# The Bochner-Riesz problem: An old approach revisited

Shaoming Guo  Department of Mathematics
University of Wisconsin Madison
USA<EMAIL_ADDRESS>,  Changkeun Oh  Department of Mathematics
University of Wisconsin Madison
USA<EMAIL_ADDRESS>,  Hong Wang  School of Mathematics
Institute for Advanced Study
USA<EMAIL_ADDRESS>,  Shukun Wu  Department of Mathematics
University of Illinois at Urbana-Champaign
Urbana, IL, 61801, USA<EMAIL_ADDRESS>and  Ruixiang Zhang  Department
of Mathematics, University of Wisconsin Madison
and School of Mathematics
Institute for Advanced Study
USA<EMAIL_ADDRESS>
###### Abstract.

We show that the recent techniques developed to study the Fourier restriction
problem apply equally well to the Bochner-Riesz problem. This is achieved via
applying a pseudo-conformal transformation and a two-parameter induction-on-
scales argument. As a consequence, we improve the Bochner-Riesz problem to the
best known range of the Fourier restriction problem in all high dimensions.

##  1\. Introduction

For $\alpha\geq 0$, the Bochner-Riesz multiplier of order $\alpha$ is defined
by

(1.1) $m^{\alpha}(\xi):=(1-|\xi|^{2})^{\alpha}_{+},$

where $(1-|\xi|^{2})_{+}$ is defined to be $1-|\xi|^{2}$ whenever $|\xi|\leq
1$, and $0$ otherwise. We define the Bochner-Riesz operator $m^{\alpha}(D)(f)$
to be

(1.2) $m^{\alpha}(D)f(x):=\int_{\mathbb{R}^{n}}e^{2\pi i\langle
x,\xi\rangle}m^{\alpha}(\xi)\hat{f}(\xi)\,d\xi.$

The Bochner-Riesz conjecture is as follows.

######  Conjecture 1.1 (Bochner-Riesz conjecture).

For every $n\geq 2$ and $p\geq 2n/(n-1)$, it holds that

(1.3)
$\|m^{\alpha}(D)f\|_{L^{p}(\mathbb{R}^{n})}\lesssim_{n,\alpha,p}\|f\|_{L^{p}(\mathbb{R}^{n})}$

whenever $\alpha>n(\frac{1}{2}-\frac{1}{p})-\frac{1}{2}$.

Let $S^{n-1}$ denote the unit sphere in $\mathbb{R}^{n}$. Let $d\sigma$ denote
the surface measure of $S^{n-1}$. Take $f\in L^{\infty}(S^{n-1},d\sigma)$.
Define

(1.4) $\widehat{fd\sigma}(x):=\int_{S^{n-1}}e^{2\pi i\langle
x,\omega\rangle}f(\omega)d\sigma(\omega),\ x\in\mathbb{R}^{n}.$

The (dual form of) Fourier restriction conjecture is as follows.

######  Conjecture 1.2 (Restriction conjecture).

Let $n\geq 2$. It holds that

(1.5)
$\|\widehat{fd\sigma}\|_{L^{p}(\mathbb{R}^{n})}\lesssim_{n,p}\|f\|_{L^{\infty}(S^{n-1},d\sigma)},$

whenever $p>2n/(n-1)$.

Let $\mathfrak{R}f=\widehat{f}|_{S^{n-1}}$ be the sphere restriction operator.
By duality and the factorization theory of Maurey, Nikishin and Pisier (see
Bourgain [Bou91a]), the restriction conjecture is equivalent to that

(1.6)
$\|\mathfrak{R}f\|_{L^{p}(S^{n-1})}\lesssim_{n,p}\|f\|_{L^{p}(\mathbb{R}^{n})},$

for every $1\leq p<2n/(n+1)$. Indeed, (1.6) is the original Fourier
restriction conjecture of Stein (see for instance page 345 of [SS11]). Here we
state the equivalent version as in Conjecture 1.2 since it is closer to the
operator under investigation in the current paper.

Tao [Tao99] proved that the Bochner-Riesz conjecture implies the restriction
conjecture. Moreover, he mentioned in his paper that these two conjectures
“are widely believed to be at least heuristically equivalent”. The information
we would like to convey in the current paper is that, after applying the
pseudo-conformal transformation (see (2.12) below), the recently developed
techniques in the Fourier restriction literature apply equally well to the
Bochner-Riesz problem. These techniques include, but are not limited to, the
broad-narrow analysis of Bourgain and Guth [BG11], the polynomial method of
Guth in [Gut16, Gut18], and the polynomial Wolff axioms obtained by Guth
[Gut16], Zahl [Zah18], and Katz and Rogers [KR18] that was applied in the
Fourier restriction setting in Guth [Gut16], Hickman and Rogers [HR19] and
Hickman and Zahl [HZ20]. To be slightly more precise, we will see that after
the pseudo-conformal transformation, the above mentioned techniques do not see
the differences between the Bochner-Riesz problem and the Fourier restriction
problem. As a consequence, we show that the Bochner-Riesz conjecture holds for
every $p$ for which the restriction conjecture has been verified in the above
mentioned papers.

To state our result, let us recall what is known about the restriction
conjecture. For $n=3$, Guth [Gut16] proved the restriction conjecture for
$p>3.25$; moreover, for $n\geq 4$, he proved in [Gut18] that the restriction
conjecture holds if

(1.7) $\begin{split}&p>2\cdot\frac{3n+1}{3n-3}\text{ for }n\text{ odd}.\\\
&p>2\cdot\frac{3n+2}{3n-2}\text{ for }n\text{ even}.\end{split}$

These results improved prior ones due to Tao [Tao03] and Bourgain and Guth
[BG11]. More recently, for certain dimensions $n$, in particular all “large”
$n$, Hickman and Rogers [HR19] and Hickman and Zahl [HZ20] applied the
polynomial Wolff axioms established by Katz and Rogers [KR18] and further
improved the results of Guth. The new ranges of $p$ in [HR19] and [HZ20] are a
bit technical to state, and we refer the interested readers to Figure 2 in
[HZ20].

Let $p_{\mathrm{restr}}$ denote the minimum of the exponent $p$ for which the
restriction conjecture has been verified in [Gut16], [Gut18], [HR19] and
[HZ20].

######  Theorem 1.3.

For every $n\geq 3$, the Bochner-Riesz conjecture holds for every $p\geq
p_{\mathrm{restr}}$.

Before we comment on the proof of Theorem 1.3, let us briefly review some
known results about the Bochner-Riesz conjecture in the literature. When
$n=2$, the Bochner-Riesz conjecture was resolved by Carleson and Sjölin in
[CS72]; see also Hörmander [H7̈3] and Fefferman [Fef73] for alternative
proofs. When $n\geq 3$, this conjecture remains open; Fefferman [Fef70],
Bourgain [Bou91a, Bou91b], Lee [Lee04] and Bourgain and Guth [BG11] made
significant partial progress towards this conjecture; see also Christ [Chr88],
Seeger [See96], Tao [Tao98] and Lee [Lee06, Lee18] for more related results
and some endpoints results. The most recent progress was made by Guth, Hickman
and Iliopoulou [GHI19]. The approach in the current paper has closer relation
with those in Bourgain and Guth [BG11] and Guth, Hickman and Iliopoulou
[GHI19], and therefore we expand a discussion on these two papers.

In these two papers, the authors there viewed the Bochner-Riesz operator as an
oscillatory integral operator of the Hörmander type with positive-definite
phase. To be more precise, they followed Carleson and Sjölin [CS72]111See also
Theorem 2.1 and Theorem 2.2 below where the same reduction is used. and
reduced the $L^{p}$ bounds of the Bochner-Riesz operator to the $L^{p}$ bounds
of oscillatory integral operators of the form

(1.8) $T^{\lambda}f(x):=\int_{\mathbb{R}^{n-1}}e^{2\pi
i\phi^{\lambda}(x;\omega)}a^{\lambda}(x;\omega)f(\omega)d\omega,\ \
x\in\mathbb{R}^{n},$

where the phase function $\phi^{\lambda}$ satisfies the Hörmander condition
(see for instance (H1) and (H2) in [GHI19, page 252]) and a positive-definite
condition (see (H2+) in [GHI19, page 254]). For these oscillatory integral
operators, it was proven in [GHI19] that

(1.9)
$\|T^{\lambda}f\|_{L^{p}(\mathbb{R}^{n})}\lesssim_{\epsilon,p,\phi}\lambda^{\epsilon}\|f\|_{L^{p}(\mathbb{R}^{n-1})},$

for every $\lambda\geq 1$, $\epsilon>0$ and every $p$ satisfying (1.7). As a
consequence, the authors there obtained the $L^{p}$ bounds (1.3) of the
Bochner-Riesz operator for the same range of $p$. It is worth mentioning that
the authors of [GHI19] also proved that their result is sharp, that is, the
range of $p$ in (1.7) is sharp for $T^{\lambda}$ with a phase function
satisfying the above-mentioned positive-definite Hörmander condition. This
also means that if one plans to prove (1.3) by viewing the Bochner-Riesz
operator as an oscillatory integral operator of the Hörmander type, then the
range (1.7) is the best that one can hope for.

The way that Guth, Hickman and Iliopoulou [GHI19] proved the sharpness of
their result is built on the work of Bourgain [Bou91b]. The sharp examples in
these two papers rely crucially on the fact that for general operators
$T^{\lambda}$ satisfying the Hörmander condition, wave packets may be curved.
One key observation of the current paper is that, after applying the pseudo-
conformal transformation (see (2.12) below) to the Bochner-Riesz operator, all
the new wave packets admit similar behavior as those in the Fourier
restriction problem, say in [Gut16], [Gut18], [HR19] and [HZ20]. Roughly
speaking, this is what allows us to apply the recent techniques developed in
the Fourier restriction theory to the Bochner-Riesz problem. Here we would
like to emphasize that the use of the pseudo-conformal transformation in the
context of the Bochner-Riesz problem is not new. Indeed, Carbery [Car92]
already used it to prove that the Fourier restriction conjecture for
paraboloids implies the Bochner-Riesz conjecture for paraboloids.

In the end, we would like to briefly mention some other interesting features
of the proof of our main theorem, and compare them with those in the
literature aforementioned, in particular, in [GHI19] where the latest progress
on the Bochner-Riesz conjecture were made. Firstly, in [GHI19], the authors
there always first reduce phase functions to normal forms (see [Bou91b, page
328] for the definition of normal forms, which are also referred as “reduced
forms” in [GHI19]) and then only work with normal forms. In our case, we work
with the phase function of the Bochner-Riesz operator directly; indeed, our
proof relies on a special parabolic rescaling structure of the phase function
(see the proof of Lemma 3.8), which makes our induction argument perhaps
different from that of [GHI19] (see also Remark 3.1 for more discussions).
Whether this is just a technical point or not remains to be understood.
Secondly, it is perhaps worth mentioning that the way we prove the transverse
equidistribution estimate (the content of Section 6) is slightly different
from that in [Gut18] and [GHI19]; we observe that instead of considering
output functions (for instance $H^{\lambda}g$ as in (2.20)) as a medium, one
can directly prove the transverse equidistribution estimate by essentially
only working with the input function $g$, which gives us a slightly simpler
proof. More details are included in Section 6.

######  Remark 1.4.

We make a remark on the Bochner-Riesz problem and the Fourier restriction
problem in the case $n=3$. In this case, so far the best result for the
restriction conjecture is due to Wang [Wan18], where she proved that this
conjecture holds for $p>3+3/13$. It is not implausible that if one combines
the argument of [Wan18] with that of the current paper, then one may be able
to improve Theorem 1.3 to the same range of $p$ when $n=3$. We do not pursue
it here. Regarding the Bochner-Riesz problem in $n=3$, recently Wu [Wu20]
proved that the Bochner-Riesz conjecture holds for $p\geq
p_{\mathrm{restr}}=3.25$ when $n=3$. His proof partially relies on some ideas
from Wang [Wan18]. Our Theorem 1.3 recovers the result in [Wu20] via a quite
different and a slightly simpler approach.

Organization of the paper. In Section 2, we make several reductions to the
Bochner-Riesz problem, including the well-known Carleson-Sjölin reduction and
a reduction via the pseudo-conformal transformation. In Section 3, we
introduce the induction hypothesis and will further reduce the desired
estimate to a broad norm estimate; the structure of our reduction argument is
similar to that of Guth, Hickman and Iliopoulou [GHI19] (see Section 11
there). In Section 4, we introduce wave packets and prove some of their
properties that will be useful in future sections. In Section 5, we compare
wave packets at different scales; this is to prepare for the use of the multi-
scale argument as in Guth [Gut18] and Guth, Hickman and Iliopoulou [GHI19]. In
Section 6, we prove a transverse equidistribution property of wave packets.
After developing the relevant tools in the previous sections, one can already
almost identify the Bochner-Riesz problem with the Fourier restriction
problem. This allows us to follow Hickman and Rogers [HR19] and Hickman and
Zahl [HZ20], as is done in Section 7, Section 8 and Section 9, to finish the
proof of the desired broad norm estimate.

Notations.

$\bullet$ We write $A(R)\leq\mathrm{RapDec}(R)B$ to mean that for any power
$\beta$, there is a constant $C_{\beta}$ such that

(1.10) $A(R)\leq C_{\beta}R^{-\beta}B\;\;\text{for all $R\geq 1$}.$

$\bullet$ The quantities $p,n$ and $\epsilon$ will be called the admissible
parameters as the estimates in the paper may be allowed to depend on these
parameters.

$\bullet$ We introduce a few other admissible parameters

(1.11)
$\epsilon^{C}\leq\delta\ll_{\epsilon}\delta_{n}\ll_{\epsilon}\delta_{n-1}\ll_{\epsilon}\cdots\ll_{\epsilon}\delta_{1}\ll_{\epsilon}\epsilon_{\circ}\ll_{\epsilon}\epsilon.$

Here $C$ is some dimensional constant and the notation $A\ll_{\epsilon}B$
indicates that $A\leq C^{-1}_{n,\epsilon}B$ for some large admissible constant
$C_{n,\epsilon}\geq 1$.

$\bullet$ For every number $R>0$ and set $S$, we denote by $N_{R}(S)$ the
$R$-neighborhood of the set $S$.

$\bullet$ We use $B({\bf x},r)$ to represent the open ball centered at ${\bf
x}$, of radius $r$, in $\mathbb{R}^{n}$. To avoid confusion, we also use
$B^{n-1}(x,r)$ to denote the $n-1$ dimensional ball centered at $x$, of radius
$r$.

Acknowledgement. The authors would like to thank Xiaochun Li, Zane Li, Andreas
Seeger, Rajula Srivastava and Terence Tao for valuable discussions. S.G. was
supported in part by the NSF grant DMS-1800274. H.W. was supported by the
National Science Foundation under Grant No. DMS-1926686. R.Z. was supported by
the NSF grant DMS-1856541, DMS-1926686 and by the Ky Fan and Yu-Fen Fan
Endowment Fund at the Institute for Advanced Study.

##  2\. Several reductions

For $\lambda\geq 1$, we define the Carleson-Sjölin operator

(2.1) $S^{\lambda}f(x):=\int_{\mathbb{R}^{n}}e^{2\pi
i\lambda|x-y|}a(x-y)f(y)\,dy,$

where $a\in C^{\infty}(\mathbb{R}^{n})$ has compact support away from the
origin. To prove Theorem 1.3, it is standard to reduce it to the following
theorem. For the reduction, we refer to Stein [Ste93, Chapter IX].

######  Theorem 2.1.

For every $n\geq 3$, it holds that

(2.2)
$\|S^{\lambda}f\|_{L^{p}(\mathbb{R}^{n})}\lesssim_{\epsilon}\lambda^{-n/p+\epsilon}\|f\|_{L^{p}(\mathbb{R}^{n})}$

for every $p\geq p_{\mathrm{restr}}$, $\lambda\geq 1$ and $\epsilon>0$.

Next, we will reduce the $L^{p}$-boundedness of the Carleson-Sjölin operator
to that of some operator $\bar{S}^{\lambda}:g\in
L^{p}(\mathbb{R}^{n-1})\mapsto L^{p}(\mathbb{R}^{n})$ by freezing one
variable. We define the operator $\bar{S}^{\lambda}g$ by

(2.3) $\bar{S}^{\lambda}g(u,t):=\int_{\mathbb{R}^{n-1}}e^{2\pi i\lambda
t^{-1}\sqrt{\lambda^{2}+|u-t\omega|^{2}}}a^{\lambda}(u,t;\omega)g(\omega)\,d\omega,$

where $u=(u_{1},\dots,u_{n-1})\in\mathbb{R}^{n-1}$,
$\omega=(\omega_{1},\dots,\omega_{n-1})\in\mathbb{R}^{n-1}$, and

(2.4)
$a^{\lambda}(u,t;\omega):=a\big{(}\frac{u}{\lambda}-\frac{t}{\lambda}\omega,\frac{t}{\lambda},\omega\big{)},$

with $a(\cdot,\cdot,\cdot)$ being a smooth function compactly supported in all
variables and away from zero in its second variable.

######  Theorem 2.2.

Under the above notation, it holds that

(2.5)
$\|\bar{S}^{\lambda}g\|_{L^{p}(\mathbb{R}^{n})}\lesssim_{\epsilon}\lambda^{\epsilon}\|g\|_{L^{p}(\mathbb{R}^{n-1})}$

for every $n\geq 3$, $p\geq p_{\mathrm{restr}}$, $\lambda\geq 1$ and
$\epsilon>0$.

Let us prove Theorem 2.1 by assuming Theorem 2.2. We fix $a$ in the definition
of the operator (2.1). Since $a$ has a compact support, by a partition of
unity and rotation, we may assume that

(2.6)
$\mathrm{supp}(a)\subset\\{(x^{\prime},x_{n},y^{\prime},y_{n}):|x_{n}-y_{n}|\simeq
1\\},$

where $x^{\prime}=(x_{1},\dots,x_{n-1})$ and
$y^{\prime}=(y_{1},\dots,y_{n-1})$. We write the Carleson-Sjölin operator as

(2.7)
$S^{\lambda}f(x)=\int_{\mathbb{R}}\bar{S}_{y_{n}}^{\lambda}f_{y_{n}}(x)\,dy_{n},$

where

(2.8) $\bar{S}_{y_{n}}^{\lambda}f_{y_{n}}(x):=\int_{\mathbb{R}^{n-1}}e^{2\pi
i\lambda|x-y|}a(x-y)f_{y_{n}}(y^{\prime})\,dy^{\prime}$

and

(2.9) $f_{y_{n}}(y^{\prime}):=f(y^{\prime},y_{n}).$

By Hölder’s inequality and by the fact that $a$ has a compact support, it
suffices to prove

(2.10)
$\|\bar{S}^{\lambda}_{y_{n}}f_{y_{n}}\|_{L^{p}(\mathbb{R}^{n})}\lesssim\lambda^{-n/p+\epsilon}\|f_{y_{n}}\|_{L^{p}_{y^{\prime}}(\mathbb{R}^{n-1})}$

uniformly for every $|y_{n}|\lesssim 1$. By a translation $x_{n}\mapsto
x_{n}-y_{n}$, we obtain an operator independent of the variable $y_{n}$.
Hence, we without loss of generality assume that $y_{n}=0$. To proceed, we
write the phase function in the oscillatory integral (2.8) as follows:

(2.11)
$\lambda\big{|}(x^{\prime},x_{n})-(y^{\prime},0)\big{|}=\lambda|x_{n}|\sqrt{1+|x^{\prime}-y^{\prime}|^{2}/x_{n}^{2}}.$

We apply a change of variables

(2.12) $(u_{1},\dots,u_{n-1},t):=(x^{\prime}/x_{n},1/x_{n}),$

which is the pseudo-conformal change of variables (see for instance Carbery
[Car92], Tao [Tao06, Section 2.3] or Rogers [Rog08]). By the support
assumption on $a$, the Jacobian is comparable to one. After this change of
variables, our operator $\bar{S}^{\lambda}_{0}f_{0}$ becomes

(2.13) $\tilde{S}^{\lambda}f_{0}(u,t):=\int_{\mathbb{R}^{n-1}}e^{2\pi i\lambda
t^{-1}\sqrt{1+|u-ty^{\prime}|^{2}}}\tilde{a}(u-ty^{\prime},t)f_{0}(y^{\prime})\,dy^{\prime},$

where $y^{\prime}=(y_{1},\dots,y_{n-1})$ and $\tilde{a}(\cdot,\cdot)$ is a
smooth function that has compact supports in all its variables and is
supported away from the origin in its second variable. At this point, due to
spatial orthogonality, we see that in (2.13) we can insert for free a smooth
cut-off function $\tilde{a}(y^{\prime})$, supported near the origin, and it is
equivalent to bound

(2.14) $\int_{\mathbb{R}^{n-1}}e^{2\pi i\lambda
t^{-1}\sqrt{1+|u-ty^{\prime}|^{2}}}\tilde{a}(u-ty^{\prime},t)\tilde{a}(y^{\prime})f_{0}(y^{\prime})\,dy^{\prime}.$

In the end, we pass from scale one to scale $\lambda$ via the change of
variables $(u,t)\mapsto\lambda^{-1}(u,t)$. This gives an operator of the form
in (2.3), and therefore finishes the proof of Theorem 2.1.

In the end, we make a reduction to the amplitude function
$a^{\lambda}(u,t;\omega)$ so that it will have a product form
$a_{1}^{\lambda}(u,t)a_{2}^{\lambda}(\omega)$. Before the reduction, let us
fix the notation. In the definition of $\bar{S}^{\lambda}$, we see that the
variable $t$ plays a distinguished role, compared with other variables.
Therefore, from now on, we will write $x=(x_{1},\dots,x_{n-1})$ and ${\bf
x}=(x,t)\in\mathbb{R}^{n}$. Moreover, define

(2.15)
$\phi^{\lambda}(x,t;\omega):=\lambda\phi\big{(}\frac{x}{\lambda},\frac{t}{\lambda};\omega\big{)}:=\frac{\lambda^{2}}{t}\sqrt{1+|\frac{x}{\lambda}-\frac{t}{\lambda}\omega|^{2}}.$

Under this notation, we can write

(2.16) $\bar{S}^{\lambda}g({\bf x})=\int_{\mathbb{R}^{n-1}}e^{2\pi
i\phi^{\lambda}({\bf x};\omega)}a^{\lambda}({\bf x};\omega)g(\omega)d\omega.$

Via a standard Fourier expansion, it suffices to consider

(2.17) $\int_{\mathbb{R}^{n-1}}e^{2\pi
i\phi^{\lambda}(x,t;\omega)}a_{1}^{\lambda}(x,t)a_{2}(\omega)g(\omega)d\omega,$

where

(2.18) $a_{1}^{\lambda}(x,t):=a_{1}(x/\lambda,t/\lambda),$

with $a_{1}(\cdot,\cdot)$ being a compactly supported function in both
variables and supported away from the origin in its second variable, and
$a_{2}(\cdot)$ is compactly supported near the origin. We need to prove

(2.19) $\Big{\|}\int_{\mathbb{R}^{n-1}}e^{2\pi
i\phi^{\lambda}(x,t;\omega)}a_{1}^{\lambda}(x,t)a_{2}(\omega)g(\omega)d\omega\Big{\|}_{L^{p}(\mathbb{R}^{n})}\lesssim\lambda^{\epsilon}\|g\|_{L^{p}(\mathbb{R}^{n-1})}.$

Such an estimate will be proven via an inductive argument on $\lambda$. In
order to set up the induction, we need to be more quantitative about the
choice of amplitude functions. Define

(2.20) $H^{\lambda}g({\bf x}):=\int_{\mathbb{R}^{n-1}}e^{2\pi
i\phi^{\lambda}(x,t;\omega)}g(\omega)d\omega.$

Let $C_{n}$ be a large enough constant such that

(2.21) $B_{C_{n}\lambda}(0)\setminus\\{{\bf x}:|t|\leq\lambda/C_{n}\\},\text{
with }B_{\lambda}(0):=[-\lambda,\lambda]^{n},$

contains the support of $a^{\lambda}_{1}$. To prove (2.19), it is elementary
to see that it suffices to prove

######  Theorem 2.3.

Under the above notation, it holds that

(2.22)
$\Big{\|}H^{\lambda}g\Big{\|}_{L^{p}(B_{C_{n}\lambda}(0)\setminus\\{{\bf
x}:|t|\leq\lambda/C_{n}\\})}\lesssim_{\epsilon}\lambda^{\epsilon}\|g\|_{L^{p}([0,1]^{n-1})},$

for every $\epsilon>0$, every $g:[0,1]^{n-1}\mapsto\mathbb{C}$, dimension
$n\geq 3$ and exponent $p\geq p_{\mathrm{restr}}$.

