# The probability that the product of two elements of finite group algebra is
zero

Haval M. Mohammed Salih∗

###### Abstract.

Let $\mathbb{F}_{q}G$ be a finite group algebra. We denote by
$P(\mathbb{F}_{q}G)$ the probability that the product of two elements of
$\mathbb{F}_{q}G$ be zero. In this paper, the general formula for computing
the $P(\mathbb{F}_{q}G)$ are established for the cyclic groups $C_{n}$, the
Quaternion group $Q_{8}$ and the symmetric group $S_{3}$, for some cases.

MSC(2010): Primary: 16S34; Secondary: 16U60.
Keywords: group algebra, probability, Wedderburn decomposition.

##  1\. Introduction

Throughout this paper, $\mathbb{F}_{q}$ denotes a finite field with $q$
elements where $q$ is a prime power of $p$. Let $G$ be a finite non trivial
group. Then $\mathbb{F}_{q}G$ is the group algebra of $G$ over the field
$\mathbb{F}_{q}$. The set of all invertible elements of $\mathbb{F}_{q}G$ form
a group called the unit group of $\mathbb{F}_{q}G$ denoted by
$\mathcal{U}(\mathbb{F}_{q}G)$. There are many known results about the group
unit of $\mathbb{F}_{q}G$ in [2, 3, 4]. The main topic of this study is to
find the nullity degree (probability) of $\mathbb{F}_{q}G$, which is the
probability that the multiplication of two randomly chosen elements of
$\mathbb{F}_{q}G$ is zero. That is

$P(\mathbb{F}_{q}G)=\frac{|\\{(a,b)\in\mathbb{F}_{q}G\times\mathbb{F}_{q}G\;:ab=0\\}|}{|\mathbb{F}_{q}G|^{2}}.$

where $a:=\sum_{g\in G}r_{g}g$ and $b:=\sum_{g\in G}\bar{r}_{g}g$
$(\bar{r}_{g},r_{g}\in\mathbb{F}_{q},g\in G)$. Since $\mathbb{F}_{q}G$ is a
finite group ring with identity, then we have that
$\mathbb{F}_{q}G=\\{0\\}\cup\mathcal{U}(\mathbb{F}_{q}G)\cup
ZD(\mathbb{F}_{q}G)$, where $ZD(\mathbb{F}_{q}G)$ is the set of nonzero zero
divisors of $\mathbb{F}_{q}G$. For $x\in\mathbb{F}_{q}G$, the left (right)
annihilator of $x$ is the set $Ann_{l}(x)=\\{\alpha\in\mathbb{F}_{q}G\;:\alpha
x=0\\}$ ($Ann_{r}(x)=\\{\alpha\in\mathbb{F}_{q}G\;:x\alpha=0\\}$). Similarly
we define the annihilator of $x$ by
$Ann(x)=\\{\alpha\in\mathbb{F}_{q}G\;:\alpha x=x\alpha=0\\}$. If $G$ is an
abelian group, then we can write $P(\mathbb{F}_{q}G)$ in term of the unit
group and zero divisor set as follows:

(1.1) $\displaystyle P(\mathbb{F}_{q}G)$
$\displaystyle=\frac{\sum_{x\in\mathbb{F}_{q}G}|Ann(x)|}{|\mathbb{F}_{q}G|^{2}}$
(1.2)
$\displaystyle=\frac{|\mathbb{F}_{q}G|+|\mathcal{U}(\mathbb{F}_{q}G)|+\sum_{0\neq
x\in ZD(\mathbb{F}_{q}G)}|Ann(x)|}{|\mathbb{F}_{q}G|^{2}},$

If $G$ is non abelian group, then

$P_{l}(\mathbb{F}_{q}G)=\frac{|\mathbb{F}_{q}G|+|\mathcal{U}(\mathbb{F}_{q}G)|+\sum_{0\neq
x\in ZD(\mathbb{F}_{q}G)}|Ann_{l}(x)|}{|\mathbb{F}_{q}G|^{2}}$

The probability of a finite commutative ring $R$ with identity can be found in
[1]. Recently in [5], Mohammed Salih derive the general formula for computing
$P(\mathbb{F}_{q}G)$ where $|G|<5$.

In this paper, $M_{n}(\mathbb{F}_{q})$ denotes the ring of all $n\times n$
matrix over $\mathbb{F}_{q}$, $R$ denotes a ring with identity. Also
$C_{n}=\langle a|a^{n}=1\rangle$ denotes the cyclic group of order $n$ and
$R_{1}\bigoplus R_{2}$ denotes direct sum of rings $R_{1}$ and $R_{2}$. The
group that we define below via its presentation is called the Quaternion group
of order 8.

$Q_{8}=\langle a,b|a^{4}=1,a^{2}=b^{2},bab^{-1}=a^{-1}\rangle.$

##  2\. Background

The following two results are well known [2]. These give us the structure of
the unit group of some group algebras $\mathbb{F}_{q}C_{n}$.

######  Theorem 2.1.

[2] If $gcd(n,p)=1$ and $q=p^{m}$, then $\mathcal{U}(\mathbb{F}_{q}C_{n})\cong
C_{q-1}\times(\displaystyle\prod_{l|n,l>1}C_{q^{d_{l}}-1}^{el})$ where $d_{l}$
is the multiplicative order of $q$ modulo $l$ and
$e_{l}=\frac{\varphi(l)}{d_{l}}$.

Now consider the case $p|n$.

######  Lemma 2.2.

[2] Let $k\in\mathbb{N}$. Then
$\mathcal{U}(\mathbb{F}_{p^{m}}C_{p^{k}})\cong\left\\{\begin{array}[]{ll}C_{p^{m}-1}\times
C_{p}^{m(p-1)}&ifk=1\\\
C_{p^{m}-1}\times\displaystyle\prod_{t=1}^{k}C_{p^{t}}^{n_{t}}&\text{
otherwise}\\\ \end{array}\right.$
where $n_{k}=m(p-1)$ and $n_{t}=mp^{k-t-1}(p-1)^{2}$, for all $t,1\leq t<k$.

The next three results tell us the group algebra $KG$ has Wedderburn
decomposition of the form.

######  Proposition 2.3.

[6] Let $G$ be an abelian group of order $n$ and $K$ a field such that the
characteristic of the field does not divide $n$. If $K$ contains a primitive
root of unity of order $n$, then
$KG\cong\underset{\text{n-times}}{K\bigoplus\dots\bigoplus K}$.

######  Proposition 2.4.

[6] Let $F$ be a finite field where the characteristic of the field is not
equal to 2. Then $FQ_{8}=F\bigoplus F\bigoplus F\bigoplus F\bigoplus M_{2}(F)$
if and only if $x^{2}+y^{2}=-1$ can be solved in $F$.

######  Proposition 2.5.

[8] Let $\mathbb{F}_{q}$ be a finite field and $gcd(q,6)=1$. Then
$\mathbb{F}_{q}S_{3}\cong\mathbb{F}_{q}\bigoplus\mathbb{F}_{q}\bigoplus
M_{2}(\mathbb{F}_{q})$.

######  Theorem 2.6.

[5] Let $RG$ be a finite group ring. Then $P(RG)\geq\frac{1}{4}$ if and only
if $RG$ is isomorphic to one of the following rings:
$\mathbb{Z}_{2}C_{2},\mathbb{Z}_{3}C_{2},\mathbb{Z}_{2}C_{3}$.

Initially, it was challenging for us to derive a computing formula for
$P(\mathbb{F}_{q}G)$, since there is no analogous concept of it. Finally, we
are able to get the formula for $P(\mathbb{F}_{q}G)$, where $|G|<5$ in [5]. In
this paper, it may be pointed out that for most of these group algebras, the
GAP program fails to compute the $P(\mathbb{F}_{q}G)$ if $|G|\geq 6$, then the
program may crash, even by using super computer. So we need to solve this
problem by using algebraic tools. Further, we provide the explicit formula of
$P(\mathbb{F}_{q}C_{n})$, if $gcd(n,p)=1$. Also we find the
$P(\mathbb{F}_{q}G)$ where $G=S_{3}$ or $Q_{8}$ for some cases by using the
Wedderburn decomposition.

##  3\. The general formula of the probability of some group algebras
$\mathbb{F}_{q}G$

In this section, we get serveral results for $P(\mathbb{F}_{q}G)$ and derive a
computing formula for $P(\mathbb{F}_{q}G)$. We begin with the following
result:

######  Lemma 3.1.

Let $R_{1},...,R_{n}$ be finite rings. Then $P(R_{1}\bigoplus
R_{2}\bigoplus...\bigoplus R_{n})=P(R_{1})P(R_{2})\dots P(R_{n})$.

###### Proof.

It is straightforward. ∎

######  Theorem 3.2.

If $gcd(n,p)=1$ and $q=p^{m}$, then
$P(\mathbb{F}_{q}C_{n})=(\frac{2q-1}{q^{2}})\times\prod_{l|n\;l>1}(\frac{2q^{d_{l}}-1}{q^{2d_{l}}})^{e_{l}}$
where $d_{l}$ is the multiplicative order of $q$ module $l$ and
$e_{l}=\frac{\phi(l)}{d_{l}}$, $\phi$ denotes the Euler Totient function.

###### Proof.

Using [10, Theorem 1] and [9, Theorem 2.21, pp.53], it follows that
$\mathbb{F}_{q}C_{n}\cong\mathbb{F}_{q}\bigoplus\;(\bigoplus_{l|n\;l>1}\mathbb{F}_{q^{d_{l}}}^{e_{l}})$.
By Lemma 3.1 and [1, Lemma 3.1], the result follows. ∎

Note that, we are going to use Perlis-Walker Theorem in order to determine how
many elements has annihilator of size $q^{l}$.

######  Theorem 3.3.

Let $\mathbb{F}_{q}$ be a field of characteristic $p$, where $q=p^{m}$. Then
$P(\mathbb{F}_{q}C_{5})\cong\left\\{\begin{array}[]{ll}\frac{q^{7}-q^{6}+q^{5}+q^{4}-q^{2}+q-1}{q^{9}}&if\;p|5.\\\
\frac{4q^{5}-2q^{4}-2q+1}{q^{10}}&q\equiv 2\text{ or }3\mod 5\text{ and
}if\;gcd(p,5)=1.\\\ \frac{q^{6}-2q^{3}+5q^{2}-2q-1}{q^{8}}&q\equiv 4\mod
5\text{ and }if\;gcd(p,5)=1.\\\
\frac{2q^{6}+5q^{5}-21q^{4}+35q^{3}-29q^{2}+10q-1}{q^{10}}&q\equiv 1\mod
5\text{ and }if\;gcd(p,5)=1.\\\ \end{array}\right.$

###### Proof.

Let $v\in ZD(\mathbb{F}_{q}C_{5})\setminus\\{0\\}$ and
$\psi\colon\mathbb{F}_{q}C_{5}\rightarrow\mathbb{F}_{q}C_{5}$ the
$\mathbb{F}_{q}$-linear transformation defined $\psi(x)=vx$, then
$Ann(x)=Ker\psi$ that a non trivial $\mathbb{F}_{q}$-linear subspace of
dimension $1\leq l\leq 4$. That is $|Ann(x)|=q,q^{2},q^{3}$ or $q^{4}$. We
have four cases to analysis:

  * Case 1,

If $q\cong 0\mod 5$, then there are $q-1,q(q-1),q^{2}(q-1),q^{3}(q-1)$
elements $0\neq x\in ZD(\mathbb{F}_{q}C_{5})$ such that
$|Ann(x)|=q,q^{2},q^{3},q^{4}$. By Lemma 2.2, we obtain
$|\mathcal{U}(\mathbb{F}_{q}C_{5})|=q^{4}(q-1)$.

  * Case 2,

if $q\cong 2\mod 5$ or $q\cong 3\mod 5$, there are $q^{4}-1,0,0,(q-1)$
elements $0\neq x\in ZD(\mathbb{F}_{q}C_{5})$ such that
$|Ann(x)|=q,q^{2},q^{3},q^{4}$. Also, by Theorem 2.1, we obtain
$|\mathcal{U}(\mathbb{F}_{q}C_{5})|=(q^{4}-1)(q-1)$.

