# Oscillatory integral operators with homogeneous phase functions

Robert Schippa  Fakultät für Mathematik, Karlsruher Institut für Technologie,
Englerstrasse 2, 76131 Karlsruhe, Germany<EMAIL_ADDRESS>
###### Abstract.

Oscillatory integral operators with $1$-homogeneous phase functions satisfying
a convexity condition are considered. For these we show the
$L^{p}-L^{p}$-estimates for the Fourier extension operator of the cone due to
Ou–Wang via polynomial partitioning. For this purpose, we combine the
arguments of Ou–Wang with the analysis of Guth–Hickman–Iliopoulou, who
previously showed sharp $L^{p}-L^{p}$-estimates for non-homogeneous phase
functions with variable coefficients under a convexity assumption.
Furthermore, we provide examples exhibiting Kakeya compression, which shows
the estimates to be sharp. We apply the oscillatory integral estimates to show
new local smoothing estimates for wave equations on compact Riemannian
manifolds $(M,g)$ with $\dim M\geq 3$. This generalizes the argument for the
Euclidean wave equation due to Gao–Liu–Miao–Xi.

##  1\. Introduction

In the following we consider oscillatory integral operators which naturally
generalize the Fourier extension operator for the cone

(1) $\mathcal{E}f(x)=\int_{A^{n-1}}e^{i(\langle
x^{\prime},\omega\rangle+x_{n}|\omega|)}f(\omega)d\omega.$

We consider operators with $\lambda\geq 1$,

(2) $T^{\lambda}f(x)=\int
e^{i\phi^{\lambda}(x;\omega)}a^{\lambda}(x;\omega)f(\omega)d\omega$

and $a\in C_{c}^{\infty}(\mathbb{R}^{n}\times\mathbb{R}^{n-1},\mathbb{R})$,
$\phi\in C^{\infty}(\mathbb{R}^{n}\times\mathbb{R}^{n-1}\backslash
0;\mathbb{R})$, $\phi^{\lambda}(x;\omega)=\lambda\phi(x/\lambda;\omega)$,
$a^{\lambda}(x;\omega)=a(x/\lambda;\omega)$. We suppose that $\phi$ is
$1$-homogeneous in $\omega$: it holds for $\mu>0$

(3) $\phi(x;\mu\omega)=\mu\phi(x;\omega).$

For the support of $a$ we suppose that

$\text{supp}(a)\subseteq A^{n-1}=B_{n-1}(0,2)\backslash B_{n-1}(0,1/2).$

Above $B_{d}(c,r)\subseteq\mathbb{R}^{d}$ denotes the open ball with center
$c$ and radius $r>0$.

We write $x=(x^{\prime},x_{n})\in\mathbb{R}^{n-1}\times\mathbb{R}$ and impose
the following conditions on $\phi$ in $\text{supp}(a)$:

$\displaystyle C1)\quad\text{rank}(\partial^{2}_{x\omega}\phi)=n-1,$
$\displaystyle
C2^{+})\quad\partial^{2}_{\omega\omega}\langle\partial_{x}\phi,G(x;\omega_{0})\rangle\big{|}_{\omega=\omega_{0}}\text{
has }n-2\text{ non-vanishing eigenvalues of the same sign},$

where $G$ denotes the Gauss map

(4)
$G_{0}(x;\omega)=\bigwedge_{j=1}^{n-1}\partial^{2}_{x\omega_{j}}\phi(x;\omega),\qquad
G=G_{0}/|G_{0}|$

of the embedded surface $\omega\mapsto\partial_{x}\phi(x;\omega)$. We identify
$\bigwedge^{n-1}\mathbb{R}^{n}\simeq\mathbb{R}^{n}$.

In this note we prove new estimates

(5)
$\|T^{\lambda}f\|_{L^{p}(\mathbb{R}^{n})}\lesssim_{\varepsilon,\phi,a}\lambda^{\varepsilon}\|f\|_{L^{p}(A^{n-1})}$

for operators (2) like described above. Firstly, we recall that the
conjectured range of $L^{p}$-estimates

(6) $\|\mathcal{E}f\|_{L^{p}(\mathbb{R}^{n})}\lesssim\|f\|_{L^{p}(A^{n-1})}$

is given by $p>\frac{2(n-1)}{n-2}$. This prominent open problem is known as
_restriction conjecture for the cone_ and goes back to Stein. The
corresponding conjecture for the paraboloid is given by

$\|\int_{B_{n-1}(0,1)}e^{i(\langle
x^{\prime},\omega\rangle+x_{n}|\omega|^{2})}f(\omega)d\omega\|_{L^{p}(\mathbb{R}^{n})}\lesssim\|f\|_{L^{p}(B_{n-1})}$

for $p>\frac{2n}{n-1}$. Note that there is a shift of dimension by one, which
is commonly explained by the number of principal curvatures being reduced by
one. More pictorially, disregarding the null direction of the cone, the
conical hypersurface in $n$ dimensions locally looks like a parabolic
hypersurface in $n-1$ dimensions.

The restriction conjecture for the cone was solved for $n=3$ by Taberner [33],
for $n=4$ by Wolff [40] via bilinear estimates, and for $n=5$ by Ou–Wang [26]
via polynomial partitioning. Let

(7) $p_{n}=\begin{cases}&4,\quad n=3,\\\ &2\cdot\frac{3n+1}{3n-3},\quad
n>3\text{ odd},\\\ &2\cdot\frac{3n}{3n-4},\quad n>3\text{ even}.\end{cases}$

Ou–Wang showed (6) for $p>p_{n}$, which is also currently the widest range in
higher dimensions to the best of the author’s knowledge. Notably, in the case
of Carleson-Sjölin phase functions (cf. [8, 19]), which are not
$1$-homogeneous anymore, where $C2^{+})$ is replaced with

$H2^{+})\quad\partial^{2}_{\omega\omega}\langle\partial_{x}\phi(x;\omega),G(x;\omega_{0})\rangle\big{|}_{\omega=\omega_{0}}\text{
has }n-1\text{ eigenvalues of the same sign},$

Guth–Hickman–Iliopoulou [14] showed the sharp range of $L^{p}-L^{p}$
estimates, in the sense that there are phase functions for which the estimate
fails for lower values of $p$. The deviation from the corresponding
generalized restriction conjecture for the paraboloid occurs due to _Kakeya
compression_. This means that wave packets, which we find after decomposing
$T^{\lambda}f$ into non-oscillating components, can cluster in low-dimensional
varieties. This is not known to happen in the constant-coefficient case: The
restriction conjecture for the sphere implies the _Kakeya conjecture_.

