s$. Summing over $s,\ell_1,\ell_2$, we obtain (<ref>).

This completes the proof of Case 3.2.3.

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