  * Case 3,

if $q\cong 4\mod 5$, then there are
$q-1,2(q^{2}-1),2(q^{2}-1)(q-1),(q^{2}-1)^{2}$ elements $0\neq x\in
ZD(\mathbb{F}_{q}C_{5})$ such that $|Ann(x)|=q,q^{2},q^{3},q^{4}$. So
$|\mathcal{U}(\mathbb{F}_{q}C_{5})|=(q^{2}-1)^{2}q$ by Theorem 2.1.

  * Case 4,

if $q\cong 1\mod 5$, then there are $5(q-1)^{4},(q-1)^{4},(q-1)^{3},5(q-1)$
elements $0\neq x\in ZD(\mathbb{F}_{q}C_{5})$ such that
$|Ann(x)|=q,q^{2},q^{3},q^{4}$. So
$|\mathcal{U}(\mathbb{F}_{q}C_{5})|=(q-1)^{5}$ by Theorem 2.1.

The rest follows from Equation (1.2). ∎

######  Remark 3.4.

We can see that the proof of the cases 2,3 and 4 in Theorem 3.3 follows from
Theorem 3.2.

######  Lemma 3.5.

Let $RG$ be a finite group algebra with identity such that
$P(RG)\geq\frac{1}{10}$. Then the structure of $RG$ and the possible values of
$P(RG)$ are given in Table 1.

###### Proof.

The proof follows from direct computations and [5, Theorems 3.1, 3.2, 3.3 and
3.4]. ∎

A direct consequence of Lemma 3.5 is the following result.

######  Corollary 3.6.

Let $RG$ be a finite group algebras with identity. Then the following holds:

  1. (1)

$P(RG)\notin(\frac{21}{64},\frac{1}{4})$.

  2. (2)

If $RG\ncong\mathbb{F}_{2}C_{2},\mathbb{F}_{2}C_{3}$, then
$P(RG)\leq\frac{21}{64}$. Moreover, the equality holds if and only if
$RG\cong\mathbb{F}_{2}C_{3}$.

  3. (3)

If $RG\ncong\mathbb{F}_{3}C_{2},\mathbb{F}_{2}C_{2},\mathbb{F}_{2}C_{3}$, then
$P(RG)<\frac{2}{10}$.

  4. (4)

$P(RG)=\frac{7}{32}$ if and only if $RG\cong\mathbb{F}_{2}(C_{2}\times
C_{2}),\mathbb{F}_{5}C_{2}$.

Table 1. All group algebras with $P(RG)\geq 0.1$ $n$ | $RG$ | $P(RG)$
---|---|---
1 | $\mathbb{F}_{2}C_{2}$ | $\frac{1}{2}=0.5$
2 | $\mathbb{F}_{3}C_{2}$ | $\frac{25}{81}\simeq 0.308$
3 | $\mathbb{F}_{5}C_{2}$ | $\frac{81}{625}\simeq 0.129$
4 | $\mathbb{F}_{2}C_{3}$ | $\frac{21}{64}\simeq 0.328$
5 | $\mathbb{F}_{2}C_{4}$ | $\frac{3}{36}\simeq 0.18$
6 | $\mathbb{F}_{3}C_{3}$ | $\frac{1}{9}\simeq 0.111$
7 | $\mathbb{F}_{4}C_{2}$ | $\frac{5}{32}\simeq 0.156$
8 | $\mathbb{F}_{2}(C_{2}\times C_{2})$ | $\frac{7}{32}\simeq 0.218$
9 | $\mathbb{Z}_{4}C_{2}$ | $\frac{7}{32}\simeq 0.218$
10 | $\mathbb{Z}_{6}C_{2}$ | $\frac{25}{162}\simeq 0.154$
11 | $\mathbb{F}_{2}S_{3}$ | $\frac{5}{64}\simeq 0.113$

######  Proposition 3.7.

If $R=M_{2}(\mathbb{F}_{q})$ is the matrix ring algebra over $\mathbb{F}_{q}$
with two generators where $q=p^{t}$, then
$P_{l}(R)=\frac{q^{4}+3q^{3}-2q(q+1)+1}{q^{7}}$.

###### Proof.

By the same idea as in proof of Theorem 3.3, we have $|Ann_{l}(x)|=q^{2}$ for
all $0\neq x\in ZD(R)$. By using [7, Theorem 3.1], we obtain
$\mathcal{U}=q(q^{3}-q^{2}-q+1)$. Since $R$ is finite ring with identity, then
$R=\\{0\\}\cup\mathcal{U}\cup ZD(R)$. So there are $q^{3}+q^{2}-q-1$ elements
$0\neq x\in ZD(R)$ of size $|Ann(x)|=q^{2}$. So $\sum_{0\neq x\in
ZD(R)}|Ann_{l}(x)|=q^{2}(q^{3}+q^{2}-q-1)$. Put these equations in Equation
(1.2) and the result follows. ∎

######  Proposition 3.8.

If $R=M_{2}(\mathbb{F}_{q})$ is the matrix ring algebra over $\mathbb{F}_{q}$
with two generators where $q=p^{t}$, then $P(R)=\frac{3q^{2}-2}{q^{6}}$.

###### Proof.

By the same idea as in proof of Theorem 3.3, we have $|Ann_{l}(x)|=q$ for all
$0\neq x\in ZD(R)$. The rest is similar as the proof of Proposition 3.7. ∎

######  Theorem 3.9.

Let $\mathbb{F}_{q}$ be a finite field of characteristic $p\neq 2$. Then

  1. (1)

$P_{l}(\mathbb{F}_{q}Q_{8})=(\frac{2q-1}{q^{2}})^{4}\times(\frac{q^{4}+3q^{3}-2q^{2}-2q+1}{q^{7}})$.

  2. (2)

$P(\mathbb{F}_{q}Q_{8})=(\frac{2q-1}{q^{2}})^{4}\times(\frac{3q^{2}-2}{q^{6}})$.

###### Proof.

Since $\mathbb{F}_{q}$ is a finite field of characteristic $p\neq 2$. Then by
Proposition 2.4, we have
$\mathbb{F}_{q}Q_{8}=\mathbb{F}_{q}\bigoplus\mathbb{F}_{q}\bigoplus\mathbb{F}_{q}\bigoplus\mathbb{F}_{q}\bigoplus
M_{2}(\mathbb{F}_{q})$.

  1. (1)

From Lemma 3.1, [1, Lemma 3.1] and Proposition 3.7, the result follows.

  2. (2)

From Lemma 3.1, [1, Lemma 3.1] and Proposition 3.8, the result follows.

∎

######  Theorem 3.10.

Let $\mathbb{F}_{q}$ be a finite field and $gcd(q,6)=1$. Then

  1. (1)

$P_{l}(\mathbb{F}_{q}S_{3})=(\frac{2q-1}{q^{2}})^{2}\times(\frac{q^{4}+3q^{3}-2q^{2}-2q+1}{q^{7}})$.

  2. (2)

$P(\mathbb{F}_{q}S_{3})=(\frac{2q-1}{q^{2}})^{2}\times(\frac{3q^{2}-2}{q^{6}})$.

###### Proof.

The proof is similar as Theorem 3.9. ∎

######  Proposition 3.11.

Let $\mathbb{F}_{q}$ be a finite field of characteristic 2. Then

  1. (1)

$P_{l}(\mathbb{F}_{q}S_{3})=\frac{3q^{5}+7q^{4}-12q^{3}-2q^{2}+7q-2}{q^{10}}$.

  2. (2)

$P(\mathbb{F}_{q}S_{3})=\frac{9q^{3}-6q^{2}-6q+4}{q^{9}}$.

  3. (3)

$P(\mathbb{F}_{q}Q_{8})=\frac{3q^{2}+3q-5}{q^{9}}$.

###### Proof.

Clear. ∎

######  Example 3.12.

Consider the finite field $\mathbb{F}_{7}$, the symmetric group $S_{3}$ and
the cyclic group $C_{6}$. The probability $P_{l}(\mathbb{F}_{7}S_{3})$ and
$P(\mathbb{F}_{7}C_{6})$ cannot compute by personal computer. So we use super
computer with property Intel®Xeon®Platinum 8280 CPU@ 2.70GHz 2.69GHz Installed
RAM 128GB (128GB usable). The computation takes nearly 129 hours (see below)
and the outputs are the following:

    [ rec(Size := [ 72576, 24192, 15840, 4608, 420, 12, 1 ],
        |ann_l|:=[ 1, 7, 49, 343, 2401, 16807, 117649 ], group := "S3", p := 560911/1977326743),
    rec(Size:= [ 46656, 46656, 19440, 4320, 540, 36, 1 ],
       |ann|:=[ 1, 7, 49, 343, 2401, 16807, 117649 ], group := "C6", p := 4826809/13841287201)]
    gap> time;
    463879313
    StringTime(463879313)
    "128:51:19.313"

On the other hand, we achieve this computation with a few seconds by using
Theorem 3.10, part 1 and Theorem 3.2.

Acknowledgments

I would like thank to the referee for careful reading of the article and
detailed report including corrections and comments; and I appreciate his/her
effort on reviewing the article.

## References

  * [1] M. A. Esmkhani and S. M. Jafarian Amiri, The probability that the multiplication of two ring elements is zero, J. Algebra and Its Applications, 17, no. 3, (2018) 1850054.
  * [2] N. Makhijani, R. K. Sharma and J. B. Srivastava, The unit group of algebra of circulant matrices, Internat. J. Group Theory, 3, no. 4, (2014) 13-16.
  * [3] J. Gildea, On the order of $U(F_{p^{k}}D_{2p^{m}})$, Int. J. Pure Appl. Math., 46, no. 2, (2008) 267-272.
  * [4] G. Tang and Y. Gao, The unit groups of FG of groups with order 12, Internat. J. Pure and Applied Mathematics, 73, no. 2, (2011) 143-158.
  * [5] H. M. Mohammed Salih, On the probability of zero divisor elements in group rings International Journal of Group Theory 11, no. 4 (2022): 253-257.
  * [6] C. P. Milies and S. K. Sehgal, An introduction to group rings, Springer Science and Business Media, 1, Kluwer Academic publishers Dordrecht/Boston/London, 2002.
  * [7] H. Cheraghpour and N. G. Ghosseiri. On the idempotents, nilpotents, units and zero-divisors of finite rings. Linear and Multilinear Algebra, 67, no. 2, (2019) 327-36.
  * [8] F. E. Brochero Martinez, On the idempotents, Structure of finite dihedral group algebra. Finite Fields and Their Applications, 35, (2015) 204-214.
  * [9] R. Lidl and H. Niederreiter, Finite Fields, Cambridge University Press, 1997
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Haval M. Mohammed Salih
Department of Mathematics, Faculty of Science, Soran University , Kawa St,
Soran, Erbil, Iraq
Email: havalmahmood07gmail.com


#  $\ell^{2}$ decoupling for certain surfaces of finite type in
$\mathbb{R}^{3}$

Zhuoran Li  The Graduate School of China Academy of Engineering Physics, P. O.
Box 2101, Beijing, China, 100088<EMAIL_ADDRESS>and  Jiqiang
Zheng  Institute of Applied Physics and Computational Mathematics, Beijing
100088<EMAIL_ADDRESS>
###### Abstract.

In this article, we establish an $\ell^{2}$ decoupling inequality for the
surface

$F_{4}^{2}:=\Big{\\{}(\xi_{1},\xi_{2},\xi_{1}^{4}+\xi_{2}^{4}):(\xi_{1},\xi_{2})\in[0,1]^{2}\Big{\\}}$

associated with the decomposition adapted to finite type geometry from our
previous work [11]. The key ingredients of the proof include the so-called
generalized rescaling technique, an $\ell^{2}$ decoupling inequality for the
surfaces

$\Big{\\{}(\xi_{1},\xi_{2},\phi_{1}(\xi_{1})+\xi_{2}^{4}):(\xi_{1},\xi_{2})\in[0,1]^{2}\Big{\\}}$

with $\phi_{1}$ being non-degenerate, reduction of dimension arguments and
induction on scales.