Kakeya compression was initially observed by Bourgain [5], see also Wisewell
[38] and Bourgain–Guth [7]. Related phenomena were discussed by
Minicozzi–Sogge [23] and Sogge [30]. In this note we point out Kakeya
compression for $1$-homogeneous phases with variable coefficients, which shows
that the following $L^{p}$-estimates are sharp up to endpoints:

######  Theorem 1.1.

Let
$\phi:\mathbb{R}^{n}\times\mathbb{R}^{n-1}\backslash\\{0\\}\rightarrow\mathbb{R}$
be a $1$-homogeneous phase satisfying $C1)$ and $C2^{+})$ and $a\in
C^{\infty}_{c}(A^{n-1})$ be an amplitude. Then, the estimate (5) holds for
$p\geq p_{n}$ with $p_{n}$ as in (7).

We remark that for $p>p_{n}$ the $\lambda^{\varepsilon}$-factor can be
dropped. Guth–Hickman–Iliopoulou showed the $\varepsilon$-removal lemma for
oscillatory integral operators in [14, Section 12], albeit with a stronger
non-degeneracy hypothesis than presently considered. The idea goes back to Tao
[35, 36]. In Section 8 we prove the following global estimates for $p>p_{n}$
by a small variation of the argument in [14]:

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

The proof of Theorem 1.1 combines ideas from the case of constant-coefficient
homogeneous phases due to Ou–Wang [26] and Gao–Liu–Miao–Xi [10] and variable-
coefficient non-homogeneous phases due to Guth–Hickman–Iliopoulou [14].

We digress for a moment to describe the tools we will use and put them into
context. Bennett–Carbery–Tao [4] delivered an important contribution with
sharp $n$-multilinear restriction estimates: The $k$-restriction conjecture
for the cone reads

(8)
$\big{\|}\prod_{i=1}^{k}|\mathcal{E}f_{i}|^{\frac{1}{k}}\big{\|}_{L^{p_{k}}(B_{n}(0,R))}\lesssim_{\varepsilon}R^{\varepsilon}\prod_{i=1}^{k}\|f_{i}\|^{\frac{1}{k}}_{2}$

with $|\mathfrak{n}(\xi_{1})\wedge\ldots\wedge\mathfrak{n}(\xi_{k})|\gtrsim 1$
with $\mathfrak{n}(\xi)=(-\frac{\xi}{|\xi|},1)$,
$\xi_{i}\in\text{supp}(f_{i})\subseteq A_{n-1}$, and
$p_{k}=\frac{2(n+k)}{n+k-2}$. For $k=2$, this is due to Wolff [40] (see Tao
[37] for the endpoint without $R^{\varepsilon}$-loss), and for $k=n$ this is
an instance of the Bennett–Carbery–Tao theorem [4]. There are only few partial
results for $3\leq k\leq n-1$, see [1] and references therein.

We note that in [4] the multilinear estimates were shown as well for constant-
coefficient phase functions as smooth perturbations thereof. Bourgain–Guth [7]
devised an iteration to deduce linear estimates from multilinear estimates.
Guth [12] observed that the full strength of $k$-multilinear estimates is not
required, but a slightly weaker variant given by $k$-broad norms suffices to
run the iteration. He used polynomial partitioning to improve on the previous
results in [12, 13]. The idea is to equipartition the broad norm with
polynomials of controlled degree: After wave packet decomposition, one finds
that either the broad norm is concentrated on “cells” or on the “wall”, which
is a neighbourhood of a variety. To oversimplify matters for a moment, if the
broad norm is concentrated on the cells, then sharp bounds follow from
induction on scales. If the broad norm is concentrated along the wall, then we
are morally dealing with a restriction problem in lower dimensions, which is
amenable to another induction hypothesis.

We introduce the $k$-broad norms in the present context: For the definition
decompose $A^{n-1}$ into finitely overlapping sectors $\tau$ of aperture $\sim
K^{-1}$ and length $\sim 1$, where $K$ is a large constant. Given
$f:A^{n-1}\to\mathbb{C}$, write $f=\sum f_{\tau}$, where $f_{\tau}$ is
supported in $\tau$. In view of the rescaling $\phi^{\lambda}$ of the phase,
we define the rescaled Gauss map

$G^{\lambda}(x;\omega)=G(\frac{x}{\lambda};\omega)\text{ for
}(x;\omega)\in\text{supp}(a^{\lambda}).$

For each $x\in B(0,\lambda)$ let

$G^{\lambda}(x;\tau)=\\{G^{\lambda}(x;\omega)\;:\;\omega\in\tau\text{ and
}(x;\omega)\in\text{supp}(a^{\lambda})\\}.$

For $V\subseteq\mathbb{R}^{n}$ a linear subspace, let
$\angle(G^{\lambda}(x;\tau),V)$ denote the smallest angle between any non-zero
vector $v\in V$ and $G^{\lambda}(x;\tau)$.

The spatial ball $B(0,\lambda)$ is decomposed into relatively small balls
$B_{K^{2}}$ of radius $K^{2}$. We fix $\mathcal{B}_{K^{2}}$ a collection of
finitely-overlapping $K^{2}$-balls, which are centred in and cover
$B(0,\lambda)$. For $B_{K^{2}}\in\mathcal{B}_{K^{2}}$ centred at $\bar{x}\in
B(0,\lambda)$, define

(9) $\mu_{T^{\lambda}f}(B_{K^{2}})=\min_{V_{1},\ldots,V_{A}\in
Gr(k-1,n)}\big{(}\max_{\tau:\angle(G^{\lambda}(\bar{x};\tau),V_{a})>K^{-2}\;\forall
a}\|T^{\lambda}f_{\tau}\|^{p}_{L^{p}(B_{K^{2}})}\big{)},$

where $Gr(k-1,n)$ denotes the Grassmannian manifold of $(k-1)$-dimensional
subspaces in $\mathbb{R}^{n}$. We stress the deviation from [14], in which the
angle threshold $K^{-1}$ was considered. In case of the Fourier extension
operator for the cone, the angle condition was narrowed to $K^{-2}$ to further
confine the narrow part in [26]. Hence, we presently use the same threshold
for the variable-coefficient operator.