##  3\. Reduction to broad norm estimates

###  3.1. The induction hypothesis

We will prove (2.22) via an inductive argument. In this subsection we set up
the induction and state the induction hypothesis. Let $1\leq R\leq\lambda$.
Let $K_{R}$ be a large number that is to be determined: It is much larger than
one, but much smaller compared with $R^{\epsilon}$.222It will be chosen to be
$R^{\delta^{\prime}}$ for some extremely small $\delta^{\prime}\ll\epsilon$.
Out of certain technical reason, we introduce a scale $\lambda_{R}$ that is
comparable to $\lambda$. For $K\geq 2$, we define $\lambda_{K,R}$: The
parameter $K$ will eventually be set to be $K_{R}$ and
$\lambda_{R}=\lambda_{K_{R},R}$. If $\lambda/R<K$, we define
$\lambda_{K,R}=C_{n}\lambda.$ If $\lambda/R\geq K$, then we define

(3.1)
$\lambda_{K,R}:=C_{n}\lambda\Big{(}2+\frac{1}{K}+\dots+\frac{1}{K^{[\log_{K}(\lambda/R)]-1}}\Big{)},$

where $[\log_{K}(\lambda/R)]$ refers to the largest integer that does not
exceed $\log_{K}(\lambda/R)$. Set $\lambda_{R}:=\lambda_{K_{R},R}$. Note that
$C_{n}\lambda\leq\lambda_{K,R}\leq 3C_{n}\lambda$. To prove (2.22), it
suffices to prove

(3.2)
$\Big{\|}H^{\lambda}g\Big{\|}_{L^{p}(B_{R})}\lesssim_{n,p,\epsilon}R^{\epsilon}\|g\|_{L^{p}([0,1]^{n-1})},$

for every $1\leq R\leq\lambda^{1-\epsilon}$, and every cube

(3.3) $B_{R}\subset[-\lambda_{R},\lambda_{R}]\times[R/C_{n},C_{n}\lambda].$

This will be proven via an inductive argument on $\lambda$ and $R$.

######  Remark 3.1.

We will see that it is crucial to run an induction on both parameters
$\lambda$ and $R$. It is worth mentioning that the smaller $R$ is, the more
“singular” our phase function $\phi^{\lambda}$ behaves, which, roughly
speaking, can be explained by that $t$ appears in the denominator in (2.15).
To deal with the case that $R$ is much smaller compared with $\lambda$, we
will need to use very particular properties (see for instance (3.29) and
(3.30)) of the phase function $\phi^{\lambda}$. That $R$ can be much smaller
compared with $\lambda$ seems to indicate that our induction is perhaps
different from that of Guth, Hickman and Iliopoulou [GHI19]. In [GHI19], the
authors there inserted (and they could as well) a compactly supported
amplitude function in the definition of $H^{\lambda}$ so that they always have
$|t|\gtrsim\lambda$.

######  Remark 3.2.

The requirement that $R\leq\lambda^{1-\epsilon}$ will be used in the proof of
Lemma 3.5.

The base case of the induction $\lambda=R=1$ is trivial. We assume that we
have proven that

(3.4) $\Big{\|}H^{\lambda^{\prime}}g\Big{\|}_{L^{p}(B_{R^{\prime}})}\leq
C_{n,p,\epsilon}(R^{\prime})^{\epsilon}\|g\|_{L^{p}([0,1]^{n-1})},$

for every $\lambda^{\prime}\leq\lambda/2$, $1\leq
R^{\prime}\leq(\lambda^{\prime})^{1-\epsilon}$, and every cube

(3.5)
$B_{R^{\prime}}\subset[-\lambda^{\prime}_{R^{\prime}},\lambda^{\prime}_{R^{\prime}}]\times[R^{\prime}/C_{n},C_{n}\lambda^{\prime}].$

Our goal is to prove the same hold with $\lambda,R$.

###  3.2. Reduction to broad norm estimate

For ${\bf x}\in[-\lambda_{R},\lambda_{R}]\times[R/C_{n},C_{n}\lambda]$ and
$\omega\in[0,1]^{n-1}$, we define Gauss maps and rescaled Gauss maps. Define

(3.6) $G_{0}({\bf x};\omega):=\partial_{\omega_{1}}\nabla_{{\bf
x}}\phi\wedge\dots\wedge\partial_{\omega_{n-1}}\nabla_{{\bf x}}\phi.$

Moreover, define

(3.7) $G({\bf x};\omega):=\frac{G_{0}({\bf x};\omega)}{|G_{0}({\bf
x};\omega)|}.$

Define rescaled Gauss map

(3.8) $G^{\lambda}({\bf x};\omega):=G({\bf x}/\lambda;\omega).$

Via some elementary computation, we obtain

######  Lemma 3.3.

Under the above notation, we have that the rescaled Gauss maps equals

(3.9) $G^{\lambda}({\bf x};\omega)=\frac{(\omega,1)}{\sqrt{1+|\omega|^{2}}}.$

In particular, the rescaled Gauss map does not depend on ${\bf x}$.

###### Proof of Lemma 3.3.

For future use, we collect a few useful computations regarding the phase
function $\phi^{\lambda}$. First of all,

(3.10)
$\nabla_{x}\phi^{\lambda}(x,t;\omega)=\frac{\lambda}{t}\frac{(x-t\omega)}{\sqrt{\lambda^{2}+|x-t\omega|^{2}}};\
\
\partial_{t}\phi^{\lambda}(x,t;\omega)=-\frac{\lambda}{t^{2}}\frac{\lambda^{2}+x\cdot(x-t\omega)}{\sqrt{\lambda^{2}+|x-t\omega|^{2}}}.$

Next, we compute the gradient in $\omega$:

(3.11)
$\partial_{\omega}\phi^{\lambda}(x,t;\omega)=\frac{\lambda(t\omega-x)}{\sqrt{\lambda^{2}+|x-t\omega|^{2}}}.$

Regarding the related second order derivatives, we have

(3.12)
$\partial_{x_{i}}\partial_{\omega_{j}}\phi^{\lambda}(x,t;\omega)=\frac{\lambda(x_{i}-t\omega_{i})(x_{j}-t\omega_{j})}{\big{(}\lambda^{2}+|x-t\omega|^{2}\big{)}^{3/2}}\text{
if }i\neq j,$ (3.13)
$\partial_{x_{i}}\partial_{\omega_{i}}\phi^{\lambda}(x,t;\omega)=-\frac{\lambda\big{(}\lambda^{2}+|x-t\omega|^{2}-(x_{i}-t\omega_{i})^{2}\big{)}}{\big{(}\lambda^{2}+|x-t\omega|^{2}\big{)}^{3/2}},$

and

(3.14)
$\partial_{t}\partial_{\omega_{j}}\phi^{\lambda}(x,t;\omega)=\frac{-x_{j}\omega\cdot(x-t\omega)+\omega_{j}x\cdot(x-t\omega)}{\big{(}\lambda^{2}+|x-t\omega|^{2}\big{)}^{3/2}}.$

Combining all these, we obtain

(3.15) $\begin{split}&\partial_{\omega_{1}}\nabla_{{\bf
x}}\phi^{\lambda}\wedge\dots\wedge\partial_{\omega_{n-1}}\nabla_{{\bf
x}}\phi^{\lambda}\end{split}$

is parallel to $(\omega,1)$. By normalization, we obtain the desired result. ∎

Let $K\geq 1$. We divide $[0,1]^{n-1}$ into caps $\tau$ of side length
$K^{-2}$. Let $g_{\tau}$ denote the restriction of $g$ to $\tau$. Moreover,

(3.16) $G^{\lambda}(\tau):=\\{G^{\lambda}({\bf x};\omega):\omega\in\tau\\}.$

Let $V\subset\mathbb{R}^{n}$ be a linear subspace. Let
$\measuredangle(G^{\lambda}(\tau),V)$ denote the smallest angle between any
non-zero vector $v\in V$ and $v^{\prime}\in G^{\lambda}(\tau)$. Moreover, we
say that $\tau\notin_{K}V$ if $\measuredangle(G^{\lambda}(\tau),V)\geq
K^{-1}$; otherwise, we say $\tau\in_{K}V$. If the value of $K$ is clear from
the context, we often abbreviate $\tau\notin_{K}V$ to $\tau\notin V$. Next,
let us introduce the notion of broad norms. Fix
$B_{K^{2}}\subset[-\lambda_{R},\lambda_{R}]\times[R/C_{n},C_{n}\lambda]$
centered at ${\bf x}_{0}$. Define

(3.17)
$\mu_{H^{\lambda}g}(B_{K^{2}}):=\min_{V_{1},\dots,V_{A}\in\text{Gr}(k-1,n)}\Big{(}\max_{\begin{subarray}{c}\tau\notin
V_{a}\\\ \text{ for any }1\leq a\leq
A\end{subarray}}\|H^{\lambda}g_{\tau}\|_{L^{p}(B_{K^{2}})}^{p}\Big{)}.$

Here $\mathrm{Gr}(k-1,n)$ is the Gressmannian manifold of all
$(k-1)$-dimensional subspaces in $\mathbb{R}^{n}$, and $k$ is to be
determined, and $A$ is a parameter that is less important and its choice will
become clear later. For $U\subset\mathbb{R}^{n}$, define

(3.18)
$\|H^{\lambda}g\|_{\mathrm{BL}_{k,A}^{p}(U)}:=\Big{(}\sum_{B_{K^{2}}}\frac{|B_{K^{2}}\cap
U|}{|B_{K^{2}}|}\mu_{H^{\lambda}g}(B_{K^{2}})\Big{)}^{1/p}.$

Next we should study and prove broad norm estimates.

######  Theorem 3.4 (Broad norm estimate).

Let $2\leq k\leq n-1$, and

(3.19) $p\geq
p_{n}(k):=2+\frac{6}{2(n-1)+(k-1)\prod_{i=k}^{n-1}\frac{2i}{2i+1}}.$

Then for every $\epsilon>0$, there exists $A$ such that

(3.20)
$\|H^{\lambda}g\|_{\mathrm{BL}_{k,A}^{p}(B_{R})}\lesssim_{K,\epsilon}R^{\epsilon}\|g\|_{L^{2}}^{2/p}\|g\|_{L^{\infty}}^{1-2/p},$

for every $K\geq 1$, $1\leq R\leq\lambda$, where $B_{R}$ is a ball of radius
$R$ satisfying
$B_{R}\subset[-3C_{n}\lambda,3C_{n}\lambda]^{n-1}\times[R/C_{n},C_{n}\lambda]$.
Moreover, the implicit constant depends polynomially on $K$.

In the rest of this section, we will finish the proof of Theorem 2.3 by
assuming Theorem 3.4. We will show that Theorem 3.4 implies (2.22) whenever
$n\geq 3$ and

(3.21) $2+\frac{4}{2n-k}\leq p\leq 2+\frac{2}{k-2}.$

The same optimization process as in Hickman and Zahl [HZ20, page 4] will give
Theorem 2.3.

Let us begin the proof. The main steps are the proofs of the following Lemma
3.5 and Lemma 3.8. After proving these two lemmas, it is standard to deduce
Theorem 2.3 whenever $p$ satisfies (3.21). For the sake of completeness, we
provide some details for this step. By a restricted type interpolation, we may
assume that $g=\chi_{E}$ for some set $E$. Let us take
$K=R^{\epsilon^{\prime}}$ for some $0<\epsilon^{\prime}\ll\epsilon$. By Lemma
3.5 and Lemma 3.8, and a standard application of the broad-narrow analysis
(see for instance [GHI19, page 358]) of Bourgain and Guth [BG11] and Guth
[Gut18], one obtains that

(3.22)
$\|H^{\lambda}g\|_{L^{p}(B_{R})}\lesssim_{n,p,\epsilon}R^{\epsilon/2}\|g\|_{2}^{p/2}\|g\|_{\infty}^{1-2/p}+R^{\epsilon}R^{-C\epsilon^{\prime}}\|g\|_{p}.$

Since $g$ is a characteristic function, the above term is bounded by a
constant multiple of

(3.23)
$\big{(}R^{-\epsilon/2}+R^{-C\epsilon^{\prime}}\big{)}R^{\epsilon}\|g\|_{L^{p}}.$

Since $R$ is a large number, this closes the induction and completes the
proof.

From now on, we will focus on the proofs of those two lemma. We start with
Lemma 3.5, a decoupling inequality.

######  Lemma 3.5.

Let $2\leq m\leq n$ and $V$ be an $m$-dimensional linear subspace in
$\mathbb{R}^{n}$. Let $K=K_{R}$ and
$B_{K^{2}}\subset[-\lambda_{R},\lambda_{R}]\times[R/C_{n},C_{n}\lambda]$. For
every $2\leq p\leq 2m/(m-1)$ and every $\varepsilon>0$, it holds that

(3.24) $\begin{split}&\Big{\|}\sum_{\tau\in
V:l(\tau)=K^{-1}}H^{\lambda}g_{\tau}\Big{\|}_{L^{p}(B_{K^{2}})}\lesssim_{N,\varepsilon,p}\mathrm{RapDec}(\lambda)\|g\|_{2}+\\\
&\sum_{\beta\in\mathbb{N}_{0}^{n-1}}\sum_{\beta^{\prime}\in\mathbb{N}_{0}^{n}}\sum_{{\bf
w}\in\mathbb{N}^{n}}\frac{2^{-N|\beta^{\prime}|}K^{(m-1)(1/2-1/p)+\varepsilon}}{(1+|\beta|)^{10n}(1+|{\bf
w}|)^{100n}}\Big{(}\sum_{\tau}\|H^{\lambda}(g_{\tau}(\omega)b_{{\bf
w},\beta^{\prime}}({\bf x}_{1};\omega)e^{2\pi
i\beta\cdot\omega})\|_{L^{p}(B_{K^{2}})}^{p}\Big{)}^{1/p}.\end{split}$

Here ${\bf x}_{1}$ is the center of $B_{R}$ (not $B_{K^{2}}$), $\tau\in V$
means $\tau\in_{K}V$, and $N$ can be as large as we would like. The function
$b_{{\bf w},\beta^{\prime}}({\bf x}_{1};\omega)$ satisfies the uniform bound
$|b_{{\bf w},\beta^{\prime}}|\lesssim 1$ in all parameters.

The proof is based on the decoupling inequality of Bourgain and Demeter [BD15]
and Taylor’s expansion. It is quite important that on the right hand side we
have $B_{K^{2}}$ without any tails.

######  Remark 3.6.

We would like to mention that it is crucial for $b_{{\bf w},\beta^{\prime}}$
not to depend on the location of $B_{K^{2}}$. This will allow us sum over all
balls $B_{K^{2}}\subset B_{R}$. If $b_{{\bf w},\beta^{\prime}}$ were allowed
to depend on the location of $B_{K^{2}}$, the proof of Lemma 3.5 would be much
simpler and we would not need the requirement that
$R\leq\lambda^{1-\epsilon}$.

###### Proof of Lemma 3.5.

Let us use ${\bf x}_{0}=(x_{0},t_{0})$ to denote the center of $B_{K^{2}}$. We
will approximate the phase function $\phi^{\lambda}$ on $B_{K^{2}}$. Write

(3.25) $\phi^{\lambda}({\bf x};\omega)-\phi^{\lambda}({\bf
x}_{0};\omega)+\nabla_{{\bf x}}\phi^{\lambda}({\bf x}_{0};\omega){\bf
x}_{0}=\nabla_{{\bf x}}\phi^{\lambda}({\bf x}_{0};\omega){\bf
x}+e^{\lambda}_{2,{\bf x}_{0}}({\bf x};\omega).$

Here $e^{\lambda}_{2,{\bf x}_{0}}$ is the second order remainder term in
Taylor’s expansion. To see what estimates it satisfies, let us first collect
some useful estimates for various derivatives. First of all, from (3.10), we
see that

(3.26) $\begin{split}&\big{|}\nabla_{x}\phi^{\lambda}({\bf
x};\omega)\big{|}\lesssim\lambda/R,\ \ \big{|}\partial_{t}\phi^{\lambda}({\bf
x};\omega)\big{|}\lesssim(\lambda/R)^{2}.\end{split}$

By taking a further derivative in ${\bf x}$, we obtain

(3.27) $\big{|}\nabla^{2}_{x}\phi^{\lambda}({\bf x};\omega)\big{|}\lesssim
1/R,\ \ \big{|}\partial_{tt}\phi^{\lambda}({\bf
x};\omega)\big{|}\lesssim\lambda^{2}/R^{3}.$

We see the second order derivative in $x$ is not bad, but that in $t$ is
always very bad when $R\ll\lambda$. In other words, we do not necessarily know
that the “error” term $e^{\lambda}_{2,{\bf x}_{0}}({\bf x};\omega)$ has
amplitude smaller than one. One way of fixing this problem is to observe that

(3.28) $\big{|}e^{\lambda}_{2,{\bf x}_{0}}({\bf x};\omega)-e^{\lambda}_{2,{\bf
x}_{0}}({\bf x};0)\big{|}\lesssim K^{4}/\lambda,$

which follows from

(3.29) $\big{|}\nabla_{x,t}^{2}\nabla_{\omega}\phi^{\lambda}({\bf
x};\omega)\big{|}\lesssim\lambda^{-1}$

and mean value theorems; the pointwise bound (3.29) follows from taking a
further derivative in ${\bf x}$ for the terms in (3.12), (3.13) and (3.14).
For later use, we record more estimates on higher order derivatives of the
phase function. Via the chain rule, we obtain

(3.30) $|\nabla_{{\bf
x}}^{\beta}\nabla_{\omega}^{\beta^{\prime}}\phi^{\lambda}({\bf
x};\omega)|\lesssim_{\beta}\lambda^{-|\beta|+1}$

for every multi-indices $\beta$ and $\beta^{\prime}$ with
$|\beta|,|\beta^{\prime}|\geq 1$. In particular, these imply

(3.31) $|\nabla^{\beta}_{\omega}e^{\lambda}_{2,{\bf x}_{0}}({\bf
x};\omega)|\lesssim_{\beta}K^{4}/\lambda,\text{ for every }|\beta|\geq 1.$

Let us write

(3.32) $\begin{split}&\sum_{\tau\in V}H^{\lambda}g_{\tau}=\sum_{\tau}\int
e^{2\pi i\phi^{\lambda}({\bf x};\omega)}g_{\tau}(\omega)d\omega\\\
&=\sum_{\tau}e^{2\pi ie^{\lambda}_{2,{\bf x}_{0}}({\bf x};0)}\int e^{2\pi
i\nabla_{{\bf x}}\phi^{\lambda}({\bf x}_{0};\omega){\bf x}}e^{2\pi
ie^{\lambda}_{2,{\bf x}_{0}}({\bf x};\omega)-2\pi ie^{\lambda}_{2,{\bf
x}_{0}}({\bf x};0)}\big{(}g_{\tau}(\omega)e^{2\pi i(\phi^{\lambda}({\bf
x}_{0};\omega)-\nabla_{{\bf x}}\phi^{\lambda}({\bf x}_{0};\omega){\bf
x}_{0})}\big{)}d\omega.\end{split}$

Denote

(3.33) $a^{\lambda}_{{\bf x}_{0}}({\bf x};\omega):=e^{2\pi
ie^{\lambda}_{2,{\bf x}_{0}}({\bf x};\omega)-2\pi ie^{\lambda}_{2,{\bf
x}_{0}}({\bf x};0)}$

and

(3.34) $g^{\lambda}_{\tau,{\bf x}_{0}}(\omega):=\big{(}g_{\tau}(\omega)e^{2\pi
i(\phi^{\lambda}({\bf x}_{0};\omega)-\nabla_{{\bf x}}\phi^{\lambda}({\bf
x}_{0};\omega){\bf x}_{0})}\big{)}.$

We apply the Fourier expansion to $a^{\lambda}_{{\bf x}_{0}}({\bf x};\omega)$
and write it as

(3.35) $\sum_{\beta\in\mathbb{N}_{0}^{n-1}}a_{\beta,{\bf
x}_{0}}^{\lambda}({\bf x})e^{2\pi i\beta\cdot\omega};$

the bound in (3.28) guarantees that such an expansion is meaningful. So far we
have

(3.36)
$\sum_{\tau}H^{\lambda}g_{\tau}=\sum_{\beta\in\mathbb{N}_{0}^{n-1}}\sum_{\tau}e^{2\pi
ie^{\lambda}_{2,{\bf x}_{0}}({\bf x};0)}\int e^{2\pi i(\nabla_{{\bf
x}}\phi^{\lambda}({\bf x}_{0};\omega))\cdot{\bf x}}a^{\lambda}_{\beta,{\bf
x}_{0}}({\bf x})g^{\lambda}_{\tau,{\bf x}_{0}}(\omega)e^{2\pi
i\beta\cdot\omega}d\omega.$