Key Words: decoupling inequality; finite type; reduction of dimension
arguments; induction on scales.
AMS Classification: 42B10

##  1\. Introduction and main result

Decoupling inequalities was introduced by Wolff [12] in connection with the
local smoothing estimates for the solution to the wave equation. Bourgain made
important progress in developing the decoupling theory for hypersurfaces in
[3] via the multilinear restriction theory of Bennett, Carbery and Tao [1],
and applied decoupling inequalities to the study of exponential sums. In 2015,
Bourgain and Demeter [4] proved the sharp $\ell^{2}$ decoupling inequality for
compact hypersurfaces with positive definite second fundamental form, which
has a wide range of important consequences. One of them is the validity of the
discrete restriction conjecture, which implies the full range of expected
$L^{p}_{t,x}$ Strichartz estimates for the Schrödinger equation on both the
rational and irrational torus. The second striking application is an
improvement on the local smoothing conjecture for the wave equation. In
particular, inspired by Bourgain-Demeter’s work [4], Guth, Wang and Zhang [10]
established a sharp reverse square function estimate for the cone in
$\mathbb{R}^{3}$, and solved the local smoothing conjecture for the wave
equation in $\mathbb{R}^{2+1}$. Thirdly, by establishing the sharp decoupling
inequality for the moment curve, Bourgain, Demeter and Guth [6] proved the
main conjecture in Vinogradov’s mean value theorem for degrees higher than
three, which is a remarkable application of decoupling methods in number
theory.

Recently, the decoupling theory for non-degenerate hypersurfaces has been well
understood. However, the decoupling theory for degenerate hypersurfaces such
as hypersurfaces of finite type is interesting but still unknown. In this
paper, we will study the $\ell^{2}$ decoupling problem for certain surfaces of
finite type in $\mathbb{R}^{3}$. More precisely, we consider the surface in
$\mathbb{R}^{3}$ given by

$F^{2}_{m}:=\Big{\\{}(\xi_{1},\xi_{2},\xi_{1}^{m}+\xi_{2}^{m}):(\xi_{1},\xi_{2})\in[0,1]^{2}\Big{\\}},\;m\geq
2.$

For any function $g\in L^{1}([0,1]^{2})$ and each subset $Q\subset[0,1]^{2}$,
we denote the corresponding Fourier extension operator by

$\mathcal{E}_{Q}g(x):=\int_{Q}g(\xi_{1},\xi_{2})e\big{(}x_{1}\xi_{1}+x_{2}\xi_{2}+x_{3}(\xi^{m}_{1}+\xi_{2}^{m})\big{)}\;d\xi_{1}\;d\xi_{2},$

where $e(t)=e^{2\pi it}$ for $t\in\mathbb{R}$, and
$x=(x_{1},x_{2},x_{3})\in\mathbb{R}^{3}.$

For $m=2$, the surface $F^{2}_{2}$ is exactly a paraboloid over the region
$[0,1]^{2}$. By making use of Bourgain-Guth methods in [7], parabolic
rescaling and induction on scales, Bourgain and Demeter [4] have established
the sharp $\ell^{2}$ decoupling inequality for the paraboloid in
$\mathbb{R}^{n}$ as follows.

######  Theorem 1.1.

For $2\leq p\leq\frac{2(n+1)}{n-1}$ and all $\varepsilon>0$, there exists a
constant $C(\varepsilon,p)$ such that

$\|\mathcal{E}^{Par}_{[0,1]^{n-1}}g\|_{L^{p}(B^{n}_{R})}\leq
C(\varepsilon,p)R^{\varepsilon}\Big{(}\sum_{\theta:\;R^{-1/2}-\text{cubes\;
in\;}[0,1]^{n-1}}\|\mathcal{E}^{Par}_{\theta}g\|^{2}_{L^{p}(w_{B^{n}_{R}})}\Big{)}^{1/2},$
(1.1)

where $\mathcal{E}^{Par}_{[0,1]^{n-1}}$ denotes the Fourier extension operator
associated with the paraboloid $P^{n-1}$ and
$w_{B^{n}_{R}}(x)=\big{(}1+\frac{|x-x_{0}|}{R}\big{)}^{-100n}$ denotes the
standard weight function adapted to the ball $B^{n}_{R}$ in $\mathbb{R}^{n}$
centered at $x_{0}$ with radius $R$.

We will also use the following equivalent version of (1.1)

$\|F\|_{L^{p}(\mathbb{R}^{n})}\leq
C(\varepsilon,p)R^{\varepsilon}\Big{(}\sum_{\bar{\theta}}\|F_{\bar{\theta}}\|^{2}_{L^{p}(\mathbb{R}^{n})}\Big{)}^{1/2},\;\;2\leq
p\leq\frac{2(n+1)}{n-1},$ (1.2)

where ${\rm supp}\;\hat{F}\subset\mathcal{N}_{\frac{1}{R}}(P^{n-1})$, the
$\bar{\theta}$’s are finitely overlapping $R^{-1/2}$-slabs in $\mathbb{R}^{n}$
which cover $\mathcal{N}_{\frac{1}{R}}(P^{n-1})$ (the
$\tfrac{1}{R}-$neighborhood of $P^{n-1}$) and
$F_{\bar{\theta}}:=\mathcal{F}^{-1}\big{(}\hat{F}\chi_{\bar{\theta}}\big{)}$.
For the equivalence of (1.1) and (1.2), we refer to [5].

For $m>2$, the Gaussian curvature of the surface $F_{m}^{2}$ vanishes when
$\xi_{1}=0$ or $\xi_{2}=0$. From now on, we focus on the case $m=4$, since the
cases for even numbers larger than four can be deduced by the same argument.
It is surprised for us that such a degenerate surface might possess the same
range of exponent as in the $\ell^{2}$ decoupling inequality for the
paraboloid. For this purpose, we adopt the notations as in the previous work
[11]. At first, given $R\gg 1$, we divide $[0,1]$ into

$[0,1]=\bigcup_{k}I_{k},$

where $I_{0}=[0,R^{-\frac{1}{4}}]$ and

$I_{k}=[2^{k-1}R^{-\frac{1}{4}},2^{k}R^{-\frac{1}{4}}],\;\;\text{for}\;\;1\leq
k\leq\big{[}\tfrac{1}{4}\log_{2}R\big{]}.$

For each $k\geq 1$, we divide $I_{k}$ further into

$I_{k}=\bigcup_{\mu=1}^{2^{2(k-1)}}I_{k,\mu},$

with

$I_{k,\mu}=\big{[}2^{k-1}{R^{-\frac{1}{4}}}+(\mu-1)2^{-(k-1)}R^{-\frac{1}{4}},2^{k-1}R^{-\frac{1}{4}}+\mu
2^{-(k-1)}R^{-\frac{1}{4}}\big{]}.$

Thus, we have the following decomposition

$[0,1]^{2}=\bigcup_{\theta\in\mathcal{F}_{3}(R,4)}\theta,$ (1.3)

where

$\displaystyle\mathcal{F}_{3}(R,4):=\big{\\{}$ $\displaystyle
I_{k_{1},\mu_{1}}\times I_{k_{2},\mu_{2}},\;I_{0}\times
I_{k_{2},\mu_{2}},\;I_{k_{1},\mu_{1}}\times I_{0},\;I_{0}\times I_{0}:$
$\displaystyle\quad 1\leq
k_{j}\leq\big{[}\tfrac{1}{4}\log_{2}R\big{]},\;1\leq\mu_{j}\leq
2^{2(k-1)},\;j=1,2\big{\\}}.$

We refer to [11] for the detailed decomposition (1.3). Buschenhenke [8]
utilized the analogous decomposition to study the restriction estimates for
certain conic surfaces of finite type.

Our main result is the following decoupling inequality associated with the
surface $F^{2}_{4}$ based on the decomposition (1.3).

######  Theorem 1.2.

For $2\leq p\leq 4$ and any $\varepsilon>0$, there exists a constant
$C(\varepsilon,p)$ such that

$\|\mathcal{E}_{[0,1]^{2}}g\|_{L^{p}(B_{R})}\leq
C(\varepsilon,p)R^{\varepsilon}\Big{(}\sum_{\theta\in\mathcal{F}_{3}(R,4)}\|\mathcal{E}_{\theta}g\|^{2}_{L^{p}(w_{B_{R}})}\Big{)}^{1/2},$
(1.4)

with $w_{B_{R}}(x)$ being the standard weight function as in Theorem 1.1.

######  Remark 1.3.

Biswas, Gilula, Li, Schwend and Xi [2] studied the $\ell^{2}$ decoupling
problem for curves of finite type in the plane. Their decoupling inequality is
based on a traditional decomposition as in the case of the parabola, which
seems ineffectively in higher dimensional cases because of failure of standard
parabolic rescaling. Recently, Yang [13] proved an uniform $\ell^{2}$
decoupling inequality for polynomials in $\mathbb{R}^{2}$. We refer the
readers to Section 2.2 in [11] for the details on the decomposition adapted to
finite type surfaces.

The paper is organized as follows. In Section 2, we give the outline of the
proof of Theorem 1.2. In Section 3, we establish an $\ell^{2}$ decoupling
inequality for the surfaces

$\Big{\\{}(\xi_{1},\xi_{2},\phi_{1}(\xi_{1})+\xi_{2}^{4}):(\xi_{1},\xi_{2})\in[0,1]^{2}\Big{\\}}$

with $\phi_{1}$ being non-degenerate as a key lemma. With this in hand, we
utilize Bourgain-Demeter’s decoupling inequality for the perturbed paraboloid,
reduction of dimension arguments and induction on scales to prove Theorem 1.2.

Notations: For nonnegative quantities $X$ and $Y$, we will write $X\lesssim Y$
to denote the estimate $X\leq CY$ for some $C>0$. If $X\lesssim Y\lesssim X$,
we simply write $X\sim Y$. Dependence of implicit constants on the power $p$
or the dimension will be suppressed; dependence on additional parameters will
be indicated by subscripts. For example, $X\lesssim_{u}Y$ indicates $X\leq CY$
for some $C=C(u)$. We denote $e(t)=e^{2\pi it}$. We denote $[x]$ to be the
greatest integer not large than $x$. We use $B(x_{0},r)$ to denote an
arbitrary ball centred at $x_{0}$ with radius $r$ in $\mathbb{R}^{3}$ and
abbreviate it by $B_{r}$ in the context. For any region
$\Omega\subset\mathbb{R}^{n}$, we denote the characteristic function on
$\Omega$ by $\chi_{\Omega}$. In $\mathbb{R}^{n}$, we denote
$R^{-1/2}\times\cdot\cdot\cdot\times R^{-1/2}\times R^{-1}$-rectangle to be an
$R^{-1/2}$-slab in $\mathbb{R}^{n}$. Define the Fourier transform on
$\mathbb{R}^{n}$ by

$\displaystyle\widehat{f}(\xi):=\int_{\mathbb{R}^{n}}e^{-2\pi
ix\cdot\xi}f(x)\,dx,$

and the inverse Fourier transform by

$\displaystyle\mathcal{F}^{-1}{f}(x):=\int_{\mathbb{R}^{n}}e^{2\pi
ix\cdot\xi}f(\xi)\,d\xi.$

##  2\. Outline of the proof of Theorem 1.2

Now, we give the sketch of the proof of Theorem 1.2. By a direct computation,
the Gaussian curvature of the surface $F^{2}_{4}$ has

$k=\frac{144\xi^{2}_{1}\xi^{2}_{2}}{(1+16\xi^{6}_{1}+16\xi^{6}_{2})^{2}}.$
(2.1)

We observe that the surface $F^{2}_{4}$ has positive definite second
fundamental form if both $\xi_{1}$ and $\xi_{2}$ are away from zero. In this
region, one can adopt Bourgain-Demeter’s decoupling inequality for the
perturbed paraboloid. However, when $\xi_{1}=0$ or $\xi_{2}=0$, the Gaussian
curvature vanishes. Our strategy is to divide $[0,1]^{2}$ into
$[0,1]^{2}=\bigcup\limits_{j=0}^{3}\Omega_{j}$ as in Fig.1,

$\xi_{1}$$\xi_{2}$O$K^{-\frac{1}{4}}$$1$$\Omega_{0}$$\Omega_{1}$$\Omega_{2}$$\Omega_{3}$$K^{-\frac{1}{4}}$$1$Fig.
1

where

$\displaystyle\Omega_{0}:=[K^{-\frac{1}{4}},1]\times[K^{-\frac{1}{4}},1],\quad\Omega_{1}:=[K^{-\frac{1}{4}},1]\times[0,K^{-\frac{1}{4}}],$
$\displaystyle\Omega_{2}:=[0,K^{-\frac{1}{4}}]\times[K^{-\frac{1}{4}},1],\quad\Omega_{3}:=[0,K^{-\frac{1}{4}}]\times[0,K^{-\frac{1}{4}}].$

In the region $\Omega_{3}$, we will use the rescaling technique. While for the
regions $\Omega_{1}$ and $\Omega_{2}$, we reduce them to a lower dimensional
decoupling problem.