We write $\tau\not\in V_{a}$ as shorthand for
$\angle(G^{\lambda}(\bar{x};\tau),V_{a})>K^{-2}$ provided that $\bar{x}$ is
clear from context. Thus, we can write as well

$\mu_{T^{\lambda}f}(B_{K^{2}})=\min_{V_{1},\ldots,V_{A}\in
Gr(k-1,n)}\big{(}\max_{\begin{subarray}{c}\tau:\tau\not\in V_{a},\\\ \text{
for }1\leq a\leq
A\end{subarray}}\|T^{\lambda}f_{\tau}\|^{p}_{L^{p}(B_{K^{2}})}\big{)}.$

For $U\subseteq\mathbb{R}^{n}$ the $k$-broad norm is defined as

$\|T^{\lambda}f\|_{BL^{p}_{k,A}(U)}=\big{(}\sum_{\begin{subarray}{c}B_{K^{2}}\in\mathcal{B}_{K^{2}},\\\
B_{K^{2}}\cap
U\neq\emptyset\end{subarray}}\mu_{T^{\lambda}f}(B_{K^{2}})\big{)}^{1/p}.$

A key step in the proof of the $L^{p}$-$L^{p}$-estimate is to show $k$-broad
estimates, which are a substitute for the $k$-restriction conjecture.

######  Theorem 1.2.

For $2\leq k\leq n$ and all $\varepsilon>0$, there exists a constant
$C_{\varepsilon}>1$ and an integer $A$ such that, whenever $T^{\lambda}$ is an
oscillatory integral operator with reduced $(\phi,a)$ and $\phi$ a
$1$-homogeneous phase satisfying $C1)$ and $C2^{+})$, the estimate

(10)
$\|T^{\lambda}f\|_{BL^{p}_{k,A}(\mathbb{R}^{n})}\lesssim_{\varepsilon}K^{C_{\varepsilon}}\lambda^{\varepsilon}\|f\|_{L^{2}(A^{n-1})}$

holds for all $\lambda\geq 1$ and $K\geq 1$ whenever

(11) $p\geq\bar{p}(k,n)=\frac{2(n+k)}{n+k-2}.$

Reduced phase functions are introduced in Section 3.1. These phases are
basically small $C^{N}$ perturbations of $1$-homogeneous phases with constant
coefficients satisfying the convexity assumption. These reduced phase
functions were previously used by Beltran–Hickman–Sogge [2] to derive
decoupling estimates. As in [2], general phase functions satisfying $C1)$ and
$C2^{+})$ are transformed by partitioning the support of the amplitude and
parabolic rescaling to reduced phases. Additionally, we impose size conditions
on the amplitude and a margin condition on its support.

The scheme to deduce Theorem 1.1 from Theorem 1.2 is essentially due to
Bourgain–Guth [7] and in the variable-coefficient context, see also
Guth–Hickman–Iliopoulou [14]. We find like in [14, p. 263]

(12)
$\|T^{\lambda}f\|^{p}_{L^{p}(B_{K^{2}})}\lesssim_{A}K^{O(1)}\mu_{T^{\lambda}f}(B_{K^{2}})+\sum_{a=1}^{A}\|\sum_{\tau\in
V_{a}}T^{\lambda}f_{\tau}\|^{p}_{L^{p}(B_{K^{2}})}.$

The first term is captured by the broad estimate; the second term is estimated
by $\ell^{p}$-decoupling (cf. [6, 2]) and induction on scales [26, 10]. A
suitable narrow decoupling, taking into account a reduced number of sectors
$\tau\in V_{a}$, is discussed in detail in the variable coefficient case. For
the constant-coefficient case we refer to [26] and [10].

Very recently, Gao–Liu–Miao–Xi [10] proved an extension of Ou–Wang’s result
for the circular cone $\phi(x,\omega)=x^{\prime}\cdot\omega+x_{n}|\omega|$ for
more general conic surfaces, but still with constant coefficients. For these
constant coefficient phase functions, the Kakeya compression described in the
present work cannot happen. Gao _et al._ [10] used $k$-broad estimates to
derive new local smoothing estimates for the wave equation in Euclidean space.
At small spatial scales, the variable coefficient phases are approximated with
extension operators for conic surfaces. Then we can use arguments from [10].
Furthermore, Hickman and Iliopoulou showed sharp $L^{p}$-estimates for non-
homogeneous phases with indefinite signature in [18]. This suggests to study
also homogeneous phase functions with indefinite signature by the present
methods.

Notably, we do not use the usual wave packet decomposition for the cone as
e.g. in [26] or [10] to prove the broad estimate. Instead, we stick to the
wave packet decomposition commonly used for the Fourier extension operator of
the paraboloid or its variable coefficient counterpart [14]. This allows us to
use many arguments from [14] without change and hints at the possibility of a
unified approach. A major change happens for the transverse equidistribution
estimates, to be analyzed in Section 5. Secondly, the narrow decoupling
requires additional considerations, see Section 7.

We remark that the idea to use the same wave packet decomposition for
homogeneous and inhomogeneous phase functions in the variable coefficient
context is not new: In [21] S. Lee considered linear and bilinear estimates
for oscillatory integral operators and could treat variable coefficient
versions of the Fourier extension operator of the paraboloid and the cone with
the same wave packet decomposition. He generalized bilinear estimates due to
Tao [34] and Wolff [40] to variable coefficient phases. Lee [21] pointed out
for the first time that a convexity condition as $H2^{+})$ or $C2^{+})$ allows
to go beyond Tomas–Stein $L^{2}-L^{p}$-estimates, which are sharp for phases
without convexity condition. Bourgain [5] showed in the context of non-
homogeneous phases without convexity conditions that the Tomas-Stein range is
sharp (see also [7]). In the present work, the $L^{p}$-$L^{p}$-estimates for
general oscillatory integral operators with phase satisfying $C1)$ and
$C2^{+})$ due to Lee [21] are improved to the sharp range up to the endpoint
for $n\geq 5$, and the previously known sharp $L^{p}$-$L^{p}$-estimates in
lower dimensions $3\leq n\leq 4$ are recovered.

