#  $L^{2}$ Schrödinger maximal estimates associated with finite type phases in
$\mathbb{R}^{2}$

Zhuoran Li1, Junyan Zhao2∗ and Tengfei Zhao3 1 The Graduate School of China
Academy of Engineering Physics, P. O. Box 2101, Beijing 100088, P.R. China.
<EMAIL_ADDRESS>2College of Mathematics and Computer Science,
Zhejiang Normal University, Jinhua 321004, P.R. China<EMAIL_ADDRESS>3School of Mathematics and Physics, University of Science and Technology
Beijing, Beijing 100083, P.R. China<EMAIL_ADDRESS>
(Date: Received: xxxxxx; Revised: yyyyyy; Accepted: zzzzzz.
∗Corresponding author)

###### Abstract.

In this paper, we establish Schrödinger maximal estimates associated with the
finite type phases

$\phi(\xi_{1},\xi_{2}):=\xi^{m}_{1}+\xi^{m}_{2},\;(\xi_{1},\xi_{2})\in[0,1]^{2},$

where $m\geq 4$ is an even number. Following [12], we prove an $L^{2}$ fractal
restriction estimate associated with the surfaces

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

as the main result, which also gives results on the average Fourier decay of
fractal measures associated with these surfaces. The key ingredients of the
proof include the rescaling technique from [16], Bourgain-Demeter’s $\ell^{2}$
decoupling inequality, the reduction of dimension arguments from [17] and
induction on scales.

###### Key words and phrases:

pointwise convergence, finite type, decoupling, reduction of dimension
arguments.

###### 2010 Mathematics Subject Classification:

Primary 42B10; Secondary 42B37.

##  1\. Introduction

The solution to the Cauchy problem of the free Schrödinger equation

$\left\\{\begin{array}[]{l}i\partial_{t}u-\Delta
u=0,\,(x,t)\in\mathbb{R}^{n}\times\mathbb{R}\,,\\\ u(x,0)=f(x)\\\
\end{array}\right.$

is given by

$e^{it\Delta}f(x)=(2\pi)^{-n}\int_{\mathbb{R}^{n}}\hat{f}(\xi)e(x\cdot\xi+t|\xi|^{2})d\xi,$

where $e(t):=e^{it}$.

Carleson in [6] proposed the problem of determining the optimal $s$ for which

$\lim_{t\rightarrow 0}e^{it\Delta}f(x)=f(x)$

almost everywhere, whenever $f\in H^{s}(\mathbb{R}^{n})$. The problem has
attracted many authors. In particular, Lee [15] proved that almost everywhere
convergence holds if $s>\frac{3}{8}$ when $n=2$. For general $n$, Bourgain [4]
gave surprising counterexamples showing that the convergence fails if
$s<\frac{n}{2(n+1)}$. Later, Du, Guth and Li [9] proved that the convergence
holds for $s>\frac{1}{3}$ when $n=2$, which combining with Bourgain’s
necessary condition solves the Carleson’s problem when $n=2,$ except at the
endpoint $s=\frac{1}{3}$. In higher dimensions ($n\geq 3$), Du and Zhang [12]
proved that almost everywhere convergence holds in the sharp range
$s>\frac{n}{2(n+1)}$. Thus, Carleson’s problem is completely solved for any
dimension $n,$ except at the endpoint $\frac{n}{2(n+1)}$. For the study of
pointwise convergence problem on 2D fractional order Schrödinger operators, we
refer to Miao, Yang and Zheng [20]. See also C. Cho and H. Ko [7].

Let $m\geq 4$ be an even number. The solution to the Cauchy problem of the
following generalized Schrödinger equation

$\left\\{\begin{array}[]{l}i\partial_{t}u-(\partial^{m}_{1}u+\partial^{m}_{2}u)=0,\,(x,t)\in\mathbb{R}^{2}\times\mathbb{R}\,,\\\
u(x,0)=f(x)\\\ \end{array}\right.$

is given by

$e^{it\phi(D)}f(x)=(2\pi)^{-2}\int_{\mathbb{R}^{2}}\hat{f}(\xi)e[x_{1}\xi_{1}+x_{2}\xi_{2}+t\phi(\xi)]d\xi,$

where $\phi(\xi_{1},\xi_{2}):=\xi^{m}_{1}+\xi^{m}_{2}$.

We note that the Gaussian curvature of the surfaces associated with
$\phi(\xi)$ vanishes when $\xi_{1}=0$ or $\xi_{2}=0$, which is different from
those non-degenerate cases in the literature. To our knowledge, there are few
works on the sufficient conditions for the almost everywhere convergence
problem associated with such degenerate phases. For the study of necessary
conditions, we refer to [1, 13].

In this article, we establish the following result.

######  Theorem 1.1.

For every $f\in H^{s}(\mathbb{R}^{2})$ with $s>\frac{1}{2}-\frac{1}{3m}$,
$\displaystyle\lim_{t\rightarrow 0}e^{it\phi(D)}f(x)=f(x)$ almost everywhere.

We use $B^{d}(x,r)$ to denote the ball centered at $x$ with radius $r$ in
$\mathbb{R}^{d}$. By a standard approximation argument, Theorem 1.1 is a
consequence of the following Schrödinger maximal estimate associated with
$\phi$:

######  Theorem 1.2.

For any $s>\frac{1}{2}-\frac{1}{3m}$, there holds

$\|\sup_{0<t\leq 1}|e^{it\phi(D)}f|\|_{L^{2}(B^{2}(0,1))}\leq
C_{s}\|f\|_{H^{s}(\mathbb{R}^{2})}$

for any function $f\in H^{s}(\mathbb{R}^{2})$.

By Lemma 2.1 in [8], the Littlewood-Paley decomposition and a rescaling
argument, Theorem 1.2 is reduced to the following result.

######  Theorem 1.3.

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

$\|\sup_{0<t\leq R}|e^{it\phi(D)}f|\|_{L^{2}(B^{2}(0,R))}\leq
C_{\varepsilon}R^{\frac{1}{2}-\frac{1}{3m}+\varepsilon}\|f\|_{2}$

holds for all $R\geq 1$ and all $f$ with $supp\hat{f}\subset
A(1):=[0,1]^{2}-[0,\frac{1}{2}]^{2}$.

######  Remark 1.4.

If $m$ is odd, we can not reduce
$supp\hat{f}\subset[-1,1]^{2}-[-\frac{1}{2},\frac{1}{2}]^{2}$ to
$supp\hat{f}\subset[0,1]^{2}-[0,\frac{1}{2}]^{2}$. To see it, let us consider
the case
$supp\hat{f}\subset[\frac{1}{2},1]\times[-1,-\frac{1}{2}]=:A(1^{\prime})$.
Taking the change of variables

$\xi_{1}=\eta_{1},\xi_{2}=-\eta_{2},$

the phase function $\phi(\eta_{1},-\eta_{2}):=\eta^{m}_{1}-\eta^{m}_{2}$ can
be viewed as small perturbations of
$\phi_{0}(\eta_{1},\eta_{2}):=\eta^{2}_{1}-\eta^{2}_{2}$, where
$(\eta_{1},\eta_{2})\in[0,1]^{2}$. But we know from [21] that for any $f\in
H^{s}(\mathbb{R}^{2})$

$\lim_{t\rightarrow 0}e^{it\phi_{0}(D)}f(x)=f(x),\quad a.e.\
x\in\mathbb{R}^{2}$

fails for $s<\frac{1}{2}$. For this reason we only consider that $m$ is even
in the current paper.

Cubes of the form $m+[0,M]^{3}$ with $m\in(M\mathbb{Z})^{3}$ are called
lattice $M-$cubes. We will prove the following $L^{2}$ restriction estimate,
from which Theorem 1.3 follows.

######  Theorem 1.5.

Suppose that $X=\bigcup_{k}B_{k}$ is a union of lattice unit cubes in
$Q_{R}:=[0,R]^{3}$ and each $R^{\frac{1}{2}}$-cube intersecting $X$ contains
$\sim\nu$ many unit cubes in $X$. Let $1\leq\alpha\leq 3$ and $\lambda$ be

$\lambda:=\max_{B^{3}(x^{\prime},r)\subset
Q_{R},x^{\prime}\in\mathbb{R}^{3},r\geq 1}\frac{\sharp\\{B_{k}:B_{k}\subset
B^{3}(x^{\prime},r)\\}}{r^{\alpha}}.$

For any $\varepsilon>0$, there exists a positive constant $C_{\varepsilon}$
such that the following inequalities hold for all $R\geq 1:$

$\|e^{it\phi(D)}f\|_{L^{2}(X)}\leq
C_{\varepsilon}\lambda^{\frac{1}{6}}\nu^{\frac{1}{6}}R^{\frac{1}{3}-\frac{4-\alpha}{6m}+\varepsilon}\|f\|_{2}$
(1.1)

for all $f$ with $supp\hat{f}\subset A(1)$, and

$\|e^{it\phi(D)}f\|_{L^{2}(X)}\leq
C_{\varepsilon}\lambda^{\frac{1}{6}}\nu^{\frac{1}{6}}R^{\frac{5}{12}-\frac{5-\alpha}{6m}+\varepsilon}\|f\|_{2}$
(1.2)

for all $f$ with $supp\hat{f}\subset[0,1]^{2}$.

######  Remark 1.6.

To deduce Theorem 1.3, (1.1) is sufficient. We only use (1.2) in Section 4.
The two different behaviors in (1.1) and (1.2) depend on whether the support
of $\hat{f}$ contains the origin or not. When the support of $\hat{f}$ does
not contain the origin, at most one principal curvature of the surface
vanishes. If the support of $\hat{f}$ contains the origin, both principal
curvatures of the surface studied vanish. Therefore, we deduce a better result
in (1.1) than in (1.2).

Clearly, one has $\nu\leq\lambda R^{\alpha/2}$ in Theorem 1.5. As a direct
result of Theorem 1.5, there holds a weaker $L^{2}$ restriction estimate:

######  Corollary 1.7.

Suppose that $X=\bigcup_{k}B_{k}$ is a union of lattice unit cubes in $Q_{R}$.
Let $1\leq\alpha\leq 3$ and $\lambda$ be

$\lambda:=\max_{B^{3}(x^{\prime},r)\subset
Q_{R},x^{\prime}\in\mathbb{R}^{3},r\geq 1}\frac{\sharp\\{B_{k}:B_{k}\subset
B^{3}(x^{\prime},r)\\}}{r^{\alpha}}.$

For any $\varepsilon>0$, there exists a positive constant $C_{\varepsilon}$
such that the following inequalities hold for all $R\geq 1$

$\|e^{it\phi(D)}f\|_{L^{2}(X)}\leq
C_{\varepsilon}\lambda^{\frac{1}{3}}R^{\frac{1}{3}+\frac{\alpha}{12}-\frac{4-\alpha}{6m}+\varepsilon}\|f\|_{2}$

for all $f$ with $supp\hat{f}\subset A(1)$, and

$\|e^{it\phi(D)}f\|_{L^{2}(X)}\leq
C_{\varepsilon}\lambda^{\frac{1}{3}}R^{\frac{5}{12}+\frac{\alpha}{12}-\frac{5-\alpha}{6m}+\varepsilon}\|f\|_{2}$

for all $f$ with $supp\hat{f}\subset[0,1]^{2}$.

Corollary 1.7 is sufficient to derive the $L^{2}$ Schrö dinger maximal
estimate (Theorem 1.3). For the details, we refer to the proof of Theorem 1.3
by Corollary 1.7 in Du-Zhang [12]. Theorem 1.5 is based on Proposition 1.8
below, which will be proved by an induction.

A collection of quantities are said to be essentially constant provided that
all the quantities lie in the same interval of the form $[2^{n},2^{n+1}]$,
where $n\in\mathbb{Z}$.

######  Proposition 1.8.

Suppose that $Y=\bigcup_{k=1}^{N}B_{k}$ is a union of lattice $K$-cubes in
$Q_{R}$ and each $R^{\frac{1}{2}}$-cube intersecting $Y$ contains $\sim\nu$
many $K$-cubes in $Y$, where $K=R^{\delta}$. Suppose that
$\|e^{it\phi(D)}f\|_{L^{6}(B_{k})}$ is essentially constant in $k=1,2,...,N$.
Let $1\leq\alpha\leq 3$ and $\lambda$ be

$\lambda:=\max_{B^{3}(x^{\prime},r)\subset
Q_{R},x^{\prime}\in\mathbb{R}^{3},r\geq K}\frac{\sharp\\{B_{k}:B_{k}\subset
B^{3}(x^{\prime},r)\\}}{r^{\alpha}}.$

For any $\varepsilon>0$, there exists a positive constant $C_{\varepsilon}$
and $\delta=\varepsilon^{100}$ such that the following inequalities hold for
all $R\geq 1:$

$\|e^{it\phi(D)}f\|_{L^{6}(Y)}\leq
C_{\varepsilon}N^{-\frac{1}{3}}\lambda^{\frac{1}{6}}\nu^{\frac{1}{6}}R^{\frac{1}{3}-\frac{4-\alpha}{6m}+\varepsilon}\|f\|_{2}$

for all $f$ with $supp\hat{f}\subset A(1)$, and

$\|e^{it\phi(D)}f\|_{L^{6}(Y)}\leq
C_{\varepsilon}N^{-\frac{1}{3}}\lambda^{\frac{1}{6}}\nu^{\frac{1}{6}}R^{\frac{5}{12}-\frac{5-\alpha}{6m}+\varepsilon}\|f\|_{2}$

for all $f$ with $supp\hat{f}\subset[0,1]^{2}$.

######  Remark 1.9.

In the proof of Proposition 1.8, we will use the $\ell^{2}$ decoupling
inequalities for the perturbed parabola and for the curve

$\\{(s,s^{m}):s\in[0,1]\\}.$

This is already known due to [3, 23]. To establish Proposition 1.8 in higher
dimensions ($n\geq 3$) with the methods of Du-Zhang [12] and the current
paper, one needs the $\ell^{2}$ decoupling inequalities for the hypersurfaces

$\Big{\\{}(\xi_{1},...,\xi_{n-1},\phi_{1}(\xi_{1})+...+\phi_{s}(\xi_{s})+\xi_{s+1}^{m}+...+\xi_{n-1}^{m}):(\xi_{1},...,\xi_{n-1})\in[0,1]^{n-1}\Big{\\}},$

$0\leq s\leq n-1$, with $\phi_{1},...,\phi_{s}$ being non-degenerate. With
these in hand, one can obtain pointwise convergence results associated with
corresponding finite type phases in higher dimensions. This is achieved in a
recent preprint [14].

From [12], we know that Proposition 1.8 implies Theorem 1.5, and therefore it
suffices to prove Proposition 1.8. Now we outline the proof of Proposition
1.8.

Outline of the proof of Proposition 1.8. We use $F_{m}^{2}$ to denote the
surface

$\\{(\xi_{2},\xi_{2},\xi_{1}^{m}+\xi_{2}^{m}):(\xi_{1},\xi_{2})\in[0,1]^{2}\\}.$

The surface $F_{m}^{2}$ is badly behaved when $\xi_{1}=0$ or $\xi_{2}=0$. The
strategy is to single out small neighborhoods of these two lines where we
apply reduction of dimension arguments. More precisely, we first divide
$[0,1]^{2}$ into $\bigcup_{j=0}^{3}\Omega_{j}$, where

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

Here $K=R^{\delta}$ is a large number. Denote $\hat{f}\mid_{\Omega_{j}}$ by
$\hat{f}_{\Omega_{j}}$ for $j=0,1,2,3$. Then, we have

$\|e^{it\phi(D)}f\|_{L^{6}(Y)}\leq\sum_{j=0}^{3}\|e^{it\phi(D)}f_{\Omega_{j}}\|_{L^{6}(Y)}.$
(1.3)

We will estimate $\|e^{it\phi(D)}f\|_{L^{6}(Y)}$ in several cases. In (1.3)
one of the

$\|e^{it\phi(D)}f_{\Omega_{j}}\|_{L^{6}(Y)}$

will dominate, and we call it $\Omega_{j}$-case.