By the triangle inequality,

(3.37)
$\begin{split}\|\sum_{\tau}H^{\lambda}g_{\tau}\|_{L^{p}(B_{K^{2}})}&\leq\sum_{\beta}\Big{\|}\sum_{\tau}\int
e^{2\pi i(\nabla_{{\bf x}}\phi^{\lambda}({\bf x}_{0};\omega))\cdot{\bf
x}}a^{\lambda}_{\beta,{\bf x}_{0}}({\bf x})g^{\lambda}_{\tau,{\bf
x}_{0}}(\omega)e^{2\pi i\beta\cdot\omega}d\omega\Big{\|}_{L^{p}(B_{K^{2}})}\\\
&\lesssim\sum_{\beta}(1+|\beta|)^{-10n}\Big{\|}\sum_{\tau}\int e^{2\pi
i(\nabla_{{\bf x}}\phi^{\lambda}({\bf x}_{0};\omega))\cdot{\bf
x}}g^{\lambda}_{\tau,{\bf x}_{0}}(\omega)e^{2\pi
i\beta\cdot\omega}d\omega\Big{\|}_{L^{p}(B_{K^{2}})},\end{split}$

where the decay in $\beta$ follows from (3.30) and (3.31). Via a direct
computation, we obtain $(\nabla_{{\bf x}}\phi^{\lambda}({\bf
x}_{0};\omega))\cdot{\bf x}$ equals

(3.38)
$\begin{split}&\frac{\lambda}{t_{0}}\frac{x_{0}-t_{0}\omega}{\sqrt{\lambda^{2}+|x_{0}-t_{0}\omega|^{2}}}\cdot
x-\frac{\lambda(\lambda^{2}+|x_{0}|^{2}-t_{0}(x_{0}\cdot\omega))}{t_{0}^{2}\sqrt{\lambda^{2}+|x_{0}-t_{0}\omega|^{2}}}\cdot
t.\end{split}$

Regarding the parametrized surface

(3.39)
$\Big{(}\frac{\lambda}{t_{0}}\frac{x_{0}-t_{0}\omega}{\sqrt{\lambda^{2}+|x_{0}-t_{0}\omega|^{2}}},-\frac{\lambda(\lambda^{2}+|x_{0}|^{2}-t_{0}(x_{0}\cdot\omega))}{t_{0}^{2}\sqrt{\lambda^{2}+|x_{0}-t_{0}\omega|^{2}}}\Big{)},$

we have that the Jacobian of the first $(n-1)$ components in $\omega$ is
comparable to $1$; more precisely, without loss of generality, we may assume
that the first component of $x_{0}-t_{0}w$ is nonzero. Then the Jacobian
matrix $\nabla_{\omega}\nabla_{x}\phi^{\lambda}({\bf x}_{0};\omega)$, which
are explicitly calculated in (3.12) and (3.13), has eigenvalue
$-\lambda^{3}(\lambda^{2}+|x_{0}-t_{0}\omega|^{2})^{-3/2}$ with the
eigenvector $x_{0}-t_{0}w$, and eigenvalue and
$-\lambda(\lambda^{2}+|x_{0}-t_{0}\omega|^{2})^{-1/2}$ with the $n-1$ linearly
independent eigenvectors

(3.40)
$(-(x_{0}-t_{0}w)_{2},(x_{0}-t_{0}w)_{1},0,\ldots,0),\ldots,(-(x_{0}-t_{0}w)_{n},0,\ldots,0,(x_{0}-t_{0}w)_{1}),$

where $(x_{0}-t_{0}w)_{i}$ is the $i$th component of the vector
$x_{0}-t_{0}w$. This can be verified by a direct computation. Therefore, we
obtain

(3.41) $\big{|}\det\nabla_{\omega}\nabla_{x}\phi^{\lambda}({\bf
x}_{0};\omega)\big{|}=\lambda^{n+2}\big{(}\lambda^{2}+|x_{0}-t_{0}\omega|^{2}\big{)}^{-(n+2)/2}\simeq
1.$

As a consequence, we obtain that the above parametrized surface is regular.
Next, we will show that it has non-vanishing Gaussian curvatures. Recall the
computation of normal directions in Lemma 3.3. Via a direct computation, we
see that the Gaussian curvature at a fixed point $\omega_{0}$ is comparable to

(3.42) $\begin{split}&\det\Big{(}\nabla^{2}_{\omega}\langle\nabla_{{\bf
x}}\phi^{\lambda}({\bf
x}_{0};\omega),(\omega_{0},1)\rangle\Big{)}\Big{|}_{\omega=\omega_{0}}\simeq
1.\end{split}$

This, combined with the fact that

(3.43) $\nabla^{2}_{\omega}\langle\nabla_{{\bf x}}\phi^{\lambda}({\bf
x}_{0};\omega),(\omega_{0},1)\rangle\big{|}_{x_{0}=0;\omega_{0}=0}$

is the identity matrix and a simple continuity argument, further implies that
the parametrized surface in (3.39) is elliptic. This allows us to apply
decoupling inequalities for surfaces of non-vanishing Gaussian curvatures, due
to Bourgain and Demeter [BD15], and bound (3.37) by

(3.44)
$\sum_{\beta}(1+|\beta|)^{-10n}K^{(m-1)(1/2-1/p)+\epsilon}\Big{(}\sum_{\tau}\Big{\|}\int
e^{2\pi i(\nabla_{{\bf x}}\phi^{\lambda}({\bf x}_{0};\omega))\cdot{\bf
x}}g^{\lambda}_{\tau,{\bf x}_{0}}(\omega)e^{2\pi
i\beta\cdot\omega}d\omega\Big{\|}_{L^{p}(w_{B_{K^{2}}})}^{p}\Big{)}^{1/p},$

where $w_{B_{K^{2}}}$ is given by

(3.45) $\big{(}1+K^{-2}|{\bf x}-{\bf x}_{0}|\big{)}^{-N(\epsilon)},$

where $N(\epsilon)$ is a large constant depending on $\epsilon$ and its choice
will become later.

######  Remark 3.7.

We will need $N(\epsilon)\to\infty$ much faster than $1/\epsilon$ as
$\epsilon\to 0$. For instance, taking $N(\epsilon)=\epsilon^{-100}$ will be
more than enough. This choice of weight will be used in the following way: If
$|{\bf x}-{\bf x}_{0}|\geq K^{2}\lambda^{\epsilon^{10}}$, then the weight
function can be controlled by $\lambda^{-\epsilon^{-90}}$, and therefore is
negligible as $\mathrm{RapDec}(\lambda)$.

The weight function $w_{B_{K^{2}}}$ has tails that are not allowed in the
study of the Bochner-Riesz problem, and we need to get rid of it. For ${\bf
w}\in\mathbb{Z}^{n}$, let $B_{K^{2},{\bf w}}$ be the translation of
$B_{K^{2}}$ by $K^{2}{\bf w}$.

(3.46) $\begin{split}&\Big{\|}\int e^{2\pi i(\nabla_{{\bf
x}}\phi^{\lambda}({\bf x}_{0};\omega))\cdot{\bf x}}g^{\lambda}_{\tau,{\bf
x}_{0}}(\omega)e^{2\pi
i\beta\cdot\omega}d\omega\Big{\|}_{L^{p}(w_{B_{K^{2}}})}\\\
&\leq\mathrm{RapDec}(\lambda)\|g\|_{2}+\sum_{\begin{subarray}{c}{\bf
w}\in\mathbb{Z}^{n}\\\ |{\bf
w}|\leq\lambda^{\epsilon^{10}}\end{subarray}}(1+|{\bf
w}|)^{-N(\epsilon)}\Big{\|}\int e^{2\pi i(\nabla_{{\bf x}}\phi^{\lambda}({\bf
x}_{0};\omega))\cdot{\bf x}}g^{\lambda}_{\tau,{\bf x}_{0}}(\omega)e^{2\pi
i\beta\cdot\omega}d\omega\Big{\|}_{L^{p}(B_{K^{2},{\bf w}})}.\end{split}$

As we have a constant coefficient operator in the last display, we can
modulate the function $g^{\lambda}_{\tau,{\bf x}_{0}}$ such that the ball
$B_{K^{2},{\bf w}}$ is shifted back to $B_{K^{2}}$. Let us use $e_{{\bf
w},K}({\bf x}_{0};\omega)$ to denote the modulation function. Therefore,

(3.47) $e_{{\bf w},K}({\bf x}_{0};\omega)=e^{2\pi iK^{2}(\nabla_{{\bf
x}}\phi^{\lambda}({\bf x}_{0};\omega))\cdot{\bf w}},$

and we have the bound

(3.48) $\begin{split}&\Big{\|}\int e^{2\pi i(\nabla_{{\bf
x}}\phi^{\lambda}({\bf x}_{0};\omega))\cdot{\bf x}}g^{\lambda}_{\tau,{\bf
x}_{0}}(\omega)e^{2\pi
i\beta\cdot\omega}d\omega\Big{\|}_{L^{p}(w_{B_{K^{2}}})}\\\
&\leq\mathrm{RapDec}(\lambda)\|g\|_{2}+\sum_{\begin{subarray}{c}{\bf
w}\in\mathbb{Z}^{n}\\\ |{\bf
w}|\leq\lambda^{\epsilon^{10}}\end{subarray}}(1+|{\bf
w}|)^{-N(\epsilon)}\Big{\|}\int e^{2\pi i(\nabla_{{\bf x}}\phi^{\lambda}({\bf
x}_{0};\omega))\cdot{\bf x}}g^{\lambda}_{\tau,{\bf x}_{0}}(\omega)e_{{\bf
w},K}({\bf x}_{0};\omega)e^{2\pi
i\beta\cdot\omega}d\omega\Big{\|}_{L^{p}(B_{K^{2}})}.\end{split}$

Unfortunately the modulation function in (3.47) depends on ${\bf x}_{0}$. Let
us try to get rid of this dependence via Taylor’s expansion; we learnt this
idea from Beltran, Hickman and Sogge [BHS20]. First of all, let us write the
$L^{p}$ norm on the right hand side of (3.48) as

(3.49) $\Big{\|}\int e^{2\pi i(\nabla_{{\bf x}}\phi^{\lambda}({\bf
x}_{0};\omega))\cdot{\bf x}}g^{\lambda}_{\tau,{\bf x}_{0}}(\omega)b_{{\bf
w},K}({\bf x}_{0};\omega)e^{2\pi
i\beta\cdot\omega}d\omega\Big{\|}_{L^{p}(B_{K^{2}})},$

with

(3.50) $b_{{\bf w},K}({\bf x}_{0};\omega):=e^{2\pi iK^{2}{\bf
w}\cdot(\nabla_{{\bf x}}\phi^{\lambda}({\bf x}_{0};\omega)-\nabla_{{\bf
x}}\phi^{\lambda}({\bf x}_{0};0))}.$

To get rid of the dependence of $b_{{\bf w},K}$ on ${\bf x}_{0}$, we carry out
a Taylor expansion of it about ${\bf x}_{1}$, an arbitrary point inside
$B_{R}$; for convenience, we take ${\bf x}_{1}$ to be the center of $B_{R}$.
Recall the bound in (3.30). As a consequence, we obtain that

(3.51) $\Big{|}\nabla_{{\bf x}}^{\beta}\Big{[}K^{2}{\bf w}\cdot(\nabla_{{\bf
x}}\phi^{\lambda}({\bf x};\omega)-\nabla_{{\bf x}}\phi^{\lambda}({\bf
x};0))\Big{]}\Big{|}\lesssim K^{2}|{\bf w}|\lambda^{-|\beta|},\text{ for every
}\beta,$

which further implies that

(3.52) $\big{|}\nabla_{{\bf x}}^{\beta}b_{{\bf w},K}({\bf
x};\omega)\big{|}\lesssim(K^{2}|{\bf w}|)^{|\beta|}\lambda^{-|\beta|}\text{
for every }|\beta|\geq 1.$

This allows us to write

(3.53) $b_{{\bf w},K}({\bf
x}_{0};\omega)=\sum_{\beta^{\prime}}2^{-N|\beta^{\prime}|}(K^{2}|{\bf
w}|)^{|\beta^{\prime}|}\lambda^{-|\beta^{\prime}|}b_{{\bf
w},\beta^{\prime}}({\bf x}_{1};\omega)({\bf x}_{0}-{\bf
x}_{1})^{\beta^{\prime}}+\mathrm{RapDec}(\lambda),$

where the dependence of $b_{{\bf w},\beta^{\prime}}$ on $K$ has been
compressed, and these Taylor coefficients satisfy the uniform bound $|b_{{\bf
w},\beta^{\prime}}({\bf x}_{1};\omega)|\lesssim 1$. We go back to (3.49) and
bound it by

(3.54)
$\sum_{\beta^{\prime}}2^{-N|\beta^{\prime}|}(K^{2}\lambda^{\epsilon^{10}})^{|\beta^{\prime}|}R^{|\beta^{\prime}|}\lambda^{-|\beta^{\prime}|}\Big{\|}\int
e^{2\pi i(\nabla_{{\bf x}}\phi^{\lambda}({\bf x}_{0};\omega))\cdot{\bf
x}}g^{\lambda}_{\tau,{\bf x}_{0}}(\omega)b_{{\bf w},\beta^{\prime}}({\bf
x}_{1};\omega)e^{2\pi i\beta\cdot\omega}d\omega\Big{\|}_{L^{p}(B_{K^{2}})}.$

Recall the assumption on $R$ that $R\leq\lambda^{1-\epsilon}$. It can be used
to control the constant factors in the last display. So far we have controlled
(3.46) by

(3.55) $\mathrm{RapDec}(\lambda)\|g\|_{2}+\sum_{\beta^{\prime}}\sum_{{\bf
w}}2^{-N|\beta^{\prime}|}(1+|{\bf w}|)^{-N(\epsilon)}\Big{\|}\int e^{2\pi
i(\nabla_{{\bf x}}\phi^{\lambda}({\bf x}_{0};\omega))\cdot{\bf
x}}g^{\lambda}_{\tau,{\bf x}_{0}}(\omega)b_{{\bf w},\beta^{\prime}}({\bf
x}_{1};\omega)e^{2\pi i\beta\cdot\omega}d\omega\Big{\|}_{L^{p}(B_{K^{2}})}.$

Now each term we have is of the form

(3.56) $\begin{split}&\int e^{2\pi i(\nabla_{{\bf x}}\phi^{\lambda}({\bf
x}_{0};\omega))\cdot{\bf x}}g^{\lambda}_{\tau,{\bf x}_{0}}(\omega)b_{{\bf
w},\beta^{\prime}}({\bf x}_{1};\omega)e^{2\pi i\beta\cdot\omega}d\omega\\\
&=\int e^{2\pi i(\nabla_{{\bf x}}\phi^{\lambda}({\bf x}_{0};\omega))\cdot{\bf
x}}g_{\tau}(\omega)e^{2\pi i(\phi^{\lambda}({\bf x}_{0};\omega)-\nabla_{{\bf
x}}\phi^{\lambda}({\bf x}_{0};\omega){\bf x}_{0})}b_{{\bf
w},\beta^{\prime}}({\bf x}_{1};\omega)e^{2\pi i\beta\cdot\omega}d\omega\\\
&=\int e^{2\pi i\phi^{\lambda}({\bf x};\omega)-2\pi ie^{\lambda}_{2,{\bf
x}_{0}}({\bf x};\omega)}g_{\tau}(\omega)b_{{\bf w},\beta^{\prime}}({\bf
x}_{1};\omega)e^{2\pi i\beta\cdot\omega}d\omega.\end{split}$

We apply a Fourier expansion “back” (compared with the one in (3.33)–(3.35))
to get rid of $e^{\lambda}_{2,{\bf x}_{0}}({\bf x};\omega)$ and obtain

(3.57) $\begin{split}&\Big{\|}\int e^{2\pi i(\nabla_{{\bf
x}}\phi^{\lambda}({\bf x}_{0};\omega))\cdot{\bf x}}g^{\lambda}_{\tau,{\bf
x}_{0}}(\omega)b_{{\bf w},\beta^{\prime}}({\bf x}_{1};\omega)e^{2\pi
i\beta\cdot\omega}d\omega\Big{\|}_{L^{p}(B_{K^{2}})}\\\
&\lesssim\sum_{\beta^{\prime\prime}\in\mathbb{N}_{0}^{n-1}}(1+|\beta^{\prime\prime}|)^{-10n}\Big{\|}\int
e^{2\pi i\phi^{\lambda}({\bf x};\omega)}g_{\tau}(\omega)b_{{\bf
w},\beta^{\prime}}({\bf x}_{1};\omega)e^{2\pi
i(\beta+\beta^{\prime\prime})\cdot\omega}d\omega\Big{\|}_{L^{p}(B_{K^{2}})},\end{split}$

which, when substituted into (3.55) and (3.44), implies the desired the
decoupling inequality. ∎

######  Lemma 3.8 (Parabolic rescaling).

Let $w=(w_{1},\dots,w_{n-1})\in[0,1]^{n-1}$ and $K=K_{R}$. Let
$\tau=[w_{1},w_{1}+K^{-1}]\times\dots\times[w_{n-1},w_{n-1}+K^{-1}]$ with
$\tau\subset[0,1]^{n-1}$. Under the induction hypothesis, for every ball
$B_{R}$ with $1\leq R\leq\lambda^{1-\epsilon}$ satisfying

(3.58)
$B_{R}\subset[-\lambda_{K,R},\lambda_{K,R}]\times[R/C_{n},C_{n}\lambda]$

we obtain

(3.59)
$\|H^{\lambda}g_{\tau}\|_{L^{p}(B_{R})}\lesssim_{\epsilon,\delta}R^{\epsilon}R^{\delta}(K^{-1})^{(n-1)-\frac{2n}{p}}\|g_{\tau}\|_{L^{p}([0,1]^{n-1})},$

for every $\delta>0$.

###### Proof of Lemma 3.8.

We use a change of variables

(3.60) $\eta:=-w+\omega.$

Recall that our phase function is given by

(3.61)
$\phi^{\lambda}(x,t;\omega)=\frac{\lambda}{t}\sqrt{\lambda^{2}+|x-t\omega|^{2}}.$

Under the change of variables, it becomes

(3.62)
$\frac{\lambda}{t}\sqrt{\lambda^{2}+|x-t(\eta+w)|^{2}}=\frac{\lambda}{t}\sqrt{\lambda^{2}+|(x-tw)-t\eta|^{2}}.$

Next we apply the change of variables

(3.63) $x-tw\mapsto x;\ t\mapsto t.$

Under this change of variables, let us assume that $B_{R}$ becomes
$\widetilde{B}_{R}$. Note that $\widetilde{B}_{R}$ is a parallelogram: The
smallest rectangle that contains it is at most twice as large as $B_{R}$. Now
our phase becomes

(3.64) $\frac{\lambda}{t}\sqrt{1+|x-t\eta|^{2}},$

and therefore we need to bound

(3.65) $\|H^{\lambda}g_{\widetilde{\tau}}\|_{L^{p}(\widetilde{B}_{R})},\text{
with }\widetilde{\tau}=[0,1/K]^{n-1}.$

Next we apply a scaling argument and set $\eta\mapsto\eta/K.$ Our phase
becomes

(3.66) $\frac{\lambda}{t}\sqrt{1+|x-t\frac{\eta}{K}|^{2}}.$

In the end, we apply the change of variables $x\mapsto Kx;t\mapsto K^{2}t.$ It
is important to keep track of the domain of evaluating the $L^{p}$ norm. Let
us use $\widetilde{D}_{R}$ to denote the smallest rectangle that contains the
image of $\widetilde{B}_{R}$ after the above change of variables. After some
elementary computation, we see that

(3.67)
$\widetilde{D}_{R}\subset[-\frac{\lambda_{K,R}}{K}-\frac{C_{n}R}{K},\frac{\lambda_{K,R}}{K}+\frac{C_{n}R}{K}]^{n-1}\times[\frac{R}{K^{2}}\frac{1}{C_{n}},\frac{\lambda}{K^{2}}C_{n}],$

and $\widetilde{D}_{R}$ is of dimension

(3.68)
$(2\frac{R}{K})\times(2\frac{R}{K})\times\dots\times(2\frac{R}{K})\times\frac{R}{K^{2}}.$

Set $\lambda^{\prime}=\lambda/K$, $R^{\prime}=R/K$ and
$R^{\prime\prime}=R/K^{2}$. What is important in (3.67) is that

(3.69)
$\frac{\lambda_{K,R}}{K}+\frac{C_{n}R}{K}\leq\lambda^{\prime}_{K,R^{\prime\prime}}\leq\lambda^{\prime}_{R^{\prime\prime}},$

which is exactly the reason of having a complicated expression of
$\lambda_{K,R}$ in our induction hypothesis. As a consequence, we know that

(3.70)
$\widetilde{D}_{R}\subset[-\lambda^{\prime}_{K,R^{\prime\prime}},\lambda^{\prime}_{K,R^{\prime\prime}}]\times[R^{\prime\prime}/C_{n},\lambda^{\prime}C_{n}].$

######  Remark 3.9.

Here we make a remark on the choice of the $t$-interval
$[R/C_{n},C_{n}\lambda]$ in (3.3) from our induction hypothesis. In our
current setup, it should not be replaced by anything like $[R/C_{n},C_{n}R]$
or $[R/C_{n},C_{n}R^{\frac{1}{1-\epsilon}}]$. For instance, if we replace it
by the latter, then the analogue of (3.70), which is

(3.71)
$\widetilde{D}_{R}\subset[-\lambda^{\prime}_{K,R^{\prime\prime}},\lambda^{\prime}_{K,R^{\prime\prime}}]\times[R^{\prime\prime}/C_{n},(R^{\prime\prime})^{\frac{1}{1-\epsilon}}C_{n}],$

does not hold, and therefore we can not apply the induction hypothesis. If one
would really like to use a time interval of the form
$[R/C_{n},C_{n}R^{\frac{1}{1-\epsilon}}]$ in the induction hypothesis, which
may bring some convenience like $t$ is always essentially comparable to $R$,
then one can cut $\widetilde{D}_{R}$ into smaller balls and then make use of
the extra gain in $K$ from (3.72), which is not explored in the current setup.

From now on, we will try to bound
$\|H^{\lambda^{\prime}}\widetilde{g}\|_{L^{p}(\widetilde{D}_{R})},$ where
$\widetilde{g}$ is now a function supported on $[0,1]^{n-1}$. It remains to
prove

(3.72)
$\|H^{\lambda^{\prime}}\widetilde{g}\|_{L^{p}(\widetilde{D}_{R})}\lesssim_{\epsilon,\delta}(R^{\prime}/K)^{\epsilon}(R^{\prime}/K)^{\delta}\|\widetilde{g}\|_{p}.$

Actually we will not need the gain in $K$, and only need to prove

(3.73)
$\|H^{\lambda^{\prime}}g\|_{L^{p}(\widetilde{D}_{R})}\lesssim_{\epsilon,\delta}R^{\epsilon}R^{\delta}\|g\|_{p},$

for every $\delta>0$ and every $g$.

Let us denote

(3.74)
$D_{R^{\prime}}\subset[-\lambda^{\prime}_{K,R^{\prime\prime}},\lambda^{\prime}_{K,R^{\prime\prime}}]\times[R^{\prime\prime}/C_{n},\lambda^{\prime}C_{n}]$

a rectangular box of dimension

(3.75) $R^{\prime}\times\dots R^{\prime}\times R^{\prime\prime}.$

We need to prove

(3.76)
$\|H^{\lambda^{\prime}}g\|_{L^{p}(D_{R^{\prime}})}\lesssim_{\epsilon,\delta}R^{\epsilon}R^{\delta}\|g\|_{p}.$

To prove such a bound, we need the following discrete version of the operator
$H^{\lambda^{\prime}}$. This step is where we apply our induction hypothesis.