Furthermore, Ou–Wang [26] proved $L^{q}$-$L^{p}$-estimates for the cone
extension operator $\mathcal{E}$; see [26, Theorem 1.2]. The conjectured
boundedness range of $\mathcal{E}:L^{q}\to L^{p}$ is given by
$p>\frac{2(n-1)}{n-2}$ and $q^{\prime}\leq\frac{n-2}{n}p$. For $n=5$ Ou–Wang
proved this more general Fourier restriction conjecture for the cone and made
progress for $n\geq 6$. The proof of [26, Theorem 1.2] is a minor variant of
the proof of $L^{p}$-$L^{p}$-estimates. It is based on the $k$-broad estimate
and narrow decoupling. At one point, also interpolation with the bilinear
estimates due to Wolff [39] is invoked. In the present work the $k$-broad
estimate and narrow decoupling are extended to the variable coefficient
setting, and Wolff’s bilinear estimate was already generalized by Lee [21].
For this reason, the arguments to show $L^{q}$-$L^{p}$-estimates in [26] can
be extended to the variable coefficient case and yield the same range as
stated in [26, Theorem 1.2] for the estimate

$\|T^{\lambda}f\|_{L^{p}}\lesssim_{\varepsilon,\phi,a}\lambda^{\varepsilon}\|f\|_{L^{q}}.$

In Section 9 we apply the new estimates for oscillatory integral operators to
prove new local smoothing estimates for solutions to wave equations on compact
Riemannian manifolds $(M,g)$ with $\dim(M)\geq 3$. To avoid confusion with the
number of space-time dimensions, which was previously denoted by $n$, we
denote the space dimension in the following by $d$. We consider

(13)
$\left\\{\begin{array}[]{clcll}\partial_{t}^{2}u-\Delta_{g}u&=0,&\quad(x,t)&\in&M\times\mathbb{R},\\\
u(\cdot,0)&=f_{0},&\quad\dot{u}(\cdot,0)&=&f_{1}.\end{array}\right.$

$\Delta_{g}$ denotes the Laplace–Beltrami operator, and the solution $u$ to
(13) is given by

$u(t)=\cos(t\sqrt{-\Delta_{g}})f_{0}+\frac{\sin(t\sqrt{-\Delta_{g}})}{\sqrt{-\Delta_{g}}}f_{1}.$

By results due to Seeger–Sogge–Stein [28] relying on the parametrix
representation (see also [27, 24] in the Euclidean case), it is known that the
fixed-time estimate

$\|u(\cdot,t)\|_{L^{p}(\mathbb{R}^{d})}\lesssim\|f_{0}\|_{L^{p}_{\bar{s}_{p}}(\mathbb{R}^{d})}+\|f_{1}\|_{L^{p}_{\bar{s}_{p}-1}(\mathbb{R}^{d})}$

with

(14) $\bar{s}_{p}=(d-1)\big{|}\frac{1}{2}-\frac{1}{p}\big{|}$

is sharp for all $1<p<\infty$ provided that $t$ avoids a discrete set. The
local smoothing conjecture due to Sogge [29] for the Euclidean wave equation,
i.e., $(M,g)=(\mathbb{R}^{d},(\delta^{ij}))$ in (13), states that

(15)
$\big{(}\int_{1}^{2}\|u(\cdot,t)\|_{L^{p}(\mathbb{R}^{d})}^{p}\big{)}^{\frac{1}{p}}\lesssim\|f_{0}\|_{L^{p}_{\bar{s}_{p}-\sigma}(\mathbb{R}^{d})}+\|f_{1}\|_{L^{p}_{\bar{s}_{p}-1-\sigma}(\mathbb{R}^{d})}$

for $\sigma<\frac{1}{p}$ and $\frac{2d}{d-1}\leq p<\infty$. (Note that
$\bar{s}_{p}-\frac{1}{p}=0$ for $p=\frac{2d}{d-1}$.) This conjecture stands on
top of prominent open problems in Harmonic Analysis as it implies as well the
restriction conjecture for the paraboloid as the Bochner–Riesz conjecture.
Initial progress was due to Sogge [29] and Mockenhaupt–Seeger–Sogge [25].
Wolff identified decoupling inequalities [39] to yield sharp local smoothing
estimates. Further progress in this direction was made in [11, 20, 22].
Bourgain–Demeter [6] covered the sharp range for decoupling inequalities,
which implies sharp local smoothing estimates for $p\geq\frac{2(d+1)}{d-1}$.
We refer to the survey by Beltran–Hickman–Sogge [3] for local smoothing
estimates for FIOs. Guth–Wang–Zhang [16] verified the Euclidean local
smoothing conjecture for $d=2$ by a sharp $L^{4}$-square function estimate.
Gao _et al._ [9] extended this to compact Riemannian surfaces. We remark that
for $d\geq 3$, counterexamples due to Minicozzi–Sogge [23] show that (15)
fails if one replaces $\mathbb{R}^{d}$ with general compact Riemannian
manifolds for $\sigma<1/p$, if $p<p_{d,+}$ with

(16) $p_{d,+}=\begin{cases}\frac{2\cdot(3d+1)}{3d-3},\text{ if }d\text{ is
odd},\\\ \frac{2\cdot(3d+2)}{3d-2},\text{ if }d\text{ is even}.\end{cases}$

Hence, local smoothing estimates for solutions to wave equations on compact
Riemannian manifolds are only conjectured for $p\geq p_{d,+}$ with
$\sigma<1/p$.

Local parametrices for $e^{it\sqrt{-\Delta_{g}}}$ are given by

$\mathcal{F}f(x^{\prime},x_{n})=\int_{\mathbb{R}^{n-1}}e^{i\phi(x^{\prime},x_{n};\omega)}a(x;\omega)\hat{f}(\omega)d\omega$

with $\phi\in
C^{\infty}(\mathbb{R}^{n}\times\mathbb{R}^{n-1}\backslash\\{0\\})$ a phase
function satisfying $C1)$ and $C2^{+})$ and $a\in S^{0}(\mathbb{R}^{n})$ with
compact support in $x$. Hence, it suffices to prove local smoothing estimates
of rescaled Fourier integral operators $\mathcal{F}^{\lambda}$. In Theorem 9.1
we extend the recent results due to Gao _et al._ [10] for the Euclidean wave
equation to wave equations on compact Riemannian manifolds. This improves on
the previously best local smoothing estimates due to Beltran–Hickman–Sogge [2]
in some range $p\leq\frac{2(d+1)}{d-1}$ for wave equations on compact
manifolds.

_Outline of the paper:_ In Section 2 we show the necessary conditions for
$L^{p}$-estimates for variable-coefficient $1$-homogeneous phases.
Preliminaries for the polynomial partitioning argument to show Theorem 1.2 are
given in Section 3. In this section we introduce the notion of a reduced
homogeneous phase function and collect geometric consequences. This will
simplify the proof of Theorem 1.2. We recall the wave packet analysis in the
context of variable coefficients [21, 14] and collect facts on the $k$-broad
norms. In Section 4 we recall the polynomial partitioning tools. In Section 5
transverse equidistribution estimates are proved. These differ from the
transverse equidistribution estimates shown in [14] for Carleson–Sjölin phase
functions. In Section 6 we deduce Theorem 1.2 from Theorem 6.1, which is
suitable for induction on dimension and radius. The proof relies on polynomial
partitioning and transverse equidistribution estimates play a key role. In
Section 7 we show how Theorem 1.2 implies Theorem 1.1. In Section 8 we show
how the $\lambda^{\varepsilon}$-factor can be removed away from the endpoint.
In Section 9 we apply the oscillatory integral estimates and narrow decoupling
to show new local smoothing estimates for solutions to wave equations on
compact Riemannian manifolds.