For technical reasons, $K^{-1/4}$, $R^{-1/4}$ and
$\big{(}\tfrac{R}{K}\big{)}^{-1/4}$ should be dyadic numbers satisfying $1\ll
K\ll R^{\varepsilon}$ for any fixed $\varepsilon>0$. Therefore, we choose
$K=2^{4s}$ and $R=2^{4l}\;(s,l\in\mathbb{N})$ to be large numbers satisfying
$K\approx\log R$. By the Minkowski inequality and Cauchy-Schwartz inequality,
we have

$\displaystyle\|\mathcal{E}_{[0,1]^{2}}g\|_{L^{p}(B_{R})}\leq$ $\displaystyle
2\Big{(}\sum_{j=0}^{3}\|\mathcal{E}_{\Omega_{j}}g\|^{2}_{L^{p}(B_{R})}\Big{)}^{1/2}.$
(2.2)

Let $\mathcal{D}_{p}(R)$ denote the least number such that

$\|\mathcal{E}_{[0,1]^{2}}g\|_{L^{p}(B_{R})}\leq\mathcal{D}_{p}(R)\Big{(}\sum_{\theta\in\mathcal{F}_{3}(R,4)}\|\mathcal{E}_{\theta}g\|^{2}_{L^{p}(w_{B_{R}})}\Big{)}^{1/2}.$
(2.3)

Case 1: The contribution comes from $\Omega_{0}$. Roughly speaking, the
strategy is to transform each small piece to a non-degenerate one. Then we
employ Bourgain-Demeter’s decoupling inequality for each non-degenerate piece.
We first divide $\Omega_{0}$ into $\Omega_{0}=\bigcup\Omega_{\lambda,\sigma}$
with

$\Omega_{\lambda,\sigma}=\big{\\{}(\xi_{1},\xi_{2}):\;\lambda\leq\xi_{1}\leq
2\lambda,\;\;\sigma\leq\xi_{2}\leq 2\sigma\big{\\}},$ (2.4)

and $\lambda,\sigma\in[K^{-\frac{1}{4}},\frac{1}{2}]$ are both dyadic numbers.
We know that the Gaussian curvature of $F^{2}_{4}$ is essentially a constant
in the region $\Omega_{\lambda,\sigma}.$ It is reasonable to divide
$\Omega_{\lambda,\sigma}$ into
$\lambda^{2}K^{\frac{1}{2}}\times\sigma^{2}K^{\frac{1}{2}}$ pieces of
dimension $\lambda^{-1}K^{-\frac{1}{2}}\times\sigma^{-1}K^{-\frac{1}{2}}$
equally

$\displaystyle\Omega_{\lambda,\sigma}=$ $\displaystyle\bigcup_{1\leq
j\leq\lambda^{2}K^{\frac{1}{2}}\atop 1\leq
m\leq\sigma^{2}K^{\frac{1}{2}}}\Big{[}\lambda+\frac{j-1}{\lambda
K^{1/2}},\lambda+\frac{j}{\lambda
K^{1/2}}\Big{]}\times\Big{[}\sigma+\frac{m-1}{\sigma
K^{1/2}},\sigma+\frac{m}{\sigma K^{1/2}}\Big{]}$ $\displaystyle=:$
$\displaystyle\bigcup\tau_{\lambda,\sigma}^{j,m}.$ (2.5)

For simplicity, we abbreviate $\tau_{\lambda,\sigma}^{j,m}$ by $\tau.$

We can implement the generalized rescaling to each $\tau$. Without loss of
generality, we may assume that

$\tau=[\lambda,\lambda+\lambda^{-1}K^{-\frac{1}{2}}]\times[\sigma,\sigma+\sigma^{-1}K^{-\frac{1}{2}}].$
(2.6)

We take the change of variables

$\xi_{1}=\lambda+\frac{\eta_{1}}{\lambda
K^{\frac{1}{2}}},\;\xi_{2}=\sigma+\frac{\eta_{2}}{\sigma K^{\frac{1}{2}}}$
(2.7)

to get

$\displaystyle|\mathcal{E}_{\tau}g(x)|=$
$\displaystyle\Big{|}\int_{[0,1]^{2}}\tilde{g}(\eta_{1},\eta_{2})e[\tilde{x}_{1}\eta_{1}+\tilde{x}_{2}\eta_{2}+\tilde{x}_{3}\psi_{0}(\eta_{1},\eta_{2})]\;d\eta_{1}\;d\eta_{2}\Big{|}$
$\displaystyle=:$ $\displaystyle|{\mathcal{E}}^{\rm
Parp}_{[0,1]^{2}}\tilde{g}(\tilde{x})|,$ (2.8)

where

$\left\\{\begin{aligned}
&\tilde{x}=\big{(}\lambda^{-1}K^{-\frac{1}{2}}x_{1}+4\lambda^{2}K^{-\frac{1}{2}}x_{3},\sigma^{-1}K^{-\frac{1}{2}}x_{2}+4\sigma^{2}K^{-\frac{1}{2}}x_{3},K^{-1}x_{3}\big{)},\\\
&\tilde{g}(\eta_{1},\eta_{2})=\lambda^{-1}\sigma^{-1}K^{-1}g\Big{(}\lambda+\frac{\eta_{1}}{\lambda
K^{\frac{1}{2}}},\sigma+\frac{\eta_{2}}{\sigma K^{\frac{1}{2}}}\Big{)},\\\
&\psi_{0}(\eta_{1},\eta_{2})=(6\eta_{1}^{2}+4\lambda^{-2}K^{-\frac{1}{2}}\eta_{1}^{3}+\lambda^{-4}K^{-1}\eta_{1}^{4})+(6\eta_{2}^{2}+4\sigma^{-2}K^{-\frac{1}{2}}\eta_{2}^{3}+\sigma^{-4}K^{-1}\eta_{2}^{4}).\end{aligned}\right.$
(2.9)

Here $\mathcal{E}^{\rm Parp}_{[0,1]^{2}}$ denotes the Fourier extension
operator associated with the phase function $\psi_{0}$. Since
$\lambda,\sigma\geq K^{-\frac{1}{4}}$ and $0\leq\xi_{i}\leq 1(i=1,2)$, it is
easy to see that $\psi_{0}$ is of the form
$\psi_{0}(\eta_{1},\eta_{2})=\phi_{1}(\eta_{1})+\phi_{2}(\eta_{2})$ satisfying

$\phi^{\prime\prime}_{j}\sim 1,\;|\phi^{(3)}_{j}|\lesssim
1,\;|\phi^{(4)}_{j}|\lesssim
1\quad\text{and}\quad\phi^{(\ell)}_{j}=0,\;\ell\geq
5\;\text{on}\;[0,1],\;j=1,\;2.$

Thus, we can adapt the argument in Section 7 of [4] to obtain the following
uniform decoupling inequality for $\Omega_{0}$:

$\|\mathcal{E}_{\Omega_{0}}g\|_{L^{p}(B_{R})}\leq
C(\varepsilon)K^{O(1)}R^{\varepsilon}\Big{(}\sum_{\theta\subset\Omega_{0}}\|\mathcal{E}_{\theta}g\|^{2}_{L^{p}(w_{B_{R}})}\Big{)}^{1/2},\;\;2\leq
p\leq 4,$ (2.10)

where $K^{O(1)}$ denotes a fixed power of $K$, and
$\theta\in\mathcal{F}_{3}(R,4)$. This will be shown in Subsection 3.1 below.

Case 2: The contribution comes from
$\Omega_{3}=[0,K^{-\frac{1}{4}}]\times[0,K^{-\frac{1}{4}}]$. To do it, we will
apply rescaling to $|\mathcal{E}_{\Omega_{3}}g|$ directly and transform
$\|\mathcal{E}_{\Omega_{3}}g\|_{L^{p}(B_{R})}$ to
$\|\mathcal{E}_{[0,1]^{2}}\tilde{g}\|_{L^{p}(B_{R/K})}$. More precisely, we
take the change of variables

$\xi_{1}=K^{-\frac{1}{4}}\eta_{1},\;\xi_{2}=K^{-\frac{1}{4}}\eta_{2},$ (2.11)

and obtain

$\displaystyle|\mathcal{E}_{\tau}g(x)|=$
$\displaystyle\Big{|}\int_{[0,1]^{2}}\tilde{g}(\eta_{1},\eta_{2})e[\tilde{x}_{1}\eta_{1}+\tilde{x}_{2}\eta_{2}+\tilde{x}_{3}\psi(\eta_{1},\eta_{2})]\;d\eta_{1}\;d\eta_{2}\Big{|}$
$\displaystyle=$ $\displaystyle|\mathcal{E}_{[0,1]^{2}}\tilde{g}(\tilde{x})|,$
(2.12)

where

$\left\\{\begin{aligned}
&\tilde{x}=\big{(}K^{-\frac{1}{4}}x_{1},K^{-\frac{1}{4}}x_{2},K^{-1}x_{3}\big{)},\\\
&\tilde{g}(\eta_{1},\eta_{2})=K^{-\frac{1}{2}}g\big{(}K^{-\frac{1}{4}}\eta_{1},K^{-\frac{1}{4}}\eta_{2}\big{)},\\\
&\psi(\eta_{1},\eta_{2})=\eta_{1}^{4}+\eta_{2}^{4}.\end{aligned}\right.$

For the contribution of $\Omega_{3}$-part, we get by rescaling and the
definition of $\mathcal{D}_{p}(\cdot)$

$\|\mathcal{E}_{\Omega_{3}}g\|_{L^{p}(B_{R})}\leq\mathcal{D}_{p}\Big{(}\frac{R}{K}\Big{)}\Big{(}\sum_{\theta\subset\Omega_{3}}\|\mathcal{E}_{\theta}g\|^{2}_{L^{p}(w_{B_{R}})}\Big{)}^{1/2},$
(2.13)

where $\theta\in\mathcal{F}_{3}(R,4)$. We refer the details in Lemma 3.2
below.

Case 3: The contribution comes from $\Omega_{1}$-part and $\Omega_{2}$-part.
By symmetry, it suffices to treat $\Omega_{1}$-part. To do it, we will
establish an auxiliary lemma (see Lemma 3.5 below) to handle the Fourier
extension operator associated with certain phase functions which are
degenerate in $\xi_{2}$-variable but non-degenerate in $\xi_{1}$-variable.
More precisely, we first divide $\Omega_{1}$ into
$\Omega_{1}=\bigcup\Omega_{1,\lambda}$ with

$\Omega_{1,\lambda}=[\lambda,2\lambda]\times[0,K^{-\frac{1}{4}}],$

where $\lambda\in\big{[}K^{-\frac{1}{4}},\tfrac{1}{2}\big{]}$ is a dyadic
number. We divide $\Omega_{1,\lambda}$ further into

$\displaystyle\Omega_{1,\lambda}=$ $\displaystyle\bigcup_{1\leq
j\leq\lambda^{2}K^{\frac{1}{2}}}\Big{[}\lambda+\frac{j-1}{\lambda
K^{1/2}},\lambda+\frac{j}{\lambda K^{1/2}}\Big{]}\times[0,K^{-\frac{1}{4}}]$
$\displaystyle=:$ $\displaystyle\bigcup\tau_{\lambda}^{j}.$ (2.14)

We abbreviate $\tau_{\lambda}^{j}$ by $\tau.$ By reduction of dimension
arguments and (1.2) with $n=2$, we can derive that

$\|\mathcal{E}_{\Omega_{1}}g\|_{L^{p}(B_{R})}\leq
C_{\varepsilon}K^{\varepsilon}\Big{(}\sum_{\tau\subset\Omega_{1}}\|\mathcal{E}_{\tau}g\|^{2}_{L^{p}(w_{B_{R}})}\Big{)}^{1/2}.$
(2.15)

This will be shown in Lemma 3.3 below.