######  Lemma 3.10 (Lemma 11.8 [GHI19]).

Let $\mathcal{D}$ be a maximal $(R^{\prime\prime})^{-1}$-separated discrete
subset of $[0,1]^{n-1}$. Then

(3.77) $\Big{\|}\sum_{\omega_{\theta}\in\mathcal{D}}e^{i\phi^{\lambda}(\cdot\
;\omega_{\theta})}F(\omega_{\theta})\Big{\|}_{L^{p}(B_{R^{\prime\prime}})}\lesssim_{\epsilon}(R^{\prime\prime})^{\epsilon}(R^{\prime\prime})^{(n-1)/p^{\prime}}\|F\|_{\ell^{p}(\mathcal{D})},$

for every $F:\mathcal{D}\rightarrow\mathbb{C}$ and every ball
$B_{R^{\prime\prime}}\subset D_{R^{\prime}}$ of radius $R^{\prime\prime}$.

###### Proof of Lemma 3.10.

The proof relies an approximation argument via Taylor’s expansion, and is
essentially the same as that of Lemma 11.8 [GHI19], see also Section 5 of
[BG11]. Let ${\bf x}_{0}$ be the center of $B_{R^{\prime\prime}}$. Let
$\psi:\mathbb{R}^{n-1}\to\mathbb{R}$ be a function supported on the ball of
radius 2 centered at the origin, satisfying $0\leq\psi\leq 1$ and
$\psi(\omega)=1$ for every $|\omega|\leq 1$. For each
$\omega_{\theta}\in\mathcal{D}$, define
$\psi_{\theta}(\omega):=\psi(10R^{\prime\prime}(\omega-\omega_{\theta}))$. For
every $x\in B_{R^{\prime\prime}}$, the sum on the left hand side of (3.77) is
a constant multiple of

(3.78) $\begin{split}(R^{\prime\prime})^{n-1}\int_{\mathbb{R}^{n-1}}e^{2\pi
i\phi^{\lambda}({\bf x};\omega)}e^{-2\pi i\phi^{\lambda}({\bf
x}_{0};\omega)+2\pi i\phi^{\lambda}({\bf
x}_{0};\omega_{\theta})}\Big{[}\sum_{\omega_{\theta}\in\mathcal{D}}e^{-2\pi
i\Omega^{\lambda}_{\theta}({\bf
x};\omega)}F(\omega_{\theta})\psi_{\theta}(\omega)\Big{]}d\omega,\end{split}$

where

(3.79) $\Omega^{\lambda}_{\theta}({\bf x};\omega):=\phi^{\lambda}({\bf
x};\omega)-\phi^{\lambda}({\bf x};\omega_{\theta})-\phi^{\lambda}({\bf
x}_{0};\omega)+\phi^{\lambda}({\bf x}_{0};\omega_{\theta}).$

By mean value theorems and the bound (3.30), we see that

(3.80) $|\Omega_{\theta}^{\lambda}({\bf x};\omega)|\lesssim|{\bf x}-{\bf
x}_{0}||\omega-\omega_{\theta}|\lesssim 1,$

whenever $x\in B_{R^{\prime\prime}}$ and $\omega$ is in the support of
$\psi_{\theta}$. Roughly speaking, this allows us to treat $e^{-2\pi
i\Omega^{\lambda}_{\theta}({\bf x};\omega)}$ as the constant function $1$,
which can be made rigorous via Fourier expansion. To apply Fourier expansion
in ${\bf x}$, we need to examine higher order derivatives of the function.
Similar to how we obtained (3.80), we obtain from (3.30) and mean value
theorems that

(3.81) $\begin{split}\big{|}\nabla_{{\bf
x}}^{\beta}\Omega_{\theta}^{\lambda}({\bf x};\omega)\big{|}\lesssim 1,\ \
\big{|}\nabla_{{\bf x}}^{\beta}e^{2\pi i\Omega_{\theta}^{\lambda}({\bf
x};\omega)}\big{|}\lesssim 1\end{split}$

for every multi-index $\beta$. As a consequence, we are able to bound (3.78)
by

(3.82)
$(R^{\prime\prime})^{n-1}\sum_{k\in\mathbb{Z}^{n}}(1+|k|)^{-(n+1)}|H^{\lambda}f_{k}(x)|,$

where

(3.83)
$f_{k}(\omega):=\sum_{\omega_{\theta}\in\mathcal{D}}F(\omega_{\theta})c_{k,\theta}(\omega)\psi_{\theta}(\omega),$

with $c_{k,\theta}$ satisfying the uniform bound
$\|c_{k,\theta}\|_{\infty}\leq 1$. We apply our induction hypothesis to
$H^{\lambda}f_{k}$ on the ball $B_{R^{\prime\prime}}$; in particular, (3.74)
guarantees that our induction hypothesis is applicable. This is gives us the
desired estimate. ∎

Let us begin the proof of (3.76). Out of certain technical reason of handling
tails, we introduce new notation. For given $R,\lambda$ satisfying
$R\leq\lambda^{1-\epsilon}$, let $a_{\lambda,R}(x,t)$ be a non-negative smooth
bump function supported on
$[-2\lambda_{R},2\lambda_{R}]\times[R/(2C_{n}),2C_{n}\lambda]$, and equal to
one on $[-\lambda_{R},\lambda_{R}]\times[R/C_{n},C_{n}\lambda]$. Define

(3.84) $H^{\lambda,R}g({\bf x}):=a_{\lambda,R}(x,t)\int e^{2\pi
i\phi^{\lambda}({\bf x};\omega)}g(\omega)d\omega.$

This notation will only be used in this section. Note that for all the ${\bf
x}$ that we care about, that is, ${\bf
x}\in[-\lambda_{R},\lambda_{R}]\times[R/C_{n},C_{n}\lambda]$, it always holds
that

(3.85) $H^{\lambda,R}g({\bf x})=H^{\lambda}g({\bf x}).$

Let us return to the proof of (3.76). Cover $[0,1]^{n-1}$ by caps $\theta$ of
side length $(R^{\prime\prime})^{-1}$. Decompose $g$ as
$g=\sum_{\theta}g_{\theta}$. Let $\omega_{\theta}$ be the center of $\theta$.
Define

(3.86) $H^{\lambda^{\prime}}_{\theta}g({\bf x}):=e^{-2\pi
i\phi^{\lambda^{\prime}}({\bf x};\omega_{\theta})}H^{\lambda^{\prime}}g({\bf
x}),$

and

(3.87) $H^{\lambda^{\prime},R^{\prime\prime}}_{\theta}g({\bf
x}):=a_{\lambda^{\prime},R^{\prime\prime}}(x,t)e^{-2\pi
i\phi^{\lambda^{\prime}}({\bf x};\omega_{\theta})}H^{\lambda^{\prime}}g({\bf
x}),$

so that for every ${\bf x}\in B_{R^{\prime\prime}}\subset D_{R^{\prime}}$, it
holds that

(3.88) $\begin{split}H^{\lambda^{\prime}}g({\bf
x})=H^{\lambda^{\prime},R^{\prime\prime}}g({\bf x})&=\sum_{\theta}e^{2\pi
i\phi^{\lambda^{\prime}}({\bf
x};\omega_{\theta})}H^{\lambda^{\prime}}_{\theta}g_{\theta}({\bf x})\\\
&=\sum_{\theta}e^{2\pi i\phi^{\lambda^{\prime}}({\bf
x};\omega_{\theta})}H^{\lambda^{\prime},R^{\prime\prime}}_{\theta}g_{\theta}({\bf
x}).\end{split}$

To proceed, we need the following lemma.

######  Lemma 3.11.

For every $\delta>0$, the following statement holds. For every ${\bf x}$ we
have

(3.89) $H^{\lambda^{\prime},R^{\prime\prime}}_{\theta}g_{\theta}({\bf
x})=H^{\lambda^{\prime},R^{\prime\prime}}_{\theta}g_{\theta}*\eta_{(R^{\prime\prime})^{1-\delta}}({\bf
x})+\mathrm{RapDec}(\lambda^{\prime})\|g\|_{2},$

for some rapidly decreasing function $\eta$ such that $|\eta|$ admits a
smooth, rapidly decreasing majorant $\zeta:\mathbb{R}^{n}\mapsto[0,\infty)$
which is locally constant at scale 1.

The proof of this lemma is pretty standard, see for instance the bottom of
page 363 in [GHI19] and Lemma 5.8 there.

######  Remark 3.12.

The parameter $\delta>0$ will be picked to be extremely small, and much
smaller compared with $\epsilon$. The use of the parameter $\delta$ and the
scale $(R^{\prime\prime})^{\delta}$ is the reason for $K$ to depend on $R$,
which further explains the necessity of the polynomial dependence of the
implicit constant on $K$ in Theorem 3.4.

One way we will be using the locally constant property of $\zeta$ is

(3.90) $\zeta_{(R^{\prime\prime})^{1-\delta}}({\bf
x})\lesssim(R^{\prime\prime})^{\delta}\zeta_{(R^{\prime\prime})^{1-\delta}}({\bf
y})\text{ if }|{\bf x}-{\bf y}|\lesssim R^{\prime\prime}.$

To control the $L^{p}$ norm of
$H^{\lambda^{\prime}}g=H^{\lambda^{\prime},R^{\prime\prime}}g$ on
$D_{R^{\prime}}$, we cover $D_{R^{\prime}}$ by finitely-overlapping
$R^{\prime\prime}$-balls, and let $B_{R^{\prime\prime}}$ be some member of
this cover. Let ${\bf x}_{0}$ denote the center of $B_{R^{\prime\prime}}$,
then

(3.91) $|H^{\lambda^{\prime},R^{\prime\prime}}g({\bf x}_{0}+{\bf z})|\lesssim
R^{\delta}\int_{\mathbb{R}^{n}}\Big{|}\sum_{\theta}e^{2\pi
i\phi^{\lambda^{\prime}}({\bf x}_{0}+{\bf
z};\omega_{\theta})}H^{\lambda^{\prime},R^{\prime\prime}}_{\theta}g_{\theta}({\bf
y})\Big{|}\zeta_{(R^{\prime\prime})^{1-\delta}}({\bf x}_{0}-{\bf y})d{\bf y},$

for every ${\bf z}$ with $|{\bf z}|\leq R^{\prime\prime}$. We follow the same
lines of proof as in [GHI19, page 364]. To be more precise, we take the
$L^{p}$ norm in ${\bf z}$ and see that
$\|H^{\lambda^{\prime}}g\|_{L^{p}(B_{R^{\prime\prime}})}$ can be bounded by

(3.92) $R^{\delta}\int_{\mathbb{R}^{n}}\Big{\|}\sum_{\theta}e^{2\pi
i\phi^{\lambda^{\prime}}({\bf x}_{0}+{\bf
z};\omega_{\theta})}H^{\lambda^{\prime},R^{\prime\prime}}_{\theta}g_{\theta}({\bf
y})\Big{\|}_{L^{p}(\\{|{\bf z}|\leq
R^{\prime\prime}\\})}\zeta_{(R^{\prime\prime})^{1-\delta}({\bf x}_{0}-{\bf
y})}d{\bf y}.$

For each fixed ${\bf y}$, we apply Lemma 3.10 and see that the $L^{p}$ norm in
the above display can be bounded by

(3.93)
$(R^{\prime\prime})^{\epsilon}(R^{\prime\prime})^{(n-1)/p}\Big{(}\sum_{\theta}|H_{\theta}^{\lambda^{\prime},R^{\prime\prime}}g_{\theta}({\bf
y})|^{p}\Big{)}^{1/p}.$

We then apply the locally constant property (3.90) again, sum over all balls
$B_{R^{\prime\prime}}$ inside $D_{R^{\prime}}$ and obtain

(3.94)
$\|H^{\lambda^{\prime}}g\|_{L^{p}(D_{R^{\prime}})}\lesssim_{\epsilon}(R^{\prime\prime})^{\epsilon+O(\delta)}(R^{\prime\prime})^{(n-1)/p^{\prime}-n/p}\Big{(}\int_{\mathbb{R}^{n}}\sum_{\theta}\|H^{\lambda^{\prime},R^{\prime\prime}}g_{\theta}\|^{p}_{L^{p}(D_{R^{\prime}}-{\bf
y})}\zeta_{(R^{\prime\prime})^{1-\delta}}({\bf y})d{\bf y}\Big{)}^{1/p},$

where $D_{R^{\prime}}-{\bf y}$ means a translation in ${\bf y}$. It remains to
prove that

(3.95)
$\Big{(}\int_{\mathbb{R}^{n}}\sum_{\theta}\|H^{\lambda^{\prime},R^{\prime\prime}}g_{\theta}\|^{p}_{L^{p}(D_{R^{\prime}}-{\bf
y})}\zeta_{(R^{\prime\prime})^{1-\delta}}({\bf y})d{\bf
y}\Big{)}^{1/p}\lesssim(R^{\prime\prime})^{-(n-1)/p^{\prime}+n/p}\|g\|_{p}.$

This is the step where it is convenient to have a cut-off function in space
and time in the definition of $H^{\lambda^{\prime},R^{\prime\prime}}f$. We
will show that

(3.96)
$\|H^{\lambda^{\prime},R^{\prime\prime}}g_{\theta}\|_{L^{p}(D_{R^{\prime}}-{\bf
y})}\lesssim(R^{\prime\prime})^{-(n-1)/p^{\prime}+n/p}\|g_{\theta}\|_{p},$

which immediately implies (3.95). To prove this bound, we use interpolation at
$p=2$ and $p=\infty$. At $p=2$, we need the following lemma.

######  Lemma 3.13.

For every $1\leq R\leq\lambda$, it holds that

(3.97) $\|H^{\lambda,R}g(x,t)\|_{L^{2}_{x}}\lesssim\|g\|_{2},$

uniformly in $t\in\mathbb{R}$.

###### Proof of Lemma 3.13.

This estimate is well-known. As it is short, we still include it here. By the
main theorem in [H7̈3], which is proven via a $TT^{*}$ argument and Schur’s
text, the desired estimate follows once we verify that

(3.98) $|\det\nabla_{x}\nabla_{\omega}\phi(x,t;\omega)|\simeq 1,$

uniformly in $|t|\lesssim 1,|x|\lesssim 1$. However this has been verified and
used in (3.41). ∎

As a consequence of Lemma 3.13, we obtain

(3.99)
$\|H^{\lambda^{\prime},R^{\prime\prime}}g_{\theta}\|_{L^{2}(D_{R^{\prime}}-{\bf
y})}\lesssim R^{-(n-1)(1/2-1/p)+1/2}\|g_{\theta}\|_{p}.$

At $p=\infty$, we have the trivial bound that

(3.100)
$\|H^{\lambda^{\prime},R^{\prime\prime}}g_{\theta}\|_{L^{\infty}(D_{R^{\prime}}-{\bf
y})}\leq R^{-(n-1)/p^{\prime}}\|g_{\theta}\|_{p}.$

Interpolation implies the desired (3.96). This finishes the proof of Lemma
3.8. ∎

##  4\. Wave packets and their essential supports

The rest of the paper is devoted to the proof of Theorem 3.4, the broad norm
estimate. When we were handling the narrow part in the previous section, we
used an inductive argument, and in order to close induction we had to be very
careful with the choice of various amplitude functions
$\bar{a}_{1},\bar{a}_{2}$, and regions of taking integrations; we also had to
introduce $H^{\lambda,R}g({\bf x})$ and distinguish it from $H^{\lambda}g({\bf
x})$ as it was very often inconvenient to carry amplitude functions in our
operator. From now on, these points are not as crucial as before, and we will
always work with $H^{\lambda,R}g({\bf x})$, a function that is compactly
supported in $[-4C_{n}\lambda,4C_{n}\lambda]\times[R/(2C_{n}),2C_{n}\lambda]$.
Moreover, to simplify notation, we always abbreviate $H^{\lambda,R}$ to
$H^{\lambda}$: The dependence on $R$ will be emphasized through the region we
evaluate the $L^{p}$ norm of $H^{\lambda}$.

###  4.1. Wave packet decomposition

Let $r\geq 1$ and take a collection $\Theta_{r}$ of dyadic cubes of side
length $\frac{9}{11}r^{-1/2}$ covering the ball $B^{n-1}(0,2)$. We take a
smooth partition of unity $(\psi_{\theta})_{\theta\in\Theta_{r}}$ with
$\mathrm{supp}\psi_{\theta}\subset\frac{11}{10}\theta$ for the ball
$B^{n-1}(0,2)$ such that

(4.1)
$\|\partial_{w}^{\alpha}\psi_{\theta}\|_{L^{\infty}}\lesssim_{\alpha}r^{|\alpha|/2}$

for any $\alpha\in\mathbb{N}_{0}^{n-1}$. We denote by $\omega_{\theta}$ the
center of $\theta$. Given a function $g$, we perform a Fourier series
decomposition to the function $g\psi_{\theta}$ on the region
$\frac{11}{9}\theta$ and obtain

(4.2)
$g(w)\psi_{\theta}(w)\cdot\mathbbm{1}_{\frac{11}{10}\theta}(\omega)=\Big{(}\frac{r^{1/2}}{2\pi}\Big{)}^{n-1}\sum_{v\in
r^{1/2}\mathbb{Z}^{n-1}}(g\psi_{\theta})^{\wedge}(v)e^{2\pi iv\cdot
w}\mathbbm{1}_{\frac{11}{10}\theta}(\omega).$

Let $\widetilde{\psi}_{\theta}$ be a non-negative smooth cutoff function
supported on $\frac{11}{9}\theta$ and equal to 1 on $\frac{11}{10}\theta$. We
can therefore write

(4.3)
$g(w)\psi_{\theta}(w)\cdot\widetilde{\psi}_{\theta}(\omega)=\Big{(}\frac{r^{1/2}}{2\pi}\Big{)}^{n-1}\sum_{v\in
r^{1/2}\mathbb{Z}^{n-1}}(g\psi_{\theta})^{\wedge}(v)e^{2\pi iv\cdot
w}\widetilde{\psi}_{\theta}(\omega).$

If we also define

(4.4)
$g_{\theta,v}(w):=\Big{(}\frac{r^{1/2}}{2\pi}\Big{)}^{n-1}(g\psi_{\theta})^{\wedge}(v)e^{2\pi
iv\cdot\omega}\widetilde{\psi}_{\theta}(\omega).$

then we have

(4.5) $g=\sum_{(\theta,v)\in\Theta_{r}\times
r^{1/2}\mathbb{Z}^{n-1}}g_{\theta,v}.$

This finishes our wave packet decomposition.

Let $1\leq r\leq R$. We fix a ball $B({\bf x}_{0},r)$ of radius $r$ satisfying

(4.6) $B({\bf
x}_{0},r)\subset[-3C_{n}\lambda,3C_{n}\lambda]^{n-1}\times[R/C_{n},\lambda
C_{n}].$

Let us fix a function $f$. We define a collection of tubes associated to the
ball $B({\bf x}_{0},r)$ by

(4.7) $\mathbb{T}[B({\bf x}_{0},r)]:=\\{T_{\theta,v}({\bf
x}_{0}):(\theta,v)\in\Theta_{r}\times r^{1/2}\mathbb{Z}^{n-1}\\},$

where $T_{\theta,v}({\bf x}_{0})$ is some set that will be determined later.
Given $T_{\theta,v}({\bf x}_{0})\in\mathbb{T}[B({\bf x}_{0},r)]$, we define a
function

(4.8) $g_{T_{\theta,v}({\bf x}_{0})}(\omega):=e^{-2\pi i\phi^{\lambda}({\bf
x}_{0};\omega)}(g(\cdot)e^{2\pi i\phi^{\lambda}({\bf x}_{0};\
\cdot)})_{\theta,v}(\omega).$

Notice that

(4.9) $H^{\lambda}g(x,t)=\sum_{T\in\mathbb{T}[B({\bf
x}_{0},r)]}H^{\lambda}g_{T}(x,t).$

In the rest of the section, we will define tubes $T$ so that
$H^{\lambda}g_{T}$ is essentially supported on $T$.

######  Remark 4.1.

It is worth mentioning the motivation and advantages of the notation
$g_{T_{\theta,v}}$. In [Gut18] and [GHI19], the following notations are
introduced:

(4.10) $\widetilde{{\bf x}}={\bf x}-{\bf
x}_{0}\;\;\;\text{and}\;\;\;\tilde{g}(w)=g(w)e^{2\pi i\phi^{\lambda}({\bf
x}_{0};w)}.$

In these notations, we have

(4.11) $H^{\lambda}g(x)=\tilde{H}^{\lambda}\tilde{g}(\widetilde{{\bf x}}),$

where

(4.12) $\tilde{H}^{\lambda}\tilde{g}(\tilde{{\bf
x}}):=\int_{\mathbb{R}^{n-1}}e^{2\pi i\phi^{\lambda}(\tilde{{\bf x}}+{\bf
x}_{0};\omega)}e^{-2\pi i\phi^{\lambda}({\bf
x}_{0};w)}\tilde{g}(\omega)d\omega.$

The motivation of their notations is that the function
$\tilde{H}^{\lambda}\tilde{g}$ is defined on the ball centered at the origin
so that we can apply the wave packet decomposition to the function
$\tilde{g}$. However, in our proof of Theorem 3.4 (in particular, Section 9
and 10), these notations cause some confusion because the function $\tilde{g}$
depends on the point ${\bf x}_{0}$, but the notation $\tilde{g}$ does not show
this dependence.

To make things clear, what we have done above is to apply the wave packet
decomposition to $\tilde{g}$ and change back the variable $\widetilde{{\bf
x}}$ to the original variable ${\bf x}$ (See (4.8) and (4.9)). Note that our
tube $T_{\theta,v}({\bf x}_{0})$ indicates the dependence on the point ${\bf
x}_{0}$.

######  Lemma 4.2 ($L^{2}$-orthogonality).

For any set $\mathbb{T}\subset\mathbb{T}[B({\bf x}_{0},r)]$, it holds that

(4.13)
$\|\sum_{T\in\mathbb{T}}g_{T}\|_{2}^{2}\lesssim\sum_{T\in\mathbb{T}}\|g_{T}\|_{2}^{2}\lesssim\|g\|_{2}^{2}.$

Moreover, if $\mathbb{T}$ is any collection of tubes with the same $\theta$,
then

(4.14)
$\|\sum_{T\in\mathbb{T}}g_{T}\|_{2}^{2}\sim\sum_{T\in\mathbb{T}}\|g_{T}\|_{2}^{2}.$

Lemma 4.2 follows immediately from Plancherel’s theorem, and the proof is
omitted.

######  Remark 4.3.

In the above lemma, without the extra assumption, (4.14) may not be correct,
due to the frequency overlapping.