Basic Notations:

  * •

For $x,y\in\mathbb{R}^{n}$ we denote the Euclidean inner product by

$\langle x,y\rangle=x\cdot y=\sum_{i=1}^{n}x_{i}y_{i}.$

  * •

$|x|=\sqrt{\langle x,x\rangle}$ denotes the Euclidean norm.

  * •

For a Lebesgue-measurable set $A\subseteq\mathbb{R}^{d}$, we denote by $|A|$
the $d$-dimensional Lebesgue measure.

  * •

For $x\in\mathbb{R}$, we denote Gauss brackets by

$\lfloor x\rfloor=\max\\{k\in\mathbb{Z}:k\leq x\\}.$

  * •

For $x,y\in\mathbb{R}^{n}\backslash\\{0\\}$ we denote the angle between $x$
and $y$ by

$\angle(x,y)=\arccos\frac{x\cdot y}{|x|\,|y|}.$

  * •

For $x\in\mathbb{R}^{n}\backslash\\{0\\}$ and $A\subseteq\mathbb{R}^{n}$ with
$A\backslash\\{0\\}\neq\emptyset$, we let

$\angle(x,A)=\inf_{y\in A\backslash\\{0\\}}\angle(x,y).$

Similarly, we define $\angle(A_{1},A_{2})$ for
$A_{1},A_{2}\subseteq\mathbb{R}^{n}$ with
$A_{i}\backslash\\{0\\}\neq\emptyset$:

$\angle(A_{1},A_{2})=\inf_{\begin{subarray}{c}x_{1}\in
A_{1}\backslash\\{0\\},\\\ x_{2}\in
A_{2}\backslash\\{0\\}\end{subarray}}\angle(x_{1},x_{2}).$

##  2\. Kakeya compression

In the following we modify the example due to Guth–Hickman–Iliopoulou [14,
Section 2] (see also [7]) for homogeneous phase functions. This yields the
necessary conditions:

######  Proposition 2.1.

Necessary for the estimate (5) to hold for $n\geq 4$ is

$p\geq p_{n}.$

Recall that $p_{n}$ was defined in (7) for $n\geq 4$ by

$p_{n}=\begin{cases}&2\cdot\frac{3n+1}{3n-3},\quad n\text{ odd},\\\
&2\cdot\frac{3n}{3n-4},\quad n\text{ even}.\end{cases}$

This shows that the $L^{p}$-$L^{p}$-estimates proved in Theorem 1.1 are sharp
up to endpoints.

We only consider $n\geq 4$ because the construction in (17) degenerates for
$n=3$ to the translation-invariant case. Moreover, the range, for which the
variable coefficient estimate holds, does not differ for $3\leq n\leq 5$ from
the translation-invariant case.

Let

$x=(\underbrace{x^{\prime\prime},x_{n-1}}_{x^{\prime}},x_{n})\in\mathbb{R}^{n-2}\times\mathbb{R}\times\mathbb{R}\text{
and
}\omega=(\omega^{\prime},\omega_{n-1})\in\mathbb{R}^{n-2}\times\mathbb{R}.$

We consider the phase functions

(17) $\phi(x;\omega)=x^{\prime}\cdot\omega+\frac{\langle
A(x_{n})\omega^{\prime},\omega^{\prime}\rangle}{2\omega_{n-1}},\quad\omega_{n-1}\in(1/2,1).$

$A(x_{n})$ denotes the $(n-2)\times(n-2)$–positive definite matrix

(18)
$A(x_{n})=\begin{cases}\bigoplus_{i=1}^{\frac{n-2}{2}}\begin{pmatrix}x_{n}&x_{n}^{2}\\\
x_{n}^{2}&x_{n}+x_{n}^{3}\end{pmatrix},&\quad n-2\text{ even},\\\
\bigoplus_{i=1}^{\frac{n-3}{2}}\begin{pmatrix}x_{n}&x_{n}^{2}\\\
x_{n}^{2}&x_{n}+x_{n}^{3}\end{pmatrix}\oplus(x_{n}),&\quad n-2\text{
odd}.\end{cases}$

The main idea is to construct many wave packets which are concentrated in the
neighbourhood of a lower dimensional algebraic variety. Whereas the direction
governed by the frequency $\omega_{\theta}$ below varies, for fixed
$\omega_{\theta}$ we consider precisely one starting position $v_{\theta}$.
This concentration in a low dimensional algebraic variety does not happen in
the linear case (1).

We consider wave packets adapted to $\phi$ as follows:
$\Xi=B^{n-2}(0,c_{1})\times(1/2,1)$ is covered by essentially disjoint
elongated caps

$\Xi_{\theta}=\\{(\omega^{\prime},\omega_{n-1})\in\Xi:|\omega^{\prime}/\omega_{n-1}-\omega_{\theta}|\lesssim\lambda^{-\frac{1}{2}}\\}$

with $\omega_{\theta}\in B^{n-2}(0,c_{1})$ for $|c_{1}|\ll 1$. These regions
will be referred to as ‘sectors’ in the following. Apparently, $\Xi$ can be
covered by $\sim\lambda^{\frac{n-2}{2}}$ finitely overlapping sets
$\Xi_{\theta}$. We consider a corresponding smooth partition of unity
$(\psi_{\theta})_{\omega_{\theta}\in\Xi}$ and wave packets

$f_{\theta,v}(\omega)=e^{-i\lambda\langle
v,\omega^{\prime}\rangle}\psi_{\theta}(\omega),\quad
v=(v_{1},\ldots,v_{n-2})\in\mathbb{R}^{n-2}.$

We have by non-oscillation of the phase

$|T^{\lambda}f_{\theta,v}(x^{\prime\prime},x_{n-1},x_{n})|\gtrsim\lambda^{-\frac{n-2}{2}}\chi_{T_{\theta,v}}(x).$

$\chi_{T_{\theta,v}}$ denotes the characteristic function of $T_{\theta,v}$.
The $T_{\theta,v}$ are curved ‘slabs’ of size
$(1\times\underbrace{\lambda^{1/2}\times\ldots\times\lambda^{1/2}}_{n-2\text{
times}}\times\lambda)$ with