Next, we estimate the $\|\mathcal{E}_{\tau}g\|_{L^{p}(B_{R})}$ for the typical
case

$\tau=[\lambda,\lambda+\lambda^{-1}K^{-\frac{1}{2}}]\times[0,K^{-\frac{1}{4}}].$

The other cases can be deduced by the same argument. We take the change of
variables

$\xi_{1}=\lambda+\frac{\eta_{1}}{\lambda
K^{\frac{1}{2}}},\;\xi_{2}=\frac{\eta_{2}}{K^{\frac{1}{4}}}$ (2.16)

to obtain

$\displaystyle|\mathcal{E}_{\tau}g(x)|=$
$\displaystyle\Big{|}\int_{[0,1]^{2}}\tilde{g}(\eta_{1},\eta_{2})e[\tilde{x}_{1}\eta_{1}+\tilde{x}_{2}\eta_{2}+\tilde{x}_{3}\psi_{1}(\eta_{1},\eta_{2})]\;d\eta_{1}\;d\eta_{2}\Big{|}$
$\displaystyle=:$
$\displaystyle|\tilde{\mathcal{E}}_{[0,1]^{2}}\tilde{g}(\tilde{x})|,$ (2.17)

where

$\left\\{\begin{aligned}
&\tilde{x}=\big{(}\lambda^{-1}K^{-\frac{1}{2}}x_{1}+4\lambda^{2}K^{-\frac{1}{2}}x_{3},K^{-\frac{1}{4}}x_{2},K^{-1}x_{3}\big{)},\\\
&\tilde{g}(\eta_{1},\eta_{2})=\lambda^{-1}K^{-3/4}g\big{(}\lambda+\frac{\eta_{1}}{\lambda
K^{\frac{1}{2}}},\frac{\eta_{2}}{K^{\frac{1}{4}}}\big{)},\\\
&\psi_{1}(\eta_{1},\eta_{2})=(6\eta_{1}^{2}+4\lambda^{-2}K^{-\frac{1}{2}}\eta_{1}^{3}+\lambda^{-4}K^{-1}\eta_{1}^{4})+\eta_{2}^{4}.\end{aligned}\right.$
(2.18)

Here $\tilde{\mathcal{E}}_{[0,1]^{2}}$ denotes the Fourier extension operator
associated with phase functions of the form
$\psi_{1}(\eta_{1},\eta_{2})=\phi_{1}(\eta_{1})+\eta_{2}^{4}$ satisfying

$\phi^{\prime\prime}_{1}\sim 1,\;|\phi^{(3)}_{1}|\lesssim
1,\;|\phi^{(4)}_{1}|\lesssim 1\;\text{and}\;\phi^{(\ell)}_{1}=0,\;\ell\geq
5\;\text{on}\;[0,1].$

To deal with the $|\tilde{\mathcal{E}}_{[0,1]^{2}}\tilde{g}|$, we divide
$[0,1]^{2}$ into $[0,1]^{2}=D_{0}\bigcup D_{1},$ where

$D_{0}:=[0,1]\times[K^{-1/4},1],\;D_{1}:=[0,1]\times[0,K^{-1/4}].$

The contribution stems from $D_{0}$ can be estimated by making use of the
argument in Section 7 of [4]. To estimate the contribution from $D_{1}$, we
divide $D_{1}$ into a disjoint union of subregions

$D_{1}=\bigcup\nu,$

where $\nu$ denotes a $K^{-1/2}\times K^{-1/4}-$rectangle. Without loss of
generality, we can assume that $\nu=[0,K^{-1/2}]\times[0,K^{-1/4}]$. The key
point is that phase functions of the form $\psi_{1}(\eta_{1},\eta_{2})$ over
the region $\nu\subset D_{1}$ are closed under the change of variable

$\xi_{1}=\frac{\eta_{1}}{\lambda
K^{\frac{1}{2}}},\;\xi_{2}=\frac{\eta_{2}}{K^{\frac{1}{4}}}$

so that induction on scales can be used to estimate
$\|\tilde{\mathcal{E}}_{\nu}\tilde{g}\|_{L^{p}(B_{R})}$. Therefore, we can
obtain the decoupling inequality for
$\|\tilde{\mathcal{E}}_{D_{1}}\tilde{g}\|_{L^{p}(B_{R})}$. From the discussion
above, we can derive

$\|\mathcal{E}_{\tau}g\|_{L^{p}(B_{R})}\leq
C_{\varepsilon}R^{\varepsilon}\Big{(}\sum_{\theta\subset\tau}\|\mathcal{E}_{\theta}g\|^{2}_{L^{p}(w_{B_{R}})}\Big{)}^{1/2},\quad
2\leq p\leq 4.$ (2.19)

This will be proved in (3.15) below.

Combining (2.15) and (2.19), one can deduce that

$\|\mathcal{E}_{\Omega_{1}}g\|_{L^{p}(B_{R})}\leq
C_{\varepsilon}R^{\varepsilon}\Big{(}\sum_{\theta\subset\Omega_{1}}\|\mathcal{E}_{\theta}g\|^{2}_{L^{p}(w_{B_{R}})}\Big{)}^{1/2},\;2\leq
p\leq 4.$ (2.20)

By inequalities (2.10), (2.13) and (2.20), we obtain for $2\leq p\leq 4$

$\|\mathcal{E}_{[0,1]^{2}}g\|_{L^{p}(B_{R})}\leq\Big{[}C(\varepsilon)K^{O(1)}R^{\varepsilon}+2C_{\varepsilon}R^{\varepsilon}+\mathcal{D}_{p}\big{(}\tfrac{R}{K}\big{)}\Big{]}\Big{(}\sum_{\theta\in\mathcal{F}_{3}(R,4)}\|\mathcal{E}_{\theta}g\|^{2}_{L^{p}(w_{B_{R}})}\Big{)}^{1/2}.$

This inequality implies the following recurrence inequality

$\mathcal{D}_{p}(R)\leq
C(\varepsilon)K^{O(1)}R^{\varepsilon}+2C_{\varepsilon}R^{\varepsilon}+\mathcal{D}_{p}\big{(}\tfrac{R}{K}\big{)}.$
(2.21)

One can iterate (2.21) to deduce
$\mathcal{D}_{p}(R)\lesssim_{\varepsilon}R^{\varepsilon}$. Therefore, we
conclude the proof of Theorem 1.2.

##  3\. Proof of Theorem 1.2

We are going to treat decoupling inequalities for different regions by making
use of different approaches. Firstly, we estimate the contribution from
$\Omega_{0}$-part.

###  3.1. Decoupling for $\Omega_{0}$

In this subsection, we will establish the decoupling inequality for
$\Omega_{0}$-part. Recall

$\Omega_{0}=\bigcup_{\lambda,\sigma}\Omega_{\lambda,\sigma}=\bigcup_{\lambda,\sigma}\Big{(}\bigcup_{j,m}\tau_{\lambda,\sigma}^{j,m}\Big{)}.$

Combining this decomposition with the Minkowski inequality and Cauchy-Schwartz
inequality, we get a trivial decoupling at scale $K$ for $2\leq p\leq 4$

$\|\mathcal{E}_{\Omega_{0}}g\|_{L^{p}(B_{K})}\lesssim
K^{1/2}\Big{(}\sum_{\tau\subset\Omega_{0}}\|\mathcal{E}_{\tau}g\|^{2}_{L^{p}(B_{K})}\Big{)}^{1/2},$
(3.1)

where $\tau$ is as in (2.5). Summing over all the balls $B_{K}\subset B_{R}$,
we obtain

$\|\mathcal{E}_{\Omega_{0}}g\|_{L^{p}(B_{R})}\lesssim
K^{1/2}\Big{(}\sum_{\tau\subset\Omega_{0}}\|\mathcal{E}_{\tau}g\|^{2}_{L^{p}(B_{R})}\Big{)}^{1/2}.$
(3.2)

For any given $\tau\subset\Omega_{0}$ of size
$\lambda^{-1}K^{-1/2}\times\sigma^{-1}K^{-1/2}$, we have

######  Lemma 3.1.

For $2\leq p\leq 4$ and each $\varepsilon>0$, there exists a positive constant
$C_{\varepsilon}$ such that

$\|\mathcal{E}_{\tau}g\|_{L^{p}(B_{R})}\leq
C_{\varepsilon}R^{\varepsilon}\Big{(}\sum_{\theta\subset\tau}\|\mathcal{E}_{\theta}g\|^{2}_{L^{p}(w_{B_{R}})}\Big{)}^{1/2},$
(3.3)

where $\theta\in\mathcal{F}_{3}(R,4).$

With Lemma 3.1 in hand, plugging (3.3) into (3.2), we get the decoupling
inequality for $\Omega_{0}$

$\|\mathcal{E}_{\Omega_{0}}g\|_{L^{p}(B_{R})}\leq
C_{\varepsilon}K^{O(1)}R^{\varepsilon}\Big{(}\sum_{\theta\subset\Omega_{0}}\|\mathcal{E}_{\theta}g\|^{2}_{L^{p}(w_{B_{R}})}\Big{)}^{1/2}.$
(3.4)

###### Proof of Lemma 3.1:.

Without loss of generality, we may assume that $B_{R}$ is centered at the
origin and $\tau$ is as in (2.6). From (2.7) in Section 2, we know that

$\|\mathcal{E}_{\tau}g\|^{p}_{L^{p}(B_{R})}=\lambda\sigma
K^{2}\|\mathcal{E}^{Parp}_{[0,1]^{2}}\tilde{g}\|^{p}_{L^{p}(\mathcal{L}_{0}(B_{R}))},$

where $\mathcal{L}_{0}$ denotes the map

$(x_{1},x_{2},x_{3})\mapsto\big{(}\lambda^{-1}K^{-\frac{1}{2}}x_{1}+4\lambda^{2}K^{-\frac{1}{2}}x_{3},\sigma^{-1}K^{-\frac{1}{2}}x_{2}+4\sigma^{2}K^{-\frac{1}{2}}x_{3},K^{-1}x_{3}\big{)},$

and $\mathcal{L}_{0}(B_{R})$ denotes the image of $B_{R}$ with size roughly as
follows

$\lambda^{-1}K^{-1/2}R\times\sigma^{-1}K^{-1/2}R\times K^{-1}R.$

We divide it into a finitely overlapping union of balls as follows

$\mathcal{L}_{0}(B_{R})=\bigcup B_{R/K}.$

For a given $\theta\subset\tau$ such as

$\theta=[\lambda,\lambda+\lambda^{-1}R^{-1/2}]\times[\sigma,\sigma+\sigma^{-1}R^{-1/2}],$

and under the change of variables (2.7), we deduce that the image of $\theta$
is

$\tilde{\theta}=[0,K^{1/2}R^{-1/2}]^{2}.$

We use Bourgain-Demeter’s decoupling inequality (1.1) with $n=3$ on each
$B_{R/K}$

$\big{\|}\mathcal{E}^{Parp}_{[0,1]^{2}}\tilde{g}\big{\|}_{L^{p}(B_{R/K})}\leq
C_{\varepsilon}R^{\varepsilon}\Big{(}\sum_{\tilde{\theta}:K^{1/2}R^{-1/2}-square}\|\mathcal{E}^{Parp}_{\tilde{\theta}}\tilde{g}\|^{2}_{L^{p}(w_{B_{R/K}})}\Big{)}^{1/2}.$
(3.5)

This can be done by the argument in Section 7 of [4]. In fact, we denote
$K_{p}(R)$ to be the least number such that

$\|F\|_{L^{p}(\mathbb{R}^{3})}\leq
K_{p}(R)\Big{(}\sum_{\bar{\theta}:R^{-1/2}-slab}\|F_{\bar{\theta}}\|^{2}_{L^{p}(\mathbb{R}^{3})}\Big{)}^{1/2}$
(3.6)

holds for each $F$ with Fourier support in $\mathcal{N}_{R^{-1}}(S)$, where
$S:=\\{(\eta_{1},\eta_{2},\psi_{0}(\eta_{1},\eta_{2})):(\eta_{1},\eta_{2})\in[0,1]^{2}\\}$
with $\psi_{0}(\eta_{1},\eta_{2})$ as in Case 1 of Section 2. It follows from
(3.6) that

$\|F\|_{L^{p}(\mathbb{R}^{3})}\leq
K_{p}\big{(}R^{2/3}\big{)}\Big{(}\sum_{\bar{\alpha}:R^{-1/3}-slab}\|F_{\bar{\alpha}}\|^{2}_{L^{p}(\mathbb{R}^{3})}\Big{)}^{1/2}.$
(3.7)

Furthermore, from Taylor’s formula, we know that on each $\bar{\alpha}$, $S$
is contained in $\mathcal{N}_{R^{-1}}(P^{2})$. By invoking (1.2) for this
paraboloid and parabolic rescaling, we get

$\|F_{\bar{\alpha}}\|_{L^{p}(\mathbb{R}^{3})}\leq
C_{\varepsilon}R^{\varepsilon}\Big{(}\sum_{\bar{\theta}:\bar{\theta}\subset\bar{\alpha}}\|F_{\bar{\theta}}\|^{2}_{L^{p}(\mathbb{R}^{3})}\Big{)}^{1/2}.$
(3.8)

Hence, we conclude

$K_{p}(R)\leq C_{\varepsilon}R^{\varepsilon}K_{p}\big{(}R^{2/3}\big{)},$

which immediately leads to $K_{p}(R)\lesssim_{\varepsilon}R^{\varepsilon}$ by
iteration. This verifies the inequality (3.5).