We first consider the $\Omega_{0}$-case. The surface

$\\{(\xi_{2},\xi_{2},\xi^{m}_{1}+\xi^{m}_{2}):(\xi_{1},\xi_{2})\in\Omega_{0}\\}$

has two positive principal curvatures with lower bound $K^{-C_{0}}$ for some
constant $C_{0}$. For $(\xi_{1},\xi_{2})\in\Omega_{0}$, our phase function
$\phi(\xi_{1},\xi_{2})$ can be viewed as perturbations of
$\xi^{2}_{1}+\xi^{2}_{2}$. By Proposition 3.1 of Du-Zhang [12], in the
$\Omega_{0}$-case we have

$\|e^{it\phi(D)}f\|_{L^{6}(Y)}\leq
C_{\varepsilon}K^{O(1)}N^{-\frac{1}{3}}\lambda^{\frac{1}{6}}\nu^{\frac{1}{6}}R^{\frac{\alpha}{12}+\varepsilon}\|f\|_{2},$

where $K^{O(1)}$ is a fixed power of $K$, and the details will be given in
Section 2. For simplicity, we assume that $supp\hat{f}\subset A(1)$. Then,
$\hat{f}_{\Omega_{3}}=0$. It remains to discuss the $\Omega_{1}$-case and the
$\Omega_{2}$-case. By symmetry, it suffices to treat the $\Omega_{1}$-case.
Since $supp\hat{f}\subset A(1)$, we only need to consider the following
subregion of $\Omega_{1}$:

$\tilde{\Omega}_{1}:=[\frac{1}{2},1]\times[0,K^{-1/m}].$

We will adapt Bourgain-Demeter’s $\ell^{2}$ decoupling inequality [5] to our
needs and reduce the problem to each $K^{-1/2}\times K^{-1/m}$ -rectangle
$\tau$. Now it is natural to employ certain rescaling technique. The main
difficulty is that the phase function

$\phi(\xi_{1},\xi_{2}):=\xi^{m}_{1}+\xi^{m}_{2},\;(\xi_{1},\xi_{2})\in\tau\subset\tilde{\Omega}_{1}$

is not closed under the change of variables

$\left\\{\begin{array}[]{l}\xi_{1}=a+K^{-1/2}\eta_{1},\\\
\xi_{2}=K^{-1/m}\eta_{2}.\end{array}\right.$

In fact, it becomes a new phase function

$\phi(a+K^{-1/2}\eta_{1},K^{-1/m}\eta_{2})\sim\frac{1}{K}(\phi_{1}(\eta_{1})+\eta_{2}^{m})=\frac{1}{K}\psi(\eta_{1},\eta_{2}),$

where

$\psi(\eta_{1},\eta_{2}):=\phi_{1}(\eta_{1})+\eta_{2}^{m},\;(\eta_{1},\eta_{2})\in[0,1]^{2}$

satisfying

$\phi_{1}^{(\prime\prime)}\sim 1;\quad|\phi_{1}^{(k)}|\lesssim 1,3\leq k\leq
m;\quad\phi_{1}^{(l)}=0,l\geq m+1$

on the interval $[0,1]$. To overcome the difficulty, we establish a fractal
$L^{2}$ restriction estimate associated with the new phase function $\psi$ as
an auxiliary proposition (Proposition 2.1), whose proof is given in Section 3.
The key observation is that the new phase function

$\psi(\eta_{1},\eta_{2}),\;(\eta_{1},\eta_{2})\in[0,K^{-1/2}]\times[0,K^{-1/m}]$

is closed under the change of variable

$\left\\{\begin{array}[]{l}\xi_{1}=K^{-\frac{1}{2}}\eta_{1},\\\
\xi_{2}=K^{-\frac{1}{m}}\eta_{2}.\end{array}\right.$

With this in hand, we can deduce the desired estimate in the $\Omega_{1}$
-case. This completes the outline of the proof of Proposition 1.8.

The paper is organized as follows. In Section 2 and Section 3, we give the
proof of Proposition 1.8 by combining the methods in [12] and the reduction of
dimension arguments from [17]. In section 4, we will give two applications of
Corollary 1.7.

Notations: For nonnegative quantities $X$ and $Y$, we will write $X\lesssim Y$
to denote the estimate $X\leq CY$ for some large constant $C$ which may vary
from line to line and depend on various parameters. 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 $C=C(u)$. For any set $E\subset\mathbb{R}^{d}$, we use
$\chi_{E}$ to denote the characteristic function on $E$. Usually, Fourier
transform on $\mathbb{R}^{d}$ is defined by

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

##  2\. The Proof of Proposition 1.8

Recall that $[0,1]^{2}$ is divided into $\bigcup_{j=0}^{3}\Omega_{j}$, where

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

We denote by

$\Sigma_{j}:=\\{\xi_{1},\xi_{2},\phi(\xi_{1},\xi_{2}):\;(\xi_{1},\xi_{2})\in\Omega_{j}\\}$

with $j\in\\{0,1,2,3\\}$. For technical reasons, $K^{-\frac{1}{m}}$,
$R^{-\frac{1}{m}}$ and $(\frac{R}{K})^{-1/m}$ should be dyadic numbers.
Therefore, we choose $K=2^{ml}$ and $R=2^{mt}$ $(l,t\in\mathbb{N})$ to be
large numbers satisfying $K\approx R^{\delta}$ with $\delta$ being in
Proposition 1.8. Recall that in (1.3) we have

$\|e^{it\phi(D)}f\|_{L^{6}(Y)}\leq\sum_{j=0}^{3}\|e^{it\phi(D)}f_{\Omega_{j}}\|_{L^{6}(Y)}.$

We will estimate $\|e^{it\phi(D)}f\|_{L^{6}(Y)}$ in several cases. If the
contribution from the region $\Omega_{j}$ dominates, we call it
$\Omega_{j}$-case. By the triangle inequality, we have

$\|e^{it\phi(D)}f_{\Omega_{j}}\|_{L^{6}(B)}\geq\frac{1}{16}\|e^{it\phi(D)}f\|_{L^{6}(B)}$

for at least one of the $j$’s.

Now we sort the $K$-cubes in $Y$. Denote

$\\{B\subset
Y:\|e^{it\phi(D)}f_{\Omega_{j}}\|_{L^{6}(B)}\geq\frac{1}{16}\|e^{it\phi(D)}f\|_{L^{6}(B)}\\}$

by $Y^{j}$. Clearly, one has $Y=\bigcup^{3}_{j=0}Y^{j}$. If
$\sharp\\{B:B\subset Y^{j}\\}\geq\frac{N}{4}$, we call it $\Omega_{j}$-case.

For the $\Omega_{0}$-case, we sort the $K$-cubes in $Y^{0}$ as follows:

  1. (1)

For any dyadic number $A^{(0)}$, let $Y^{0}_{A^{(0)}}$ be the union of the
$K$-cubes in $Y^{0}$ satisfying

$\|e^{it\phi(D)}f_{\Omega_{0}}\|_{L^{6}(B)}\approx A^{(0)}.$

  2. (2)

Fix $A^{(0)}$, for any dyadic number $\nu^{(0)}$, let
$Y^{0}_{A^{(0)},\nu^{(0)}}$ be the union of the $K$-cubes in $Y^{0}_{A^{(0)}}$
such that for each $B\subset Y^{0}_{A^{(0)},\nu^{(0)}}$, the $R^{1/2}$-cube
intersecting $B$ contains $\approx\nu^{(0)}$ cubes from $Y^{0}_{A^{(0)}}$.

Without loss of generality, we may assume $\|f\|_{2}=1$. Here $\nu^{(0)}$
satisfies $1\leq\nu^{(0)}\lesssim R^{3}$, and the dyadic number $A^{(0)}$
making significant contributions can be assumed to be between $R^{-C}$ and
$R^{C}$ for a large constant $C$. In fact, if we denote

$\\{B\subset Y^{0}:\|e^{it\phi(D)}f_{\Omega_{0}}\|_{L^{6}(B)}\leq R^{-100}\\}$

by $Y_{-1}^{0}$, then

$\|e^{it\phi(D)}f_{\Omega_{0}}\|_{L^{6}(Y_{-1}^{0})}\lesssim
R^{-90}\|f\|_{2}.$

Note that

$N^{(0)}\lesssim R^{3},\lambda^{(0)}\geq R^{-3},\nu^{(0)}\geq 1,\alpha>0$

and $K=R^{\delta},\delta=\varepsilon^{100}$, where

$N^{(0)}:=\sharp\\{B:B\subset Y_{A^{(0)},\nu^{(0)}}^{0}\\}$

and

$\lambda^{(0)}:=\max_{B^{3}(x^{\prime},r)\subset
Q_{R},x^{\prime}\in\mathbb{R}^{3},r\geq 1}\frac{\sharp\\{B_{k}\subset
Y^{0}:B_{k}\subset B^{3}(x^{\prime},r)\\}}{r^{\alpha}}.$

So we have

$\|e^{it\phi(D)}f_{\Omega_{0}}\|_{L^{6}(Y_{-1}^{0})}\leq
C_{\varepsilon}K^{O(1)}(N^{(0)})^{-\frac{1}{3}}(\lambda^{(0)})^{\frac{1}{6}}(\nu^{(0)})^{\frac{1}{6}}R^{\frac{\alpha}{12}+\varepsilon}\|f\|_{2}.$

Therefore, there exist some dyadic numbers $A^{(0)},\nu^{(0)}$ such that

$N^{(0)}\gtrsim\frac{N}{(\log R)^{2}}.$

Fix that choice of $A^{(0)},\nu^{(0)}$ and denote $Y_{A^{(0)},\nu^{(0)}}^{0}$
by $Y^{0}$ for convenience. Since $\|e^{it\phi(D)}f\|_{L^{6}(B_{k})}$ is
essentially constant in $k=1,2,...,N$ and $\frac{N}{(\log R)^{2}}\lesssim
N^{(0)}$, we have

$\|e^{it\phi(D)}f\|_{L^{6}(Y)}\lesssim(\log
R)^{O(1)}\|e^{it\phi(D)}f\|_{L^{6}(Y^{0})}.$ (2.1)

Note that

$\|e^{it\phi(D)}f_{\Omega_{0}}\|_{L^{6}(B)}\geq\frac{1}{16}\|e^{it\phi(D)}f\|_{L^{6}(B)}$

for $B\subset Y^{0}.$ It follows that

$\|e^{it\phi(D)}f\|_{L^{6}(Y^{0})}\lesssim\|e^{it\phi(D)}f_{\Omega_{0}}\|_{L^{6}(Y^{0})}.$

This together with (2.1) gives

$\|e^{it\phi(D)}f\|_{L^{6}(Y)}\lesssim_{\varepsilon}R^{\varepsilon}\|e^{it\phi(D)}f_{\Omega_{0}}\|_{L^{6}(Y^{0})}$
(2.2)

in the $\Omega_{0}$-case. We also have

$\frac{N}{(\log R)^{2}}\lesssim N^{(0)}\leq
N,\quad\nu^{(0)}\leq\nu,\quad\lambda^{(0)}\leq\lambda.$ (2.3)

Using the argument from [12] as well as the rescaling trick from Case (c) in
Subsection 2.2 of [16], we claim

$\|e^{it\phi(D)}f_{\Omega_{0}}\|_{L^{6}(Y^{0})}\leq
C_{\varepsilon}K^{O(1)}(N^{(0)})^{-\frac{1}{3}}(\lambda^{(0)})^{\frac{1}{6}}(\nu^{(0)})^{\frac{1}{6}}R^{\frac{\alpha}{12}+\varepsilon}\|f\|_{2}.$
(2.4)

Note that
$\min\\{\frac{5}{12}-\frac{5-\alpha}{6m},\frac{1}{3}-\frac{4-\alpha}{6m}\\}\geq\frac{\alpha}{12}$
for any $\alpha\in(0,3]$ and each $m\geq 2$. The factor $K^{O(1)}$ appears on
the right-hand side of (2.4) because the principal curvature (in the direction
of $\xi_{2}$-axis) of the surface

$\Sigma_{0}:=\\{(\xi_{1},\xi_{2},\phi(\xi_{1},\xi_{2})):(\xi_{1},\xi_{2})\in\Omega_{0}\\}$

has a lower bound $K^{-C}$. Inserting (2.3) into (2.4), we complete the proof
of Proposition 1.8 in the $\Omega_{0}$-case.

Now we turn to prove (2.4). We divide the region $\Omega_{0}$ into a family of
subregions

$\Omega_{0}=\bigcup_{\sigma_{1},\sigma_{2}}\Omega_{\sigma_{1},\sigma_{2}}$

for dyadic numbers $\sigma_{i}\in[K^{-1/m},\frac{1}{2}]$ with $i\in\\{1,2\\}$.
Here $\Omega_{\sigma_{1},\sigma_{2}}:=I_{\sigma_{1}}\times I_{\sigma_{2}}$
with $I_{\sigma_{i}}=[\sigma_{i},2\sigma_{i}]$. We divide $I_{\sigma_{i}}$
further into

$I_{\sigma_{i}}=\bigcup_{j=1}^{\sigma_{i}^{\frac{m}{2}}K^{1/2}}I_{\sigma_{i},j}.$

Each $I_{\sigma_{i},j}$ has length of $\sigma_{i}^{-\frac{m-2}{2}}K^{-1/2}$ .
So we can write

$\Omega_{\sigma_{1},\sigma_{2}}=\bigcup_{\tau}\tau,$

where each $\tau$ in the above union has the form

$\tau=[a_{1},a_{1}+\sigma_{1}^{-\frac{m-2}{2}}K^{-1/2}]\times[a_{2},a_{2}+\sigma_{2}^{-\frac{m-2}{2}}K^{-1/2}]$

with $\sigma_{i}\leq a_{i}\leq 2\sigma_{i}\;(i=1,2)$. Therefore, we have

$e^{it\phi(D)}f_{\Omega_{0}}=\sum_{\sigma_{1},\sigma_{2}}e^{it\phi(D)}f_{\Omega_{\sigma_{1},\sigma_{2}}},$
(2.5)

and

$e^{it\phi(D)}f_{\Omega_{\sigma_{1},\sigma_{2}}}=\sum_{\tau\subset\Omega_{\sigma_{1},\sigma_{2}}}e^{it\phi(D)}f_{\tau}.$

We will estimate $\|e^{it\phi(D)}f_{\Omega_{0}}\|_{L^{6}(Y^{0})}$ in several
cases. If the contribution from the region $\Omega_{\sigma_{1},\sigma_{2}}$
dominates, we call it $\Omega_{\sigma_{1},\sigma_{2}}$ -case. More precisely,
given a $K$-cube $B$, by the triangle inequality, we have

$\|e^{it\phi(D)}f_{\Omega_{\sigma_{1},\sigma_{2}}}\|_{L^{6}(B)}\gtrsim\frac{\|e^{it\phi(D)}f_{\Omega_{0}}\|_{L^{6}(B)}}{(\log
K)^{2}}$

for at least one pair $(\sigma_{1},\sigma_{2})$.

We sort the $K$-cubes in $Y^{0}$ as follows. Denote by

$Y^{\sigma_{1},\sigma_{2}}:=\\{B\subset
Y^{0}:\;\|e^{it\phi(D)}f_{\Omega_{\sigma_{1},\sigma_{2}}}\|_{L^{6}(B)}\gtrsim\frac{1}{(\log
K)^{2}}\|e^{it\phi(D)}f_{\Omega_{0}}\|_{L^{6}(B)}\\}.$

Clearly, one has $Y=\bigcup_{\sigma_{1},\sigma_{2}}Y^{\sigma_{1},\sigma_{2}}$.
If

$\sharp\\{B:B\subset Y^{\sigma_{1},\sigma_{2}}\\}\gtrsim\frac{N^{(0)}}{(\log
K)^{2}},$

we call it $\Omega_{\sigma_{1},\sigma_{2}}$-case.

For the $\Omega_{\sigma_{1},\sigma_{2}}$-case, we sort the $K$-cubes in
$Y^{\sigma_{1},\sigma_{2}}$ as follows:

  1. (1)

For any dyadic number $A^{(\sigma_{1},\sigma_{2})}$, let
$Y^{\sigma_{1},\sigma_{2}}_{A^{(\sigma_{1},\sigma_{2})}}$ be the union of the
$K$ -cubes in $Y^{\sigma_{1},\sigma_{2}}$ satisfying

$\|e^{it\phi(D)}f_{\Omega_{\sigma_{1},\sigma_{2}}}\|_{L^{6}(B)}\approx
A^{(\sigma_{1},\sigma_{2})}.$

  2. (2)

Fix $A^{(\sigma_{1},\sigma_{2})}$, for any dyadic numbers
$\nu^{(\sigma_{1},\sigma_{2})}$, let
$Y^{\sigma_{1},\sigma_{2}}_{A^{(\sigma_{1},\sigma_{2})},\nu^{(\sigma_{1},\sigma_{2})}}$
be the union of the $K$-cubes in
$Y^{\sigma_{1},\sigma_{2}}_{A^{(\sigma_{1},\sigma_{2})}}$ such that for each
$B\subset
Y^{\sigma_{1},\sigma_{2}}_{A^{(\sigma_{1},\sigma_{2})},\nu^{(\sigma_{1},\sigma_{2})}}$,
the $R^{1/2}$-cube intersecting $B$ contains
$\approx\nu^{(\sigma_{1},\sigma_{2})}$ cubes from
$Y^{\sigma_{1},\sigma_{2}}_{A^{(\sigma_{1},\sigma_{2})}}$.