$T_{\theta,v}\subseteq\\{x\in
B(0,\lambda)\,:\,|x^{\prime\prime}-\lambda\gamma_{\theta,v}\left(\frac{x_{n}}{\lambda}\right)|<c_{2}\lambda^{\frac{1}{2}+\varepsilon}\text{
and
}|x_{n-1}-\lambda\gamma^{\prime}_{\theta}(x_{n}/\lambda)|<c_{2}\lambda^{\frac{1}{2}+\varepsilon}\\},$

for any $\varepsilon>0$, which follows from non-stationary phase; $c_{2}$
denotes a small constant and $\gamma_{\theta,v}$, $\gamma_{\theta}^{\prime}$
denote curves:

$\gamma_{\theta,v}(x_{n})=v-A(x_{n})\omega_{\theta},\quad\gamma^{\prime}_{\theta}(x_{n})=\frac{1}{2}\langle
A(x_{n})\omega_{\theta},\omega_{\theta}\rangle.$

Furthermore, note that the conditions on $\omega^{\prime}$ and $\omega_{n-1}$

$\big{|}\frac{\omega^{\prime}}{\omega_{n-1}}-\omega_{\theta}\big{|}\lesssim\lambda^{-\frac{1}{2}},\;\omega_{n-1}\in(1/2,1),\;\omega^{\prime}\in
B_{n-2}(0,c_{1})$

correspond to considering $\lambda^{-\frac{1}{2}}$-sectors in direction
$(\omega_{\theta},1)$. The degeneracy of $\partial_{\omega\omega}^{2}\phi$ in
the radial direction, which is immediate from the $1$-homogeneity of $\phi$,
gives the localization of slabs to size $\lambda^{\varepsilon}$ in this
direction: We have

$\partial_{\omega}\phi(x;\omega)\cdot\frac{(\omega_{\theta},1)}{|(\omega_{\theta},1)|}=\partial_{\omega}\phi(x;(\omega_{\theta},1))\cdot\frac{(\omega_{\theta},1)}{|(\omega_{\theta},1)|}+O(\lambda^{-1})\text{
for
}|\frac{\omega}{|\omega|}-\frac{(\omega_{\theta},1)}{|(\omega_{\theta},1)|}|\lesssim\lambda^{-\frac{1}{2}}.$

The non-degeneracy of $\partial_{\omega\omega}^{2}\phi$ gives localization to
size less than $\lambda^{\frac{1}{2}+\varepsilon}$ in the remaining
directions. We argue in the following why the curved tubes
$\chi_{T_{\theta,v}}$ are in fact of size
$1\times\lambda^{\frac{1}{2}}\times\ldots\times\lambda^{\frac{1}{2}}\times\lambda$
(and not significantly less): Consider the oscillatory integral

$F(x)=\int
e^{i(x^{\prime}\cdot\omega+\lambda\tilde{\phi}(x_{n}/\lambda,\omega))}\psi_{\theta}(\omega)d\omega$

with $\psi_{\theta}\in C^{\infty}_{c}(A^{n-1})$ localizing to a slab into
direction $\theta\in\mathbb{S}^{n-2}$ and

$\tilde{\phi}(x_{n},\mu\omega)=\mu\tilde{\phi}(x_{n},\omega)\text{ for
}\mu>0.$

We use Taylor expansion in $\omega$ to write

$\begin{split}\lambda\tilde{\phi}(x_{n}/\lambda,\omega)&=|\omega|(\lambda\tilde{\phi}(x_{n}/\lambda,\omega/|\omega|))\\\
&=|\omega|(\lambda\tilde{\phi}(x_{n}/\lambda,\theta)+\lambda\nabla_{\omega}\tilde{\phi}(x_{n}/\lambda,\theta)(\frac{\omega}{|\omega|}-\theta))+O(c).\end{split}$

For $\omega\in\text{supp}(\psi_{\theta})$ we have

$|\omega|=\omega\cdot\theta+O(c\lambda^{-1}).$

Hence, we can write

$\lambda\tilde{\phi}(x_{n}/\lambda,\omega)=\lambda\tilde{\phi}(x_{n}/\lambda,\theta)(\omega\cdot\theta)+\lambda\nabla_{\omega}\tilde{\phi}(x_{n}/\lambda,\theta)(\omega-(\omega\cdot\theta)\theta)+O(c).$

Let $\\{\theta^{1}_{\perp},\ldots,\theta_{\perp}^{n-2},\theta\\}$ be an
orthonormal basis of $\mathbb{R}^{n-1}$. Then,

$\lambda\tilde{\phi}(x_{n}/\lambda,\omega)=\lambda\tilde{\phi}(x_{n}/\lambda,\theta)(\omega\cdot\theta)+\lambda\sum_{i=1}^{n-2}(\nabla_{\omega}\tilde{\phi}(x_{n}/\lambda,\theta)\cdot\theta_{\perp}^{i})(\omega\cdot\theta_{\perp}^{i})+O(c).$

Consequently,

$\begin{split}x^{\prime}\cdot\omega+\lambda\tilde{\phi}(x_{n}/\lambda,\omega)&=(x^{\prime}\cdot\theta+\lambda\tilde{\phi}(x_{n}/\lambda,\theta))(\omega\cdot\theta)\\\
&\quad+\sum_{i=1}^{n-2}(x^{\prime}\cdot\theta_{\perp}^{i}+\lambda\nabla_{\omega}\tilde{\phi}(x_{n}/\lambda,\theta)\cdot\theta_{\perp}^{i})(\omega\cdot\theta_{\perp}^{i})+O(c).\end{split}$

And for $|x^{\prime}\cdot\theta+\lambda\tilde{\phi}(x_{n}/\lambda,\theta)|\ll
c$ and
$|x^{\prime}\cdot\theta_{\perp}^{i}+\lambda\nabla_{\omega}\tilde{\phi}(x_{n}/\lambda,\theta)\cdot\theta_{\perp}^{i}|\ll
c\lambda^{1/2}$, we see that the modulus of the whole phase is $O(c)$. Hence,
there is no oscillation within $\text{supp}(\psi_{\theta})$ and for fixed
$x_{n}$ this defines a region $A_{x_{n}}$ for $x^{\prime}$ of size
$1\times\lambda^{1/2}\times\ldots\times\lambda^{1/2}$, for which
$|T^{\lambda}f(x^{\prime},x_{n})|$ is roughly constant. Taking
$T_{\theta}=\bigcup_{x_{n}}A_{x_{n}}$ yields the
$1\times\lambda^{1/2}\times\ldots\times\lambda^{1/2}\times\lambda$-tube. Note
that the factor $e^{-i\lambda\langle v,\omega^{\prime}\rangle}$ amounts to a
shift in $x^{\prime}$ by $\lambda v$, but does not change the size of the
tube.