The definition of $T_{\theta,v}({\bf x}_{0})$ relies on which of the following
two cases we are in:

(4.15) $|v-\nabla_{\omega}\phi^{\lambda}({\bf
x}_{0};\omega_{\theta})|\geq\frac{10C_{n}\lambda}{\sqrt{1+(10C_{n})^{2}}}\text{
or }|v-\nabla_{\omega}\phi^{\lambda}({\bf
x}_{0};\omega_{\theta})|\leq\frac{10C_{n}\lambda}{\sqrt{1+(10C_{n})^{2}}}.$

In the former case, we define $T_{\theta,v}$ to be an empty set, and show that
$H^{\lambda}f_{T}$ always decays rapidly, that is, it is bounded by
$\lambda^{-N}\|f\|_{2}$ pointwise. In the latter case, we define
$T_{\theta,v}$ to be as follows: Define a line $l_{\theta,v}$ as

(4.16)
$l_{\theta,v}:=\big{\\{}(x,t):\omega_{\theta}t-x=\frac{\lambda(\nabla_{\omega}\phi^{\lambda}({\bf
x}_{0};\omega_{\theta})-v)}{\sqrt{\lambda^{2}-|\nabla_{\omega}\phi^{\lambda}({\bf
x}_{0};\omega_{\theta})-v|^{2}}}\big{\\}}$

and a $r^{1/2+\delta}$ tube $T_{\theta,v}$ as
$T_{\theta,v}=\mathcal{N}_{r^{1/2+\delta}}(l_{\theta,v})$. Here
$0<\delta\ll\epsilon$ is a small parameter whose choice will become clear
later, and $\mathcal{N}_{r^{1/2}+\delta}$ means the
$r^{1/2+\delta}$-neighbourhood. Next, we show that the essential support of
$H^{\lambda}f_{T}$ is $T$. The precise meaning of essential supports will be
made clear in the statements of the following two lemmas.

######  Lemma 4.4.

If $T=T_{\theta,v}({\bf x}_{0})$ is such that the former case in (4.15) holds,
then we have

(4.17) $\|H^{\lambda}g_{T}\|_{L^{\infty}(B({\bf
x}_{0},r))}\lesssim_{N}\lambda^{-N}\|g\|_{2},$

for every $N\in\mathbb{N}$, whenever $B({\bf x}_{0},r)$ satisfies (4.6).

###### Proof of Lemma 4.4.

For given ${\bf x}_{0}$, we denote $g_{{\bf x}_{0}}(\omega)=g(\omega)e^{2\pi
i\phi^{\lambda}({\bf x}_{0};\omega)}$. Recall the definition of $g_{T}$ that

(4.18) $g_{T}(\omega)=e^{-2\pi i\phi^{\lambda}({\bf x}_{0};\omega)}(g_{{\bf
x}_{0}}\psi_{\theta})^{\wedge}(v)\Big{(}\frac{r^{1/2}}{2\pi}\Big{)}^{n-1}e^{2\pi
iv\cdot\omega}\widetilde{\psi}_{\theta}(\omega).$

Let $H^{\lambda}$ act on $f_{T}$ and we obtain

(4.19) $H^{\lambda}g_{T}(x,t)=\Big{(}\frac{r^{1/2}}{2\pi}\Big{)}^{n-1}(g_{{\bf
x}_{0}}\psi_{\theta})^{\wedge}(v)a_{\lambda,R}(x,t)\int e^{2\pi i\phi_{{\bf
x}_{0}}^{\lambda}({\bf x};\omega)+2\pi
iv\cdot\omega}\widetilde{\psi}_{\theta}(\omega)d\omega,$

where

(4.20) $\phi_{{\bf x}_{0}}^{\lambda}({\bf x};\omega):=\phi^{\lambda}({\bf
x};\omega)-\phi^{\lambda}({\bf x}_{0};\omega).$

We make the change of variable $\omega\to\omega+\omega_{\theta}$ and consider
the oscillatory integral

(4.21) $\begin{split}&\int e^{2\pi i\big{(}\phi^{\lambda}_{{\bf
x}_{0}}(x,t;\omega+\omega_{\theta})+v\cdot\omega\big{)}}\widetilde{\psi}(r^{1/2}\omega)d\omega\\\
&=r^{-(n-1)/2}\int e^{2\pi i\big{(}\phi^{\lambda}_{{\bf
x}_{0}}(x,t;r^{-1/2}\omega+\omega_{\theta})+r^{-1/2}v\cdot\omega\big{)}}\widetilde{\psi}(\omega)d\omega.\end{split}$

Denote $\omega^{\prime}:=r^{-1/2}\omega$. First of all, $\nabla_{\omega}$ of
the phase function equals

(4.22) $r^{-1/2}\Big{(}\nabla_{\omega}\phi^{\lambda}_{{\bf
x}_{0}}(x,t;\omega^{\prime}+\omega_{\theta})+v\Big{)}.$

We write

(4.23)
$\begin{split}&\eqref{2011227_13}=r^{-1/2}\Big{(}\nabla_{\omega}\phi^{\lambda}_{{\bf
x}_{0}}(x,t;\omega^{\prime}+\omega_{\theta})-\nabla_{\omega}\phi^{\lambda}_{{\bf
x}_{0}}(x,t;\omega_{\theta})\Big{)}\\\
&+r^{-1/2}\Big{(}\nabla_{\omega}\phi^{\lambda}_{{\bf
x}_{0}}(x,t;\omega_{\theta})+v\Big{)}.\end{split}$

We first look at the second term. By the definition of $\phi^{\lambda}_{{\bf
x}_{0}}$, it equals to

(4.24)
$\begin{split}&r^{-1/2}\Big{(}\nabla_{\omega}\phi^{\lambda}(x,t;\omega_{\theta})-\nabla_{\omega}\phi^{\lambda}(x_{0},t_{0};\omega_{\theta})+v\Big{)}.\end{split}$

Recall we are in the case that

(4.25)
$|\nabla_{\omega}\phi^{\lambda}(x_{0},t_{0};\omega_{\theta})-v|\geq\frac{10C_{n}\lambda}{\sqrt{1+(10C_{n})^{2}}}.$

Moreover, recall that $|t|\leq 2C_{n}\lambda$ and $|x|\leq 4C_{n}\lambda,$ and
therefore we obtain

(4.26)
$\begin{split}&|\nabla_{\omega}\phi^{\lambda}(x,t;\omega_{\theta})|=\lambda\Big{|}\frac{t\omega_{\theta}-x}{\sqrt{\lambda^{2}+|x-t\omega_{\theta}|^{2}}}\Big{|}\leq\frac{6C_{n}\lambda}{\sqrt{1+(6C_{n})^{2}}}\end{split}$

By the triangle inequality, we obtain that

(4.27)
$\eqref{201122e7_15}\gtrsim\Big{(}\frac{10C_{n}\lambda}{\sqrt{1+(10C_{n})^{2}}}-\frac{6C_{n}\lambda}{\sqrt{1+(6C_{n})^{2}}}\Big{)}r^{-1/2}\gtrsim\lambda/r^{1/2}.$

Let us continue to compute the first term in (4.23). First,

(4.28) $\displaystyle\nabla_{\omega}\phi^{\lambda}_{{\bf
x}_{0}}(x,t;\omega)=\lambda\Big{(}\frac{t\omega-x}{\sqrt{\lambda^{2}+|x-t\omega|^{2}}}-\frac{t_{0}\omega-
x_{0}}{\sqrt{\lambda^{2}+|x_{0}-t_{0}\omega|^{2}}}\Big{)}.$

Therefore

(4.29) $\begin{split}&\nabla_{\omega}\phi^{\lambda}_{{\bf
x}_{0}}(x,t;\omega^{\prime}+\omega_{\theta})-\nabla_{\omega}\phi^{\lambda}_{{\bf
x}_{0}}(x,t;\omega_{\theta})\\\
&=\lambda\Big{(}\frac{t(\omega^{\prime}+\omega_{\theta})-x}{\sqrt{\lambda^{2}+|x-t(\omega^{\prime}+\omega_{\theta})|^{2}}}-\frac{t_{0}(\omega^{\prime}+\omega_{\theta})-x_{0}}{\sqrt{\lambda^{2}+|x_{0}-t_{0}(\omega^{\prime}+\omega_{\theta})|^{2}}}\Big{)}\\\
&-\lambda\Big{(}\frac{t\omega_{\theta}-x}{\sqrt{\lambda^{2}+|x-t\omega_{\theta}|^{2}}}-\frac{t_{0}\omega_{\theta}-x_{0}}{\sqrt{\lambda^{2}+|x_{0}-t_{0}\omega_{\theta}|^{2}}}\Big{)}\end{split}$

We use the bound (3.30) and mean value theorems, and see that the above
expression can be bounded from above by $r^{1/2}$, which, together with the
leading coefficient $r^{-1/2}$, produces the bound $1$ for the first term.
Putting everything together, we see that

(4.30) $\eqref{2011227_14}\gtrsim\lambda/r^{1/2}-1\gtrsim\lambda/r^{1/2}.$

The desired rapid decay follows immediately from integration by parts. ∎

######  Lemma 4.5.

For $T=T_{\theta,v}$ and $(x,t)\in B({\bf x}_{0},r)\setminus T_{\theta,v}$
with $B({\bf x}_{0},r)$ satisfying (4.6), it holds that

(4.31)
$|H^{\lambda}g_{T}(x,t)|\leq(1+r^{-1/2}|\nabla_{\omega}\phi^{\lambda}_{{\bf
x}_{0}}(x,t;\omega_{\theta})+v|)^{-N}{\rm RapDec}(r)\|g\|_{2}.$

Here $\phi^{\lambda}_{{\bf x}_{0}}$ is defined as in (4.20).

This lemma describes the essential support of a wave packet, and it is the
right hand side of (4.31) that motivates the definition of the core line in
(4.16); see (4.36) below that connects the expression of the core line in
(4.16) with the right hand side of (4.31).

###### Proof of Lemma 4.5.

Note that each piece $H^{\lambda}g_{T}$ has the expression

(4.32) $H^{\lambda}g_{T}(x,t)=\Big{(}\frac{r^{1/2}}{2\pi}\Big{)}^{n-1}(g_{{\bf
x}_{0}}\psi_{\theta})^{\wedge}(v)a_{\lambda,R}(x,t)\int
e^{i2\pi\big{(}\phi_{{\bf x}_{0}}^{\lambda}({\bf
x};\omega)+v\cdot\omega\big{)}}\widetilde{\psi}_{\theta}(\omega)d\omega,$

We will see that the proof is quite similar to that of Lemma 4.4. We make the
change of variable $\omega\to\omega+\omega_{\theta}$ and consider the
oscillatory integral

(4.33) $\begin{split}&\int e^{i2\pi\big{(}\phi^{\lambda}_{{\bf
x}_{0}}(x,t;\omega+\omega_{\theta})+v\cdot\omega\big{)}}\widetilde{\psi}(r^{1/2}\omega)d\omega\\\
&=r^{-(n-1)/2}\int e^{i2\pi\big{(}\phi^{\lambda}_{{\bf
x}_{0}}(x,t;r^{-1/2}\omega+\omega_{\theta})+r^{-1/2}v\cdot\omega\big{)}}\widetilde{\psi}(\omega)d\omega.\end{split}$

Let us compute the derivative of the phase function. We write the derivative
as in (4.23). There are two terms there. The first term contributes $1$, which
has been shown in (4.28) and (4.29). We therefore have

(4.34) $|\eqref{2011227_14}|\gtrsim
r^{-1/2}\big{|}\nabla_{\omega}\phi^{\lambda}_{{\bf
x}_{0}}(x,t;\omega_{\theta})+v\big{|}-1.$

This gives us the desired rapid decay in the complement of the set

(4.35) $E_{\theta,v}=\\{(x,t):|\nabla_{\omega}\phi^{\lambda}_{{\bf
x}_{0}}(x,t;\omega_{\theta})+v|\lesssim r^{(1+\delta)/2}\\}.$

It remains to show $E_{\theta,v}\subset T_{\theta,v}$. We solve the equation
$\nabla_{\omega}\phi^{\lambda}_{{\bf x}_{0}}(x,t;\omega_{\theta})+v=0$ for the
variables $(x,t)$ to have

(4.36)
$\frac{\lambda\big{(}t\omega_{\theta}-x\big{)}}{\sqrt{\lambda^{2}+|x-t\omega_{\theta}|^{2}}}=-v+\frac{\lambda(t_{0}\omega_{\theta}-x_{0})}{\sqrt{\lambda^{2}+|x_{0}-t_{0}\omega_{\theta}|^{2}}}=-v+\nabla_{\omega}\phi^{\lambda}({\bf
x}_{0};\omega_{\theta}),$

which indeed is the straight line $l_{\theta,v}$. Moreover, we can solve
(4.35) directly, and see that it indeed lies inside our tube. Another way of
seeing it is to notice that the Jacobian of
$\nabla_{\omega}\phi^{\lambda}_{{\bf x}_{0}}(x,t;\omega_{\theta})$ in $x$ is
comparable to 1, as has been verified in (3.41). ∎

###  4.2. $L^{2}$ properties for wave packets

In this subsection, we prove two lemmas about $L^{2}$ properties of wave
packets. One of these two lemmas, Lemma 4.6, will not be directly used in our
proof; however we would still like to include it here and show that wave
packets in our setting have similar properties to those in the Fourier
restriction problem.

######  Lemma 4.6.

Let $B({\bf x}_{0},r)$ satisfy (4.6). Given a tube
$T_{1}=T_{\theta_{1},v_{1}}\in\mathbb{T}[B({\bf x}_{0},r)]$, all but
$r^{O(\delta)}$ many tubes $T_{2}=T_{\theta_{2},v_{2}}\in\mathbb{T}[B({\bf
x}_{0},r)]$ satisfy

(4.37) $\int_{|t-t_{0}|\leq r}\int
H^{\lambda}g_{T_{1}}(x,t)\overline{H^{\lambda}g_{T_{2}}}(x,t)dxdt={\rm{RapDec}}(r)\|g\|_{2}^{2}.$

###### Proof of Lemma 4.6.

Note that each piece $T^{\lambda}g_{T}$ has the expression

(4.38) $H^{\lambda}g_{T}(x,t)=\Big{(}\frac{r^{1/2}}{2\pi}\Big{)}^{n-1}(g_{{\bf
x}_{0}}\psi_{\theta})^{\wedge}(v)a_{\lambda,R}(x,t)\int
e^{i2\pi\big{(}\phi_{{\bf x}_{0}}^{\lambda}({\bf
x};\omega)+v\cdot\omega\big{)}}\widetilde{\psi}_{\theta}(\omega)d\omega,$

To estimate (4.37), it suffices to consider the oscillatory integral

(4.39) $\int_{|t-t_{0}|\leq r}\int e^{2\pi i\big{(}\phi^{\lambda}_{{\bf
x}_{0}}(x,t;\omega_{1})-\phi^{\lambda}_{{\bf
x}_{0}}(x,t;\omega_{2})\big{)}}a_{\lambda,R}(x,t)dxdt,$

with $|\omega_{1}-\omega_{\theta_{1}}|\leq r^{-1/2}$ and
$|\omega_{2}-\omega_{\theta_{2}}|\leq r^{-1/2}$, and show that it decays
rapidly. As we will apply the argument of integration by parts in $x$, we do
the change of variable $x\mapsto\lambda x$ and obtain

(4.40) $\lambda^{n-1}\int_{|t-t_{0}|\leq r}\int e^{2\pi
i\lambda\big{(}\phi(x,t^{\prime};\omega_{1})-\phi(x^{\prime}_{0},t^{\prime}_{0};\omega_{1})-\phi(x,t^{\prime};\omega_{2})+\phi(x^{\prime}_{0},t^{\prime}_{0};\omega_{2})\big{)}}a_{\lambda,R}(\lambda
x,t)dxdt,$

where we have also set $t^{\prime}:=t/\lambda$,
$x^{\prime}_{0}:=x_{0}/\lambda$ and $t^{\prime}_{0}:=t_{0}/\lambda$.

Let us first assume that we are in the case $|\omega_{1}-\omega_{2}|\geq
r^{-1/2+\delta}$. We compute the partial derivative of the phase function in
the $x$ variable and obtain

(4.41)
$\begin{split}&\frac{\lambda}{t^{\prime}}\frac{x-t^{\prime}\omega_{1}}{\sqrt{1+|x-t^{\prime}\omega_{1}|^{2}}}-\frac{\lambda}{t^{\prime}}\frac{x-t^{\prime}\omega_{2}}{\sqrt{1+|x-t^{\prime}\omega_{2}|^{2}}}.\end{split}$

Recall that the mixed Hessian of $\phi(x,t;\omega)$ in $x$ and $\omega$ is
non-vanishing, that is,

(4.42) $|\det\nabla_{\omega}\nabla_{x}\phi(x,t;\omega)|\simeq 1,$

as has been verified in (3.41). This, combined with mean value theorems,
implies that

(4.43) $|\eqref{210202e4_38}|\gtrsim\lambda|\omega_{1}-\omega_{2}|.$

Now we can apply integration by parts and finish the proof of the case
$|\omega_{1}-\omega_{2}|\geq r^{-1/2+\delta}$.

In the end, we consider the case $|\omega_{1}-\omega_{2}|\leq r^{-1/2+\delta}$
and $|v_{1}-v_{2}|\geq r^{1/2+\delta}$. However, this case follows immediately
from the pointwise estimate in Lemma 4.5. ∎

##  5\. Comparing wave packets at different scales

In the previous section, we built up a wave packet decomposition for a
function $g$ on the ball $B({\bf x}_{0},r)$. Since our proof of Theorem 3.4
relies on the multiscale argument as in [Gut18] and [GHI19], it is necessary
to compare wave packets at two different scales. In this section, we will
prove lemmas in Section 9 of [GHI19] in our setting. It is worth mentioning
that in their paper they consider more general operators. However, their
lemmas are valid only when the operators are of “reduced form”. Since we do
not reduce our operator to reduced form, we are unable to simply cite their
lemmas. Instead, we follow their arguments and prove the lemmas in our
setting. We include the details for the completeness of the paper.

Consider another ball $B(\widetilde{{\bf x}}_{0},\rho)\subset B({\bf
x}_{0},r)$ for some $\widetilde{{\bf
x}}_{0}=(\widetilde{x}_{0},\widetilde{t}_{0})\in\mathbb{R}^{n}$ and
$r^{1/2}<\rho<r$. Since we are considering wave packets at different scales,
it would be convenient to introduce some notations to distinguish wave packets
from different scales. We will use the notation
$\widetilde{\mathbb{T}}[B(\widetilde{{\bf x}}_{0},\rho)]$ for the collection
${\mathbb{T}}[B(\widetilde{{\bf x}}_{0},\rho)]$ in (4.7). We will denote the
elements of $\widetilde{\mathbb{T}}[B(\widetilde{{\bf x}}_{0},\rho)]$ by
$\widetilde{T}_{\widetilde{\theta},\widetilde{v}}$. Here, the symbol $\sim$
indicates that the objects are generated at a smaller scale.

We find it appropriate to introduce three more definitions here.

######  Definition 5.1.

We say a function $h$ is concentrated on wave packets from a tube set
$\mathbb{T}_{\alpha}$, if

(5.1) $h=\sum_{T\in\mathbb{T}_{\alpha}}h_{T}+\mathrm{RapDec}(r)\|h\|_{2}.$

######  Definition 5.2.

For a ball $B$ and $\mathbb{W}\subset\mathbb{T}[B]$, we define

(5.2) $h|_{\mathbb{W}}:=\sum_{T\in\mathbb{W}}h_{T}.$

######  Definition 5.3.

Let $({\theta},v)\in\Theta_{r}\times r^{1/2}\mathbb{Z}^{n-1}$ and let
$(\tilde{\theta},\widetilde{v})\in\Theta_{\rho}\times\rho^{1/2}\mathbb{Z}^{n-1}$.
we define a collection of smaller tubes
$\widetilde{T}_{\tilde{\theta},\widetilde{v}}$ that are “close to” the bigger
tube $T_{\theta,v}$ as

(5.3) $\widetilde{\mathbb{T}}_{\theta,v}[B(\widetilde{{\bf
x}}_{0},\rho)]:=\big{\\{}\widetilde{T}_{\widetilde{\theta},\widetilde{v}}\in\widetilde{\mathbb{T}}[B(\widetilde{{\bf
x}}_{0},\rho)]:{\rm{dist}}(\theta,\widetilde{\theta})\lesssim\rho^{-1/2},\;|\tilde{v}-(\nabla_{\omega}\phi_{{\bf
x}_{0}}^{\lambda}(\widetilde{{\bf
x}}_{0};\omega_{\theta})+v)|\lesssim{r^{(1+\delta)/2}}\big{\\}}.$

We sometimes abbreviate the collection
$\widetilde{\mathbb{T}}_{\theta,v}[B(\widetilde{{\bf x}}_{0},\rho)]$ to
$\widetilde{\mathbb{T}}_{\theta,v}$ for simplicity.

Notice that we can write

(5.4) $\tilde{v}-(\nabla_{\omega}\phi^{\lambda}_{{\bf x}_{0}}(\widetilde{{\bf
x}}_{0};\omega_{\theta})+v)=\big{(}\nabla_{\omega}\phi^{\lambda}(\widetilde{{\bf
x}}_{0};\omega_{\theta})+\tilde{v}\big{)}-\big{(}\nabla_{\omega}\phi^{\lambda}({\bf
x}_{0};\omega_{\theta})+v\big{)}.$

Heuristically, on the ball $B(\widetilde{{\bf x}}_{0},\rho)$, only those new
wave packets concentrated in
$\widetilde{\mathbb{T}}_{\theta,v}[B(\widetilde{{\bf x}}_{0},\rho)]$ would
make significant contribution to our old wave packet $g_{T_{\theta,v}}$. This
will be stated rigorously in the following lemma.