The dyadic numbers $A^{(\sigma_{1},\sigma_{2})}$ and
$\nu^{(\sigma_{1},\sigma_{2})}$ making significant contribution can be assumed
to be between $R^{-C}$ and $R^{C}$. Therefore, there exist some dyadic numbers
$A^{(\sigma_{1},\sigma_{2})}$ and $\nu^{(\sigma_{1},\sigma_{2})}$ such that

$\sharp\\{B:B\subset
Y_{A^{(\sigma_{1},\sigma_{2})},\nu^{(\sigma_{1},\sigma_{2})}}^{\sigma_{1},\sigma_{2}}\\}=:N^{(\sigma_{1},\sigma_{2})}\gtrsim\frac{N^{(0)}}{(\log
K)^{2}(\log R)^{2}}.$

Fix that choice of $A^{(\sigma_{1},\sigma_{2})},\nu^{(\sigma_{1},\sigma_{2})}$
and denote
$Y_{A^{(\sigma_{1},\sigma_{2})},\nu^{(\sigma_{1},\sigma_{2})}}^{\sigma_{1},\sigma_{2}}$
by $Y^{\sigma_{1},\sigma_{2}}$ for convenience. Then, in the
$\Omega_{\sigma_{1},\sigma_{2}}$ -case we have that

$\|e^{it\phi(D)}f_{\Omega_{0}}\|_{L^{6}(Y^{0})}\lesssim_{\varepsilon}R^{\varepsilon}\|e^{it\phi(D)}f_{\Omega_{\sigma_{1},\sigma_{2}}}\|_{L^{6}(Y^{\sigma_{1},\sigma_{2}})},$

and

$\frac{N^{(0)}}{(\log K)^{2}(\log R)^{2}}\lesssim
N^{(\sigma_{1},\sigma_{2})}\leq
N^{(0)},\quad\nu^{(\sigma_{1},\sigma_{2})}\leq\nu^{(0)},\quad\lambda^{(\sigma_{1},\sigma_{2})}\leq\lambda^{(0)},$

where

$\lambda^{(\sigma_{1},\sigma_{2})}:=\max_{B^{3}(x^{\prime},r)\subset
Q_{R},x^{\prime}\in\mathbb{R}^{3},r\geq K}\frac{\sharp\\{B_{k}\subset
Y^{\sigma_{1},\sigma_{2}}:B_{k}\subset B^{3}(x^{\prime},r)\\}}{r^{\alpha}}.$

We are going to estimate
$\|e^{it\phi(D)}f_{\Omega_{\sigma_{1},\sigma_{2}}}\|_{L^{6}(Y^{\sigma_{1},\sigma_{2}})}$.
By the triangle inequality, one has

$\|e^{it\phi(D)}f_{\Omega_{\sigma_{1},\sigma_{2}}}\|_{L^{6}(B)}\lesssim\sum_{\tau\subset\Omega_{\sigma_{1},\sigma_{2}}}\|e^{it\phi(D)}f_{\tau}\|_{L^{6}(B)}.$
(2.6)

For each $\tau$, we deal with $e^{it\phi(D)}f_{\tau}$ by the rescaling trick
from Case (c) in Subsection 2.2 of [16]. To do it, we need to further
decompose $f_{\tau}$ in physical space and perform dyadic pigeonholing several
times. First we divide the physical square $[0,R]^{2}$ into
$\frac{R}{K^{\frac{1}{2}}\sigma_{1}^{-\frac{m-2}{2}}}\times\frac{R}{K^{\frac{1}{2}}\sigma_{2}^{-\frac{m-2}{2}}}$-rectangles
$D$. For each pair $(\tau,D)$, let $f_{\Box_{\tau,D}}$ be the function formed
by cutting off $f$ on the rectangle $D$ (with a Schwartz tail) in physical
space and the rectangle $\tau$ in Fourier space. Note that
$e^{it\phi(D)}f_{\Box_{\tau,D}}$ is essentially supported on an
$\frac{R}{K^{\frac{1}{2}}\sigma_{1}^{-\frac{m-2}{2}}}\times\frac{R}{K^{\frac{1}{2}}\sigma_{2}^{-\frac{m-2}{2}}}\times
R$-box, which is denoted by $\Box_{\tau,D}$. The long axis of $\Box_{\tau,D}$
is parallel to the normal direction of the surface $\Sigma_{0}$ at the left
bottom corner of $\tau$. For a fixed $\tau$, the different boxes
$\Box_{\tau,D}$ tile $Q_{R}$. We have

$f_{\Omega_{\sigma_{1},\sigma_{2}}}=\sum_{\tau\subset{\Omega_{\sigma_{1},\sigma_{2}}}}\sum_{D}f_{\Box_{\tau,D}},$

and write

$f_{\Omega_{\sigma_{1},\sigma_{2}}}=\sum_{\Box}f_{\Box}$

for simplicity. For each $\tau$, a given $K$-cube $B$ lies in exactly one box
$\Box_{\tau,D}$. Recall that $K=R^{\delta}$, where $\delta=\varepsilon^{100}$.
Denote

$R_{1}:=\frac{R}{K}=R^{1-\delta},\quad
K_{1}=R_{1}^{\delta}=R^{\delta-\delta^{2}}.$

Tile $\Box$ by
$\sigma_{1}^{\frac{m-2}{2}}K^{1/2}K_{1}\times\sigma_{2}^{\frac{m-2}{2}}K^{1/2}K_{1}\times
KK_{1}$-tube $S$, and also tile $\Box$ by
$\frac{R^{1/2}}{\sigma_{2}^{-\frac{m-2}{2}}}\times\frac{R^{1/2}}{\sigma_{2}^{-\frac{m-2}{2}}}\times
K^{\frac{1}{2}}R^{\frac{1}{2}}$-tubes $S^{\prime}$ (all running parallel to
the long axis of $\Box$). After rescaling, the $\Box$ becomes an $R_{1}$-cube,
the tubes $S^{\prime}$ and $S$ become lattice $R_{1}^{\frac{1}{2}}$-cubes and
$K_{1}$-cubes, respectively.

We regroup tubes $S$ and $S^{\prime}$ inside each $\Box$ as follows:

  1. 1)

Sort those tubes $S$ which intersect $Y^{0}$ according to the value
$\|e^{it\phi(D)}f_{\Box}\|_{L^{6}(S)}$ and the number of $K$-cubes contained
in it. For dyadic numbers $\eta,\beta_{1}$, we use
$\mathbb{S}_{\Box,\eta,\beta_{1}}$ to stand for the collection of tubes
$S\subset\Box$ each of which containing $\sim\eta$ $K$-cubes and
$\|e^{it\phi(D)}f_{\Box}\|_{L^{6}(S)}\sim\beta_{1}$.

  2. 2)

For fixed $\eta,\beta_{1}$, we sort the tubes $S^{\prime}\subset\Box$
according to the number of the tubes $S\in\mathbb{S}_{\Box,\eta,\beta_{1}}$
contained in it. For dyadic number $\nu_{1}$, let
$\mathbb{S}_{\Box,\eta,\beta_{1},\nu_{1}}$ be the subcollection of
$\mathbb{S}_{\Box,\eta,\beta_{1}}$ such that for each
$S\in\mathbb{S}_{\Box,\eta,\beta_{1},\nu_{1}}$, the tube $S^{\prime}$
containing $S$ contains $\sim\nu_{1}$ tubes from
$\mathbb{S}_{\Box,\eta,\beta_{1}}$.

  3. 3)

For fixed $\eta,\beta_{1},\nu_{1}$, we sort the boxes $\Box$ according to the
value $\|f\|_{2}$, the number $\sharp\mathbb{S}_{\Box,\eta,\beta_{1},\nu_{1}}$
and the value $\lambda_{1}$ defined below. For dyadic numbers
$\beta_{2},N_{1},\lambda_{1}$, let
$\mathbb{B}_{\eta,\beta_{1},\nu_{1},\beta_{2},M_{1},\lambda_{1}}$ denote the
collection of boxes $\Box$ satisfying that

$\|f\|_{2}\sim\beta_{2},\quad\sharp\mathbb{S}_{\Box,\eta,\beta_{1},\nu_{1}}\sim
N_{1}$

and

$\max_{T_{r}\subset\Box:r\geq
K_{1}}\frac{\sharp\\{S\in\mathbb{S}_{\Box,\eta,\beta_{1},\nu_{1}}:S\subset
T_{r}\\}}{r^{2}}\sim\lambda_{1},$ (2.7)

where $T_{r}$ are
$\sigma_{2}^{\frac{m-2}{2}}K^{\frac{1}{2}}r\times\sigma_{2}^{\frac{m-2}{2}}K^{\frac{1}{2}}r\times
Kr$-tubes in $\Box$ running parallel to the long axis of $\Box$.

Let $Y_{\Box,\eta,\beta_{1},\nu_{1}}^{\sigma_{1},\sigma_{2}}$ denote
$\\{S:S\subset\mathbb{S}_{\Box,\eta,\beta_{1},\nu_{1}}\\}$ and
$\chi_{Y_{\Box,\eta,\beta_{1},\nu_{1}}^{\sigma_{1},\sigma_{2}}}$ be the
corresponding characteristic function. Without loss of generality, we can
assume $\|f\|_{2}=1$. Therefore, there are only $O(logR)$ significant choices
for each dyadic number. By pigeonholing, we can choose
$\eta,\beta_{1},\nu_{1},\beta_{2},N_{1},\lambda_{1}$ such that

$\begin{split}&\quad\|e^{it\phi(D)}f_{\Omega_{\sigma_{1},\sigma_{2}}}\|_{L^{6}(B)}\\\
&\lesssim(logR)^{6}K^{1/2}\Big{(}\sum_{\Box\in\mathbb{B}_{\eta,\beta_{1},\nu_{1},\beta_{2},N_{1},\lambda_{1}},B\subset
Y_{\Box,\eta,\beta_{1},\nu_{1}}}\|e^{it\phi(D)}f_{\Box}\|_{L^{6}(\omega_{B})}^{2}\Big{)}^{1/2}\end{split}$
(2.8)

holds for a fraction $\gtrsim(logR)^{-6}$ of all $K$-cubes $B$, where we have
used the fact that

$\sharp\\{\tau:\tau\subset\Omega_{\sigma_{1},\sigma_{2}}\\}\lesssim K.$

For brevity, we denote by

$Y_{\Box}^{\sigma_{1},\sigma_{2}}:=Y_{\Box,\eta,\beta_{1},\nu_{1}}^{\sigma_{1},\sigma_{2}},\quad\mathbb{B}:=\mathbb{B}_{\eta,\beta_{1},\nu_{1},\beta_{2},N_{1},\lambda_{1}}.$

Finally, we sort the $K$-cubes $B$ satisfying (2.8) by
$\sharp\\{\Box\in\mathbb{B}:B\subset Y_{\Box}^{\sigma_{1},\sigma_{2}}\\}$. Let
$Y^{\prime\sigma_{1},\sigma_{2}}$ be a union of $K$-cubes $B$ obeying

$\|e^{it\phi(D)}f_{\Omega_{\sigma_{1},\sigma_{2}}}\|_{L^{6}(B)}\lesssim(logR)^{6}K^{1/2}\Big{(}\sum_{\Box\in\mathbb{B},B\subset
Y_{\Box}^{0}}\|e^{it\phi(D)}f_{\Box}\|_{L^{6}(\omega_{B})}^{2}\Big{)}^{1/2}$
(2.9)

and

$\sharp\\{\Box\in\mathbb{B}:B\subset
Y_{\Box}^{\sigma_{1},\sigma_{2}}\sim\mu\\}$ (2.10)

for some dyadic number $1\leq\mu\leq K^{O(1)}$. Moreover, the number of $K$
-cubes $B$ in $Y^{\prime}$ is $\gtrsim(logR)^{-7}N$.

By our assumption that
$\|e^{it\phi(D)}f_{\Omega_{\sigma_{1},\sigma_{2}}}\|_{L^{6}(B_{k})}$ is
essentially constant in $k=1,2,\cdots,N^{0}$, we have

$\|e^{it\phi(D)}f_{\Omega_{\sigma_{1},\sigma_{2}}}\|^{6}_{L^{6}(Y^{\sigma_{1},\sigma_{2}})}\lesssim
K^{3}(logR)^{7}\sum_{B\subset
Y^{\prime}}\|e^{it\phi(D)}f_{\Omega_{\sigma_{1},\sigma_{2}}}\|^{6}_{L^{6}(B)}.$
(2.11)

For each $B\subset Y^{\prime}$, it follows from (2.9), (2.10) and Hölder’s
inequality that

$\|e^{it\phi(D)}f_{\Omega_{\sigma_{1},\sigma_{2}}}\|^{6}_{L^{6}(B)}\lesssim(logR)^{36}K^{3}\mu^{2}\Big{(}\sum_{\Box\in\mathbb{B},B\subset
Y^{\sigma_{1},\sigma_{2}}_{\Box}}\|e^{it\phi(D)}f_{\Box}\|^{6}_{L^{6}(\omega_{B})}\Big{)}^{1/2}.$
(2.12)

Combining (2.11) with (2.12), one has

$\|e^{it\phi(D)}f_{\Omega_{\sigma_{1},\sigma_{2}}}\|_{L^{6}(Y^{\sigma_{1},\sigma_{2}})}\lesssim(logR)^{C}K^{1/2}\mu^{1/3}\Big{(}\sum_{\Box\in\mathbb{B}}\|e^{it\phi(D)}f_{\Box}\|^{6}_{L^{6}(Y^{\sigma_{1},\sigma_{2}}_{\Box})}\Big{)}^{1/6}.$
(2.13)

Next, we apply rescaling to each
$\|e^{it\phi(D)}f_{\Box}\|_{L^{6}(Y^{\sigma_{1},\sigma_{2}}_{\Box})}.$ For
each
$\sigma_{1}^{-\frac{m-2}{2}}K^{-1/2}\times\sigma_{2}^{-\frac{m-2}{2}}K^{-1/2}$-rectangle
$\tau=\tau_{\Box}$ in $\Omega_{\sigma_{1},\sigma_{2}}$, we write

$\left\\{\begin{array}[]{l}\xi_{1}=a_{1}+\sigma_{1}^{-\frac{m-2}{2}}K^{-1/2}\eta_{1},\\\
\xi_{2}=a_{2}+\sigma_{2}^{-\frac{m-2}{2}}K^{-1/2}\eta_{2}.\end{array}\right.$

Then

$|e^{it\phi(D)}f_{\Box}(x)|=\sigma_{1}^{-\frac{m-2}{4}}\sigma_{2}^{-\frac{m-2}{4}}K^{-1/2}|e^{i\tilde{t}\tilde{\phi}(D)}g(\tilde{x})|$

with $\|g\|_{2}=\|f\|_{2}$ and

$\hat{g}(\eta_{1},\eta_{2}):=\sigma_{1}^{-\frac{m-2}{4}}\sigma_{2}^{-\frac{m-2}{4}}K^{-1/2}\hat{f}(a_{1}+\sigma_{1}^{-\frac{m-2}{2}}K^{-1/2}\eta_{1},a_{2}+\sigma_{2}^{-\frac{m-2}{2}}K^{-1/2}\eta_{2}),$

where

$\begin{split}&\quad\tilde{\phi}(\eta_{1},\eta_{2})\\\
&:=\sum_{i=1}^{2}\Big{(}\frac{m(m-1)}{2}a_{i}^{m-2}\sigma_{i}^{-(m-2)}\eta^{2}_{i}\\\
&\quad+\frac{m(m-1)(m-2)}{6}a_{i}^{m-3}\sigma_{i}^{-\frac{3(m-2)}{2}}K^{-1/2}\eta^{3}_{i}+\dots+\sigma_{i}^{-\frac{m(m-2)}{2}}K^{-\frac{m-2}{2}}\eta^{m}_{i}\Big{)},\end{split}$

and

$\left\\{\begin{array}[]{l}\tilde{x}_{1}:=\sigma_{1}^{-\frac{m-2}{2}}K^{-1/2}(x_{2}+ma_{1}^{m-1}t),\\\
\tilde{x}_{2}:=\sigma_{2}^{-\frac{m-2}{2}}K^{-1/2}(x_{2}+ma_{2}^{m-1}t),\\\
\tilde{t}:=K^{-1}t.\end{array}\right.$

For brevity, we denote the above relation by
$(\tilde{x},\tilde{t})=\mathcal{L}_{0}(x,t)$. Therefore, we have

$\|e^{it\phi(D)}f_{\Box}\|_{L^{6}(Y^{\sigma_{1},\sigma_{2}}_{\Box})}=\rho^{-\frac{m-2}{6}}K^{-1/6}\|e^{i\tilde{t}\tilde{\phi}(D)}g(\tilde{x})\|_{L^{6}(\tilde{Y}^{\sigma_{1},\sigma_{2}})},$
(2.14)

where
$\tilde{Y}^{\sigma_{1},\sigma_{2}}=\mathcal{L}_{0}(Y^{\sigma_{1},\sigma_{2}}_{\Box})$.
Since $a_{i}\sim\sigma_{i}$ and $\frac{1}{2}\geq\sigma_{i}\geq K^{-1/m}$, it
is easy to check that $\tilde{\phi}$ is a phase function of elliptic type,
namely,

$\partial^{2}_{\eta_{i}}\tilde{\phi}\sim_{m}1,\quad i=1,2,$

and

$|\partial^{l}_{\eta_{2}}\tilde{\phi}|\lesssim_{m}1,\quad 3\leq l\leq m.$

Applying Proposition 3.1 in [12] to the term on the right-hand side of (2.14)
at scale $R_{1}$, we deduce

$\|e^{it\phi(D)}f_{\Box}\|_{L^{6}(Y^{\sigma_{1},\sigma_{2}}_{\Box})}\lesssim\sigma_{1}^{-\frac{m-2}{6}}\sigma_{2}^{-\frac{m-2}{6}}K^{-1/6}N_{1}^{-\frac{1}{3}}\lambda_{1}^{\frac{1}{6}}\nu_{1}^{\frac{1}{6}}(\frac{R}{K})^{\frac{\alpha}{12}+\varepsilon}\|f_{\Box}\|_{2}.$
(2.15)

Using a similar argument as in the proof of inequality (3.24) in [12], one has

$\frac{\mu}{\sharp\mathbb{B}}\lesssim\frac{(logR)^{7}N_{1}\eta}{N^{0}}$ (2.16)

and

$\lambda_{1}\eta\lesssim\max_{T_{r}\subset\Box:r\geq
K_{1}}\frac{\sharp\\{B\subset Y^{0}:B\subset
T_{r}\\}}{r^{\alpha}}\leq\frac{\lambda^{(0)}(K^{O(1)}r)^{{\alpha}}}{r^{\alpha}},$

where we have used the fact that one can cover a
$\sigma_{1}^{\frac{m-2}{2}}K^{1/2}r\times\sigma_{2}^{\frac{m-2}{2}}K^{1/2}r\times
Kr$-tube $T_{r}$ by $K^{O(1)}$ finitely overlapping
$(\min\\{\sigma_{1},\sigma_{2}\\})^{\frac{m-2}{2}}K^{1/2}r$-balls. Hence, we
get

$\eta\lesssim\frac{\lambda^{0}K^{O(1)}}{\lambda_{1}}.$ (2.17)

Now we relate $\nu_{1}$ and $\nu^{(0)}$ by considering the number of $K$
-cubes in each relevant
$\frac{R^{1/2}}{\sigma_{1}^{-\frac{m-2}{2}}}\times\frac{R^{1/2}}{\sigma_{2}^{-\frac{m-2}{2}}}\times
K^{\frac{1}{2}}R^{\frac{1}{2}}$-tube $S^{\prime}$. Recall that each relevant
$S^{\prime}$ contains $\sim\nu_{1}$ tubes $S$ in $Y_{\Box}^{0}$ and each such
$S$ contains $\sim\eta$ $K$-cubes. On the other hand, we can cover
$S^{\prime}$ by $K^{O(1)}$ finitely overlapping $R^{\frac{1}{2}}$-cubes, and
each $R^{\frac{1}{2}}$-cube contains $\lesssim\nu^{(0)}$ many $K$-cubes in
$Y^{0}$. Thus, it follows that

$\nu_{1}\lesssim\frac{K^{\frac{1}{2}}\nu^{(0)}}{\eta}.$ (2.18)

By inserting (2.16), (2.17) and (2.18) into (2.13), we derive

$\begin{split}&\quad\|e^{it\phi(D)}f_{\Omega_{\sigma_{1},\sigma_{2}}}\|_{L^{6}(Y^{\sigma_{1},\sigma_{2}})}\\\
&\leq
C_{\varepsilon}K^{O(1)}\sigma_{1}^{-\frac{m-2}{6m}}\sigma_{2}^{-\frac{m-2}{6m}}(N^{(0)})^{-\frac{1}{3}}(\nu^{(0)})^{\frac{1}{6}}(\lambda^{(0)})^{\frac{1}{6}}R^{\frac{\alpha}{12}+\varepsilon}\|f\|_{2}.\end{split}$
(2.19)

Recall that $\sigma_{i}\geq K^{-1/m}$. Combining (2.19) with (2.5), we obtain

$\|e^{it\phi(D)}f_{\Omega_{0}}\|_{L^{6}(Y^{0})}\leq
C_{\varepsilon}K^{O(1)}(N^{(0)})^{-\frac{1}{3}}(\nu^{(0)})^{\frac{1}{6}}(\lambda^{(0)})^{\frac{1}{6}}R^{\frac{\alpha}{12}+\varepsilon}\|f\|_{2},$

which verifies inequality (2.4).