We prepare the initial data with randomized signs:

$f=\sum_{\theta}\varepsilon_{\theta}f_{\theta,v}.$

By Khintchine’s theorem, the expected value of $|T^{\lambda}f(x)|$ is given by
the square sum:

$\mathbf{E}[|T^{\lambda}f(x)|]\sim\big{(}\sum_{\theta}|T^{\lambda}f_{\theta,v_{\theta}}|^{2}\big{)}^{1/2}\gtrsim\lambda^{-\frac{n-2}{2}}\big{(}\sum_{\theta}\chi_{T_{\theta,v_{\theta}}}(x)\big{)}^{1/2}.$

Taking $L^{p}$-norms yields by Minkowski’s inequality

$\lambda^{-\frac{n-2}{2}}\big{(}\int\big{(}\sum_{\theta}\chi_{T_{\theta,v_{\theta}}}\big{)}^{\frac{p}{2}}\big{)}^{\frac{1}{p}}\lesssim\mathbf{E}[\|T^{\lambda}f\|_{L^{p}}].$

Next, we find by applying Hölder’s inequality

$\displaystyle\lambda^{-\frac{n-2}{2}}\big{(}\int\sum_{\theta}\chi_{T_{\theta,v_{\theta}}}\big{)}^{1/2}$
$\displaystyle\lesssim\left|\bigcup_{\theta}T_{\theta,v_{\theta}}\right|^{1/2-1/p}\mathbf{E}[\|T^{\lambda}f\|_{L^{p}}]$
$\displaystyle\lesssim\left|\bigcup_{\theta}T_{\theta,v_{\theta}}\right|^{1/2-1/p}\|f\|_{p}\lesssim\left|\bigcup_{\theta}T_{\theta,v_{\theta}}\right|^{1/2-1/p}.$

The penultimate estimate is by hypothesis, and the final estimate follows from
$|f|=1$ and $|\text{supp }f|\sim 1$. Since the tubes $T_{\theta,v_{\theta}}$
are
$(1\times\lambda^{1/2}\times\ldots\times\lambda^{1/2}\times\lambda)$-slabs,
$\int\chi_{T_{\theta,v_{\theta}}}\sim\lambda^{\frac{n}{2}}$. Moreover, there
are about $\lambda^{\frac{n-2}{2}}$ slabs. Hence,

$\lambda^{-\frac{n-2}{2}}\big{(}\int\sum\chi_{T_{\theta,v_{\theta}}}\big{)}^{1/2}\sim\lambda^{\frac{1}{2}}.$

Thus, we arrive at

(19)
$1\lesssim\left|\bigcup_{\theta}T_{\theta,v_{\theta}}\right|^{1/2-1/p}\lambda^{-\frac{1}{2}}.$

Next, we shall see how to choose $v_{\theta}$ such that the curved slabs are
concentrated in a neighbourhood of a low-dimensional algebraic variety
inspired by [14].
For $\Xi_{\theta}$, we set

(20) $v_{\theta,2j-1}=-(\omega_{\theta})_{2j}\text{ and
}v_{\theta,2j}=v_{\theta,n-2}=0\text{ for }1\leq
j\leq\lfloor\frac{n-2}{2}\rfloor.$

Let $d=n-1-\lfloor\frac{n-2}{2}\rfloor$ and $Z=Z(P_{1},\ldots,P_{n-1-d})$ be
the common zero set of the polynomials

(21) $P_{j}(x_{1},\ldots,x_{n-2},x_{n})=\lambda x_{2j}-x_{2j-1}x_{n}\text{ for
}1\leq j\leq\lfloor\frac{n-2}{2}\rfloor.$

It is straight-forward to show that the image of
$x_{n}\mapsto(\lambda\gamma_{\theta,v_{\theta}}(x_{n}/\lambda),x_{n})$ is
contained in $Z(P_{1},\ldots,P_{n-1-d})$. $Z$ is an algebraic variety of
dimension

(22) $d=(n-1)-\lfloor\frac{n-2}{2}\rfloor$

in $\mathbb{R}^{n-1}$ and of degree $O_{n}(1)$. Thus, Wongkew’s theorem (cf.
[41]) on the size of neighbourhoods of algebraic varieties applies, and we
find

(23) $\tilde{N}_{\lambda^{\frac{1}{2}}}=|N_{\lambda^{\frac{1}{2}}}(Z)\cap
B_{n-1}(0,\lambda)|\lesssim\lambda^{d+\frac{n-1-d}{2}}.$

We find by (22) and (23)

(24)
$\tilde{N}_{\lambda^{\frac{1}{2}}}\lesssim\begin{cases}\lambda^{\frac{3n-2}{4}},\quad
n\text{ even},\\\ \lambda^{\frac{3n-1}{4}},\quad n\text{ odd.}\end{cases}$

Moreover, for $(x_{1},\ldots,x_{n})\in T_{\theta,v_{\theta}}$ we have
$x_{n-1}\in
B(\lambda\gamma^{\prime}_{\theta}(x_{n}/\lambda),\lambda^{\frac{1}{2}+\varepsilon})$.

This yields

(25)
$\left|\bigcup_{\theta}T_{\theta,v_{\theta}}\right|^{1/2-1/p}\lesssim(\tilde{N}_{\lambda^{\frac{1}{2}}}\cdot\lambda^{\frac{1}{2}})^{\frac{1}{2}-\frac{1}{p}}.$

Plugging (25) into (19) with the estimate from (24), we find

$p\geq\begin{cases}&2\cdot\frac{3n}{3n-4},\quad n\text{ even},\\\
&2\cdot\frac{3n+1}{3n-3},\quad n\text{ odd}.\end{cases}$

This finishes the proof of Proposition 2.1. $\hfill\Box$

##  3\. Preliminaries

###  3.1. Basic reductions of the phase and amplitude function

In this paragraph we shall normalize $1$-homogeneous phase functions, which
satisfy $C1)$ and $C2)$, and the amplitude. This will highlight that the class
of considered phase functions are indeed $C^{N}$-perturbations of the
translation-invariant case

$\phi_{*}(x;\omega)=\langle
x^{\prime},\omega\rangle+\frac{x_{n}(\omega^{\prime})^{2}}{2\omega_{n-1}},\quad\omega^{\prime}\in
B(0,c),\;\omega_{n-1}\in(1,2).$

Constant-coefficient perturbations were analyzed in [10].