######  Lemma 5.4.

Let $T_{\theta,v}\in\mathbb{T}[B({\bf x}_{0},r)]$. Then it holds that

(5.5)
$g_{T_{\theta,v}}=(g_{T_{\theta,v}})|_{\widetilde{\mathbb{T}}_{\theta,v}[B(\widetilde{{\bf
x}}_{0},\rho)]}+{\rm{RapDec}}(r)\|g\|_{2}.$

###### Proof.

By the definition of wave packets,

(5.6) $\begin{split}(g_{T_{\theta,v}({\bf
x}_{0})})_{\widetilde{T}_{\tilde{\theta},\tilde{v}}(\widetilde{{\bf
x}}_{0})}(\omega)&=e^{-2\pi i\phi^{\lambda}(\widetilde{{\bf
x}}_{0};\omega)}(g_{T_{\theta,v}}(\cdot)e^{2\pi
i\phi^{\lambda}(\widetilde{{\bf
x}}_{0};\,\cdot\,)})_{\tilde{\theta},\tilde{v}}(\omega)\\\ &=e^{-2\pi
i\phi^{\lambda}(\widetilde{{\bf x}}_{0};\omega)}\Big{(}e^{2\pi
i(\phi^{\lambda}(\widetilde{{\bf x}}_{0};\,\cdot\,)-\phi^{\lambda}({{\bf
x}}_{0};\,\cdot\,))}(g(\cdot)e^{2\pi i\phi^{\lambda}({\bf
x}_{0};\,\cdot\,)})_{{\theta,v}}(\cdot)\Big{)}_{\tilde{\theta},\tilde{v}}(\omega).\end{split}$

Since a function $g_{\theta,v}$ is supported near $\theta$ for every function
$g$, by the above expression, we see that

(5.7) $(g_{T_{\theta,v}({\bf
x}_{0})})_{\widetilde{T}_{\tilde{\theta},\tilde{v}}(\widetilde{{\bf
x}}_{0})}\equiv
0\;\;\text{unless}\;\;\text{dist}(\theta,\widetilde{\theta})\lesssim\rho^{-1/2}.$

By renaming the function $g(\omega)e^{2\pi i\phi^{\lambda}({\bf
x}_{0},\omega)}$ by $g(\omega)$, it remains to show that

(5.8) $\big{(}e^{2\pi i\phi^{\lambda}_{{\bf x}_{0}}(\widetilde{{\bf
x}}_{0};\,\cdot\,)}g_{{\theta,v}}(\cdot)\big{)}_{\tilde{\theta},\tilde{v}}=\mathrm{RapDec}(r)\|g\|_{2},$

whenever $|-v+\tilde{v}-\nabla_{\omega}\phi_{{\bf
x}_{0}}^{\lambda}(\widetilde{{\bf
x}}_{0};\omega_{\theta})|\gtrsim{r^{(1+\delta)/2}}.$ By the definition, it
amounts to proving

(5.9) $\big{(}e^{2\pi i\phi_{{\bf x}_{0}}^{\lambda}(\widetilde{{\bf
x}}_{0};\,\cdot\,)}g_{\theta,v}(\cdot)\psi_{\tilde{\theta}}(\cdot)\big{)}^{\wedge}(\tilde{v})=\mathrm{RapDec}(r)\|g\|_{2}.$

The left hand side can be written as

(5.10) $\big{(}\widehat{\psi}_{\tilde{\theta}}*\big{(}e^{2\pi i\phi_{{\bf
x}_{0}}^{\lambda}(\widetilde{{\bf
x}}_{0};\,\cdot\,)}g_{\theta,v}(\cdot)\big{)}^{\wedge}\big{)}(\tilde{v}).$

Since the function $\widehat{\psi}_{\tilde{\theta}}$ is essentially supported
in $B(0,\rho^{1/2})$, it suffices to show that

(5.11) $\begin{split}\big{(}e^{2\pi i\phi_{{\bf
x}_{0}}^{\lambda}(\widetilde{{\bf
x}}_{0};\,\cdot\,)}g_{\theta,v}(\cdot)\big{)}^{\wedge}(z)=(1+r^{-1/2}|z-v-\nabla_{\omega}\phi_{{\bf
x}_{0}}^{\lambda}(\widetilde{\bf
x}_{0};\omega_{\theta})|)^{-(n+1)}\mathrm{RapDec}(r)\|g\|_{2}\end{split}$

for every $|z-v-\nabla_{\omega}\phi_{{\bf x}_{0}}^{\lambda}(\widetilde{\bf
x}_{0};\omega_{\theta})|\gtrsim r^{(1+\delta)/2}.$ We take a function
$\widetilde{\widetilde{\psi}}_{\theta}(\omega)=\widetilde{\widetilde{\psi}}(r^{1/2}(\omega-\omega_{\theta}))$
for some compactly supported function $\widetilde{\widetilde{\psi}}$ so that
this function is adapted to $\theta$ but is equal to one on the support of
$\widetilde{\psi}_{\theta}$. Since $g_{\theta,v}$ is supported on the support
of $\widetilde{\psi}_{\theta}$, the left hand side of (5.11) can be written as

(5.12) $\big{(}e^{2\pi i\phi_{{\bf x}_{0}}^{\lambda}(\widetilde{{\bf
x}}_{0};\,\cdot\,)}\widetilde{\widetilde{\psi}}_{\theta}(\cdot)g_{\theta,v}(\cdot)\big{)}^{\wedge}(z)=\big{(}e^{2\pi
i\phi_{{\bf x}_{0}}^{\lambda}(\widetilde{{\bf
x}}_{0};\,\cdot\,)}\widetilde{\widetilde{\psi}}_{\theta}(\cdot)\big{)}^{\wedge}*\widehat{g_{\theta,v}}(z).$

Since the function $\widehat{g_{\theta,v}}$ is concentrated on
$B(v,r^{(1+\delta)/2})$, by the above expression, the claim (5.11) is reduced
to

(5.13) $\begin{split}\int_{\mathbb{R}^{n}}e^{2\pi i(-z\cdot\omega+\phi_{{\bf
x}_{0}}^{\lambda}(\widetilde{{\bf
x}}_{0};\omega))}\widetilde{\widetilde{\psi}}_{\theta}(\omega)\,d\omega=\big{(}1+r^{-1/2}|z-\nabla_{\omega}\phi_{{\bf
x}_{0}}^{\lambda}(\widetilde{{\bf
x}}_{0};\omega_{\theta})|\big{)}^{-(n+1)}\mathrm{RapDec}(r)\end{split}$

whenever $|z-\nabla_{\omega}\phi_{{\bf x}_{0}}^{\lambda}(\widetilde{{\bf
x}}_{0};\omega_{\theta})|\gtrsim r^{(1+\delta)/2}$. We apply the change of
variables: $\omega\mapsto r^{-1/2}\omega+\omega_{\theta}$ and the above
integral becomes

(5.14) $r^{-(n-1)/2}e^{-2\pi
iz\cdot\omega_{\theta}}\int_{\mathbb{R}^{n}}e^{2\pi
i(-r^{-1/2}z\cdot\omega+\phi_{{\bf x}_{0}}^{\lambda}(\widetilde{{\bf
x}}_{0};\omega_{\theta}+r^{-1/2}\omega))}\widetilde{\widetilde{\psi}}(\omega)\,d\omega.$

By the stationary phase method, the estimate (5.13) follows from

(5.15) $|z-\nabla_{\omega}\phi_{{\bf x}_{0}}^{\lambda}(\widetilde{{\bf
x}}_{0};\omega_{\theta}+r^{-1/2}\omega)|\gtrsim r^{(1+\delta)/2}$

for every $|z-\nabla_{\omega}\phi_{{\bf x}_{0}}^{\lambda}(\widetilde{{\bf
x}}_{0};\omega_{\theta})|\gtrsim r^{(1+\delta)/2}$. By using a mean value
theorem with $|{\bf x}_{0}-\widetilde{{\bf x}}_{0}|\lesssim r$, this follows
from

(5.16) $|\nabla^{2}_{\omega}\nabla_{{\bf
x}}\phi^{\lambda}(x,t;\omega)|\lesssim 1$

for every $|x|+|t|\lesssim\lambda$, which has been verified in (3.30). ∎

######  Lemma 5.5.

Suppose that $T_{\theta,v}\in\mathbb{T}[B({\bf x}_{0},r)]$. If
$\widetilde{T}_{\tilde{\theta},\tilde{v}}\in\widetilde{\mathbb{T}}_{\theta,v}[B(\widetilde{{\bf
x}}_{0},\rho)]$, then it holds that

(5.17)
$\mathrm{HausDist}(\widetilde{T}_{\tilde{\theta},\tilde{v}},T_{\theta,v}\cap
B(\widetilde{{\bf x}}_{0},\rho))\lesssim r^{1/2+\delta}$

and

(5.18)
$\measuredangle(G(w_{\theta}),G(w_{\tilde{\theta}}))\lesssim\rho^{-1/2}.$

###### Proof.

The bound (5.18) is trivial, and we only need to prove (5.17). Let us assume
that neither $\widetilde{T}_{\tilde{\theta},\tilde{v}}$ nor $T_{\theta,v}\cap
B(\widetilde{{\bf x}}_{0},\rho)$ is empty. Let $l_{\theta,v}$ and
$l_{\tilde{\theta},\tilde{v}}$ be the core lines of $T_{\theta,v}$ and
$\widetilde{T}_{\tilde{\theta},\tilde{v}}$ respectively. The two tubes
involved in (5.17) have width $\lesssim r^{1/2+\delta}$, and therefore to
prove (5.17), it suffices to consider the core lines of the tubes only. From
the definition of core lines (see also (4.36)), we can write $l_{\theta,v}$ as

(5.19)
$\nabla_{\omega}\phi^{\lambda}(x,t;\omega_{\theta})-\nabla_{\omega}\phi^{\lambda}(x_{0},t_{0};\omega_{\theta})=-v;$

moreover, we can write the core line of $l_{\tilde{\theta},\tilde{v}}$ as

(5.20)
$\nabla_{\omega}\phi^{\lambda}(x,t;\omega_{\tilde{\theta}})-\nabla_{\omega}\phi^{\lambda}(\widetilde{x}_{0},\widetilde{t}_{0};\omega_{\tilde{\theta}})=-\tilde{v}.$

Suppose that $l_{\theta,v}$ passes through a point $(x_{1},\widetilde{t}_{0})$
and that $l_{\tilde{\theta},\tilde{v}}$ passes through
$(x_{2},\widetilde{t}_{0})$. Note that the angle between $l_{\theta,v}$ and
$l_{\tilde{\theta},\tilde{v}}$ is at most $\rho^{-1/2}$ and that we are
computing a Hausdorff distance within a ball of radius $\rho$. To prove
(5.17), it therefore suffices to show that $|x_{1}-x_{2}|\lesssim
r^{1/2+\delta}$, which, by (3.41), is the same as saying that

(5.21)
$\Big{|}\nabla_{\omega}\phi^{\lambda}(x_{1},\widetilde{t}_{0};\omega_{\theta})-\nabla_{\omega}\phi^{\lambda}(x_{2},\widetilde{t}_{0};\omega_{\theta})\Big{|}\lesssim
r^{1/2+\delta}.$

We consider two cases $\rho\leq r^{1/2+\delta}$ and $\rho\geq r^{1/2+\delta}$
separately.

Assume we are in the former case. By the assumption that $T_{\theta,v}\cap
B(\widetilde{{\bf x}}_{0},\rho)$ is not empty, we obtain
$|x_{1}-\widetilde{x}_{0}|\lesssim r^{1/2+\delta}$. Moreover, by (5.3), (5.4),
(5.19), (5.20) and the triangle inequality, we obtain that

(5.22)
$\begin{split}&\big{|}\nabla_{\omega}\phi^{\lambda}(x_{2},\widetilde{t}_{0};\omega_{\tilde{\theta}})-\nabla_{\omega}\phi^{\lambda}(\widetilde{x}_{0},\widetilde{t}_{0};\omega_{\tilde{\theta}})\big{|}\\\
&\lesssim
r^{1/2+\delta}+\big{|}\nabla_{\omega}\phi^{\lambda}(x_{1},\widetilde{t}_{0};\omega_{\theta})-\nabla_{\omega}\phi^{\lambda}(\widetilde{x}_{0},\widetilde{t}_{0};\omega_{\theta})\big{|}.\end{split}$

This, together with (3.41) and the mean value theorem, implies that

(5.23) $|x_{2}-\widetilde{x}_{0}|\lesssim r^{1/2+\delta},$

which further leads to the desired bound for $|x_{1}-x_{2}|$.

Assume we are in the latter case. The starting point of the proof of this is
similar as above. By the assumption that $T_{\theta,v}\cap B(\widetilde{{\bf
x}}_{0},\rho)$ is not empty, we obtain
$|x_{1}-\widetilde{x}_{0}|\lesssim\rho$. This, combined with (5.22), implies
that $|x_{2}-\widetilde{x}_{0}|\lesssim\rho$. Next, the proof starts to be
different.

(5.24)
$\begin{split}&\nabla_{\omega}\phi^{\lambda}(x_{1},\widetilde{t}_{0};\omega_{\theta})-\nabla_{\omega}\phi^{\lambda}(x_{2},\widetilde{t}_{0};\omega_{\theta})\\\
&=\nabla_{\omega}\phi^{\lambda}(x_{1},\widetilde{t}_{0};\omega_{\theta})-\nabla_{\omega}\phi^{\lambda}(x_{2},\widetilde{t}_{0};\omega_{\tilde{\theta}})+\nabla_{\omega}\phi^{\lambda}(x_{2},\widetilde{t}_{0};\omega_{\tilde{\theta}})-\nabla_{\omega}\phi^{\lambda}(x_{2},\widetilde{t}_{0};\omega_{\theta})\end{split}$

By (5.19) and (5.20), the last expression can be written as

(5.25)
$\nabla_{\omega}\phi^{\lambda}(\widetilde{x}_{0},\widetilde{t}_{0};\omega_{\theta})-\nabla_{\omega}\phi^{\lambda}(\widetilde{x}_{0},\widetilde{t}_{0};\omega_{\tilde{\theta}})+\nabla_{\omega}\phi^{\lambda}(x_{2},\widetilde{t}_{0};\omega_{\tilde{\theta}})-\nabla_{\omega}\phi^{\lambda}(x_{2},\widetilde{t}_{0};\omega_{\theta}).$

By the mean value theorem and the bound (3.30) with $\beta=1$ and
$\beta^{\prime}=2$, the absolute value of the last display can be bounded by

(5.26)
$|x_{2}-\widetilde{x}_{0}||\omega_{\tilde{\theta}}-\omega_{\theta}|\lesssim\rho\rho^{-1/2}\lesssim\rho^{1/2}.$

Recall that $\rho\leq r$. This finishes the proof of latter case, thus the
proof of the whole lemma. ∎

##  6\. The transverse equidistribution property

The proof of the transverse equidistribution estimate requires us to study
wave packets from different scales. To make ourselves clear, we sometimes use
“large wave packet” to mean the scale $r$ wave packet, and use “small wave
packet” to mean the scale $\rho$ one.

Recall the admissible parameters in (1.11). Let us introduce more notations
for the next several sections. The first notation is about the transverse
complete intersection. The second definition is about the tangency between
tubes and a transverse complete intersection. The third definition is about
collections of tangent tubes at two different scales.

######  Definition 6.1.

Let $P_{1},\ldots,P_{n-m}:\mathbb{R}^{n}\rightarrow\mathbb{R}$ be polynomials.
We consider the common zero set

(6.1)
$Z(P_{1},\ldots,P_{n-m}):=\\{x\in\mathbb{R}^{n}:P_{1}(x)=\cdots=P_{n-m}(x)=0\\}.$

Suppose that for all $z\in Z(P_{1},\ldots,P_{n-m})$, one has

(6.2) $\bigwedge_{j=1}^{n-m}\nabla P_{j}(z)\neq 0.$

Then a connected branch of this set, or a union of connected branches of this
set, is called an $m$-dimensional transverse complete intersection. Given a
set $Z$ of the form (6.1), the degree of $Z$ is defined by

(6.3) $\min\Big{(}{\prod_{i=1}^{n-m}\mathrm{deg}(P_{i})}\Big{)},$

where the minimum is taken over all possible representations of
$Z=Z(P_{1},\ldots,P_{n-m})$.

######  Definition 6.2.

Let $r\geq 1$ and $Z$ be an $m$-dimensional transverse complete intersection.
A tube $T_{\theta,v}({\bf x}_{0})\in\mathbb{T}[B({\bf x}_{0},r)]$ is said to
be $r^{-1/2+\delta_{m}}$-tangent to $Z$ in $B({\bf x}_{0},r)$ if it satisfies

  * •

$T_{\theta,v}({\bf x}_{0})\subset N_{r^{1/2+\delta_{m}}}(Z)\cap B({\bf
x}_{0},r)$;

  * •

For every $z\in Z\cap B({\bf x}_{0},r)$, if there is $y\in T_{\theta,v}({\bf
x}_{0})$ with $|z-y|\lesssim r^{1/2+\delta_{m}}$, then one has

(6.4) $\measuredangle(G(\theta),T_{z}Z)\lesssim r^{-1/2+\delta_{m}}.$

Here, $T_{z}Z$ is the tangent space of $Z$ at $z$.

######  Definition 6.3.

Let $r\geq\rho\geq 1$ and $Z$ be an $m$-dimensional transverse complete
intersection and let $B(\tilde{{\bf x}}_{0},\rho)\subset B({\bf x}_{0},r)$.
Define a collection of bigger tangent tubes inside a smaller ball as

(6.5) $\begin{split}\mathbb{T}_{Z}[B({\bf x}_{0},r)]:=\\{T\in\mathbb{T}[B({\bf
x}_{0},r)]:T\text{ is $r^{-1/2+\delta_{m}}$-tangent to }Z\text{ in }B({\bf
x}_{0},r)\\}.\end{split}$

Given an arbitrary translation $b\in\mathbb{R}^{n}$, define

(6.6) $\begin{split}\widetilde{\mathbb{T}}_{b}[B(\tilde{{\bf
x}}_{0},\rho)]:=\\{\widetilde{T}\in\widetilde{\mathbb{T}}[B(\tilde{{\bf
x}}_{0},\rho)]:\widetilde{T}\text{ is $\rho^{-1/2+\delta_{m}}$-tangent to
$Z+b$ in $B(\tilde{{\bf x}}_{0},\rho)$}\\}.\end{split}$

For simplicity, we sometimes use the notation $\mathbb{T}_{Z}$ and
$\widetilde{\mathbb{T}}_{b}$ in short for $\mathbb{T}_{Z}[B({\bf x}_{0},r)]$
and $\widetilde{\mathbb{T}}_{b}[B(\tilde{{\bf x}}_{0},\rho)]$, respectively.

Let $1\leq r^{1/2}\leq\rho\leq r$ and $\widetilde{\bf
x}_{0}=(\widetilde{x}_{0},\widetilde{t}_{0})\in\mathbb{R}^{n}$. For every pair
$(\tilde{\theta},w)\in\Theta_{\rho}\times r^{1/2}\mathbb{Z}^{n-1}$, we define
a collection of bigger tubes

(6.7) $\mathbb{T}_{\tilde{\theta},w}:=\\{T_{\theta,v}\in\mathbb{T}[B({\bf
x}_{0},r)]:\mathrm{dist}(\theta,\tilde{\theta})\lesssim\rho^{-1/2}\;\;\mathrm{and}\;\;|v+\nabla_{\omega}\phi_{{\bf
x}_{0}}^{\lambda}(\widetilde{\bf x}_{0};\omega_{\theta})-w|\lesssim
r^{1/2}\\}.$

Then for every $T\in\mathbb{T}_{\tilde{\theta},w}\cap\mathbb{T}_{Z}$, if
$T\cap B(\widetilde{\bf x}_{0},2\rho)\not=\varnothing$, the intersection of
$T$ and the horizontal plane $\\{t=\widetilde{t}_{0}\\}$ is contained in the
ball $B:=B^{n-1}(w_{H},Cr^{1/2+\delta})\subset
B^{n-1}(\widetilde{x}_{0},3\rho)$, where the center $w_{H}$ is defined as

(6.8)
$w_{H}:=\widetilde{t}_{0}\omega_{\tilde{\theta}}+\frac{\lambda(\nabla_{\omega}\phi^{\lambda}(\widetilde{\bf
x}_{0};\omega_{\tilde{\theta}})-w)}{(\lambda^{2}-|\nabla_{\omega}\phi^{\lambda}(\widetilde{\bf
x}_{0};\omega_{\tilde{\theta}})-w|^{2})^{1/2}}.$

This is due to the coreline equation (4.16) for the tube $T$, and Lemma 5.5.
We remark that if we perturb $\widetilde{t}_{0}$ by an extremely small factor
$\beta$, the intersection between $T$ and the horizontal plane
$\\{t=\bar{t}_{0}\\}$ with $\bar{t}_{0}=\widetilde{t}_{0}+\beta$ is still
contained in the ball $B$.

The main goal of this section is to prove the following lemma.

######  Lemma 6.4.

Let $|b|\lesssim r^{1/2+\delta_{m}}$ and recall (5.2). Suppose that $h$ is
concentrated on large wave packets from
$\mathbb{T}_{Z}\cap\mathbb{T}_{\tilde{\theta},w}$ for some
$(\tilde{\theta},w)\in\Theta_{\rho}\times r^{1/2}\mathbb{Z}^{n-1}$. Then for
every $\widetilde{\mathbb{W}}\subset\widetilde{\mathbb{T}}_{b}$,

(6.9) $\big{\|}h|_{\widetilde{\mathbb{W}}}\big{\|}_{2}^{2}\lesssim
r^{O(\delta_{m})}(r/\rho)^{-(n-m)/2}\|h\|_{2}^{2}.$

As a consequence, for every function $h$ concentrated on $\mathbb{T}_{Z}$,

(6.10) $\big{\|}h|_{\widetilde{\mathbb{W}}}\big{\|}_{2}^{2}\lesssim
r^{O(\delta_{m})}(r/\rho)^{-(n-m)/2}\|h\|_{2}^{2}.$

We first show that (6.10) follows from the local one (6.9). Decompose the
function $h$ as

(6.11) $h=\sum_{(\tilde{\theta},w)}h_{\tilde{\theta},w}$

such that $h_{\tilde{\theta},w}$ is concentrated on wave packets from
$\mathbb{T}_{Z}\cap\mathbb{T}_{\tilde{\theta},w}$. Since when $|w-x|\geq
Cr^{1/2+\delta}$,
$\widehat{h}_{\tilde{\theta},w}(x)=\mathrm{RapDec}(r)\|h\|_{2}$, we have the
almost $L^{2}$ orthogonality

(6.12) $\big{\|}h|_{\widetilde{\mathbb{W}}}\big{\|}_{2}^{2}\lesssim
r^{O(\delta)}\sum_{(\tilde{\theta},w)}\big{\|}h_{\tilde{\theta},w}|_{\widetilde{\mathbb{W}}}\big{\|}_{2}^{2}.$

Finally, we use the local estimate (6.9) and Lemma 4.2 to conclude the global
estimate (6.10).

To prove (6.9), note that $\widetilde{\mathbb{T}}_{b}$ only consists smaller
tubes $\widetilde{T}$ that $\widetilde{T}\cap B(\widetilde{\bf
x}_{0},3\rho/2)\not=\varnothing$. By Lemma 4.5, Lemma 5.4 and Lemma 5.5, we
can assume without loss of generality that for every bigger tube
$T\in\mathbb{T}_{\tilde{\theta},w}\cap\mathbb{T}_{Z}$, one has $T\cap
B(\widetilde{\bf x}_{0},2\rho)\not=\varnothing$.

We break the proof of the local transverse equidistribution estimate (6.9)
into several smaller lemmas. The first thing we would like to find out is the
location of $(h|_{\widetilde{\mathbb{W}}})^{\wedge}$. Consider $Z_{0}$, the
intersection between of variety $Z+b$ and the horizontal hyperplane
$\\{(x,t):t=\bar{t}_{0}\\}$. By the transversality theorem (For example,
Theorem 10.2), we can choose $\bar{t}_{0}=\widetilde{t}_{0}+\beta$ for an
extremely small number $\beta$, so that $Z_{0}$ is a transverse complete
intersection. Since the small perturbation $\beta$ is harmless in our proof,
to save us from abundant notations, let us assume
$\bar{t}_{0}=\widetilde{t}_{0}$. Hence
$Z_{0}=Z+b\cap\\{t=\widetilde{t}_{0}\\}$ can be considered as a transverse
complete intersection in $\mathbb{R}^{n-1}$, and $\deg(Z_{0})=O(\deg(Z))$.