For the $\Omega_{3}$-case, we sort the $K$-cubes in $Y^{3}$ as follows:

  1. (1)

For any dyadic number $A^{(3)}$, let $Y^{3}_{A^{(3)}}$ be the union of the
$K$-cubes in $Y^{3}$ satisfying

$\|e^{it\phi(D)}f_{\Omega_{3}}\|_{L^{6}(B)}\approx A^{(3)}.$ (2.20)

  2. (2)

Fix $A^{(3)}$, for any dyadic number $\nu^{(3)}$, let
$Y^{3}_{A^{(3)},\nu^{(3)}}$ be the union of the $K$-cubes in $Y^{3}_{A^{(3)}}$
such that for each $B\subset Y^{3}_{A^{(3)},\nu^{(3)}}$, the $R^{1/2}$-cube
intersecting $B$ contains $\approx\nu^{(3)}$ cubes from $Y^{3}_{A^{(3)}}$.

The dyadic numbers $A^{(3)}$ and $\nu^{(3)}$ making significant contribution
can be assumed to be between $R^{-C}$ and $R^{C}$. Therefore, there exist some
dyadic numbers $A^{(3)}$ and $\nu^{(3)}$ such that

$\sharp\\{B:B\subset
Y^{3}_{A^{(3)},\nu^{(3)}}\\}=:N^{(3)}\gtrsim\frac{N}{(\log R)^{2}}.$

Fix that choice of $A^{(3)},\nu^{(3)}$ and denote $Y^{3}_{A^{(3)},\nu^{(3)}}$
by $Y^{3}$ for convenience. Then in the $\Omega_{3}$-case, we have

$\|e^{it\phi(D)}f\|_{L^{6}(Y)}\lesssim_{\varepsilon}R^{\varepsilon}\|e^{it\phi(D)}f_{\Omega_{3}}\|_{L^{6}(Y^{3})},$
(2.21)

and

$\frac{N}{(\log R)^{2}}\lesssim N^{(3)}\leq
N,\quad\nu^{(3)}\leq\nu,\quad\lambda^{(3)}\leq\lambda,$

where

$\lambda^{(3)}:=\max_{B^{3}(x^{\prime},r)\subset
Q_{R},x^{\prime}\in\mathbb{R}^{3},r\geq 1}\frac{\sharp\\{B_{k}\subset
Y^{3}:B_{k}\subset B^{3}(x^{\prime},r)\\}}{r^{\alpha}}.$

We are going to estimate $\|e^{it\phi(D)}f_{\Omega_{3}}\|_{L^{6}(Y^{3})}$ . We
deal with $e^{it\phi(D)}f_{\Omega_{3}}$ by rescaling and induction on scales.
To do it, we further decompose $f_{\Omega_{3}}$ in physical space and perform
dyadic pigeonholing several times.

Firstly, we divide the physical square $[0,R]^{2}$ into
$\frac{R}{K^{1-\frac{1}{m}}}\times\frac{R}{K^{1-\frac{1}{m}}}$-rectangles $D$.
For each pair $D$, let $f_{\Box_{\Omega_{3},D}}$ be the function formed by
cutting off $f_{\Omega_{3}}$ on the rectangle $D$ (with a Schwartz tail) in
physical space. Thus, we have

$f_{\Omega_{3}}=\sum_{D}f_{\Box_{\Omega_{3},D}},$

and write $f_{\Omega_{3}}=\sum_{\Box}f_{\Box}$ for abbreviation. Note that
$e^{it\phi(D)}f_{\Box_{\Omega_{3},D}}$ is essentially supported on an
$\frac{R}{K^{1-\frac{1}{m}}}\times\frac{R}{K^{1-\frac{1}{m}}}\times R$ -box,
which is denoted by $\Box_{\Omega_{3},D}$. The long axis of
$\Box_{\Omega_{3},D}$ is parallel to the vector $(0,0,1)$. The different boxes
$\Box_{\Omega_{3},D}$ tile $Q_{R}$. In particular, a given $K$-cube $B$ lies
in exactly one box $\Box_{\Omega_{3},D}$. For each $K$-cube $B$, it holds
trivially that

$\|e^{it\phi(D)}f_{\Omega_{3}}\|_{L^{6}(B)}\lesssim\Big{(}\sum_{\Box}\|e^{it\phi(D)}f_{\Box}\|^{6}_{L^{6}(B)}\Big{)}^{1/6},$
(2.22)

and

$\sharp\\{\Box\in\mathbb{B}:B\subset Y^{3}_{\Box}\\}\sim 1.$ (2.23)

Recall that $K=R^{\delta}$, where $\delta=\varepsilon^{100}$. Denote

$R_{1}:=\frac{R}{K}=R^{1-\delta},\quad
K_{1}=R^{\delta}_{1}=R^{\delta-\delta^{2}}.$

Tile $\Box$ by $K^{\frac{1}{m}}K_{1}\times K^{\frac{1}{m}}K_{1}\times KK_{1}$
-tube $S$, and also tile $\Box$ by
$\frac{R^{1/2}}{K^{\frac{1}{2}-\frac{1}{m}}}\times\frac{R^{1/2}}{K^{\frac{1}{2}-\frac{1}{m}}}\times
K^{\frac{1}{2}}R^{\frac{1}{2}}$-tubes $S^{\prime}$ (all running parallel to
the long axis of $\Box$). After rescaling the $\Box$ becomes an $R_{1}$-cube,
the tubes $S^{\prime}$ and $S$ becomes lattice $R_{1}^{\frac{1}{2}}$-cubes and
$K_{1}$ -cubes, respectively. We regroup the tubes $S$ and $S^{\prime}$ inside
each $\Box$ as follows:

  1. (1)

Sort those tubes $S$ which intersect $Y^{3}$ according to the value
$\|e^{it\phi(D)}f_{\Box}\|_{L^{6}(S)}$ and the number of $K$-cubes contained
in it. For dyadic numbers $\eta,\beta_{1}$, we use
$\mathbb{S}_{\Box,\eta,\beta_{1}}$ to stand for the collection of tubes
$S\subset\Box$ each of which contains $\sim\eta$ $K$-cubes and
$\|e^{it\phi(D)}f_{\Box}\|_{L^{6}(S)}\sim\beta_{1}$.

  2. (2)

For fixed $\eta,\beta_{1}$, we sort the tubes $S^{\prime}\subset\Box$
according to the number of the tubes $S\in\mathbb{S}_{\Box,\eta,\beta_{1}}$
contained in it. For each dyadic number $\nu_{3}$, let
$\mathbb{S}_{\Box,\eta,\beta_{1},\nu_{3}}$ be the subcollection of
$\mathbb{S}_{\Box,\eta,\beta_{1}}$ such that for each
$S\in\mathbb{S}_{\Box,\eta,\beta_{1},\nu_{3}}$, the tube $S^{\prime}$
containing $S$ contains $\sim\nu_{3}$ tubes from
$\mathbb{S}_{\Box,\eta,\beta_{1}}$.

  3. (3)

For fixed $\eta,\beta_{1},\nu_{3}$, we sort the boxes $\Box$ according to the
value $\|f_{\Box}\|_{2}$, the number
$\sharp\mathbb{S}_{\Box,\eta,\beta_{1},\nu_{3}}$ and the value $\lambda_{3}$
defined below. For dyadic numbers $\beta_{2},N_{3},\lambda_{3}$, let
$\mathbb{B}_{\eta,\beta_{1},\nu_{3},\beta_{2},N_{3},\lambda_{3}}$ denote the
collection of boxes $\Box$ each of which satisfies

$\|f_{\Box}\|_{2}\sim\beta_{2},\quad\sharp\mathbb{S}_{\Box,\eta,\beta_{1},\nu_{3}}\sim
N_{3}$

and

$\max_{T_{r}\subset\Box:r\geq
K_{1}}\frac{\sharp\\{S\in\mathbb{S}_{\Box,\eta,\beta_{1},\nu_{3}}:S\subset
T_{r}\\}}{r^{\alpha}}\sim\lambda_{3},$ (2.24)

where $T_{r}$ are $K^{\frac{1}{m}}r\times K^{\frac{1}{m}}r\times Kr$-tubes in
$\Box$ running parallel to the long axis of $\Box$.

Let $Y^{3}_{\Box,\eta,\beta_{1},\nu_{3}}$ denote
$\\{S:S\subset\mathbb{S}_{\Box,\eta,\beta_{1},\nu_{3}}\\}$. By (2.22), we have

$\|e^{it\phi(D)}f_{\Omega_{3}}\|^{6}_{L^{6}(B)}\lesssim\sum_{\eta,\beta_{1},\nu_{3},\beta_{2},N_{3},\lambda_{3}}\Big{(}\sum_{\Box\in\mathbb{B}_{\eta,\beta_{1},\nu_{3},\beta_{2},N_{3},\lambda_{3}},B\subset
Y^{3}_{\Box,\eta,\beta_{1},\nu_{3}}}\|e^{it\phi(D)}f_{\Box}\|^{6}_{L^{6}(\omega_{B})}\Big{)}.$

Thus, for each $B\subset Y^{3}$, we can choose
$\eta,\beta_{1},\nu_{3},\beta_{2},N_{3},\lambda_{3}$ depending on $B$ such
that

$\|e^{it\phi(D)}f_{\Omega_{3}}\|^{6}_{L^{6}(B)}\lesssim(\log
R)^{6}\Big{(}\sum_{\Box\in\mathbb{B}_{\eta,\beta_{1},\nu_{3},\beta_{2},N_{3},\lambda_{3}},B\subset
Y^{3}_{\Box,\eta,\beta_{1},\nu_{3}}}\|e^{it\phi(D)}f_{\Box}\|^{6}_{L^{6}(\omega_{B})}\Big{)}.$

Since there are only $O(\log R)$ significant choices for each dyadic number,
by pigeonholing, we can choose
$\eta,\beta_{1},\nu_{3},\beta_{2},N_{3},\lambda_{3}$ such that

$\begin{split}&\quad\|e^{it\phi(D)}f_{\Omega_{3}}\|^{6}_{L^{6}(B)}\\\
&\lesssim(\log
R)^{6}\Big{(}\sum_{\Box\in\mathbb{B}_{\eta,\beta_{1},\nu_{3},\beta_{2},N_{3},\lambda_{3}},B\subset
Y^{3}_{\Box,\eta,\beta_{1},\nu_{3}}}\|e^{it\phi(D)}f_{\Box}\|^{6}_{L^{6}(\omega_{B})}\Big{)}\end{split}$
(2.25)

holds for a fraction $\gtrsim(\log R)^{-6}$ of all $K$-cubes $B$ in $Y^{3}$.
For brevity, we denote by

$Y^{3}_{\Box}:=Y^{3}_{\Box,\eta,\beta_{1},\nu_{3}},\quad\mathbb{B}:=\mathbb{B}_{\eta,\beta_{1},\nu_{3},\beta_{2},N_{3},\lambda_{3}}.$

Let $Y^{\prime}$ be the union of $K$-cubes $B$ each of which obeys

$\|e^{it\phi(D)}f_{\Omega_{3}}\|_{L^{6}(B)}\lesssim\log
R\Big{(}\sum_{\Box\in\mathbb{B},B\subset
Y^{3}_{\Box}}\|e^{it\phi(D)}f_{\Box}\|^{6}_{L^{6}(B)}\Big{)}^{1/6}.$ (2.26)

The number of $K$-cubes $B$ in $Y^{\prime}$ is $\gtrsim(\log R)^{-6}N^{(3)}$.
This together with (2.20) yields

$\|e^{it\phi(D)}f_{\Omega_{3}}\|^{6}_{L^{6}(Y^{3})}\lesssim(\log
R)^{C}\sum_{B\subset
Y^{\prime}}\|e^{it\phi(D)}f_{\Omega_{3}}\|^{6}_{L^{6}(B)}.$ (2.27)

Putting (2.27) and (2.26) together, one has

$\|e^{it\phi(D)}f_{\Omega_{3}}\|_{L^{6}(Y^{3})}\lesssim(\log
R)^{C}\Big{(}\sum_{\Box\in\mathbb{B}}\|e^{it\phi(D)}f_{\Box}\|^{6}_{L^{6}(Y^{3}_{\Box})}\Big{)}^{1/6}.$
(2.28)

Next, we apply rescaling to each
$\|e^{it\phi(D)}f_{\Box}\|_{L^{6}(Y_{\Box}^{3})}$ and induction on scales.
Taking the change of variables

$\left\\{\begin{array}[]{l}\xi_{1}=K^{-\frac{1}{m}}\eta_{1},\\\
\xi_{2}=K^{-\frac{1}{m}}\eta_{2},\end{array}\right.$

we have

$|e^{it\phi(D)}f_{\Box}(x)|=K^{-\frac{1}{m}}|e^{i\tilde{t}\phi(D)}g(\tilde{x})|,$

where

$\hat{g}(\eta_{1},\eta_{2}):=K^{-\frac{1}{m}}\hat{f_{\Box}}(K^{-\frac{1}{m}}\eta_{1},K^{-\frac{1}{m}}\eta_{2}),\quad\|g\|_{2}=\|f_{\Box}\|_{2},$

and

$\left\\{\begin{array}[]{l}\tilde{x}_{1}:=K^{-\frac{1}{m}}x_{1},\\\
\tilde{x}_{2}:=K^{-\frac{1}{m}}x_{2},\\\
\tilde{t}:=K^{-1}t.\end{array}\right.$

For brevity, we denote the above relation by
$(\tilde{x},\tilde{t})=\mathcal{L}_{3}(x,t)$. Therefore, we have

$\|e^{it\phi(D)}f_{\Box}\|_{L^{6}(Y_{\Box}^{3})}=K^{\frac{1}{6}-\frac{2}{3m}}\|e^{i\tilde{t}\phi(D)}g(\tilde{x})\|_{L^{6}(\tilde{Y}^{3})},$
(2.29)

where $\tilde{Y}^{3}=\mathcal{L}_{3}(Y_{\Box}^{3})$. Hence, by (2.29) and the
inductive hypothesis at scale $R_{1}$, we deduce

$\|e^{it\phi(D)}f_{\Box}\|_{L^{6}(Y_{\Box}^{3})}\lesssim
K^{\frac{1}{6}-\frac{2}{3m}}N_{3}^{-\frac{1}{3}}\lambda_{3}^{\frac{1}{6}}\nu_{3}^{\frac{1}{6}}(\tfrac{R}{K})^{\frac{5}{12}-\frac{5-\alpha}{6m}+\varepsilon}\|f_{\Box}\|_{2}.$
(2.30)

By inserting (2.30) into (2.28), we get

$\|e^{it\phi(D)}f_{\Omega_{3}}\|_{L^{6}(Y^{3})}\lesssim(N^{(3)})^{-\frac{1}{3}}(\nu^{(3)})^{\frac{1}{6}}(\lambda^{(3)})^{\frac{1}{6}}(\frac{R}{K})^{\frac{5}{12}-\frac{5-\alpha}{6m}+\varepsilon}\big{(}\sum_{\Box\in\mathbb{B}}\|f_{\Box}\|_{2}^{6}\big{)}^{1/6}.$
(2.31)

Note that $\|f_{\Box}\|_{2}$ is essentially constant for $\Box\in\mathbb{B}$.
It follows that

$\|e^{it\phi(D)}f_{\Omega_{3}}\|_{L^{6}(Y^{3})}\lesssim(\sharp\mathbb{B})^{-1/3}(N^{(3)})^{-\frac{1}{3}}(\nu^{(3)})^{\frac{1}{6}}(\lambda^{(3)})^{\frac{1}{6}}R^{\frac{5}{12}-\frac{5-\alpha}{6m}+\varepsilon}\|f\|_{2}.$
(2.32)

Consider the cardinality of the set $\\{(\Box,B):\Box\in\mathbb{B},B\subseteq
Y_{\Box}^{3}\cap Y^{\prime}\\}$. It is easy to see that there is a lower bound

$\sharp\\{(\Box,B):\Box\in\mathbb{B},B\subseteq Y_{\Box}^{3}\cap
Y^{\prime-6}N^{(3)}.$

On the other hand, by our choices of $N_{3}$ and $\eta$, for each
$\Box\in\mathbb{B}$, $Y_{\Box}^{3}$ contains $\sim N_{3}$ tubes $S$ and each
$S$ contains $\sim\eta$ cubes in $Y^{3}$, so