Summing over all the balls $B_{R/K}\subset\mathcal{L}_{0}(B_{R})$ on both
sides of (3.5) and using Minkowski’s inequality, we have

$\|\mathcal{E}^{Parp}_{[0,1]^{2}}\tilde{g}\|_{L^{p}(\mathcal{L}_{0}(B_{R}))}\leq
C_{\varepsilon}R^{\varepsilon}\Big{(}\sum_{\tilde{\theta}:K^{1/2}R^{-1/2}-square}\|\mathcal{E}^{Parp}_{\tilde{\theta}}\tilde{g}\|^{2}_{L^{p}(\mathcal{L}_{0}(B_{R}))}\Big{)}^{1/2}.$

Taking the inverse change of variables, it follows that

$\|\mathcal{E}_{\tau}g\|_{L^{p}(B_{R})}\leq
C_{\varepsilon}R^{\varepsilon}\Big{(}\sum_{\theta\subset\tau}\|\mathcal{E}_{\theta}g\|^{2}_{L^{p}(w_{B_{R}})}\Big{)}^{1/2}.$

Therefore, we complete the proof of Lemma 3.1. ∎

Next, we turn to discuss the contribution from $\Omega_{3}$-part.

###  3.2. Decoupling for $\Omega_{3}$

By rescaling and induction on scales, we can show

######  Lemma 3.2.

For $2\leq p\leq 4$, there holds

$\|\mathcal{E}_{\Omega_{3}}g\|_{L^{p}(B_{R})}\leq\mathcal{D}_{p}\big{(}\tfrac{R}{K}\big{)}\Big{(}\sum_{\theta\subset\Omega_{3}}\|\mathcal{E}_{\theta}g\|^{2}_{L^{p}(w_{B_{R}})}\Big{)}^{1/2},$
(3.9)

where $\theta\in\mathcal{F}_{3}(R,4).$

###### Proof.

Taking the change of variables

$\eta_{i}=K^{1/4}\xi_{i}\;(i=1,2),$

it follows

$|\mathcal{E}_{\Omega_{3}}g(x)|=|\mathcal{E}_{[0,1]^{2}}\widetilde{g}(\widetilde{x})|,$

where

$\tilde{x}:=(K^{-1/4}x_{1},K^{-1/4}x_{2},K^{-1}x_{3}),$

and

$\tilde{g}(\eta):=K^{-1/2}g(K^{-1/4}\eta_{1},K^{-1/4}\eta_{2}).$

The $\tilde{x}$-variables belong to a rectangular box $\square$ with size

$\frac{R}{K^{1/4}}\times\frac{R}{K^{1/4}}\times\frac{R}{K},$

which can be written into a finitely overlapping union of balls

$\square=\bigcup B_{R/K}.$

By the definition of $\mathcal{D}_{p}(\cdot)$, we obtain

$\|\mathcal{E}_{[0,1]^{2}}\tilde{g}\|_{L^{p}(B_{R/K})}\leq\mathcal{D}_{p}\big{(}\tfrac{R}{K}\big{)}\Big{(}\sum_{\tilde{\theta}\in\mathcal{F}_{3}(\frac{R}{K},4)}\|\mathcal{E}_{\tilde{\theta}}\tilde{g}\|^{2}_{L^{p}(w_{B_{R/K}})}\Big{)}^{1/2}.$

Summing over all the cubes $B_{R/K}\subset\square$ and using Minkowski’s
inequality, we have

$\|\mathcal{E}_{[0,1]^{2}}\tilde{g}\|_{L^{p}(\square)}\leq\mathcal{D}_{p}\big{(}\tfrac{R}{K}\big{)}\Big{(}\sum_{\tilde{\theta}\in\mathcal{F}_{3}(\frac{R}{K},4)}\|\mathcal{E}_{\tilde{\theta}}\tilde{g}\|^{2}_{L^{p}(\square)}\Big{)}^{1/2}.$

Taking the inverse change of variables, one has

$\|\mathcal{E}_{\Omega_{3}}g\|_{L^{p}(B_{R})}\leq\mathcal{D}_{p}\big{(}\tfrac{R}{K}\big{)}\Big{(}\sum_{\theta\subset\Omega_{3}}\|\mathcal{E}_{\theta}g\|^{2}_{L^{p}(w_{B_{R}})}\Big{)}^{1/2}.$

Thus, we complete the proof of Lemma 3.2. ∎

Finally, we deal with the contribution from $\Omega_{1}$-part.

###  3.3. Decoupling for $\Omega_{1}$

Recall the decomposition

$\displaystyle\Omega_{1}=$
$\displaystyle\bigcup_{\lambda}\Omega_{1,\lambda},\quad\Omega_{1,\lambda}=\bigcup_{\lambda}\bigcup_{j}\tau_{\lambda}^{j}=:\bigcup\tau_{\lambda},$

where we abbreviate $\tau_{\lambda}^{j}$ by $\tau_{\lambda}$. First, by
reduction of dimension arguments, we are able to prove the following result.

######  Lemma 3.3.

For $2\leq p\leq 6$ and any $\varepsilon>0$, there exists a constant
$C_{\varepsilon}$ such that

$\|\mathcal{E}_{\Omega_{1}}g\|_{L^{p}(B_{R})}\leq
C_{\varepsilon}K^{\varepsilon}\Big{(}\sum_{\lambda}\sum_{\tau_{\lambda}\subset\Omega_{1,\lambda}}\|\mathcal{E}_{\tau_{\lambda}}g\|^{2}_{L^{p}(w_{B_{R}})}\Big{)}^{1/2}.$
(3.10)

To prove Lemma 3.3, we employ the following lemma from [13].

######  Lemma 3.4.

Let $\Gamma_{\lambda}=\\{(t,t^{4}):t\in[\lambda,2\lambda]\\}$ and $2\leq p\leq
6$. For any $\varepsilon>0$, there holds

$\|G\|_{L^{p}(\mathbb{R}^{2})}\leq
C_{\varepsilon}K^{\varepsilon}\Big{(}\sum_{\tau}\|G_{\tau}\|^{2}_{L^{p}(\mathbb{R}^{2})}\Big{)}^{1/2},\;G_{\tau}:=\mathcal{F}^{-1}(\widehat{G}\chi_{\tau}),$
(3.11)

where ${\rm supp}\;\hat{G}\subset\mathcal{N}_{K^{-1}}(\Gamma_{\lambda})$ and
$\tau$’s are $\lambda^{-1}K^{-\frac{1}{2}}\times K^{-1}$-rectangles.

With Lemma 3.4 in hand, we prove Lemma 3.3 by freezing the $x_{2}$ variable as
follows. Fix a bump function $\varphi\in C^{\infty}_{c}(\mathbb{R}^{3})$ with
${\rm supp}\;\varphi\subset B(0,1)$ and $|\check{\varphi}(x)|\geq 1$ for all
$x\in B(0,1)$. Define $F:=\check{\varphi}_{K^{-1}}\mathcal{E}_{\Omega_{1}}g$,
where $\varphi_{K^{-1}}(\xi):=K^{3}\varphi(K\xi),\;\xi\in\mathbb{R}^{3}$. We
denote $F(\cdot,x_{2},\cdot)$ by $G$. From Guth [9], it is easy to see that
${\rm supp}\;\hat{G}$ is contained in the projection of ${\rm supp}\;\hat{F}$
on the plane $\xi_{2}=0$, that is, in the $K^{-1}$-neighborhood of
$\Gamma_{\lambda}$. We employ Lemma 3.4 to get

$\|G\|_{L^{p}(\mathbb{R}^{2})}\leq
C_{\varepsilon}K^{\varepsilon}\Big{(}\sum_{\tau}\|G_{\tau}\|^{2}_{L^{p}(\mathbb{R}^{2})}\Big{)}^{1/2},$

i.e.,

$\|F(\cdot,x_{2},\cdot)\|_{L^{p}(\mathbb{R}^{2})}\leq
C_{\varepsilon}K^{\varepsilon}\Big{(}\sum_{\tau}\|F_{\tau}(\cdot,x_{2},\cdot)\|^{2}_{L^{p}(\mathbb{R}^{2})}\Big{)}^{1/2},$

where $F_{\tau}(x):=\check{\varphi}_{K^{-1}}\mathcal{E}_{\tau}g$. Integrating
on both sides of the above inequality with respect to $x_{2}$-variable from
$-\infty$ to $\infty$, we derive

$\|F\|_{L^{p}(\mathbb{R}^{3})}\leq
C_{\varepsilon}K^{\varepsilon}\Big{(}\sum_{\tau}\|F_{\tau}\|^{2}_{L^{p}(\mathbb{R}^{3})}\Big{)}^{1/2}.$

Thus, we have

$\displaystyle\|\mathcal{E}_{\Omega_{1,\lambda}}g\|_{L^{p}(B_{K})}\lesssim$
$\displaystyle\|F\|_{L^{p}(\mathbb{R}^{3})}\leq
C_{\varepsilon}K^{\varepsilon}\Big{(}\sum_{\tau}\|F_{\tau}\|^{2}_{L^{p}(\mathbb{R}^{3})}\Big{)}^{1/2}$
$\displaystyle\leq$ $\displaystyle
C_{\varepsilon}K^{\varepsilon}\Big{(}\sum_{\tau_{\lambda}\subset\Omega_{1,\lambda}}\|\mathcal{E}_{\tau}g\|^{2}_{L^{p}(w_{B_{K}})}\Big{)}^{1/2}.$

It follows that

$\|\mathcal{E}_{\Omega_{1}}g\|_{L^{p}(B_{K})}\leq
C_{\varepsilon}K^{\varepsilon}\Big{(}\sum_{\lambda}\sum_{\tau_{\lambda}\subset\Omega_{1,\lambda}}\|\mathcal{E}_{\tau_{\lambda}}g\|^{2}_{L^{p}(w_{B_{K}})}\Big{)}^{1/2}.$

Summing over all the balls $B_{K}\subset B_{R}$, we get the inequality (3.10),
as required.

Next, we estimate the $\|\mathcal{E}_{\tau_{\lambda}}g\|_{L^{p}(B_{R})}$ on
the right hand side of (3.10). For this purpose, we consider the following
surfaces:

$F^{2}_{2,4}:=\\{(\xi_{1},\xi_{2},\phi_{1}(\xi_{1})+\xi_{2}^{4}):(\xi_{1},\xi_{2})\in[0,1]^{2}\\},$
(3.12)

where $\phi_{1}(\xi_{1})$ satisfies $\phi_{1}^{\prime\prime}\sim 1$,
$|\phi_{1}^{(3)}|\lesssim 1,|\phi_{1}^{(4)}|\lesssim 1$ and
$\phi_{1}^{(\ell)}=0,\;\ell\geq 5$ on $[0,1]$. Now, we construct the
decomposition of $[0,1]^{2}$ associated with the surfaces $F^{2}_{2,4}$. We
first divide $[0,1]$ into

$[0,1]=\bigcup_{k}I_{k},$

where $I_{0}=[0,R^{-\frac{1}{4}}]$ and

$I_{k}=[2^{k-1}R^{-\frac{1}{4}},2^{k}R^{-\frac{1}{4}}],\;\;\text{for}\;\;1\leq
k\leq\big{[}\tfrac{1}{4}\log_{2}R\big{]}.$

Furthermore, for each $k\geq 1$, we divide $I_{k}$ into

$I_{k}=\bigcup_{\mu=1}^{2^{2(k-1)}}I_{k,\mu},$

with

$I_{k,\mu}=\big{[}2^{k-1}{R^{-\frac{1}{4}}}+(\mu-1)2^{-(k-1)}R^{-\frac{1}{4}},2^{k-1}R^{-\frac{1}{4}}+\mu
2^{-(k-1)}R^{-\frac{1}{4}}\big{]}.$

Then, we have the following decomposition

$[0,1]^{2}=\bigcup\tilde{\theta},$ (3.13)

where $\tilde{\theta}\in\mathcal{F}_{3}(R,2,4)$ and

$\displaystyle\mathcal{F}_{3}(R,2,4):=\Big{\\{}$
$\displaystyle[a,a+R^{-1/2}]\times I_{k,\mu},\;[a,a+R^{-1/2}]\times I_{0}:$
$\displaystyle a\in[0,1-R^{-1/2}]\cap R^{-1/2}\mathbb{Z},\quad 1\leq
k\leq\big{[}\tfrac{1}{4}\log_{2}R\big{]},\;1\leq\mu\leq 2^{2(k-1)}\Big{\\}}.$

We establish the decoupling inequality for the surfaces $F_{2,4}^{2}$
associated with the decomposition (3.13) as follows.