If $H^{\lambda}$ was the Fourier extension operator,
$(h|_{\widetilde{\mathbb{W}}})^{\wedge}$ is just contained in a thin
neighborhood of $Z_{0}$. While in our case,
$(h|_{\widetilde{\mathbb{W}}})^{\wedge}$ is roughly contained in a thin
neighborhood of $\Phi(Z_{0})$, where
$\Phi:\mathbb{R}^{n-1}\to\mathbb{R}^{n-1}$ is a smooth map defined as

(6.13)
$\Phi(x):=-\nabla_{\omega}\phi^{\lambda}(x,\widetilde{t}_{0};\omega_{\tilde{\theta}})=-\frac{\lambda\big{(}\widetilde{t}_{0}\omega_{\tilde{\theta}}-x\big{)}}{(\lambda^{2}+|x-\bar{t}_{0}\omega_{\tilde{\theta}}|^{2})^{1/2}}.$

Before proving something about the support of
$(h|_{\widetilde{\mathbb{W}}})^{\wedge}$ rigorously, let us take a look at the
smooth function $\Phi$. The next lemma shows that $\Phi$ indeed looks like the
identity map of $\mathbb{R}^{n-1}$.

######  Lemma 6.5.

For any $x\in\mathbb{R}^{n-1}$ and $r\geq\ell>0$ that $B^{n-1}(x,\ell)\subset
B^{n-1}(0,3C_{n}\lambda)$, one has

(6.14) $B(\Phi(x),\ell/C)\subset\Phi(B(x,\ell))\subset B(\Phi(x),C\ell).$

The constant $C$ only depends on the choice of $C_{n}$ in Theorem 3.4.

###### Proof.

By Taylor’s theorem, we have that for any $y\in B(x,r)$,

(6.15) $\Phi(y)-\Phi(x)=(y-x)J_{\Phi}(x)+O(\lambda^{-1}|y-x|^{2}).$

The Jacobian $J_{\Phi}$ is a $(n-1)\times(n-1)$ symmetric matrix $A=(a_{kl})$,
with

(6.16)
$a_{kl}:=\begin{cases}\dfrac{\lambda(\widetilde{t}\omega_{\tilde{\theta}}-x)_{k}(\widetilde{t}\omega_{\tilde{\theta}}-x)_{l}}{(\lambda^{2}+|x-\widetilde{t}\omega_{\tilde{\theta}}|^{2})^{3/2}}&\text{
when }k\not=l,\\\\[12.91663pt]
\dfrac{\lambda[\lambda^{2}+\sum_{j\not=k}(\widetilde{t}\omega_{\tilde{\theta}}-x)_{j}^{2}]}{(\lambda^{2}+|x-\widetilde{t}\omega_{\tilde{\theta}}|^{2})^{3/2}}&\text{
when }k=l.\end{cases}$

Similar to the argument in (3.39), we can simplify $A$ as

(6.17)
$A=\frac{\lambda}{(\lambda^{2}+|x-\widetilde{t}\omega_{\tilde{\theta}}|^{2})^{1/2}}\Big{(}\lambda^{2}I_{n}-\frac{(\widetilde{t}\omega_{\tilde{\theta}}-x)^{T}(\widetilde{t}\omega_{\tilde{\theta}}-x)}{\lambda^{2}+|x-\widetilde{t}\omega_{\tilde{\theta}}|^{2}}\Big{)}.$

Inside the bracket, the first matrix has eigenvalues $1$, while the second
matrix has only one eigenvalue
$|x-\widetilde{t}\omega_{\tilde{\theta}}|^{2}/(\lambda^{2}+|x-\widetilde{t}\omega_{\tilde{\theta}}|^{2})\leq
4C_{n}/(1+4C_{n})<1$ since $x\in B(0,3C_{n}\lambda)$ and since $t\leq
C_{n}\lambda$. This proves that the eigenvalues of $A$ are all positive and
have lower bound $C^{-1}$ and upper bound $C$ for an absolute constant that
depends only on $C_{n}$, uniformly in $x$. Thus,

(6.18) $|\Phi(y)-\Phi(x)|\sim|y-x|+O(\lambda^{-1}|y-x|^{2}),$

which yields $\Phi(B(x,\ell))\subset B(\Phi(x),C\ell)$ since $|y-x|\leq r\leq
R\leq\lambda^{1-\varepsilon}$ as assumed in (3.3). For the other side, by the
implicit function theorem, we know that $\Phi^{-1}$ is well define in the
domain $\Phi(B^{n-1}(0,3C_{n}\lambda))$ and
$J_{\Phi^{-1}}(x)=(J_{\Phi}(x))^{-1}$. So the eigenvalues of
$J_{\Phi^{-1}}(x)$ have an upper bound $C^{-1}$ and a lower bound $C$. Hence
we can prove $B(\Phi(x),\ell/C)\subset\Phi(B(x,\ell))$ similarly. ∎

A direct corollary of this lemma is the following:

######  Corollary 6.6.

Let $Z_{0}$ and $B$ be as above. Then

(6.19)
$N_{\rho^{1/2+\delta_{m}}/C}(\Phi(Z_{0}))\cap\Phi((1/C)B)\subset\Phi(N_{\rho^{1/2+\delta_{m}}}(Z_{0})\cap
B)\subset N_{C\rho^{1/2+\delta_{m}}}(\Phi(Z_{0}))\cap\Phi(CB).$

Now via Lemma 6.5 and Corollary 6.6, we can say something rigorously related
to the support of $(h|_{\widetilde{\mathbb{W}}})^{\wedge}$. This is shown in
the next lemma.

######  Lemma 6.7.

Let $h$ be concentrated on bigger wave packets from
$\mathbb{T}_{\tilde{\theta},w}\cap\mathbb{T}_{Z}$. Recall that $Z_{0}$ was
defined in above (6.13), and $h|_{\widetilde{\mathbb{W}}}$ was introduced in
Lemma 6.4. Then

(6.20)
$\big{\|}(h|_{\widetilde{\mathbb{W}}})^{\wedge}\big{\|}_{2}\lesssim\big{\|}\widehat{h}\cdot{\bf
1}_{\\{N_{C\rho^{1/2+\delta_{m}}}(\Phi(Z_{0}))\cap\Phi(CB)\\}}\big{\|}_{2}+\mathrm{RapDec}(\rho)\|h\|_{2}.$

###### Proof.

By Lemma 4.2, one has

(6.21)
$\|(h|_{\widetilde{\mathbb{W}}})^{\wedge}\|_{2}^{2}\lesssim\sum_{\widetilde{T}\in\widetilde{\mathbb{W}}}\|{h}_{\widetilde{T}}\|_{2}^{2}.$

The scale $\rho$ wave packet $\widehat{h}_{\tilde{T}}$ was defined in (4.18).
We use its definition to have

(6.22)
$\sum_{\widetilde{T}\in\widetilde{\mathbb{W}}}\|\widehat{h}_{\widetilde{T}}\|_{2}^{2}=\rho^{n-1}\sum_{\widetilde{T}\in\widetilde{\mathbb{W}}}|(h_{\widetilde{\bf
x}_{0}}\psi_{\tilde{\theta}})^{\wedge}(v)|^{2}\|\widetilde{\psi}_{\tilde{\theta}}\|_{2}^{2}=\rho^{(n-1)/2}\sum_{\widetilde{T}\in\widetilde{\mathbb{W}}}|(h_{\widetilde{\bf
x}_{0}}\psi_{\tilde{\theta}})^{\wedge}(v)|^{2}.$

Since we can write $(h_{\widetilde{\bf
x}_{0}}\psi_{\tilde{\theta}})^{\wedge}(v)=\widehat{h}\ast(e^{-2\pi
i\phi^{\lambda}(\widetilde{\bf x}_{0};\
\cdot)}\widetilde{\psi}_{\tilde{\theta}}(\cdot))^{\wedge}(v)$, and since the
$L^{1}$ norm of the second function $(e^{-2\pi i\phi^{\lambda}(\widetilde{\bf
x}_{0};\ \cdot)}\widetilde{\psi}_{\tilde{\theta}}(\cdot))^{\wedge}$ is bounded
above by $O(1)$, one can use Hölder’s inequality for $|(h_{\widetilde{\bf
x}_{0}}\psi_{\tilde{\theta}})^{\wedge}(v)|^{2}$ and obtain

(6.23)
$\displaystyle\sum_{\widetilde{T}\in\widetilde{\mathbb{W}}}\|\widehat{h}_{\widetilde{T}}\|_{2}^{2}\lesssim\int|\widehat{h}(y)|^{2}\Big{(}\sum_{\widetilde{T}_{\tilde{\theta},\tilde{v}}\in\widetilde{\mathbb{W}}}|\rho^{(n-1)/2}(e^{-2\pi
i\phi^{\lambda}(\widetilde{\bf x}_{0};\
\cdot)}\widetilde{\psi}_{\tilde{\theta}}(\cdot))^{\wedge}(y-\widetilde{v})|\Big{)}dy.$

We claim that the sum inside the bracket is $O(1)$, and it decays rapidly
outside of the set $N_{C\rho^{1/2+\delta_{m}}}(\Phi(Z_{0}))\cap\Phi(CB)$. This
proves the lemma.

To prove the $O(1)$ upper bound, one just needs to notice that there are
$O(1)$ caps $\tilde{\theta}$ making contribution in $\widetilde{\mathbb{W}}$.
It remains to prove the rapidly-decaying property.

To prove the rapidly-decaying property, we first notice that
$h_{\widetilde{T}_{\tilde{\theta},\widetilde{v}}}=\mathrm{RapDec}(\rho)\|h\|_{2}$
unless
$\widetilde{T}_{\tilde{\theta},\widetilde{v}}\cap\\{t=\widetilde{t}_{0}\\}\subset
N_{2{\rho^{1/2+\delta_{m}}}}(Z_{0})\cap 2B$ for every smaller tube
$\widetilde{T}_{\tilde{\theta},\widetilde{v}}\in\widetilde{\mathbb{W}}$. This
is because on one hand
$\widetilde{T}_{\tilde{\theta},\widetilde{v}}\cap\\{t=\widetilde{t}_{0}\\}\subset
N_{2{\rho^{1/2+\delta_{m}}}}(Z_{0})$ since
$\widetilde{T}_{\tilde{\theta},\widetilde{v}}$ is tangent to $Z+b$; on the
other hand, since $h$ is concentrated on $\mathbb{T}_{\tilde{\theta},w}$, by
Lemma 5.4 we know that
$h_{\widetilde{T}_{\tilde{\theta},\widetilde{v}}}=\mathrm{RapDec}(\rho)\|h\|_{2}$
unless
$\widetilde{T}_{\tilde{\theta},\widetilde{v}}\in\widetilde{\mathbb{T}}_{\theta,v}[B(\widetilde{{\bf
x}}_{0},\rho)]$ for some $(\theta,v)$ with
$T_{\theta,v}\in\mathbb{T}_{\widetilde{\theta},w}$. But for such
$T_{\theta,v}$, one has $T_{\theta,v}\cap\\{t=\widetilde{t}_{0}\\}\subset B$
as shown in above (6.8). Hence by Lemma 5.5 we have that
$h_{\widetilde{T}_{\tilde{\theta},\widetilde{v}}}=\mathrm{RapDec}(\rho)\|h\|_{2}$
unless
$\widetilde{T}_{\tilde{\theta},\widetilde{v}}\cap\\{t=\widetilde{t}_{0}\\}\subset
2B$.

Now we only need to consider those smaller tubes
$\widetilde{T}_{\tilde{\theta},\widetilde{v}}$ with
$\widetilde{T}_{\tilde{\theta},\widetilde{v}}\cap\\{t=\widetilde{t}_{0}\\}\subset
N_{2{\rho^{1/2+\delta_{m}}}}(Z_{0})\cap 2B$. Using this information, we would
like to find out the location of $\widetilde{v}$ in the bracket of (6.23).
Recall that the coreline of the smaller tube
$\widetilde{T}_{\tilde{\theta},\widetilde{v}}$ satisfies (4.16), which indeed
is (4.36). On the hyperplane $\\{t=\widetilde{t}_{0}\\}$, we can rewrite
(4.36) as
$\widetilde{v}=-\nabla_{\omega}\phi^{\lambda}(x,\widetilde{t};\omega_{\tilde{\theta}})+\nabla_{\omega}\phi^{\lambda}(\widetilde{\bf
x}_{0};\omega_{\tilde{\theta}})=:\Psi(x)$. Hence by Corollary 6.6 one has

(6.24)
$\widetilde{v}\in\Psi(T_{\tilde{\theta},\widetilde{v}}\cap\\{t=\widetilde{t}_{0}\\})\subset\Psi(N_{2{\rho^{1/2+\delta_{m}}}}(Z_{0})\cap
2B)\subset N_{C\rho^{1/2+\delta_{m}}}\Psi(Z_{0})\cap\Psi(2B).$

Finally, we know from Lemma 4.5 that the function $(e^{-2\pi
i\phi^{\lambda}(\widetilde{\bf x}_{0};\
\cdot)}\widetilde{\psi}_{\tilde{\theta}}(\cdot))^{\wedge}(y-\widetilde{v})$ in
(6.23) decays rapidly unless
$|y-\widetilde{v}+\nabla_{\omega}\phi^{\lambda}(\widetilde{\bf
x}_{0},\omega_{\tilde{\theta}})|\lesssim\rho^{1/2+\delta_{m}}$, that is,
$y-\widetilde{v}\in B^{n-1}(-\nabla_{\omega}\phi^{\lambda}(\widetilde{\bf
x}_{0},\omega_{\tilde{\theta}}),C\rho^{1/2+\delta_{m}})$. Plugging this back
into (6.24) and noticing $\Phi+\nabla_{\omega}\phi^{\lambda}(\widetilde{\bf
x}_{0};\omega_{\tilde{\theta}})=\Psi$, we therefore can conclude that
$(e^{-2\pi i\phi^{\lambda}(\widetilde{\bf x}_{0};\
\cdot)}\widetilde{\psi}_{\tilde{\theta}}(\cdot))^{\wedge}(y-\widetilde{v})$
decays rapidly unless

(6.25) $y\in N_{C\rho^{1/2+\delta_{m}}}\Phi(Z_{0})\cap\Phi(CB),$

where $\Phi$ was defined in (6.13). This proves the claim and hence the lemma.
∎

To conclude (6.9), we need to prove the transverse equidistribution estimate
stated below.

######  Proposition 6.8.

Recall $Z_{0}=(Z+b)\cap\\{t=\widetilde{t}_{0}\\}$ and
$B=B^{n-1}(w,r^{1/2+\delta_{m}})$. Suppose that $h$ is concentrated on scale
$r$ wave packets in $\mathbb{T}_{\tilde{\theta},w}\cap\mathbb{T}_{Z}$. Then

(6.26) $\int|\widehat{h}|^{2}\cdot{\bf
1}_{\\{N_{C\rho^{1/2+\delta_{m}}}(\Phi(Z_{0}))\cap\Phi(CB)\\}}\lesssim
r^{O(\delta_{m})}\Big{(}\frac{\rho}{r}\Big{)}^{(n-m)/2}\|h\|_{2}^{2}.$

Note that the desired estimate (6.9) is a direct corollary of (6.20) and
Proposition 6.8.

The proof of Proposition 6.8 relies on an auxiliary lemma. Let us introduce
some more definitions. First, define

(6.27)
$\mathbb{T}_{Z,B,\tilde{\theta}}:=\\{T_{\theta,v}\in\mathbb{T}_{Z}:\mathrm{dist}(\theta,\tilde{\theta})\lesssim\rho^{-1/2}~{}\text{and}~{}T_{\theta,v}\cap\\{t=\widetilde{t}_{0}\\}\cap
B\not=\varnothing\\},$

so as proved in above (6.8), one has
$\mathbb{T}_{\tilde{\theta},w}\cap\mathbb{T}_{Z}\subset\mathbb{T}_{Z,B,\tilde{\theta}}$.
We will prove Proposition 6.8 with
$\mathbb{T}_{\tilde{\theta},w}\cap\mathbb{T}_{Z}$ replaced by
$\mathbb{T}_{Z,B,\tilde{\theta}}$. Next, for every (linear) subspace $V$ in
$\mathbb{R}^{n}$, we define a collection of bigger wave packets
$\mathbb{T}_{V,B,\tilde{\theta}}$ as

(6.28) $\mathbb{T}_{V,B,\tilde{\theta}}:=\\{(\theta,v):T_{\theta,v}\cap
B\not=\varnothing,~{}\measuredangle(G(\omega_{\theta}),V)\lesssim
r^{-1/2+\delta_{m}}~{}\text{and}~{}\mathrm{dist}(\theta,\tilde{\theta})\lesssim\rho^{-1/2}\\}.$

######  Lemma 6.9.

Suppose that $V$ is a subspace of $\mathbb{R}^{n}$. Let
$V_{0}:=V\cap\\{t=\widetilde{t}_{0}\\}$ and define $V^{\prime}$ as
$V_{0}^{\perp}$ in $\mathbb{R}^{n-1}$. If $g$ is concentrated on bigger wave
packets from $\mathbb{T}_{V,B,\tilde{\theta}}$, if
$\Pi\subset\\{t=\widetilde{t}_{0}\\}$ is any affine subspace parallel to
$V^{\prime}$ and if $y_{1}\in\Pi\cap\Phi(CB)$, then

(6.29) $\int_{\Pi\cap B(y_{1},\rho^{1/2+\delta_{m}})}|\widehat{g}|^{2}\lesssim
r^{O(\delta_{m})}\Big{(}\frac{\rho^{1/2}}{r^{1/2}}\Big{)}^{\dim(V^{\prime})}\int_{\Pi}|\widehat{g}|^{2}.$

###### Proof.

Since $g$ is concentrated on large wave packets from
$\mathbb{T}_{V,B,\tilde{\theta}}$ and since $G(\omega)$ is linear (up to a
scalar depending on $\omega$) as shown in (3.9), there is a shift
$\omega_{V}\in B^{n-1}(0,1)$ such that

(6.30) $\\{\omega:\measuredangle(G(\omega),V)\lesssim
r^{-1/2+\delta_{m}}\\}\subset\\{\omega:\text{dist}(\omega,V_{0}+\omega_{V})\lesssim
r^{-1/2+\delta_{m}}\\}.$

It implies that $g$ is supported in the $r^{-1/2+\delta_{m}}$ neighborhood of
$V_{0}+\omega_{V}$ inside the unit ball $B^{n-1}(0,1)$. As a result, the
Fourier transform of $(\widehat{g}|_{\Pi})^{\vee}$ is supported in an $n-m$
dimensional $r^{-1/2+\delta_{m}}$ ball centered at
$\text{proj}_{V^{\prime}}(\omega_{V})$, which implies (See also Lemma 3.11)

(6.31)
$|(\widehat{g}|_{\Pi})|\lesssim|(\widehat{g}|_{\Pi})|\ast\eta_{r^{1/2-\delta_{m}}}.$

Finally, we integrate $|(\widehat{g}|_{\Pi})|^{2}$ inside the ball
$B(y,\rho^{1/2+\delta_{m}})$ and invoke Hölder’s inequality to conclude
(6.29). ∎

###### Proof of Proposition 6.8.

We prove Proposition 6.8 via Lemma 6.9. Our proof is similar to the proof of
Lemma 6.2 in [Gut18] and the proof of Lemma 8.4 in [GHI19].

Since wave packets in $\mathbb{T}_{Z,B,\tilde{\theta}}$ are tangent to the
variety $Z$ inside the ball $B$, by the angular condition (6.4) we have

(6.32) $\measuredangle(G(\theta),T_{z}Z)\lesssim r^{-1/2+\delta_{m}}$

for every $z\in Z\cap 2B$ and
$T_{\theta,v}\in\mathbb{T}_{Z,B,\tilde{\theta}}$. One thus can find a subspace
$V\subset\mathbb{R}^{n}$ of minimal dimension and $\dim V\leq\dim Z$ such that
for all $\theta$ making contribution in $\mathbb{T}_{Z,B,\tilde{\theta}}$,

(6.33) $\measuredangle(G(\theta),V)\lesssim r^{-1/2+\delta_{m}}.$

It implies that the function $h$ is indeed concentrated on wave packets from
$\mathbb{T}_{V,B,\tilde{\theta}}$. So we can apply Lemma 6.9 to obtain a
subspace $V^{\prime}$ that

(6.34) $\int_{\Pi\cap B(y_{1},\rho^{1/2+\delta_{m}})}|\widehat{h}|^{2}\lesssim
r^{O(\delta_{m})}\Big{(}\frac{\rho^{1/2}}{r^{1/2}}\Big{)}^{\dim(V^{\prime})}\int_{\Pi}|\widehat{h}|^{2}$

for any $\Pi\subset\\{t=\widetilde{t}_{0}\\}$ being parallel to $V^{\prime}$
and $y_{1}\in\Pi\cap\Phi(CB)$. To finish the proof, we need three additional
claims.

Claim 1. The pushfoward $\Phi(Z_{0})$ is quantitatively transverse to
$V^{\prime}$ at every point $z\in\Phi(Z_{0})\cap\Phi(B(0,3C_{n}\lambda))$.

Claim 2. $\Phi^{-1}(\Pi)$ is an $n-1-\dim(V^{\prime})$ dimensional transverse
complete intersection in $\mathbb{R}^{n-1}$.

Claim 3. $\Pi\cap N_{C\rho^{1/2+\delta_{m}}}(\Phi(Z_{0}))\cap\Phi(CB)$ can be
covered by $r^{O(\delta_{m})}(r^{1/2}/\rho^{1/2})^{\dim Z_{0}-\dim V_{0}}$
many balls in $\Pi$ of radius $\rho^{1/2+\delta_{m}}$.

Assume at first that the three claims were verified. We plug Claim 3 back to
(6.34) so that

(6.35) $\int_{\Pi}|\widehat{h}|^{2}\cdot{\bf
1}_{\\{N_{C\rho^{1/2+\delta_{m}}}(\Phi(Z_{0}))\cap\Phi(CB)\\}}\lesssim
r^{O(\delta_{m})}\Big{(}\frac{\rho}{r}\Big{)}^{(n-m)/2}\int_{\Pi}|\widehat{h}|^{2}.$

Integrate over all (generic) affine subspaces $\Pi$ that are parallel to
$V^{\prime}$ to conclude the proof of Proposition 6.8.