$\sharp\\{(\Box,B):\Box\in\mathbb{B},B\subseteq Y_{\Box}^{3}\cap
Y^{\prime}\\}\lesssim(\sharp\mathbb{B})N_{3}\eta.$

Therefore, we get

$\frac{1}{\sharp\mathbb{B}}\lesssim\frac{(\log R)^{6}N_{3}\eta}{N^{(3)}}.$
(2.33)

Then by our choices of $\lambda_{3}$ as in (2.24) and $\eta$, we have

$\begin{split}\lambda_{3}\eta&\sim\max_{T_{r}\subset\Box:r\geq
K_{1}}\frac{\sharp\\{S:S\subseteq Y_{\Box}^{3}\cap
T_{r}\\}}{r^{\alpha}}\cdot\sharp\\{B:B\subseteq S\cap Y^{3}\\}\\\
&\lesssim\max_{T_{r}\subset\Box:r\geq K_{1}}\frac{\sharp\\{B\subset
Y^{3}:B\subset T_{r}\\}}{r^{\alpha}}\\\
&\leq\frac{K^{1-\frac{1}{m}}\lambda^{(3)}(K^{\frac{1}{m}}r)^{\alpha}}{r^{\alpha}},\end{split}$

where the last inequality follows from the fact that we can cover a
$K^{\frac{1}{m}}r\times K^{\frac{1}{m}}r\times Kr$-tube $T_{r}$ by $\sim
K^{1-\frac{1}{m}}$ finitely overlapping $K^{\frac{1}{m}}r$-balls. Hence, we
obtain

$\eta\lesssim\frac{\lambda^{(3)}K^{1+\frac{\alpha-1}{m}}}{\lambda_{3}}.$
(2.34)

Finally, we relate $\nu_{3}$ and $\nu^{(3)}$ by considering the number of
$K$-cubes in each relevant
$\frac{R^{1/2}}{K^{\frac{1}{2}-\frac{1}{m}}}\times\frac{R^{1/2}}{K^{\frac{1}{2}-\frac{1}{m}}}\times
K^{\frac{1}{2}}R^{\frac{1}{2}}$-tube $S^{\prime}$. Recall that each relevant
$S^{\prime}$ contains $\sim\nu_{3}$ tubes $S$ in $Y_{\Box}$ and each such $S$
contains $\sim\eta$ $K$-cubes. On the other hand, we can cover $S^{\prime}$ by
$\sim K^{1/2}$ finitely overlapping $R^{\frac{1}{2}}$-cubes, and each
$R^{\frac{1}{2}}$-cube contains $\lesssim\nu^{(3)}$ many $K$-cubes in $Y^{3}$.
Thus, it follows that

$\nu_{3}\lesssim\frac{K^{\frac{1}{2}}\nu^{(3)}}{\eta}.$ (2.35)

By inserting (2.33), (2.34) and (2.35) into (2.32), we derive

$\|e^{it\phi(D)}f_{\Omega_{3}}\|_{L^{6}(Y^{3})}\lesssim
K^{-\varepsilon}(N^{(3)})^{-\frac{1}{3}}(\nu^{(3)})^{\frac{1}{6}}(\lambda^{(3)})^{\frac{1}{6}}R^{\frac{5}{12}-\frac{5-\alpha}{6m}+\varepsilon}\|f\|_{2}.$
(2.36)

Since $K=R^{\delta}$ and $R$ can be assumed to be sufficiently large compared
to any constant depending on $\varepsilon$, we have $K^{-\varepsilon}\ll 1$,
and the induction closes. Recall that $\frac{N}{\log R}\lesssim N^{(3)}\leq
N,\nu^{(3)}\leq\nu$ and $\lambda^{(3)}\leq\lambda$. It follows that

$\|e^{it\phi(D)}f_{\Omega_{3}}\|_{L^{6}(Y^{3})}\lesssim_{\varepsilon}N^{-\frac{1}{3}}\lambda^{\frac{1}{6}}\nu^{\frac{1}{6}}R^{\frac{5}{12}-\frac{5-\alpha}{6m}+\varepsilon}\|f\|_{2}.$

For the $\Omega_{1}$-case, we will employ the following proposition, whose
proof is given in Section 4. Let
$\psi(\xi_{1},\xi_{2}):=\phi_{1}(\xi_{1})+\xi^{m}_{2}$ be a class of smooth
phase functions satisfying

$\phi_{1}^{(\prime\prime)}\sim 1;\quad|\phi_{1}^{(k)}|\lesssim 1,3\leq k\leq
m;\quad\phi_{1}^{(l)}=0,l\geq m+1$ (2.37)

on $[0,1]$.

######  Proposition 2.1.

For any $0<\varepsilon<\frac{1}{100}$, there exist constants $C_{\varepsilon}$
and $\delta=\varepsilon^{100}$ such that the following holds for all $R\geq 1$
and all $f$ with $supp\hat{f}\subset[0,1]^{2}$. Suppose that
$Y=\bigcup_{k=1}^{N}B_{k}$ is a union of lattice $K$ -cubes in $Q_{R}$ and
each $R^{\frac{1}{2}}$-cube intersecting $Y$ contains $\sim\nu$ many $K$-cubes
in $Y$, where $K=R^{\delta}$. Suppose that $\|e^{it\psi(D)}f\|_{L^{6}(B_{k})}$
is essentially constant in $k=1,2,...,N$. Given $\lambda$ by

$\lambda:=\max_{B^{3}(x^{\prime},r)\subset
Q_{R},x^{\prime}\in\mathbb{R}^{3},r\geq K}\frac{\sharp\\{B_{k}:B_{k}\subset
B^{3}(x^{\prime},r)\\}}{r^{\alpha}},$

then

$\|e^{it\psi(D)}f\|_{L^{6}(Y)}\leq
C_{\varepsilon}N^{-\frac{1}{3}}\lambda^{\frac{1}{6}}\nu^{\frac{1}{6}}R^{\frac{1}{3}-\frac{4-\alpha}{6m}+\varepsilon}\|f\|_{2}.$

Assume that Proposition 2.1 holds for a while, we estimate
$\|e^{it\phi(D)}f_{\Omega_{1}}\|_{L^{6}(Y^{1})}$ as follows. We divide the
region $\Omega_{1}$ into a family of subregions

$\Omega_{1}=\bigcup_{\rho}\Omega_{1,\rho},$

where $\Omega_{1,\rho}:=I_{\rho}\times[0,K^{-\frac{1}{m}}]$ with
$I_{\rho}=[\rho,2\rho]$. Here $\rho$ is a dyadic number in
$[K^{-\frac{1}{m}},\frac{1}{2}]$. We abbreviate $\Omega_{1,\rho}$ by
$\Omega_{\rho}$ and write

$e^{it\phi(D)}f_{\Omega_{1}}=\sum_{\rho}e^{it\phi(D)}f_{\Omega_{\rho}}.$

We will estimate $\|e^{it\phi(D)}f\|_{L^{6}(Y^{1})}$ in several cases. Loosely
speaking, if the contribution from the region $\Omega_{\rho}$ dominates, we
call it $\Omega_{\rho}$ case. More precisely, given a $K$-cube $B$, by the
triangle inequality, we have

$\|e^{it\phi(D)}f_{\Omega_{\rho}}\|_{L^{6}(B)}\gtrsim\frac{\|e^{it\phi(D)}f_{\Omega_{1}}\|_{L^{6}(B)}}{\log
K}$

for at least one of the $\rho$’s. We sort the $K$-cubes in $Y^{1}$ as follows:
Denote

$\\{B\subset
Y^{1}:\|e^{it\phi(D)}f_{\Omega_{\rho}}\|_{L^{6}(B)}\gtrsim\frac{\|e^{it\phi(D)}f_{\Omega_{1}}\|_{L^{6}(B)}}{\log
K}\\}$

by $Y^{\rho}$. Clearly, one has $Y=\bigcup_{\rho}Y^{\rho}$. If
$\sharp\\{B:B\subset Y^{\rho}\\}\gtrsim\frac{N}{\log K}$, we call it
$\Omega_{\rho}$ -case.

For the $\Omega_{\rho}$-case, we sort the $K$-cubes in $Y^{\rho}$ as follows:

  1. (1)

For any dyadic number $A^{(\rho)}$, let $Y^{\rho}_{A^{(\rho)}}$ be the union
of the $K$-cubes in $Y^{\rho}$ satisfying

$\|e^{it\phi(D)}f_{\Omega_{\rho}}\|_{L^{6}(B)}\approx A^{(\rho)}.$

  2. (2)

Fix $A^{(\rho)}$, for any dyadic numbers $\nu^{(\rho)}$, let
$Y^{\rho}_{A^{(\rho)},\nu^{(\rho)}}$ be the union of the $K$-cubes in
$Y^{\rho}_{A^{(\rho)}}$ such that for each $B\subset
Y^{\rho}_{A^{(\rho)},\nu^{(\rho)}}$, the $R^{1/2}$-cube intersecting $B$
contains $\approx\nu^{(\rho)}$ cubes from $Y^{\rho}_{A^{(\rho)}}$.

The dyadic numbers $A^{(\rho)},\nu^{(\rho)}$ making significant contributions
can be assumed to be between $R^{-C}$ and $R^{C}$. Therefore, there exist some
dyadic numbers $A^{(\rho)},\nu^{(\rho)}$ such that $\sharp\\{B:B\subset
Y_{A^{(\rho)},\nu^{(\rho)}}^{\rho}\\}=:N^{(\rho)}\gtrsim\frac{N}{\log K(\log
R)^{4}}$ cubes $B$. Fix a choice of $A^{(\rho)},\nu^{(\rho)}$ and denote
$Y_{A^{(\rho)},\nu^{(\rho)}}^{\rho}$ by $Y^{\rho}$ for convenience. Then, with
a similar procedure as in the proof of (2.2), in the $\Omega_{\rho}$-case we
have

$\|e^{it\phi(D)}f\|_{L^{6}(Y)}\lesssim_{\varepsilon}R^{\varepsilon}\|e^{it\phi(D)}f_{\Omega_{\rho}}\|_{L^{6}(Y^{\rho})}.$
(2.38)

We also have

$\frac{N}{\log K(\log R)^{4}}\lesssim N^{(\rho)}\leq
N,\quad\nu^{(\rho)}\leq\nu,\quad\lambda^{(\rho)}\leq\lambda,$

where

$\lambda^{(\rho)}:=\max_{B^{3}(x^{\prime},r)\subset
Q_{R},x^{\prime}\in\mathbb{R}^{3},r\geq K}\frac{\sharp\\{B_{k}\subset
Y^{\rho}:B_{k}\subset B^{3}(x^{\prime},r)\\}}{r^{\alpha}}.$

We are going to estimate
$\|e^{it\phi(D)}f_{\Omega_{\rho}}\|_{L^{6}(Y^{\rho})}$. For this purpose, we
divide each $I_{\rho}$ further into

$I_{\rho}=\bigcup_{j=1}^{\rho^{\frac{m}{2}}K^{1/2}}I_{\rho,j}.$

Each $I_{\rho,j}$ has length of $\rho^{-\frac{m-2}{2}}K^{-1/2}$. We write

$\Omega_{\rho}=\bigcup\tau,$

where each $\tau$ in the above union has the form

$I_{\rho,j}\times[0,K^{-\frac{1}{m}}].$

Therefore, we have

$e^{it\phi(D)}f_{\Omega_{\rho}}=\sum_{\tau\subset\Omega_{\rho}}e^{it\phi(D)}f_{\tau}.$

For each cube $B\subset Y^{\rho}$, we employ the following decoupling
inequality from [23], which has root in the original paper of Bourgain and
Demeter [5].

######  Lemma 2.2.

Suppose that $B$ is a $K$-cube contained in $Y^{\rho}$ . Then for any
$\varepsilon>0$, there exists a constant $C_{\varepsilon}$ such that

$\|e^{it\phi(D)}f_{\Omega_{\rho}}\|_{L^{6}(B)}\leq
C_{\varepsilon}K^{\varepsilon}\Big{(}\sum_{\tau}\|e^{it\phi(D)}f_{\tau}\|^{2}_{L^{6}(\omega_{B})}\Big{)}^{1/2},$

where $\omega_{B}$ is a weight function essentially supported on $B$.

######  Remark 2.3.

The authors of [12] use a broad-narrow analysis to deal with the Schrödinger
case. In particular, they apply Bourgain-Demeter’s decoupling inequality in
[5] on each narrow ball. In the current paper, to estimate
$\|e^{it\phi(D)}f\|_{L^{6}(Y)}$, we sort the cubes in $Y$ into
$Y=\bigcup^{3}_{j=0}Y^{j}$, which can be regarded as a “non-degenerate-
degenerate” analysis. For the convenience of readers, in this remark we call
the cubes in $Y^{0}$ and $Y^{1}$ non-degenerate cubes and degenerate cubes,
respectively. Our strategy is to apply decoupling on each degenerate cube,
which has a similar flavor as the narrow ball in [12].

For each $\tau$, we will handle $e^{it\phi(D)}f_{\tau}$ by rescaling and
induction on scales. To do it, we further decompose $f_{\Omega_{\rho}}$ in
physical space and perform dyadic pigeonholing several times.

Firstly, we divide the physical square $[0,R]^{2}$ into
$\frac{R}{\rho^{-\frac{m-2}{2}}K^{1/2}}\times\frac{R}{K^{1-\frac{1}{m}}}$-rectangles
$D$. For each pair $(\tau,D)$, let $f_{\Box_{\tau,D}}$ be the function formed
by cutting off $f$ on the rectangle $D$ (with a Schwartz tail) in physical
space and the rectangle $\tau$ in Fourier space. We have

$f_{\Omega_{\rho}}=\sum_{\tau}\sum_{D}f_{\Box_{\tau,D}},$

and write $f_{\Omega_{\rho}}=\sum_{\Box}f_{\Box}$ for abbreviation. Note that
$e^{it\phi(D)}f_{\Box_{\tau,D}}$ is essentially supported on an
$\frac{R}{\rho^{-\frac{m-2}{2}}K^{1/2}}\times\frac{R}{K^{1-\frac{1}{m}}}\times
R$-box, which is denoted by $\Box_{\tau,D}$. The box $\Box_{\tau,D}$ is
parallel to the normal direction of the surface $\Sigma_{\tau}$ at the left
bottom corner of $\tau$. For a fixed $\tau$, the different boxes
$\Box_{\tau,D}$ tile $Q_{R}$. Note that a given $K$-cube $B$ lies in exactly
one box $\Box_{\tau,D}$ for each $\tau$. By Lemma 2.2, for each $K$-cube $B$,
there holds

$\|e^{it\phi(D)}f_{\Omega_{\rho}}\|_{L^{6}(B)}\leq
C_{\varepsilon}K^{\varepsilon^{2}}\Big{(}\sum_{\Box}\|e^{it\phi(D)}f_{\Box}\|_{L^{6}(\omega_{B})}^{2}\Big{)}^{1/2}.$

We denote

$R_{1}:=\frac{R}{K}=R^{1-\delta},\quad
K_{1}=R_{1}^{\delta}=R^{\delta-\delta^{2}}.$

Tile $\Box$ by $\rho^{\frac{m-2}{2}}K^{\frac{1}{2}}K_{1}\times
K^{\frac{1}{m}}K_{1}\times KK_{1}$-tubes $S$, and also tile $\Box$ by
$\frac{R^{1/2}}{\rho^{-\frac{m-2}{2}}}\times\frac{R^{1/2}}{K^{\frac{1}{2}-\frac{1}{m}}}\times
K^{\frac{1}{2}}R^{\frac{1}{2}}$-tubes $S^{\prime}$ (all running parallel to
the long axis of $\Box$). After rescaling the $\Box$ becomes an $R_{1}$-cube,
the tubes $S^{\prime}$ and $S$ become lattice $R_{1}^{\frac{1}{2}}$-cubes and
$K_{1}$-cubes, respectively. We regroup the tubes $S$ and $S^{\prime}$ inside
each $\Box$ as follows:

  1. (1)

Sort those tubes $S$ which intersect $Y^{\rho}$ according to the value
$\|e^{it\phi(D)}f_{\Box}\|_{L^{6}(S)}$ and the number of $K$-cubes contained
in it. For dyadic numbers $\eta,\beta_{1}$, we use
$\mathbb{S}_{\Box,\eta,\beta_{1}}$ to stand for the collection of tubes
$S\subset\Box$ each of which containing $\sim\eta$ $K$-cubes and
$\|e^{it\phi(D)}f_{\Box}\|_{L^{6}(S)}\sim\beta_{1}$.

  2. (2)

For fixed $\eta,\beta_{1}$, we sort the tubes $S^{\prime}\subset\Box$
according to the number of the tubes $S\in\mathbb{S}_{\Box,\eta,\beta_{1}}$
contained in it. For dyadic number $\nu_{\rho}$, let
$\mathbb{S}_{\Box,\eta,\beta_{1},\nu_{\rho}}$ be the sub-collection of
$\mathbb{S}_{\Box,\eta,\beta_{1}}$ such that for each
$S\in\mathbb{S}_{\Box,\eta,\beta_{1},\nu_{\rho}}$, the tube $S^{\prime}$
containing $S$ contains $\sim\nu_{\rho}$ tubes from
$\mathbb{S}_{\Box,\eta,\beta_{1}}$.