######  Lemma 3.5.

For $2\leq p\leq 4$ and any $\varepsilon>0$, there exists a constant
$C_{\varepsilon}$ such that

$\|\tilde{\mathcal{E}}_{[0,1]^{2}}f\|_{L^{p}(B_{R})}\leq
C_{\varepsilon}R^{\varepsilon}\Big{(}\sum_{\tilde{\theta}\in\mathcal{F}_{3}(R,2,4)}\|\tilde{\mathcal{E}}_{\tilde{\theta}}f\|^{2}_{L^{p}(w_{B_{R}})}\Big{)}^{1/2},$
(3.14)

where $\tilde{\mathcal{E}}_{[0,1]^{2}}$ denotes the Fourier extension operator
associated with the surfaces $F^{2}_{2,4}$.

Assume that Lemma 3.5 holds for a while, for each $\lambda$ and any
$\tau_{\lambda}\subset\Omega_{1,\lambda}$, we claim that

$\|\mathcal{E}_{\tau_{\lambda}}g\|_{L^{p}(B_{R})}\leq
C_{\varepsilon}\big{(}\tfrac{R}{K}\big{)}^{\varepsilon}\Big{(}\sum_{\theta\subset\tau_{\lambda}}\|\mathcal{E}_{\theta}g\|^{2}_{L^{p}(w_{B_{R}})}\Big{)}^{1/2},\quad
2\leq p\leq 4,$ (3.15)

where $\theta\in\mathcal{F}_{3}(R,4).$ Without loss of generality, we may
assume that
$\tau_{\lambda}=[\lambda,\lambda+\lambda^{-1}K^{-1/2}]\times[0,K^{-1/4}]$.
Taking the change of variables

$\xi_{1}=\lambda+\lambda^{-1}K^{-1/2}\eta_{1},\;\xi_{2}=K^{-1/4}\eta_{2},$
(3.16)

we get

$\|\mathcal{E}_{\tau}g\|^{p}_{L^{p}(B_{R})}=\lambda
K^{7/4}\|\tilde{\mathcal{E}}_{[0,1]^{2}}\tilde{g}\|^{p}_{L^{p}(\mathcal{L}_{\lambda}(B_{R}))},$

where

$\tilde{g}(\eta_{1},\eta_{2})=\lambda^{-1}K^{-3/4}g(\lambda+\lambda^{-1}K^{-1/2}\eta_{1},K^{-1/4}\eta_{2}).$

Here $\mathcal{L}_{\lambda}$ denotes the map

$(x_{1},x_{2},x_{3})\mapsto\big{(}\lambda^{-1}K^{-\frac{1}{2}}x_{1}+4\lambda^{2}K^{-\frac{1}{2}}x_{3},K^{-\frac{1}{4}}x_{2},K^{-1}x_{3}\big{)},$

and $\mathcal{L}_{\lambda}(B_{R})$ denotes the image of $B_{R}$ with size
roughly

$\lambda^{-1/2}K^{-1/2}R\times K^{-1/4}R\times K^{-1}R.$

We rewrite $\mathcal{L}_{\lambda}(B_{R})$ into a finitely overlapping union of
balls as follows

$\mathcal{L}_{\lambda}(B_{R})=\bigcup B_{R/K}.$

Now, we employ Lemma 3.5 on each $B_{R/K}$ to estimate the

$\|\tilde{\mathcal{E}}_{[0,1]^{2}}\tilde{g}\|_{L^{p}(B_{R/K})}.$

To do this, we need verify that for $\theta\subset\tau_{\lambda}$, its image
$\tilde{\theta}$ belongs to $\mathcal{F}_{3}(\frac{R}{K},2,4)$ under the
change of variables (3.16). We only need to consider the following two cases:

Case(a): $\theta\subset[\lambda,2\lambda]\times[0,R^{-1/4}]$. In this case, we
may assume that

$\theta=[\lambda,\lambda+\lambda^{-1}R^{-1/2}]\times[0,R^{-1/4}].$

Under the change of variables (3.16), we see that

$\tilde{\theta}=[0,K^{1/2}R^{-1/2}]\times[0,K^{1/4}R^{-1/4}]\in\mathcal{F}_{3}\big{(}\tfrac{R}{K},2,4\big{)}.$

Case(b): $\theta\subset[\lambda,2\lambda]\times[\sigma,2\sigma].$ In this
case, we may assume that

$\theta=[\lambda,\lambda+\lambda^{-1}R^{-1/2}]\times[\sigma,\sigma+\sigma^{-1}R^{-1/2}],$

where $\sigma$ is a dyadic number with $R^{-1/4}\leq\sigma\leq\frac{1}{2}$.
Under the change of variables (3.16), one has

$\tilde{\theta}=\big{[}0,\big{(}\tfrac{R}{K}\big{)}^{-1/2}\big{]}\times\big{[}\tilde{\sigma},\tilde{\sigma}+\tilde{\sigma}^{-1}\big{(}\tfrac{R}{K}\big{)}^{-1/2}\big{]}\in\mathcal{F}_{3}\big{(}\tfrac{R}{K},2,4\big{)},$

where $\tilde{\sigma}:=K^{1/4}\sigma$ is also a dyadic number.

By Lemma 3.5, we obtain

$\|\tilde{\mathcal{E}}_{[0,1]^{2}}\tilde{g}\|_{L^{p}(B_{R/K})}\leq
C_{\varepsilon}\big{(}\tfrac{R}{K}\big{)}^{\varepsilon}\Big{(}\sum_{\tilde{\theta}\in\mathcal{F}_{3}(\frac{R}{K},2,4)}\|\tilde{\mathcal{E}}_{\tilde{\theta}}\tilde{g}\|^{2}_{L^{p}(w_{B_{R/K}})}\Big{)}^{1/2}.$

Summing over all the balls $B_{R/K}\subset\mathcal{L}_{\lambda}(B_{R})$ and
using Minkowski’s inequality, we have

$\|\tilde{\mathcal{E}}_{[0,1]^{2}}\tilde{g}\|_{L^{p}(\mathcal{L}_{\lambda}(B_{R}))}\leq
C_{\varepsilon}\big{(}\tfrac{R}{K}\big{)}^{\varepsilon}\Big{(}\sum_{\tilde{\theta}\in\mathcal{F}_{3}(\frac{R}{K},2,4)}\|\tilde{\mathcal{E}}_{\tilde{\theta}}\tilde{g}\|^{2}_{L^{p}(\mathcal{L}_{\lambda}(B_{R}))}\Big{)}^{1/2}.$

Taking the inverse change of variables, we deduce that

$\|\mathcal{E}_{\tau_{\lambda}}g\|_{L^{p}(B_{R})}\leq
C_{\varepsilon}\big{(}\tfrac{R}{K}\big{)}^{\varepsilon}\Big{(}\sum_{\theta\subset\tau}\|\mathcal{E}_{\theta}g\|^{2}_{L^{p}(w_{B_{R}})}\Big{)}^{1/2}.$

Thus, we prove the claim (3.15).

Plugging (3.15) into (3.10), one has

$\displaystyle\|\mathcal{E}_{\Omega_{1}}g\|_{L^{p}(B_{R})}\leq$ $\displaystyle
C_{\varepsilon}K^{\varepsilon}\big{(}\tfrac{R}{K}\big{)}^{\varepsilon}\Big{(}\sum_{\lambda}\sum_{\tau_{\lambda}}\sum_{\theta\subset\tau_{\lambda}}\|\mathcal{E}_{\theta}g\|^{2}_{L^{p}(w_{B_{R}})}\Big{)}^{1/2}$
$\displaystyle=$ $\displaystyle
C_{\varepsilon}R^{\varepsilon}\Big{(}\sum_{\theta\subset\Omega_{1}}\|\mathcal{E}_{\theta}g\|^{2}_{L^{p}(w_{B_{R}})}\Big{)}^{1/2},$
(3.17)

where $\theta\in\mathcal{F}_{3}(R,4)$. Therefore, we obtain the contribution
from $\Omega_{1}$-part under the assumption of Lemma 3.5.

Now, we return to prove Lemma 3.5.

###### Proof of Lemma 3.5:.

Denote the least constant such that (3.14) holds by
$\tilde{\mathcal{D}}_{p}(R)$. Divide $[0,1]^{2}$ into $[0,1]^{2}=D_{0}\bigcup
D_{1}$, where

$D_{0}:=[0,1]\times[K^{-1/4},1],\;D_{1}:=[0,1]\times[0,K^{-1/4}].$

It is easy to see that

$\|\tilde{\mathcal{E}}_{[0,1]^{2}}f\|_{L^{p}(B_{R})}\leq\|\tilde{\mathcal{E}}_{D_{0}}f\|_{L^{p}(B_{R})}+\|\tilde{\mathcal{E}}_{D_{1}}f\|_{L^{p}(B_{R})}.$
(3.18)

Step 1: The contribution stems from $D_{0}$. We have a trivial decoupling at
scale $K$ for $p\geq 2$

$\|\tilde{\mathcal{E}}_{D_{0}}f\|_{L^{p}(B_{K})}\lesssim
K^{O(1)}\Big{(}\sum_{\nu\subset
D_{0}}\|\tilde{\mathcal{E}}_{\nu}f\|^{2}_{L^{p}(B_{K})}\Big{)}^{1/2},$ (3.19)

where $\nu\in\mathcal{F}_{3}(K,2,4).$ Summing over all the balls $B_{K}\subset
B_{R}$, we obtain

$\|\tilde{\mathcal{E}}_{D_{0}}f\|_{L^{p}(B_{R})}\leq
CK^{O(1)}\Big{(}\sum_{\nu\subset
D_{0}}\|\tilde{\mathcal{E}}_{\nu}f\|^{2}_{L^{p}(B_{R})}\Big{)}^{1/2},\;\nu\in\mathcal{F}_{3}(K,2,4).$
(3.20)

For any given $\nu\subset D_{0}$ of size $K^{-1/2}\times\sigma^{-1}K^{-1/2}$,
where $\sigma$ is a dyadic number with
$K^{-\frac{1}{4}}\leq\sigma\leq\frac{1}{2}$, we claim that

$\|\tilde{\mathcal{E}}_{\nu}f\|_{L^{p}(B_{R})}\leq
C_{\varepsilon}\big{(}\tfrac{R}{K}\big{)}^{\varepsilon}\Big{(}\sum_{\vartheta\subset\nu}\|\tilde{\mathcal{E}}_{\vartheta}f\|^{2}_{L^{p}(w_{B_{R}})}\Big{)}^{1/2},\;\;2\leq
p\leq 4,$ (3.21)

where $\vartheta\in\mathcal{F}_{3}(R,2,4)$. Plugging (3.21) into (3.20), it
follows that

$\|\tilde{\mathcal{E}}_{D_{0}}f\|_{L^{p}(B_{R})}\leq
C_{\varepsilon}K^{O(1)}R^{\varepsilon}\Big{(}\sum_{\vartheta\subset
D_{0}}\|\tilde{\mathcal{E}}_{\vartheta}f\|^{2}_{L^{p}(w_{B_{R}})}\Big{)}^{1/2},\;\;2\leq
p\leq 4.$ (3.22)