_Proof of Claim 1._ We follow essentially the proof strategy in [Gut18].
Suppose that Claim 1 fails. It means that there is a point
$\Phi(z)\in\Phi(Z_{0})\cap\Phi(B(0,C\lambda))$ and a subspace $W_{\Phi}\subset
T_{\Phi(z)}(\Phi(Z_{0}))$ with $\dim Z_{0}-\dim W_{\Phi}+\dim V^{\prime}<n-1$,
such that for any non-zero vector $w\in W_{\Phi}$, there is a big constant
$C^{\prime}$ only depending on the $C_{n}$ in Theorem 3.4 that

(6.36) $\measuredangle(w,V^{\prime})\leq(C^{\prime})^{-2}.$

Let $(\Phi^{-1})^{\ast}_{z}:T_{\Phi(z)}\Phi(Z_{0})\to T_{z}Z_{0}$ be the map
between tangent spaces. Then
$\measuredangle(v,(\Phi^{-1})_{z}^{\ast}(v))<\pi/2-(C^{\prime})^{-1}$ for any
$v\in T_{\Phi(z)}\Phi(Z_{0})$ because of the positivity of $J_{\Phi^{-1}}$
(See Lemma 6.5). Hence if we define $W$ as the pullback
$W:=(\Phi^{-1})^{\ast}_{z}(W_{\Phi})\subset T_{z}Z_{0}$, then
$\measuredangle(w,V_{0})\gtrsim 1$ since $V_{0}$ is perpendicular to
$V^{\prime}$. As we will show below, this indeed implies for arbitrary non-
zero vector $w\in W$,

(6.37) $\measuredangle(w,V)\gtrsim 1.$

To prove (6.37), let $w=w_{1}+w_{2}$ with $w_{1}\perp V_{0}$ and $w_{2}\in
V_{0}$, so $\text{orth}_{V}(w)=\text{orth}_{V}(w_{1})$ (here we use
$\text{orth}_{V}(w):=w-\text{proj}_{V}(w)$ in convention). Note that the lower
bound $\measuredangle(w,V_{0})\gtrsim 1$ gives $|w_{1}|\gtrsim|w|$. Now for
any unit vector $v\in V$, we write $v=v_{1}+v_{2}$ where $v_{1}\perp V_{0}$
and $v_{2}\in V_{0}$, so $|v_{1}|\leq 1$ and
$\measuredangle(v_{1},\mathbb{R}^{n-1})\gtrsim 1$ due to (6.33). Hence
$|w_{1}\cdot v|=|w_{1}\cdot v_{1}|\leq c|w_{1}|$ for some absolute constant
$c<1$, which implies
$|\text{orth}_{V}(w)|=|\text{orth}_{V}(w_{1})|\sim|w_{1}|\gtrsim|w|$ and hence
(6.37). Finally, since $\dim(W)=\dim(W_{\Phi})$, one has
$\dim(Z_{0})-\dim(W)<\dim(V)$. Hence the angle estimate (6.37) contradicts the
minimality of $\dim(V)$, which can be seen via repeating the proof in page 115
of [Gut18]. This proves Claim 1.

_Proof of Claim 2._ Since $\Pi$ is an affine subspace, it is the intersection
of $\kappa:=n-1-\dim(V^{\prime})$ mutually orthogonal hyperplanes
$L_{1},\ldots,L_{\kappa}$. Suppose that each $L_{j}$ is parameterized as

(6.38) $L_{j}:m_{j}\cdot x+l_{j}=0,$

where $m_{j}$ is the normal vector of $L_{j}$ and $l_{j}\in\mathbb{R}$. Then
$\Phi^{-1}(L_{j})$ is indeed a branch of the quadratic variety

(6.39) $\Phi^{-1}(L_{j}):|\lambda
m_{j}\cdot(\widetilde{t}_{0}\omega_{\tilde{\theta}}-x)|^{2}-(\lambda^{2}+|x-\widetilde{t}_{0}\omega_{\tilde{\theta}}|^{2})l_{j}=0.$

Similar to the proof of Lemma 6.5, we can show $J_{\Phi^{-1}}(x)$ is not
degenerate for all $x\in\text{Range}(\Phi)$. Hence
$\Phi^{-1}(L_{1})\cap\ldots\cap\Phi^{-1}(L_{\kappa})$ is a transverse complete
intersection.

_Proof of Claim 3._ Similar to Claim 1, we know that $\Phi^{-1}(\Pi)\cap
Z_{0}$ is a transverse complete intersection inside $B(0,C_{n}\lambda)$ (In
fact, by Lemma 10.3, the pullback $\Phi^{-1}(\Pi)$ is transverse to $Z$ for
generic affine spaces $\Pi$). Note that the dimension of the variety
$\Phi^{-1}(\Pi)\cap Z_{0}$ is $\dim Z_{0}-\dim V_{0}$. By Wongkew’s theorem
[Won93], the set $N_{C\rho^{1/2+\delta_{m}}}(\Phi^{-1}(\Pi)\cap Z_{0})\cap CB$
can be covered by

(6.40) $r^{O(\delta_{m})}\Big{(}\frac{r^{1/2}}{\rho^{1/2}}\Big{)}^{\dim
Z_{0}-\dim V_{0}}$

many $(n-1)$-dimensional balls. Hence via Lemma 6.5 we know that
$N_{C\rho^{1/2+\delta_{m}}}(\Pi\cap\Phi(Z_{0}))\cap\Phi(CB)$ can be covered
the the same amount of $(n-1)$-dimensional balls, up to a constant.

Since $T_{z}\Phi(Z_{0})$ is quantitatively transverse to $\Pi$ at every point
$z\in\Phi(Z_{0})\cap\Phi(2CB)$ by Claim 1, one can argue similarly as in Lemma
8.13 in [GHI19] to conclude

(6.41) $\Pi\cap N_{C\rho^{1/2+\delta_{m}}}(\Phi(Z_{0}))\cap\Phi(CB)\subset{\rm
Proj}_{\Pi}[N_{C\rho^{1/2+\delta_{m}}}(\Pi\cap\Phi(Z_{0}))\cap\Phi(CB)].$

Therefore $\Pi\cap N_{C\rho^{1/2+\delta_{m}}}(\Phi(Z_{0}))\cap\Phi(CB)$ can be
covered by $r^{O(\delta_{m})}(r^{1/2}/\rho^{1/2})^{\dim Z_{0}-\dim V_{0}}$
many lower dimensional balls in $\Pi$ of radius $\rho^{1/2+\delta_{m}}$. This
proves Claim 3.

∎

##  7\. Multigrains and functions concentrated near a variety

In this section, following [HR19] and [HZ20], we introduce some definitions
and propositions for later use in Section 8 and 9. In contrast to [HZ20], the
definitions and propositions are stated in terms of the “dimension” of the
transverse complete intersection instead of the codimension so that the
notations are consistent with those in the previous sections.

###  7.1. Multigrain and nested structure

Here we introduce some definitions and lemmas about grains and multigrains,
following Section 3 of [HZ20].

######  Definition 7.1.

A grain is defined to be a pair $(S,B_{r})$ where $S\subset\mathbb{R}^{n}$ is
a transverse complete intersection and $B_{r}\subset\mathbb{R}^{n}$ is a ball
of some radius $r>0$. The dimension of a grain $(S,B_{r})$ is the dimension of
the transverse complete intersection $S$, and its degree is the degree of $S$.

######  Definition 7.2.

Let $(S,B({\bf x}_{0},r))$ be a grain of dimension $m$. A function $f$ is said
to be tangent to $(S,B({\bf x}_{0},r))$ if it is concentrated on wave packets
belonging to the collection

(7.1) $\\{T_{\theta,v}({\bf x}_{0})\in\mathbb{T}[B({\bf
x}_{0},r)]:T_{\theta,v}({\bf
x}_{0})\;\mathrm{is}\;r^{-1/2+\delta_{m}}\mathrm{-tangent\;to}\;S\;\mathrm{in}\;B({\bf
x}_{0},r)\\}.$

######  Definition 7.3.

A multigrain $\vec{S}_{m}$ is an $(n-m+1)$-tuple of grains

$\vec{S}_{m}=(\mathcal{G}_{n},\dots,\mathcal{G}_{m}),\qquad\mathcal{G}_{i}=(S_{i},B_{r_{i}})\qquad\textrm{
for $m\leq i\leq n$}$

satisfying

  * •

$\mathrm{dim}(S_{i})=i$ for $m\leq i\leq n$,

  * •

$S_{n}\supset S_{n-1}\supset\cdots\supset S_{m}$,

  * •

$B_{r_{n}}\supset B_{r_{n-1}}\supset\cdots\supset B_{r_{m}}$.

The parameter $n-m$ is referred to as the level of the multigrain
$\vec{S}_{m}$. The complexity of the multigrain is defined to be the maximum
of the degrees $\deg S_{i}$ over all $m\leq i\leq n$. Finally, the multiscale
of $\vec{S}_{m}$ is the tuple $\vec{r}=(r_{n},r_{n-1},\dots,r_{m})$. For two
multigrains $\vec{S}_{l}$ and $\vec{S}_{m}$ with $m\leq l$, we write

$\vec{S}_{m}\preceq\vec{S}_{l}$

if the first $n-l+1$ components of $\vec{S}_{m}$ agree those of $\vec{S}_{l}$.

######  Definition 7.4.

Let $\vec{S}_{m}=(\mathcal{G}_{n},\dots,\mathcal{G}_{m})$ be a multigrain and

$\mathcal{G}_{i}=\big{(}S_{i},B({\bf x}_{i},r_{i})\big{)}\quad\textrm{for
$m\leq i\leq n$.}$

Define $\mathbb{T}[\vec{S}_{m}]$ to be the set of scale $R:=r_{n}$ tubes
$T_{\theta_{n},v_{n}}({\bf x}_{n})\in\mathbb{T}[B({\bf x}_{n},r_{n})]$
satisfying the following hypothesis:

Nested tube hypothesis. There exists $T_{\theta_{i},v_{i}}({\bf
x}_{i})\in\mathbb{T}[B({\bf x}_{i},r_{i})]$ for $m\leq i<n$ such that

  1. (1)

$\mathrm{dist}(\theta_{i},\theta_{j})\lesssim r_{j}^{-1/2}$,

  2. (2)

$\mathrm{dist}\big{(}T_{\theta_{j},v_{j}}({\bf
x}_{j}),\,T_{\theta_{i},v_{i}}({\bf x}_{i})\cap B({\bf
x}_{j},r_{j})\big{)}\lesssim r_{i}^{(1+\delta)/2}$,

  3. (3)

$T_{\theta_{j},v_{j}}({\bf x}_{j})\subset N_{r_{j}^{1/2+\delta_{j}}}S_{j}$

hold true for all $i,j$ with $m\leq j\leq i\leq n$.

The direction set of $\mathbb{T}[\vec{S}_{m}]$ is defined by

(7.2)
$\Theta[\vec{S}_{m}]:=\\{\theta\in\Theta_{R}:T_{\theta,v}\in\mathbb{T}[\vec{S}_{m}]\text{
for some $v\in R^{1/2}\mathbb{Z}^{n-1}$}\\}.$

A main ingredient of the proof of Theorem 3.4 is the following lemma.

######  Lemma 7.5 (Lemma 3.7 of [HZ20]).

Let $\vec{S}_{m}$ be a level $n-m$ multigrain with multiscale
$\vec{r}_{m}=(r_{n},\ldots,r_{m})$ and complexity at most $d$. If $R=r_{n}$
and the constants in (1.11) are chosen appropriately, then

(7.3)
$\\#\Theta[\vec{S}_{m}]\lesssim_{\epsilon_{\circ},d}\Big{(}\prod_{i=m}^{n-1}r_{i}^{-1/2}\Big{)}R^{(n-1)/2+\epsilon_{\circ}}.$

Since our tubes $T_{\theta,v}$ are straight, and the tubes $T_{\theta,v}$
corresponding to the same $\theta$ indicate the same direction (see the
definition of the tube $T_{\theta,v}$ (4.16)), the above lemma can be proved
by using the nested polynomial Wolff axioms of Zahl [Zah21] and Hickman,
Rogers and Zhang [HRZ19]; for instance, one can follow the proof of Lemma 3.7
of [HZ20]. We leave out the details.

###  7.2. Some lemmas

There are more definitions and lemmas that we will need in Section 8 and 9.
Let us state them here, following Section 8 of [HR19] and Section 5 of [HZ20].
In this subsection, we fix a scale $r\geq 1$ and a smaller scale
$r^{1/2}\leq\rho\leq r$, and we consider balls

(7.4) $B(\widetilde{{\bf x}}_{0},\rho)\subset B({\bf
x}_{0},r)\subset[-3C_{n}\lambda,3C_{n}\lambda]^{n-1}\times[R/C_{n},C_{n}R].$

Recall some definitions in Section 6:

(7.5) $\begin{split}\mathbb{T}_{Z}[B({\bf x}_{0},r)]&=\\{T\in\mathbb{T}[B({\bf
x}_{0},r)]:T\text{ is $r^{-1/2+\delta_{m}}$-tangent to }Z\text{ in }B({\bf
x}_{0},r)\\},\\\ \widetilde{\mathbb{T}}_{b}[B(\widetilde{{\bf
x}}_{0},\rho)]&=\\{\widetilde{T}\in\widetilde{\mathbb{T}}[B(\widetilde{{\bf
x}}_{0},\rho)]:\widetilde{T}\text{ is $\rho^{-1/2+\delta_{m}}$-tangent to
}Z+b\text{ in }B(\widetilde{{\bf x}}_{0},\rho)\\}.\end{split}$

For some technical issue and rigorousness, we introduce a “thickening” of an
arbitrary subset of $\mathbb{T}[B({\bf x}_{0},r)]$: Given any
$\mathbb{W}\subset\mathbb{T}[B({\bf x}_{0},r)]$, we define

(7.6) $\mathbb{W}^{*}:=\\{T_{\theta,v}\in\mathbb{T}[B({\bf
x}_{0},r)]:\mathrm{dist}(\theta,{\theta}^{\prime})\lesssim
r^{-1/2}\,\mathrm{and}\,|v-{v}^{\prime}|\lesssim
r^{(1+\delta)/2}\,\mathrm{for\;some}\,({\theta}^{\prime},{v}^{\prime})\subset\mathbb{W}\\}.$

Note that $\mathbb{W}^{*}$ is a set slightly larger than $\mathbb{W}$.
Intuitively, one can identify $\mathbb{W}^{\ast}$ with $\mathbb{W}$.

######  Definition 7.6.

For $\widetilde{\mathbb{W}}\subset\widetilde{\mathbb{T}}[B(\widetilde{{\bf
x}}_{0},\rho)]$ we define $\uparrow\\!\\!\\!\widetilde{\mathbb{W}}$ to be a
collection of all pairs $T_{\theta,v}({\bf x}_{0})\in\mathbb{T}[B({\bf
x}_{0},r)]$ such that there exists a non-empty set
$\widetilde{T}_{\tilde{\theta},\tilde{v}}(\widetilde{{\bf
x}}_{0})\in\widetilde{\mathbb{W}}$ satisfying

  1. (1)

$\mathrm{dist}(\tilde{\theta},{\theta})\lesssim\rho^{-1/2}$

  2. (2)

$\mathrm{dist}(\widetilde{T}_{\tilde{\theta},\tilde{v}}(\widetilde{{\bf
x}}_{0}),T_{\theta,v}({\bf x}_{0})\cap B(\widetilde{{\bf
x}}_{0},\rho))\lesssim r^{1/2+\delta}$.

This definition naturally appears when comparing wave packets at different
scales by the following property: For every
$\widetilde{\mathbb{W}}\subset\widetilde{\mathbb{T}}[B(\widetilde{{\bf
x}}_{0},\rho)]$ and $g\in L^{1}$

(7.7)
$g|_{\widetilde{\mathbb{W}}}=(g|_{\uparrow\widetilde{\mathbb{W}}})|_{\widetilde{\mathbb{W}}}+\mathrm{RapDec}(r)\|g\|_{2}.$

It can be thought of as a reverse version of Lemma 5.4. Since the proof is
straightforward, we leave out the details.

######  Lemma 7.7.

For every set $\mathbb{W}_{1},\mathbb{W}_{2}\subset\mathbb{T}[B({\bf
x}_{0},r)]$ and function $g$, it holds that

(7.8)
$\|(g|_{\mathbb{W}_{1}})|_{\mathbb{W}_{2}}\|_{L^{2}}\lesssim\|g|_{{\mathbb{W}_{1}}\cap{\mathbb{W}_{2}^{*}}}\|_{L^{2}}+\mathrm{RapDec}(r)\|g\|_{2}.$

###### Proof.

We split our function into two parts:

(7.9)
$(g|_{\mathbb{W}_{1}})|_{\mathbb{W}_{2}}=(g|_{\mathbb{W}_{1}\cap\mathbb{W}_{2}^{*}})|_{\mathbb{W}_{2}}+(g|_{\mathbb{W}_{1}\setminus\mathbb{W}_{2}^{*}})|_{\mathbb{W}_{2}}.$

We apply Lemma 4.2 to the function $g|_{\mathbb{W}_{1}\cap\mathbb{W}_{2}^{*}}$
and obtain

(7.10)
$\|(g|_{\mathbb{W}_{1}\cap\mathbb{W}_{2}^{*}})|_{\mathbb{W}_{2}}\|_{2}\lesssim\|g|_{\mathbb{W}_{1}\cap\mathbb{W}_{2}^{*}}\|_{2}.$

On the other hand, by the proof of Lemma 5.4, we see that

(7.11)
$\|(g|_{\mathbb{W}_{1}\setminus\mathbb{W}_{2}^{*}})|_{\mathbb{W}_{2}}\|_{2}=\mathrm{RapDec}(r)\|g\|_{2}.$

This completes the proof. ∎

######  Lemma 7.8.

Let $Z=Z(P_{1},\ldots,P_{n-m})$ be a transverse complete intersection, with
$\mathrm{deg}P_{j}\leq d$. Let $b\in\mathbb{R}^{n}$ with $|b|\lesssim
r^{1/2+\delta_{m}}$. Suppose that $B(\widetilde{{\bf x}}_{0},\rho)\subset
B({\bf x}_{0},r)$. If $g$ is concentrated on wave packets from
$\mathbb{T}_{Z}[B({\bf x}_{0},r)]$ and
$\widetilde{\mathbb{W}}\subset\widetilde{\mathbb{T}}_{b}[B(\widetilde{{\bf
x}}_{0},\rho)]$, then

(7.12)
$\|{g}|_{\widetilde{\mathbb{W}}}\|_{2}^{2}\lesssim_{d}r^{O(\delta_{m})}\Big{(}\frac{r}{\rho}\Big{)}^{-\frac{n-m}{2}}\|g|_{\uparrow\widetilde{\mathbb{W}}}\|_{2}^{2}+\mathrm{RapDec}(r)\|g\|_{2}^{2}.$

The above lemma is a corollary of Lemma 6.4 and 7.7. The same lemma for the
Fourier extension operator for paraboloid is stated and proved in [HZ20]. We
refer to Lemma 5.3 of [HZ20] for the details of the proof.

######  Lemma 7.9.

Let $Z=Z(P_{1},\ldots,P_{n-m})$ be a transverse complete intersection, with
$\mathrm{deg}P_{j}\leq d$. Suppose that $B(\widetilde{{\bf
x}}_{0},\rho)\subset B({\bf x}_{0},r)$ and $B(\widetilde{{\bf
x}}_{0},\rho)\cap N_{\rho^{1/2+\delta_{m}}}Z\neq\varnothing$. Let $g$ be
concentrated on wave packets from $\mathbb{T}_{Z}[B({\bf x}_{0},r)]$. Then
there are a set of translates $\mathcal{B}\subset B(0,2r^{1/2+\delta_{m}})$
and functions $\\{g_{b}\\}_{b\in\mathcal{B}}$ such that

(7.13) $\|H^{\lambda}g\|_{\mathrm{BL}_{k,A}^{p}(B(\tilde{{\bf
x}}_{0},\rho))}^{p}\lesssim_{d}(\log{r})^{2p}\sum_{b\in\mathcal{B}}\|{H}^{\lambda}{g}_{b}\|_{\mathrm{BL}_{k,A}^{p}(B({\tilde{{\bf
x}}_{0}},\rho)\cap
N_{\rho^{1/2+\delta_{m}}}(Z+b))}^{p}+\mathrm{RapDec}(r)\|g\|_{2}^{p},$ (7.14)
$\sum_{b\in\mathcal{B}}\|{g}_{b}\|_{2}^{2}\lesssim_{d}\|g\|_{2}^{2},$ (7.15)
$\\#\mathcal{B}\lesssim_{d}(r/\rho)^{n(1/2+\delta_{m})},$

and

(7.16) $g_{b}:=g|_{\widetilde{\mathbb{T}}^{\prime}_{b}[B(\widetilde{{\bf
x}}_{0},\rho)]}\text{ \, for some set \,
$\widetilde{\mathbb{T}}^{\prime}_{b}[B(\widetilde{{\bf
x}}_{0},\rho)]\subset\widetilde{\mathbb{T}}_{b}$}.$

The proof of Lemma 7.9 requires the lemma below.

######  Lemma 7.10.

Let $\rho\leq r/2$ and $Z\subset\mathbb{R}^{n}$ be a transverse complete
intersection. Let $T_{\theta,v}\in\mathbb{T}_{Z}[B({\bf x}_{0},r)]$ and $b\in
B(0,2r^{1/2+\delta_{m}})$. If
$\widetilde{T}_{\tilde{\theta},\tilde{v}}(\widetilde{{\bf
x}}_{0})\in\widetilde{\mathbb{T}}_{\theta,v}[B(\widetilde{{\bf x}}_{0},\rho)]$
satisfies

(7.17) $\widetilde{T}_{\tilde{\theta},\tilde{v}}(\widetilde{{\bf x}}_{0})\cap
N_{\rho^{1/2+\delta_{m}}/2}(Z+b)\neq\varnothing,$

then
$\widetilde{T}_{\tilde{\theta},\tilde{v}}\in\widetilde{\mathbb{T}}_{b}[B(\widetilde{{\bf
x}}_{0},\rho)]$.

The rigorous proof of Lemma 7.10 is quite technical, and we refer to the proof
of Proposition 9.2 of [GHI19] for the details. We do not reproduce the proof
here.

###### Proof of Lemma 7.9.

We first apply Lemma 10.5 of [GHI19]444We cannot directly apply the lemma
because our operator is not of the normal form. However, one can prove the
lemma for our operator by following the same argument. We leave out the
details here.. Then there exist a finite set $\mathcal{B}\subset
B(0,2r^{1/2+\delta_{m}})$ with cardinality at most
$O((r/\rho)^{n(1/2+\delta_{m})})$ and a collection $\mathcal{B}^{\prime}$ of
finitely overlapping $K^{2}$-balls $B_{K^{2}}$ intersecting $B(\widetilde{{\bf
x}}_{0},\rho)$ such that

(7.18) $\|H^{\lambda}g\|_{\mathrm{BL}_{k,A}^{p}(B(\widetilde{{\bf
x}}_{0},\rho))}^{p}\lesssim(\log{r})^{2p}\sum_{B_{K^{2}}\in\mathcal{B}^{\prime}}\mu_{H^{\lambda}g}(B_{K^{2}}),$