  3. (3)

For fixed $\eta,\beta_{1},\nu_{\rho}$, we sort the boxes $\Box$ according to
the value $\|f_{\Box}\|_{2}$, the number
$\sharp\mathbb{S}_{\Box,\eta,\beta_{1},\nu_{\rho}}$ and the value
$\lambda_{\rho}$ defined below. For dyadic numbers
$\beta_{2},N_{\rho},\lambda_{\rho}$, let
$\mathbb{B}_{\eta,\beta_{1},\nu_{\rho},\beta_{2},N_{\rho},\lambda_{\rho}}$
denote the collection of boxes $\Box$ satisfying

$\|f_{\Box}\|_{2}\sim\beta_{2},\quad\sharp\mathbb{S}_{\Box,\eta,\beta_{1},\nu_{\rho}}\sim
N_{\rho}$

and

$\max_{T_{r}\subset\Box:r\geq
K_{1}}\frac{\sharp\\{S\in\mathbb{S}_{\Box,\eta,\beta_{1},\nu_{\rho}}:S\subset
T_{r}\\}}{r^{\alpha}}\sim\lambda_{\rho},$ (2.39)

where $T_{r}$ are $K^{\frac{1}{2}}\rho^{\frac{m-2}{2}}r\times
K^{\frac{1}{m}}r\times Kr$-tubes in $\Box$ running parallel to the long axis
of $\Box$.

Let $Y_{\Box,\eta,\beta_{1},\nu_{\rho}}^{\rho}$ denote
$\\{S:S\subset\mathbb{S}_{\Box,\eta,\beta_{1},\nu_{\rho}}\\}$. Thus, there are
only $O(\log R)$ significant choices for each dyadic number. By pigeonholing,
we can choose $\eta,\beta_{1},\nu_{\rho},\beta_{2},N_{\rho},\lambda_{\rho}$ so
that

$\begin{split}&\quad\|e^{it\phi(D)}f_{\Omega_{\rho}}\|_{L^{6}(B)}\\\
&\lesssim(\log
R)^{6}K^{\varepsilon^{2}}\Big{(}\sum_{\Box\in\mathbb{B}_{\eta,\beta_{1},\nu_{\rho},\beta_{2},N_{\rho},\lambda_{\rho}},B\subset
Y_{\Box,\eta,\beta_{1},\nu_{\rho}}}\|e^{it\phi(D)}f_{\Box}\|_{L^{6}(\omega_{B})}^{2}\Big{)}^{1/2}\end{split}$
(2.40)

holds for a fraction $\gtrsim(\log R)^{-6}$ of all $K$-cubes $B$. For brevity,
we denote by

$Y_{\Box}^{\rho}:=Y_{\Box,\eta,\beta_{1},\nu_{\rho}}^{\rho},\quad\mathbb{B}:=\mathbb{B}_{\eta,\beta_{1},\nu_{\rho},\beta_{2},N_{\rho},\lambda_{\rho}}.$

We sort the $K$-cubes $B$ satisfying (2.40) by
$\sharp\\{\Box\in\mathbb{B}:B\subset Y_{\Box}^{\rho}\\}$. Let
$Y_{\mu}^{\prime}\subset Y^{\rho}$ be a union of $K$-cubes $B$ that obey

$\|e^{it\phi(D)}f_{\Omega_{\rho}}\|_{L^{6}(B)}\lesssim(\log
R)^{6}K^{\varepsilon^{2}}\Big{(}\sum_{\Box\in\mathbb{B},B\subset
Y_{\Box}^{\rho}}\|e^{it\phi(D)}f_{\Box}\|_{L^{6}(\omega_{B})}^{2}\Big{)}^{1/2}$
(2.41)

and

$\sharp\\{\Box\in\mathbb{B}:B\subset Y_{\Box}^{\rho}\\}\sim\mu.$ (2.42)

Since the dyadic number $\mu$ must satisfy $1\leq\mu\leq R^{C}$, there are
$\sim\log R$ different choices for $\mu$, so by pigeonholing there exists one
such $\mu$ such that the number of cubes in $Y_{\mu}^{\prime}$ is
$\gtrsim\frac{\sharp\\{B:B\subset Y^{\rho}\\}}{\log
R}\gtrsim\frac{N^{(\rho)}}{(\log R)^{7}}$. And then for simplicity we rename
$Y_{\mu}^{\prime}$ as $Y^{\prime}$. We have

$\|e^{it\phi(D)}f_{\Omega_{\rho}}\|_{L^{6}(Y^{\rho})}^{6}\lesssim(\log
R)^{C}\sum_{B\subset
Y^{\prime}}\|e^{it\phi(D)}f_{\Omega_{\rho}}\|_{L^{6}(B)}^{6}.$ (2.43)

For each $B\subset Y^{\prime}$, it follows from (2.41), (2.42) and Hölder’s
inequality that

$\|e^{it\phi(D)}f_{\Omega_{\rho}}\|_{L^{6}(B)}^{6}\lesssim(\log
R)^{C}K^{6\varepsilon^{2}}\mu^{2}\sum_{\Box\in\mathbb{B},B\subset
Y_{\Box}^{\rho}}\|e^{it\phi(D)}f_{\Box}\|_{L^{6}(\omega_{B})}^{6}.$ (2.44)

Putting (2.43) and (2.44) together, one has

$\|e^{it\phi(D)}f_{\Omega_{\rho}}\|_{L^{6}(Y^{\rho})}\lesssim(\log
R)^{C}K^{\varepsilon^{2}}\mu^{1/3}\Big{(}\sum_{\Box\in\mathbb{B}}\|e^{it\phi(D)}f_{\Box}\|_{L^{6}(Y_{\Box}^{\rho})}^{6}\Big{)}^{1/6}.$
(2.45)

Next, we apply rescaling to each
$\|e^{it\phi(D)}f_{\Box}\|_{L^{6}(Y^{\rho}_{\Box})}$ and run induction on
scales. For each $\rho^{-\frac{m-2}{2}}K^{-1/2}\times K^{-1/m}$-rectangle
$\tau=\tau_{\Box}$ in $\Omega_{\rho}$, we write

$\left\\{\begin{array}[]{l}\xi_{1}=a+\rho^{-\frac{m-2}{2}}K^{-\frac{1}{2}}\eta_{1},\\\
\xi_{2}=K^{-\frac{1}{m}}\eta_{2},\end{array}\right.$

for some $\rho\leq a\leq 2\rho$. Then

$|e^{it\phi(D)}f_{\Box}(x)|=\rho^{-\frac{m-2}{4}}K^{-\frac{1}{4}-\frac{1}{2m}}|e^{i\tilde{t}\psi(D)}g(\tilde{x})|,$

with $\|g\|_{2}=\|f_{\Box}\|_{2},$ and

$\hat{g}(\eta_{1},\eta_{2}):=\rho^{-\frac{m-2}{4}}K^{-\frac{m+2}{4m}}\hat{f_{\Box}}(a+K^{-\frac{1}{2}}\eta_{1},K^{-\frac{1}{m}}\eta_{2}),$

where

$\begin{split}\psi(\eta_{1},\eta_{2})&:=\Big{(}\frac{m(m-1)}{2}a^{m-2}\rho^{-(m-2)}\eta^{2}_{1}\\\
&\quad+\frac{m(m-1)(m-2)}{6}a^{m-3}\rho^{-\frac{3(m-2)}{2}}K^{-1/2}\eta^{3}_{1}\\\
&\quad+...+\rho^{-\frac{m(m-2)}{2}}K^{-\frac{m-2}{2}}\eta^{m}_{1}\Big{)}+\eta^{m}_{2}\\\
&=:\phi_{1}(\eta_{1})+\eta^{m}_{2},\end{split}$

and

$\left\\{\begin{array}[]{l}\tilde{x}_{1}:=\rho^{-\frac{m-2}{2}}K^{-\frac{1}{2}}(x_{1}+ma^{m-1}t),\\\
\tilde{x}_{2}:=K^{-\frac{1}{m}}x_{2},\\\
\tilde{t}:=K^{-1}t.\end{array}\right.$

For brevity, we denote the above relation by
$(\tilde{x},\tilde{t})=\mathcal{L}_{1}(x,t)$. We write

$\|e^{it\phi(D)}f_{\Box}\|_{L^{6}(Y^{\rho}_{\Box})}=\rho^{-\frac{m-2}{6}}K^{-\frac{1}{3m}}\|e^{i\tilde{t}\psi(D)}g(\tilde{x})\|_{L^{6}(\tilde{Y}^{\rho})},$
(2.46)

where $\tilde{Y}^{\rho}=\mathcal{L}_{1}(Y^{\rho}_{\Box})$. Note that
$a\sim\rho$ and $K^{-1/m}\leq\rho\leq\frac{1}{2}$. It is easy to see that the
new phase function $\psi(\eta_{1},\eta_{2})=\phi_{1}(\eta_{1})+\eta^{m}_{2}$
satisfies

$\phi_{1}^{(\prime\prime)}\sim 1;\quad|\phi_{1}^{(k)}|\lesssim 1,3\leq k\leq
m;\quad\phi_{1}^{(l)}=0,l\geq m+1$

on $[0,1]$. Therefore, applying Proposition 2.1 to the term

$\|e^{i\tilde{t}\psi(D)}g(\tilde{x})\|_{L^{6}(\tilde{Y}^{\rho})}$

at scale $R_{1}$, we deduce

$\|e^{it\phi(D)}f_{\Box}\|_{L^{6}(Y^{\rho}_{\Box})}\lesssim\rho^{-\frac{m-2}{6}}K^{-\frac{1}{3m}}N_{\rho}^{-\frac{1}{3}}\lambda_{\rho}^{\frac{1}{6}}\nu_{\rho}^{\frac{1}{6}}(\tfrac{R}{K})^{\frac{1}{3}-\frac{4-\alpha}{6m}+\varepsilon}\|f_{\Box}\|_{2}.$

Note that
$\frac{1}{3}-\frac{4-\alpha}{6m}\leq\frac{5}{12}-\frac{5-\alpha}{6m}$ when
$m\geq 2$. Consider the cardinality of the set
$\\{(\Box,B):\Box\in\mathbb{B},B\subseteq Y^{\rho}_{\Box}\cap Y^{\prime}\\}$.
By our choice of $\mu$ as in (2.42), there is a lower bound

$\sharp\\{(\Box,B):\Box\in\mathbb{B},B\subseteq Y^{\rho}_{\Box}\cap
Y^{\prime-7}N^{({\rho})}\mu.$

On the other hand, by our choices of $N_{\rho}$ and $\eta$, for each
$\Box\in\mathbb{B}$, $Y^{\rho}_{\Box}$ contains $\sim N_{\rho}$ tubes $S$ and
each $S$ contains $\sim\eta$ cubes in $Y^{\rho}$, so

$\sharp\\{(\Box,B):\Box\in\mathbb{B},B\subseteq Y^{\rho}_{\Box}\cap
Y^{\prime}\\}\lesssim(\sharp\mathbb{B})N_{\rho}\eta.$

Therefore, we get

$\frac{\mu}{\sharp\mathbb{B}}\lesssim\frac{(\log
R)^{7}N_{\rho}\eta}{N^{({\rho})}}.$ (2.47)

Then by our choices of $\lambda_{\rho}$ as in (2.39) and $\eta$, we have

$\begin{split}\lambda_{\rho}\eta&\sim\max_{T_{r}\subset\Box:r\geq
K_{1}}\frac{\sharp\\{S:S\subseteq Y^{\rho}_{\Box}\cap
T_{r}\\}}{r^{\alpha}}\cdot\sharp\\{B:B\subseteq S\cap Y^{\rho}\\}\\\
&\lesssim\max_{T_{r}\subset\Box:r\geq K_{1}}\frac{\sharp\\{B\subset
Y^{\rho}:B\subset T_{r}\\}}{r^{\alpha}}\\\
&\leq\frac{\rho^{\frac{m-2}{2}}K^{\frac{3}{2}-\frac{2}{m}}\lambda^{({\rho})}(K^{\frac{1}{m}}r)^{\alpha}}{r^{\alpha}},\end{split}$

where the last inequality follows from the fact that we can cover a
$\rho^{\frac{m-2}{2}}K^{\frac{1}{2}}r\times K^{\frac{1}{m}}r\times Kr$-tube
$T_{r}$ by $\sim\rho^{\frac{m-2}{2}}K^{\frac{3}{2}-\frac{2}{m}}$ finitely
overlapping $K^{\frac{1}{m}}r$-balls. Hence, we obtain

$\eta\lesssim\frac{\lambda^{(\rho)}\rho^{\frac{m-2}{2}}K^{\frac{3}{2}+\frac{\alpha-2}{m}}}{\lambda_{\rho}}.$
(2.48)

Finally, we relate $\nu_{\rho}$ and $\nu^{({\rho})}$ by considering the number
of $K$-cubes in each relevant
$\frac{R^{1/2}}{\rho^{-\frac{m-2}{2}}}\times\frac{R^{1/2}}{K^{\frac{1}{2}-\frac{1}{m}}}\times
K^{\frac{1}{2}}R^{\frac{1}{2}}$-tube $S^{\prime}$. Recall that each relevant
$S^{\prime}$ contains $\sim\nu_{\rho}$ tubes $S$ in $Y_{\Box}$ and each such
$S$ contains $\sim\eta$ $K$-cubes. On the other hand, we can cover
$S^{\prime}$ by $\sim K^{1/2}$ finitely overlapping $R^{\frac{1}{2}}$-cubes,
and each $R^{\frac{1}{2}}$-cube contains $\lesssim\nu^{({\rho})}$ many
$K$-cubes in $Y^{\rho}$. Thus, it follows that

$\nu_{\rho}\lesssim\frac{K^{\frac{1}{2}}\nu^{({\rho})}}{\eta}.$ (2.49)

By inserting (2.47), (2.48) and (2.49) into (2.45), we derive

$\|e^{it\phi(D)}f_{\Omega_{\rho}}\|_{L^{6}(Y^{\rho})}\lesssim
K^{2\varepsilon^{2}-\varepsilon}(N^{({\rho})})^{-\frac{1}{3}}(\nu^{({\rho})})^{\frac{1}{6}}(\lambda^{({\rho})})^{\frac{1}{6}}R^{\frac{5}{12}-\frac{5-\alpha}{6m}+\varepsilon}\|f\|_{2}.$

Since $K=R^{\delta}$ and $R$ can be assumed to be sufficiently large compared
to any constant depending on $\varepsilon$, we have
$K^{2\varepsilon^{2}-\varepsilon}\ll 1$, and the induction closes. Recall that
$\frac{N}{\log K(\log R)^{4}}\lesssim N^{(\rho)}\leq N,\nu^{(\rho)}\leq\nu$
and $\lambda^{(\rho)}\leq\lambda$. This yields

$\|e^{it\phi(D)}f_{\Omega_{\rho}}\|_{L^{6}(Y^{\rho})}\lesssim_{\varepsilon}N^{-\frac{1}{3}}\lambda^{\frac{1}{6}}\nu^{\frac{1}{6}}R^{\frac{5}{12}-\frac{5-\alpha}{6m}+\varepsilon}\|f\|_{2}.$

Combining it with (2.38), we get

$\|e^{it\phi(D)}f_{\Omega_{1}}\|_{L^{6}(Y^{1})}\lesssim_{\varepsilon}N^{-\frac{1}{3}}\lambda^{\frac{1}{6}}\nu^{\frac{1}{6}}R^{\frac{5}{12}-\frac{5-\alpha}{6m}+\varepsilon}\|f\|_{2}.$

##  3\. The Proof of Proposition 2.1

To prove Proposition 2.1, we divide $[0,1]^{2}$ into two parts

$[0,1]^{2}=\Omega_{0}\bigcup\Omega_{1},$

where $\Omega_{0}:=[0,1]\times[K^{-\frac{1}{m}},1]$,
$\Omega_{1}:=[0,1]\times[0,K^{-\frac{1}{m}}]$. We denote
$\hat{f}\mid_{\Omega_{0}}$ and $\hat{f}\mid_{\Omega_{1}}$ by
$\hat{f}_{\Omega_{0}}$ and $\hat{f}_{\Omega_{1}}$, respectively. Given a
$K$-cube $B$, by the triangle inequality, we have either

$\|e^{it\psi(D)}f_{\Omega_{0}}\|_{L^{6}(B)}\geq\frac{1}{8}\|e^{it\psi(D)}f\|_{L^{6}(B)}$

or

$\|e^{it\psi(D)}f_{\Omega_{1}}\|_{L^{6}(B)}\geq\frac{1}{8}\|e^{it\psi(D)}f\|_{L^{6}(B)}.$

We sort the $K$-cubes in $Y$ as follows. Denote by

$\\{B\subset
Y:\|e^{it\psi(D)}f_{\Omega_{0}}\|_{L^{6}(B)}\geq\frac{1}{8}\|e^{it\psi(D)}f\|_{L^{6}(B)}\\}$

and

$\\{B\subset
Y:\|e^{it\psi(D)}f_{\Omega_{1}}\|_{L^{6}(B)}\geq\frac{1}{8}\|e^{it\psi(D)}f\|_{L^{6}(B)}\\}$

by $Y^{0}$ and $Y^{1}$, respectively. Clearly, one has $Y=Y^{0}\cup Y^{1}$. If
$\sharp\\{B:B\subset Y^{0}\\}\geq\frac{N}{2}$, we call it $\Omega_{0}$-case.
If $\sharp\\{B:B\subset Y^{1}\\}\geq\frac{N}{2}$, we call it
$\Omega_{1}$-case.

For the $\Omega_{0}$-case, we further sort the $K$-cubes in $Y^{0}$ as
follows:

  1. (1)

For any dyadic number $A^{(0)}$, let $Y^{0}_{A^{(0)}}$ be the union of the
$K$-cubes in $Y^{0}$ satisfying

$\|e^{it\psi(D)}f_{\Omega_{0}}\|_{L^{6}(B)}\approx A^{(0)}.$

  2. (2)

Fix $A^{(0)}$, for any dyadic number $\nu^{(0)}$, let
$Y^{0}_{A^{(0)},\nu^{(0)}}$ be the union of the $K$-cubes in $Y^{0}_{A^{(0)}}$
such that for each $B\subset Y^{0}_{A^{(0)},\nu^{(0)}}$, the $R^{1/2}$-cube
intersecting $B$ contains $\approx\nu^{(0)}$ cubes from $Y^{0}_{A^{(0)}}$.