Now, we turn to prove the claim (3.21). Without loss of generality, we may
assume that $\nu=[0,K^{-1/2}]\times[\sigma,\sigma+\sigma^{-1}K^{-1/2}]$.
Taking the change of variables

$\xi_{1}=K^{-1/2}\eta_{1},\;\xi_{2}=\sigma+\sigma^{-1}K^{-1/2}\eta_{2},$
(3.23)

we get

$\|\tilde{\mathcal{E}}_{\nu}f\|^{p}_{L^{p}(B_{R})}=\sigma
K^{2}\big{\|}\mathcal{E}^{Parp}_{[0,1]^{2}}\tilde{f}\big{\|}^{p}_{L^{p}(\mathcal{T}_{0}(B_{R}))},$

where
$\tilde{f}(\eta_{1},\eta_{2}):=\sigma^{-1}K^{-1}f(K^{-1/2}\eta_{1},\sigma+\sigma^{-1}K^{-1/2}\eta_{2})$,
$\mathcal{T}_{0}$ denotes the map

$\mathcal{T}_{0}:\;(x_{1},x_{2},x_{3})\rightarrow\big{(}K^{-\frac{1}{2}}x_{1},\sigma^{-1}K^{-\frac{1}{2}}x_{2}+4\sigma^{2}K^{-\frac{1}{2}}x_{3}K^{-1}x_{3}\big{)},$

and $\mathcal{T}_{0}(B_{R})$ denotes the image of $B_{R}$ with size roughly

$K^{-1/2}R\times\sigma^{-1}K^{-1/2}R\times K^{-1}R.$

We rewrite it into a finitely overlapping union of balls as following

$\mathcal{T}_{0}(B_{R})=\bigcup B_{R/K}.$

Using a similar argument as in the proof of (3.5), we get

$\big{\|}\mathcal{E}^{Parp}_{[0,1]^{2}}\tilde{f}\big{\|}_{L^{p}(B_{R/K})}\leq
C_{\varepsilon}\big{(}\tfrac{R}{K}\big{)}^{\varepsilon}\Big{(}\sum_{\vartheta:K^{1/2}R^{-1/2}-square}\|\mathcal{E}^{Parp}_{\vartheta}\tilde{f}\|^{2}_{L^{p}(w_{B_{R/K}})}\Big{)}^{1/2}.$

Summing over all the balls $B_{R/K}\subset\mathcal{T}_{0}(B_{R})$ and using
Minkowski’s inequality, we have

$\|\mathcal{E}^{Parp}_{[0,1]^{2}}\tilde{f}\|_{L^{p}(\mathcal{T}_{0}(B_{R}))}\leq
C_{\varepsilon}\big{(}\tfrac{R}{K}\big{)}^{\varepsilon}\Big{(}\sum_{\vartheta:K^{1/2}R^{-1/2}-square}\|\mathcal{E}^{Parp}_{\vartheta}\tilde{f}\|^{2}_{L^{p}(\mathcal{T}_{0}(B_{R}))}\Big{)}^{1/2}.$

Taking the inverse change of variables, we obtain

$\|\tilde{\mathcal{E}}_{\nu}f\|_{L^{p}(B_{R})}\leq
C_{\varepsilon}\big{(}\tfrac{R}{K}\big{)}^{\varepsilon}\Big{(}\sum_{\vartheta\subset\nu}\|\tilde{\mathcal{E}}_{\vartheta}f\|^{2}_{L^{p}(w_{B_{R}})}\Big{)}^{1/2}.$

Hence, we conclude the claim (3.21).

Step 2: The contribution stems from $D_{1}$. We first prove

$\|\tilde{\mathcal{E}}_{D_{1}}f\|_{L^{p}(B_{R})}\leq
C_{\varepsilon}K^{\varepsilon}\Big{(}\sum_{\nu\subset
D_{1}}\|\tilde{\mathcal{E}}_{\nu}f\|^{2}_{L^{p}(w_{B_{R}})}\Big{)}^{1/2},\;\;2\leq
p\leq 6,$ (3.24)

where each $\nu$ denotes a $K^{-1/2}\times K^{-1/4}$-rectangle contained in
the region $D_{1}$. By the same argument as in the proof of Lemma 3.3, it
suffices to prove

######  Lemma 3.6.

Let $\Gamma_{\phi_{1}}:=\\{(t,\phi_{1}(t)):t\in[0,1]\\}$ satisfy

$\phi_{1}^{\prime\prime}\sim 1,\;|\phi_{1}^{(3)}|\lesssim
1,\;|\phi_{1}^{(4)}|\lesssim 1\;\;\text{and}\;\;\phi^{(\ell)}_{1}=0,\;\ell\geq
5\;\text{on}\;[0,1].$

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

$\|G\|_{L^{p}(\mathbb{R}^{2})}\leq
C_{\varepsilon}K^{\varepsilon}\Big{(}\sum_{\nu}\|G_{\nu}\|^{2}_{L^{p}(\mathbb{R}^{2})}\Big{)}^{1/2},\;2\leq
p\leq 6,$ (3.25)

where ${\rm supp}\;\hat{G}\subset\mathcal{N}_{K^{-1}}(\Gamma_{\phi_{1}})$,
$\nu$’s are $K^{-\frac{1}{2}}\times K^{-1}$-rectangles and
$G_{\nu}:=\mathcal{F}^{-1}(\widehat{G}\chi_{\nu})$.

By the modification as in the verification of (3.5), Lemma 3.6 follows by
Bourgain-Demeter’s decoupling inequality (1.1) for $n=2$.

Now, we turn to estimate each term
$\|\tilde{\mathcal{E}}_{\nu}f\|_{L^{p}(w_{B_{R}})}$ in the right hand side of
(3.24). For each $\nu\subset D_{1}$, we claim that

$\|\tilde{\mathcal{E}}_{\nu}f\|_{L^{p}(B_{R})}\leq\tilde{\mathcal{D}}_{p}\big{(}\tfrac{R}{K}\big{)}\Big{(}\sum_{\vartheta\subset\nu}\|\tilde{\mathcal{E}}_{\vartheta}f\|^{2}_{L^{p}(w_{B_{R}})}\Big{)}^{1/2},\quad
2\leq p\leq 4,$ (3.26)

where $\vartheta\in\mathcal{F}_{3}(R,2,4).$ Plugging (3.26) into (3.24), we
obtain

$\|\tilde{\mathcal{E}}_{D_{1}}f\|_{L^{p}(B_{R})}\leq
C_{\varepsilon}K^{\varepsilon}\tilde{\mathcal{D}}_{p}\big{(}\tfrac{R}{K}\big{)}\Big{(}\sum_{\vartheta\subset
D_{1}}\|\tilde{\mathcal{E}}_{\vartheta}f\|^{2}_{L^{p}(w_{B_{R}})}\Big{)}^{1/2},\;\;2\leq
p\leq 4.$ (3.27)

Next, we turn to show the claim (3.26). Without loss of generality, we may
assume that $\nu=[0,K^{-1/2}]\times[0,K^{-1/4}]$. Taking the change of
variables

$\xi_{1}=K^{-1/2}\eta_{1},\;\xi_{2}=K^{-1/4}\eta_{2},$

we get

$\|\tilde{\mathcal{E}}_{\nu}f\|^{p}_{L^{p}(B_{R})}=\sigma
K^{2}\|\tilde{\mathcal{E}}_{[0,1]^{2}}\tilde{f}\|^{p}_{L^{p}(\mathcal{T}_{1}(B_{R}))},$

where
$\tilde{f}(\eta_{1},\eta_{2}):=K^{-3/4}f(K^{-1/2}\eta_{1},K^{-1/4}\eta_{2})$.
Here $\mathcal{T}_{1}$ denotes the map

$(x_{1},x_{2},x_{3})\rightarrow(K^{-1/2}x_{1},K^{-1/4}x_{2},K^{-1}x_{3})$

and $\mathcal{T}_{1}(B_{R})$ denotes the image of $B_{R}$ with size roughly as
follows

$K^{-1/2}R\times K^{-1/4}R\times K^{-1}R.$

Using the similar argument as in Case(a) and Case(b) in the proof of (3.15),
one can show that the image of $\vartheta$ belongs to the decomposition
$\mathcal{F}_{3}(\frac{R}{K},2,4)$. We write $\mathcal{T}_{1}(B_{R})$ into a
finitely overlapping union of balls as follows

$\mathcal{T}_{1}(B_{R})=\bigcup B_{R/K}.$

By the definition of $\tilde{\mathcal{D}}_{p}(\cdot),$ we obtain

$\|\tilde{\mathcal{E}}_{[0,1]^{2}}\tilde{f}\|_{L^{p}(B_{R/K})}\leq\tilde{\mathcal{D}}_{p}\big{(}\tfrac{R}{K}\big{)}\Big{(}\sum_{\vartheta\in\mathcal{F}_{3}(\frac{R}{K},2,4)}\|\tilde{\mathcal{E}}_{\vartheta}\tilde{f}\|^{2}_{L^{p}(w_{B_{R/K}})}\Big{)}^{1/2}.$

Summing over all the balls $B_{R/K}\subset\mathcal{T}_{1}(B_{R})$ and using
Minkowski’s inequality, we have

$\|\tilde{\mathcal{E}}_{[0,1]^{2}}\tilde{f}\|_{L^{p}(\mathcal{T}_{1}(B_{R}))}\leq
C_{\varepsilon}R^{\varepsilon}\Big{(}\sum_{\vartheta\in\mathcal{F}_{3}(\frac{R}{K},2,4)}\|\tilde{\mathcal{E}}_{\vartheta}\tilde{f}\|^{2}_{L^{p}(\mathcal{T}_{1}(B_{R}))}\Big{)}^{1/2}.$

Taking the inverse change of variables, we obtain

$\|\tilde{\mathcal{E}}_{\nu}f\|_{L^{p}(B_{R})}\leq\tilde{\mathcal{D}}_{p}\big{(}\tfrac{R}{K}\big{)}\Big{(}\sum_{\vartheta\subset\nu}\|\tilde{\mathcal{E}}_{\vartheta}f\|^{2}_{L^{p}(w_{B_{R}})}\Big{)}^{1/2}.$

Hence, we obtain the claim (3.26).

Using (3.18), (3.22) and (3.27), we derive

$\|\tilde{\mathcal{E}}_{[0,1]^{2}}f\|_{L^{p}(B_{R})}\leq\Big{(}C(\varepsilon)K^{O(1)}R^{\varepsilon}+C_{\varepsilon}K^{\varepsilon}\tilde{\mathcal{D}}_{p}\big{(}\tfrac{R}{K}\big{)}\Big{)}\Big{(}\sum_{\vartheta\subset\mathcal{F}_{3}(R,2,4)}\|\tilde{\mathcal{E}}_{\vartheta}f\|^{2}_{L^{p}(w_{B_{R}})}\Big{)}^{1/2}.$

This inequality together with the definition of $\tilde{\mathcal{D}}_{p}(R)$
yields

$\tilde{\mathcal{D}}_{p}(R)\leq
C(\varepsilon)K^{O(1)}R^{\varepsilon}+C_{\varepsilon}K^{\varepsilon}\tilde{\mathcal{D}}_{p}\big{(}\tfrac{R}{K}\big{)}.$

Iterating the above inequality $m=[\log_{K}R]$ times, we derive that
$\tilde{\mathcal{D}}_{p}(R)\lesssim_{\varepsilon}R^{\varepsilon}$.

Therefore, we complete the proof of Lemma 3.5. ∎

######  Remark 3.7.

The argument in this paper also implies the following result. Let $2\leq p\leq
4$ and $m\geq 4$ be an even number. For any $\varepsilon>0$ , there holds

$\|\mathcal{E}_{[0,1]^{2}}g\|_{L^{p}(B_{R})}\leq
C(\varepsilon,p)R^{\varepsilon}\Big{(}\sum_{\theta\in\mathcal{F}_{3}(R,m)}\|\mathcal{E}_{\theta}g\|^{2}_{L^{p}(w_{B_{R}})}\Big{)}^{1/2},$

where

$\mathcal{E}_{Q}g(x)=\int_{Q}g(\xi_{1},\xi_{2})e(x_{1}\xi_{1}+x_{2}\xi_{2}+x_{3}(\xi_{1}^{m}+\xi_{2}^{m}))\;d\xi_{1}\;d\xi_{2}.$

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