With the assumption $\|f\|_{2}=1$ the dyadic numbers $A^{(0)},\nu^{(0)}$
making significant contributions are between $R^{-C}$ and $R^{C}$. Therefore,
there exist some dyadic numbers $A^{(0)},\nu^{(0)}$ such that
$\sharp\\{B:B\subset
Y^{0}_{A^{(0)},\nu^{(0)}}\\}=:N^{(0)}\gtrsim\frac{N}{(\log R)^{2}}$. Fix a
choice of $A^{(0)},\nu^{(0)}$ and denote $Y^{0}_{A^{(0)},\nu^{(0)}}$ by
$Y^{0}$ for convenience. Then, in the $\Omega_{0}$ -case we have

$\|e^{it\psi(D)}f\|_{L^{6}(Y)}\lesssim_{\varepsilon}R^{\varepsilon}\|e^{it\psi(D)}f_{\Omega_{0}}\|_{L^{6}(Y^{0})},$

and

$\frac{N}{(\log R)^{2}}\lesssim N^{(0)}\leq
N,\quad\nu^{(0)}\leq\nu,\quad\lambda^{(0)}\leq\lambda,$

where

$\lambda^{(0)}:=\max_{B^{3}(x^{\prime},r)\subset
Q_{R},x^{\prime}\in\mathbb{R}^{3},r\geq 1}\frac{\sharp\\{B_{k}\subset
Y^{0}:B_{k}\subset B^{3}(x^{\prime},r)\\}}{r^{\alpha}}.$

By a similar argument as in the proof of (2.4), we have

$\|e^{it\psi(D)}f_{\Omega_{0}}\|_{L^{6}(Y^{0})}\leq
C_{\varepsilon}K^{O(1)}(N^{(0)})^{-\frac{1}{3}}(\lambda^{(0)})^{\frac{1}{6}}(\nu^{(0)})^{\frac{1}{6}}R^{\frac{\alpha}{12}+\varepsilon}\|f\|_{2}.$

For the $\Omega_{1}$-case, we sort the $K$-cubes in $Y^{1}$ as similar as in
the $\Omega_{0}$-case:

  1. (1)

For any dyadic number $A^{(1)}$, let $Y^{1}_{A^{(1)}}$ be the union of the
$K$-cubes in $Y^{1}$ satisfying

$\|e^{it\psi(D)}f_{\Omega_{1}}\|_{L^{6}(B)}\approx A^{(1)}.$

  2. (2)

Fix $A^{(1)}$, for any dyadic number $\nu^{(1)}$, let
$Y^{1}_{A^{(1)},\nu^{(1)}}$ be the union of the $K$-cubes in $Y^{1}_{A^{(1)}}$
such that for each $B\subset Y^{1}_{A^{(1)},\nu^{(1)}}$, the $R^{1/2}$-cube
intersecting $B$ contains $\approx\nu^{(1)}$ cubes from $Y^{1}_{A^{(1)}}$.

The dyadic numbers $A^{(1)},\nu^{(1)}$ making significant contributions can be
assumed to be between $R^{-C}$ and $R^{C}$. Therefore, there exist some dyadic
numbers $A^{(1)},\nu^{(1)}$ such that $\sharp\\{B:B\subset
Y^{1}_{A^{(1)},\nu^{(1)}}\\}=:N^{(1)}\gtrsim\frac{N}{(\log R)^{2}}$ many cubes
$B$. Fix that choice of $A^{(1)},\nu^{(1)}$ and denote
$Y^{1}_{A^{(1)},\nu^{(1)}}$ by $Y^{1}$ for convenience. Then, in the
$\Omega_{1}$-case we have

$\|e^{it\psi(D)}f\|_{L^{6}(Y)}\lesssim_{\varepsilon}R^{\varepsilon}\|e^{it\psi(D)}f_{\Omega_{1}}\|_{L^{6}(Y^{1})},$

and

$\frac{N}{(\log R)^{2}}\lesssim N^{(1)}\leq
N,\quad\nu^{(1)}\leq\nu,\quad\lambda^{(1)}\leq\lambda,$

where

$\lambda^{(1)}:=\max_{B^{3}(x^{\prime},r)\subset
Q_{R},x^{\prime}\in\mathbb{R}^{3},r\geq K}\frac{\sharp\\{B_{k}\subset
Y^{1}:B_{k}\subset B^{3}(x^{\prime},r)\\}}{r^{\alpha}}.$

We are going to estimate $\|e^{it\psi(D)}f_{\Omega_{1}}\|_{L^{6}(Y^{1})}$. To
this end, we partition $\Omega_{1}$ into disjoint rectangles $\tau$ of
dimensions $K^{-\frac{1}{2}}\times K^{-\frac{1}{m}}$, and write
$f_{\Omega_{1}}=\sum_{\tau}f_{\tau}$. For each $K$-cube $B$, we have the
following decoupling inequality due to Bourgain-Demeter [5] (see also Lemma
3.3 in [17]).

######  Lemma 3.1.

Suppose that $B$ is a $K$-cube in $\mathbb{R}^{3}$. Then for any
$\varepsilon>0$, there exists a constant $C_{\varepsilon}$ such that

$\|e^{it\psi(D)}f_{\Omega_{1}}\|_{L^{6}(B)}\leq
C_{\varepsilon}K^{\varepsilon}\Big{(}\sum_{\tau}\|e^{it\psi(D)}f_{\tau}\|^{2}_{L^{6}(\omega_{B})}\Big{)}^{1/2},$

where $\omega_{B}$ is a weight function essentially supported on $B$.

We decompose the function $f_{\Omega_{1}}$ in physical space further and
perform some dyadic pigeonholing like in Section 2.

First we divide the physical square $[0,R]^{2}$ into
$\frac{R}{K^{1/2}}\times\frac{R}{K^{1-\frac{1}{m}}}$-rectangles $D$. For each
pair $(\tau,D)$, let $f_{\Box_{\tau,D}}$ be the function formed by cutting off
$f$ on the rectangle $D$ (with a Schwartz tail) in physical space and the
rectangle $\tau$ in Fourier space. We see that
$e^{it\psi(D)}f_{\Box_{\tau,D}}$ is essentially supported on an
$\frac{R}{K^{1/2}}\times\frac{R}{K^{1-\frac{1}{m}}}\times R$-box, which is
denoted by $\Box_{\tau,D}$.

######  Remark 3.2.

In $x_{2}$ the wave packets follow the evolution of the phase $\xi^{m}_{2}$,
but in $x_{1}$ they essentially follow the Schrödinger evolution because the
phase $\phi_{1}(\xi_{1})$ is basically like $\xi^{2}_{1}$. So the scaling in
both directions is different.

The box $\Box_{\tau,D}$ is parallel to the normal direction of the surface
$\Sigma_{\tau}$ at the left bottom corner of $\tau$. For any fixed $\tau$, the
different boxes $\Box_{\tau,D}$ tile $Q_{R}$. In particular, for each $\tau$,
a given $K$-cube $B$ lies in exactly one box $\Box_{\tau,D}$. We have

$f_{\Omega_{1}}=\sum_{\tau}\sum_{D}f_{\Box_{\tau,D}}$

and write $f_{\Omega_{1}}=\sum_{\Box}f_{\Box}$ for abbreviation. By Lemma 3.1,
for each $K$-cube $B$, there holds

$\|e^{it\psi(D)}f_{\Omega_{1}}\|_{L^{6}(B)}\leq
C_{\varepsilon}K^{\varepsilon^{2}}\Big{(}\sum_{\Box}\|e^{it\psi(D)}f_{\Box}\|^{2}_{L^{6}(\omega_{B})}\Big{)}^{1/2}.$

Recall that $K=R^{\delta}$, where $\delta=\varepsilon^{100}$. We denote by

$R_{1}:=\frac{R}{K}=R^{1-\delta},\quad
K_{1}=R^{\delta}_{1}=R^{\delta-\delta^{2}}.$

Tile $\Box$ by $K^{\frac{1}{2}}K_{1}\times K^{\frac{1}{m}}K_{1}\times KK_{1}$
-tube $S$, and also tile $\Box$ by
$R^{\frac{1}{2}}\times\frac{R^{1/2}}{K^{\frac{1}{2}-\frac{1}{m}}}\times
K^{\frac{1}{2}}R^{\frac{1}{2}}$-tubes $S^{\prime}$ (all running parallel to
the long axis of $\Box$). After rescaling the $\Box$ becomes an $R_{1}$-cube,
the tubes $S^{\prime}$ and $S$ become lattice $R_{1}^{\frac{1}{2}}$-cubes and
$K_{1}$-cubes, respectively. We regroup tubes $S$ and $S^{\prime}$ inside each
$\Box$ as follows:

  1. (1)

Sort those tubes $S$ which intersect $Y^{1}$ according to the value
$\|e^{it\psi(D)}f_{\Box}\|_{L^{6}(S)}$ and the number of $K$-cubes contained
in it. For dyadic numbers $\eta,\beta_{1}$, we use
$\mathbb{S}_{\Box,\eta,\beta_{1}}$ to stand for the collection of tubes
$S\subset\Box$ each of which containing $\sim\eta$ $K$-cubes and
$\|e^{it\psi(D)}f_{\Box}\|_{L^{6}(S)}\sim\beta_{1}$.

  2. (2)

For fixed $\eta,\beta_{1}$, we sort the tubes $S^{\prime}\subset\Box$
according to the number of the tubes $S\in\mathbb{S}_{\Box,\eta,\beta_{1}}$
contained in it. For dyadic number $\nu_{1}$, let
$\mathbb{S}_{\Box,\eta,\beta_{1},\nu_{1}}$ be the subcollection of
$\mathbb{S}_{\Box,\eta,\beta_{1}}$ such that for each
$S\in\mathbb{S}_{\Box,\eta,\beta_{1},\nu_{1}}$, the tube $S^{\prime}$
containing $S$ contains $\sim\nu_{1}$ tubes from
$\mathbb{S}_{\Box,\eta,\beta_{1}}$.

  3. (3)

For fixed $\eta,\beta_{1},\nu_{1}$, we sort the boxes $\Box$ according to the
value $\|f_{\Box}\|_{2}$, the number
$\sharp\mathbb{S}_{\Box,\eta,\beta_{1},\nu_{1}}$ and the value $\lambda_{1}$
defined below. For dyadic numbers $\beta_{2},N_{1},\lambda_{1}$, let
$\mathbb{B}_{\eta,\beta_{1},\nu_{1},\beta_{2},N_{1},\lambda_{1}}$ denote the
collection of boxes $\Box$ each of which satisfying that

$\|f_{\Box}\|_{2}\sim\beta_{2},\quad\sharp\mathbb{S}_{\Box,\eta,\beta_{1},\nu_{1}}\sim
N_{1}$

and

$\max_{T_{r}\subset\Box:r\geq
K_{1}}\frac{\sharp\\{S\in\mathbb{S}_{\Box,\eta,\beta_{1},\nu_{1}}:S\subset
T_{r}\\}}{r^{\alpha}}\sim\lambda_{1},$ (3.1)

where $T_{r}$ are $K^{\frac{1}{2}}r\times K^{\frac{1}{m}}r\times Kr$-tubes in
$\Box$ running parallel to the long axis of $\Box$.

Let $Y^{1}_{\Box,\eta,\beta_{1},\nu_{1}}$ denote
$\\{S:S\subset\mathbb{S}_{\Box,\eta,\beta_{1},\nu_{1}}\\}$. By pigeonholing,
we can choose $\eta,\beta_{1},\nu_{1},\beta_{2},N_{1},\lambda_{1}$ such that

$\begin{split}&\quad\|e^{it\psi(D)}f_{\Omega_{1}}\|_{L^{6}(B)}\\\
&\lesssim(\log
R)^{6}K^{\varepsilon^{4}}\Big{(}\sum_{\Box\in\mathbb{B}_{\eta,\beta_{1},\nu_{1},\beta_{2},N_{1},\lambda_{1}},B\subset
Y_{\Box,\eta,\beta_{1},\nu_{1}}}\|e^{it\psi(D)}f_{\Box}\|^{2}_{L^{6}(\omega_{B})}\Big{)}^{1/2}\end{split}$
(3.2)

holds for a fraction $\gtrsim(\log R)^{-6}$ of all $K$-cubes $B$. For brevity,
we denote by

$Y^{1}_{\Box}:=Y^{1}_{\Box,\eta,\beta_{1},\nu_{1}},\quad\mathbb{B}:=\mathbb{B}_{\eta,\beta_{1},\nu_{1},\beta_{2},N_{1},\lambda_{1}}.$

Finally, we sort the $K$-cubes $B$ satisfying (3.2) by
$\sharp\\{\Box\in\mathbb{B}:B\subset Y^{1}_{\Box}\\}$. Let
$Y^{\prime}_{\mu}\subset Y^{1}$ be a union of cubes that obey

$\|e^{it\psi(D)}f_{\Omega_{1}}\|_{L^{6}(B)}\lesssim(\log
R)^{6}K^{\varepsilon^{2}}\Big{(}\sum_{\Box\in\mathbb{B},B\subset
Y^{1}_{\Box}}\|e^{it\psi(D)}f_{\Box}\|^{2}_{L^{6}(\omega_{B})}\Big{)}^{1/2}$
(3.3)

and

$\sharp\\{\Box\in\mathbb{B}:B\subset Y^{1}_{\Box}\\}\sim\mu.$ (3.4)

Since the dyadic number $\mu$ must satisfy $1\leq\mu\leq R^{C}$, there are
$\sim\log R$ different choices for $\mu$, so by pigeonholing there exists one
such $\mu$ such that the number of cubes in $Y^{\prime}_{\mu}$ is
$\gtrsim\frac{N}{(\log R)^{7}}$. We have

$\|e^{it\psi(D)}f_{\Omega_{1}}\|^{6}_{L^{6}(Y)}\lesssim(\log
R)^{7}\sum_{B\subset
Y^{\prime}}\|e^{it\psi(D)}f_{\Omega_{1}}\|^{6}_{L^{6}(B)}.$ (3.5)

For each $B\subset Y^{\prime}$, it follows from (3.3), (3.4) and Hölder’s
inequality that

$\|e^{it\psi(D)}f_{\Omega_{1}}\|^{6}_{L^{6}(B)}\lesssim(\log
R)^{36}K^{6\varepsilon^{2}}\mu^{2}\sum_{\Box\in\mathbb{B},B\subset
Y_{\Box}}\|e^{it\psi(D)}f_{\Box}\|^{6}_{L^{6}(\omega_{B})}.$ (3.6)

Putting (3.5) and (3.6) together, one has

$\|e^{it\psi(D)}f_{\Omega_{1}}\|_{L^{6}(Y^{1})}\lesssim(\log
R)^{C}K^{\varepsilon^{2}}\mu^{1/3}\Big{(}\sum_{\Box\in\mathbb{B}}\|e^{it\psi(D)}f_{\Box}\|^{6}_{L^{6}(Y^{1}_{\Box})}\Big{)}^{1/6}.$
(3.7)

The key point is that phase functions of the form $\psi(\xi_{1},\xi_{2})$ are
closed over each region $\tau=\tau_{\Box}$ in $\Omega_{1}$ under the change of
variable

$\left\\{\begin{array}[]{l}\xi_{1}=a+K^{-\frac{1}{2}}\eta_{1},\;a\in[\frac{1}{2},1-K^{-\frac{1}{2}}]\cap
K^{-1/2}\mathbb{Z},\\\ \xi_{2}=K^{-\frac{1}{m}}\eta_{2}.\end{array}\right.$

More precisely, under the change of variables $\psi(\xi_{1},\xi_{2})$ becomes

$\tilde{\psi}(\eta_{1},\eta_{2}):=K[\phi_{1}(a+K^{-1/2}\eta_{1})-\phi_{1}(a)-K^{-1/2}\eta_{1}\phi^{\prime}_{1}(a)]+\eta^{m}_{2},$

which satisfies the condition (2.37). We write

$|e^{it\psi(D)}f_{\Box}(x)|=K^{-\frac{m+2}{4m}}|e^{i\tilde{t}\tilde{\psi}(D)}g(\tilde{x})|,$

where

$\hat{g}(\eta_{1},\eta_{2}):=K^{-\frac{m+2}{4m}}\hat{f}(a+K^{-\frac{1}{2}}\eta_{1},K^{-\frac{1}{m}}\eta_{2}),\quad\|g\|_{2}=\|f\|_{2},$

and

$\left\\{\begin{array}[]{l}\tilde{x}_{1}:=K^{-\frac{1}{2}}(x_{1}+\phi^{\prime}_{1}(a)t),\\\
\tilde{x}_{2}:=K^{-\frac{1}{m}}x_{2},\\\
\tilde{t}:=K^{-1}t.\end{array}\right.$

We denote the above relation by $(\tilde{x},\tilde{t})=\mathcal{L}(x,t)$ and
write

$\|e^{it\psi(D)}f_{\Box}\|_{L^{6}(Y^{1}_{\Box})}=K^{-\frac{1}{3m}}\|e^{i\tilde{t}\tilde{\psi}(D)}g(\tilde{x})\|_{L^{6}(\tilde{Y}^{1})},$
(3.8)

where $\tilde{Y}^{1}=\mathcal{L}(Y^{1}_{\Box})$. By (3.8) and the inductive
hypothesis at scale $R_{1}$, we derive

$\|e^{it\psi(D)}f_{\Box}\|_{L^{6}(Y^{1}_{\Box})}\lesssim
K^{-\frac{1}{3m}}N_{1}^{-\frac{1}{3}}\lambda_{1}^{\frac{1}{6}}\nu_{1}^{\frac{1}{6}}(\tfrac{R}{K})^{\frac{1}{3}-\frac{4-\alpha}{6m}+\varepsilon}\|f_{\Box}\|_{2}.$

Now we consider the cardinality of the set
$\\{(\Box,B):\Box\in\mathbb{B},B\subseteq Y^{1}_{\Box}\cap Y^{\prime}\\}$. On
one hand, by the choice of $\mu$ as in (3.4), there is a lower bound

$\sharp\\{(\Box,B):\Box\in\mathbb{B},B\subseteq Y^{1}_{\Box}\cap
Y^{\prime-7}N^{(1)}\mu.$

On the other hand, by the choices of $N_{1}$ and $\eta$, for each
$\Box\in\mathbb{B}$, $Y^{1}_{\Box}$ contains $\sim N_{1}$ tubes $S$ and each
$S$ contains $\sim\eta$ cubes in $Y^{1}$, so

$\sharp\\{(\Box,B):\Box\in\mathbb{B},B\subseteq Y^{1}_{\Box}\cap
Y^{\prime}\\}\lesssim(\sharp\mathbb{B})N_{1}\eta.$

Therefore, we get

$\frac{\mu}{\sharp\mathbb{B}}\lesssim\frac{(\log R)^{7}N_{1}\eta}{N^{(1)}}.$
(3.9)

Next, by our choices of $\lambda_{1}$ as in (3.1) and $\eta$, it holds that

$\begin{split}\lambda_{1}\eta&\sim\max_{T_{r}\subset\Box:r\geq
K_{1}}\frac{\sharp\\{S:S\subseteq Y_{\Box}^{1}\cap
T_{r}\\}}{r^{\alpha}}\cdot\sharp\\{B:B\subseteq S\cap Y^{1}\\}\\\
&\lesssim\max_{T_{r}\subset\Box:r\geq K_{1}}\frac{\sharp\\{B\subset Y:B\subset
T_{r}\\}}{r^{\alpha}}\\\
&\leq\frac{K^{\frac{3}{2}-\frac{2}{m}}\lambda^{(1)}(K^{\frac{1}{m}}r)^{\alpha}}{r^{\alpha}},\end{split}$

where the last inequality follows from the fact that one can cover a
$K^{\frac{1}{2}}r\times K^{\frac{1}{m}}r\times Kr$-tube $T_{r}$ by $\sim
K^{\frac{3}{2}-\frac{2}{m}}$ finitely overlapping $K^{\frac{1}{m}}r$-balls.
Thus, we obtain

$\eta\lesssim\frac{\lambda^{(1)}K^{\frac{3}{2}+\frac{\alpha-2}{m}}}{\lambda_{1}}.$
(3.10)

Finally, we relate $\nu_{1}$ and $\nu^{(1)}$ by considering the number of
$K$-cubes in each relevant
$R^{\frac{1}{2}}\times\frac{R^{1/2}}{K^{\frac{1}{2}-\frac{1}{m}}}\times
K^{\frac{1}{2}}R^{\frac{1}{2}}$-tube $S^{\prime}$. On one hand, each relevant
$S^{\prime}$ contains $\sim\nu_{1}$ tubes $S$ in $Y_{\Box}$ and each such $S$
contains $\sim\eta$ $K$-cubes. On the other hand, a relevant $S^{\prime}$ can
be covered by $\sim K^{1/2}$ finitely overlapping $R^{\frac{1}{2}}$-cubes and
each $R^{\frac{1}{2}}$-cube contains $\lesssim\nu^{(1)}$ many $K$-cubes in
$Y^{1}$. It follows that

$\nu_{1}\lesssim\frac{K^{\frac{1}{2}}\nu^{(1)}}{\eta}.$ (3.11)

Using the similar argument as in the proof of (2.36), we derive

$\|e^{it\psi(D)}f_{\Omega_{1}}\|_{L^{6}(Y^{1})}\lesssim
K^{2\varepsilon^{2}-\varepsilon}(N^{(1)})^{-\frac{1}{3}}(\nu^{(1)})^{\frac{1}{6}}(\lambda^{(1)})^{\frac{1}{6}}R^{\frac{1}{3}-\frac{4-\alpha}{6m}+\varepsilon}\|f\|_{2}.$

Since $K=R^{\delta}$ and $R$ can be assumed to be sufficiently large compared
to any constant depending on $\varepsilon$, we have
$K^{2\varepsilon^{2}-\varepsilon}\ll 1$ and the induction closes. Recall that
$\frac{N}{\log R}\lesssim N^{(1)}\leq N,\nu^{(1)}\leq\nu$ and
$\lambda^{(1)}\leq\lambda$. This yields

$\|e^{it\psi(D)}f_{\Omega_{1}}\|_{L^{6}(Y^{1})}\lesssim_{\varepsilon}N^{-\frac{1}{3}}\lambda^{\frac{1}{6}}\nu^{\frac{1}{6}}R^{\frac{1}{3}-\frac{4-\alpha}{6m}+\varepsilon}\|f\|_{2}.$

We complete the proof of Lemma 2.1.

##  4\. Applications of Corollary 1.7

###  4.1. Application to the average Fourier decay of fractal measures
associated with the surfaces $F^{2}_{m}$.

We recall the definition of $\alpha$-dimensional measure as follows.

######  Definition 4.1.

Let $\alpha\in(0,d]$. We say that $\mu$ is an $\alpha$ -dimensional measure in
$\mathbb{R}^{d}$ if it is a probability measure supported in the unit ball
$B^{d}(0,1)$ and satisfies

$\mu(B(x,r))\leq C_{\mu}r^{\alpha},\;\forall r>0,\;\forall
x\in\mathbb{R}^{d}.$

We denote $d\mu_{R}(\cdot):=R^{\alpha}d\mu(\frac{\cdot}{R})$. Let
$\gamma_{d}(\alpha)$ denote the supremum of the numbers $\gamma$ for which

$\|\hat{\mu}(R\cdot)\|_{L^{2}(\mathbb{S}^{d-1})}^{2}\leq
C_{\alpha,\mu}R^{-\gamma}$

whenever $R>1$ and $\mu$ is an $\alpha$-dimensional measure in
$\mathbb{R}^{d}$. The problem of identifying the precise value of
$\gamma_{d}(\alpha)$ was proposed by Mattila [18]. The lower bound of
$\gamma_{d}(\alpha)$ has been studied by several authors. For instance, one
can see [22, 11, 12].

Let $\beta_{2,m}(\alpha)$ denote the supremum of the numbers $\beta$ for which

$\|\hat{\mu}(R\cdot)\|_{L^{2}(F_{m}^{2})}^{2}\leq C_{\alpha,\mu}R^{-\beta}$

whenever $R>1$ and $\mu$ is an $\alpha$-dimensional measure in
$\mathbb{R}^{3}$. We will study the lower bound of $\beta_{2,m}(\alpha)$. For
$m=2$, the surface $F_{m}^{2}$ is exactly the paraboloid $P^{2}$ over the
region $[0,1]^{2}$. Note that the Gaussian curvature of the surface
$F_{m}^{2}$ vanishes when $\xi_{1}=0$ or $\xi_{2}=0$, which is different from
the sphere or the paraboloid case studied in the literature.

We are going to derive the following result.

######  Theorem 4.2.

Let $m\geq 4$ be an even number and $0<\alpha\leq 3$. Then

$\beta_{2,m}(\alpha)\geq\big{(}\frac{5}{6}-\frac{1}{3m}\big{)}\alpha+\frac{5}{3m}-\frac{5}{6}.$

By the arguments in Remark 2.5 from [11], Theorem 4.2 can be reduced to the
weighted restriction estimate in Theorem 4.3. In fact, one can obtain

$\beta_{2,m}(\alpha)\geq 2\big{(}\frac{\alpha}{2}-\gamma\big{)},$

if there holds that for any $\varepsilon>0,$

$\|e^{it\phi(D)}f\|_{L^{2}(B^{3}(0,R);d\mu_{R}(x,t))}\leq
C_{\varepsilon}R^{\gamma+\varepsilon}\|f\|_{2}.$

######  Theorem 4.3.

Let $m\geq 4$ be an even number and $0<\alpha\leq 3$. Suppose that $\mu$ is an
$\alpha$-dimensional measure in $\mathbb{R}^{3}$. For all $R>1$ and any
$\varepsilon>0$, there exists a positive constant $C_{\varepsilon}$ such that

$\|e^{it\phi(D)}f\|_{L^{2}(B^{3}(0,R);d\mu_{R}(x,t))}\leq
C_{\varepsilon}R^{\frac{5}{12}-\frac{5}{6m}+(\frac{1}{12}+\frac{1}{6m})\alpha+\varepsilon}\|f\|_{2},$

holds for all $f$ with Fourier supports in $[0,1]^{2}$.

Using Theorem 1.5 and the argument in the proof of Theorem 2.2 from [12] we
may prove Theorem 4.3 as follows.

Denote $e^{it\phi(D)}f(x)$ by $\mathcal{E}f(x,t)$, and $(x,t)$ by $\tilde{x}$.
Since $supp\;\hat{f}\subseteq B^{2}(0,1)$, we have
$supp\;\widehat{\mathcal{E}f}\subseteq B^{3}(0,1)$. Thus there exists a
Schwartz bump function $\psi$ on $\mathbb{R}^{3}$ such that
$(\mathcal{E}f)^{2}=(\mathcal{E}f)^{2}\ast\psi$. The function
$\max_{|\tilde{y}-\tilde{x}|\leq e^{300}}|\psi(\tilde{y})|,$ which we denote
it by $\psi_{1}(\tilde{x}),$ rapidly decays. Note also that any $(x,t)$ in
$\mathbb{R}^{3}$ belongs to a unique integer lattice cube whose center we
denote by $\tilde{m}=(m,m_{3})=(m_{1},m_{2},m_{3})=\tilde{m}(x,t).$ Then we
have

$\begin{split}&\quad\left\|e^{it\phi(D)}f\right\|_{L^{2}(B^{3}(0,R);d\mu_{R}(x,t))}^{2}\\\
&=\int_{B^{3}(0,R)}|\mathcal{E}f(x,t)|^{2}d\mu_{R}(x,t)\\\
&\leq\int_{B^{3}(0,R)}(|\mathcal{E}f|^{2}\ast\psi)(x,t)d\mu_{R}(x,t)\\\
&\lesssim\int_{B^{3}(0,R)}(|\mathcal{E}f|^{2}\ast\psi_{1})(\tilde{m}(x,t))d\mu_{R}(x,t)\\\
&\leq\sum_{\tilde{m}\in\mathbb{Z}^{3},|m_{i}|\leq
R}\Big{(}\int_{|(x,t)-\tilde{m}|\leq
10}d\mu_{R}(x,t)\Big{)}\cdot(|\mathcal{E}f|^{2}\ast\psi_{1})(\tilde{m}).\end{split}$

Without loss of generality, we may assume $\|f\|_{2}=1$. For each
$\tilde{m}\in\mathbb{Z}^{3}$ we define

$\nu_{\tilde{m}}:=\int_{|(x,t)-\tilde{m}|\leq 10}d\mu_{R}(x)\lesssim 1.$

From the discussion above, we have
$(|\mathcal{E}f|^{2}\ast\psi_{1})(\tilde{m})\lesssim 1,$ which implies

$\|e^{it\phi(D)}f\|_{L^{2}(B^{3}(0,R);d\mu_{R}(x,t))}^{2}\lesssim\sum_{\nu\in[R^{-300},1]}\sum_{\tilde{m}:\nu_{\tilde{m}\sim\nu}}\nu\cdot(|\mathcal{E}f|^{2}\ast\psi_{1})(\tilde{m})+R^{-270},$

where the first summation is over all dyadic numbers $\nu$. For each dyadic
$\nu$, denote $A_{\nu}=\\{\tilde{m}\in\mathbb{Z}^{3}:\;|m_{i}|\leq
R,\nu_{\tilde{m}}\sim\nu\\}$. Performing a dyadic pigeonholing over $\nu$ we
see that there exists a dyadic $\nu\in[R^{-300},1]$ such that for any small
$\varepsilon>0$,

$\begin{split}&\quad\left\|e^{it\phi(D)}f\right\|_{L^{2}(B^{3}(0,R);d\mu_{R}(x,t))}^{2}\\\
&\lesssim_{\varepsilon}R^{\varepsilon}\sum_{\tilde{m}\in
A_{\nu}}\nu\left(\int_{B^{3}(\tilde{m},R^{\varepsilon})}|\mathcal{E}f|^{2}\right)+R^{-240}\\\
&\lesssim_{\varepsilon}R^{\varepsilon}\nu\cdot\int_{\bigcup_{\tilde{m}\in
A_{\nu}}B^{3}(\tilde{m},R^{\varepsilon})}|\mathcal{E}f|^{2}+R^{-240}.\end{split}$

Consider the set $X_{\nu}=\bigcup_{m\in
A_{\nu}}B^{3}(\tilde{m},R^{\varepsilon})$. For every $r>R^{2\varepsilon}$, by
the definition of $A_{\nu}$ and the assumption of $\mu_{R}$, for any
$\tilde{x}\in B^{3}(0,r)$, we have

$\nu\sum_{\tilde{m}\in A_{\nu}\atop B^{3}(\tilde{m},R^{\varepsilon})\subset
B^{3}(\tilde{x},r)}1\leq\sum_{\tilde{m}\in
A_{\nu}}\int_{B^{3}(\tilde{m},R^{\varepsilon})\cap
B^{3}(\tilde{x},r)}d\mu_{R}\leq\int_{B^{3}(\tilde{x},r)}d\mu_{R}\leq
r^{\alpha}.$

Thus, the intersection of $X_{\nu}$ and any $r-$ball can be contained in no
more than $\nu^{-1}r^{\alpha}$ disjoint $R^{\varepsilon}$-balls. Hence, we can
decompose the set $\bigcup_{\tilde{m}\in
A_{\nu}}B^{3}(\tilde{m},R^{\varepsilon})$ further into a union of unit cubes
$X=\cup B_{k}$ satisfying that

$\\#\\{B_{k}\subset X:B_{k}\subset B^{3}(\tilde{x},r)\\}\lesssim
R^{30\epsilon}\nu^{-1}r^{\alpha}\text{ for any
}\tilde{x}\in\mathbb{R}^{3}\text{ and }r\geq 1.$

Now, we can apply Corollary 1.7 to $X$ with $\lambda\lesssim
R^{30\varepsilon}\nu^{-1}$ and $\alpha$, which yields

$\begin{split}\quad\left\|e^{it\phi(D)}f\right\|_{L^{2}(B^{3}(0,R);d\mu_{R}(x,t))}&\lesssim\nu^{\frac{1}{2}}\left\|e^{it\phi(D)}f\right\|_{L^{2}(X)}\\\
&\lesssim
C_{\varepsilon}\nu^{\frac{1}{2}-\frac{1}{3}}R^{\frac{5}{6}+\frac{\alpha}{6}-\frac{5-\alpha}{3m}+20\varepsilon}\|f\|_{L^{2}}\\\
&\lesssim
C_{\varepsilon}R^{\frac{5}{6}+\frac{\alpha}{6}-\frac{5-\alpha}{3m}+20\varepsilon}\|f\|_{L^{2}}.\end{split}$

Here, we used the fact $\nu\lesssim 1$ in the last inequality.

It completes the proof of Theorem 4.3.

###  4.2. Application to the almost everywhere convergence problem associated
to fractal dimensional measures

One can also consider the best Sobolev exponent for the almost everywhere
convergence problems associated to $\alpha$-dimensional Hausdorff measures.
Results of this direction are widely studied by many authors, and one can
refer to [2, 10, 12, 19, 13] for more details.

We denote

$s_{m}(\alpha)=\inf\Big{\\{}s\geq 0:\lim_{t\rightarrow
0}e^{it\phi(D)}f=f~{}\alpha\text{-a.e.}\text{ for every }f\in
H^{s}(\mathbb{R}^{2})\Big{\\}}.$

Combining Corollary 1.1 and results of Eceizabarrena and Ponce-Vanegas [13]
and the trivial dispersive estimates

$\|e^{it\phi(D)}\varphi(x)\|_{L_{x}^{\infty}(\mathbb{R}^{2})}\leq
C_{\varphi}~{}|t|^{-\frac{1}{2}-\frac{1}{m}}$

for any Schwartz function $\varphi$ with support in $A(1)$ 111 In fact, we
need Theorem 1.1, Theorem 1.2, and Lemma 2.4 in [13]., we have

######  Corollary 4.4.

For $m\geq 4$ being an even number, we have

$s_{m}(\alpha)\left\\{\begin{aligned}
=&~{}\frac{2-\alpha}{2},&0\leq\alpha\leq\frac{1}{2}+\frac{1}{m};\\\
\leq&~{}\frac{3}{4}-\frac{1}{2m},&\frac{1}{2}+\frac{1}{m}\leq\alpha\leq\frac{7m-2}{5m-2};\\\
\leq&~{}\frac{4}{3}-\frac{2}{3m}-\frac{\alpha}{12}\big{(}5-\frac{2}{m}\big{)},&\frac{7m-2}{5m-2}\leq\alpha\leq
2.\\\ \end{aligned}\right.$

######  Remark 4.5.

For general $0<\alpha\leq 2$, we collect the known results of upper bound and
lower bound of $s_{m}(\alpha)$ and show them in Figure 1 (we take $m=4$ for
simplicity). Eceizabarrena and Ponce-Vanegas [13] proved that the light grey
region is the divergence region and the dark grey region is the convergence
region. Our results show that the yellow region is also a convergence region.
To the authors’ best knowledge, the convergence of the remaining regions
remains open.

Figure 1. Convergence and Divergence regions($m=4$).

Acknowledgments. We thank Daniel Eceizabarrena for pointing out their results
in [13] to us. J. Zhao is supported by National Natural Science Foundation of
China (Grant No. 12101562) and Natural Science Foundation of Zhejiang (No.
LQ20A010003). T. Zhao is supported by the Fundamental Research Funds for the
Central Universities (FRF-TP-20-076A1).

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