The arguments were provided in [2, Section 2], and details are omitted here
(see also [21]). It is important to work with a class of phase functions,
which is stable under rescaling. After localisation and translation, we may
assume that $a$ is supported inside $X\times\Xi$, where $X=X^{\prime}\times T$
for $X^{\prime}\subseteq B(0,1)\subseteq\mathbb{R}^{n-1}$ and
$T\subseteq(-1,1)\subseteq\mathbb{R}$ are small open neighbourhoods of the
origin and $\Xi\subseteq A^{n-1}$ is a small sector of dimensions $1\times
c\times\ldots\times c$ with $0<c\ll 1$ centred at
$e_{n-1}=(0,\ldots,0,1)\in\mathbb{R}^{n-1}$.

Firstly, we can suppose that

$\displaystyle C1^{\prime})\quad\det\partial^{2}_{\omega
x^{\prime}}\phi(x;\omega)\neq 0\text{ for all }(x,\omega)\in X\times\Xi,$
$\displaystyle
C2^{\prime})\quad\partial^{2}_{\omega^{\prime}\omega^{\prime}}\partial_{x_{n}}\phi(x,\omega)\text{
has eigenvalues of the same sign for all }(x,\omega)\in X\times\Xi.$

This follows as in [2]. By rotation in the $x$-variables, we can also suppose
that

$G(0;e_{n-1})=e_{n}\text{ and }\partial^{2}_{x_{n}\omega}\phi(0;e_{n-1})=0.$

Hence, by making $\Xi$ small enough, we find that

(26) $|\partial^{2}_{x_{n}\omega}\phi(x;\omega)|\leq c_{cone}\text{ for
}(x,\omega)\in X\times\Xi.$

By non-degeneracy $C1^{\prime})$ and the implicit function theorem, we find a
smooth mapping $\Phi_{x_{n},\omega}:X^{\prime}\to\mathbb{R}^{n-1}$ such that

$\partial_{\omega}\phi(\Phi_{x_{n},\omega}(x^{\prime}),x_{n};\omega)=x^{\prime}.$

We shall also write
$\Phi_{x_{n},\omega}(x^{\prime})=\Phi(x^{\prime},x_{n};\omega)$. There is also
a smooth mapping $\Psi(x,\cdot)$ with

$\partial_{x^{\prime}}\phi(x;\Psi(x;\omega))=\omega.$

For $\lambda\geq 1$, we consider the rescaled versions
$\Phi^{\lambda}(x;\omega)=\lambda\Phi(x/\lambda;\omega)$ and
$\Psi^{\lambda}(x;\omega)=\Psi(x/\lambda;\omega)$. We assume that $X$ and
$\Xi$ are such that the above mappings are defined on the whole support of
$a$.

In the following we shall quantify the deviation from $\phi_{*}$ by
restricting the values of second and third derivatives and bounding higher
order derivatives: Let $c_{cone}>0$ denote a small constant and
$I_{m}\in\mathbb{R}^{m\times m}$, $(I_{m})_{ij}=\delta_{ij}$ denote the unit
matrix. Firstly note that there are (possibly large) constants
$A_{1},A_{2},A_{3}\geq 1$ such that

$\displaystyle C1^{\prime\prime})\quad|\partial^{2}_{\omega
x^{\prime}}\phi(x;\omega)-I_{n-1}|\leq c_{cone}A_{1}\text{ for }(x;\omega)\in
X\times\Xi,$ $\displaystyle
C2^{\prime\prime})\quad|\partial^{2}_{\omega^{\prime}\omega^{\prime}}\partial_{x_{n}}\phi(x;\omega)-\frac{I_{n-2}}{\omega_{n-1}}|\leq
c_{cone}A_{2}\text{ for }(x;\omega)\in X\times\Xi.$

In the above display and in the following we abuse notation and denote the
Euclidean norm of a vector $v\in\mathbb{R}^{m}$ or the Hilbert-Schmidt norm of
a matrix $A\in\mathbb{R}^{m\times m}$ by $|v|$ or $|A|$.

For technical reasons, we shall impose in the following for the higher order
derivatives:

$\displaystyle
D1)\;\|\partial^{\beta}_{\omega}\partial_{x_{k}}\phi\|_{L^{\infty}(X\times\Xi)}\leq
c_{cone}A_{1}\text{ for }1\leq k\leq n-1\text{ and
}\beta=(\beta^{\prime},\beta_{n-1})\in\mathbb{N}_{0}^{n-1},$
$\displaystyle\qquad\text{ which satisfies }2\leq|\beta|\leq 3\text{ and
}|\beta^{\prime}|\geq 2;$
$\displaystyle\qquad\|\partial^{\beta}_{\omega}\partial_{x_{n}}\phi\|_{L^{\infty}(X\times\Xi)}\leq\frac{c_{cone}}{2n}A_{1}\text{
for all }\beta\in\mathbb{N}_{0}^{n-1}\text{ with }|\beta^{\prime}|\geq 3\text{
and }|\beta|\leq N.$ $\displaystyle D2)\;\text{For some large integer
}N\in\mathbb{N},\text{ one has}$
$\displaystyle\qquad\|\partial_{\omega}^{\beta}\partial_{x}^{\alpha}\phi\|_{L^{\infty}(X\times\Xi)}\leq\frac{c_{cone}}{2n}A_{3}\text{
for all }(\alpha,\beta)\in\mathbb{N}_{0}^{n}\times\mathbb{N}_{0}^{n-1}\text{
with }2\leq|\alpha|\leq 4N$ $\displaystyle\qquad\text{and }1\leq|\beta|\leq
4N+2\text{ satisfying }1\leq|\beta|\leq 4N\text{ or }|\beta^{\prime}|\geq 2.$

By parabolic rescaling (cf. Lemma 7.9), we see that we can reduce to phases
with $A_{i}=1$; these phases are said to be reduced.

Moreover, we can suppose that uniform bounds hold for the amplitude $a\in
C^{\infty}_{c}(X\times\Xi)$:

(27) $\|\partial_{\omega}^{\alpha}a\|_{L^{\infty}(Z\times\Xi)}\leq
C_{\text{amp}}\text{ for }0\leq|\alpha|\leq N.$

$C_{\text{amp}}$ denotes a universal constant. (27) is accomplished by
dividing $a$ through a large constant depending on $a$ and its derivatives.
Furthermore, we can suppose by Fourier series expansion (see [2]) that $a$ is
of product form

$a(x,\omega)=a_{1}(x)a_{2}(\omega).$

For the spatial part, we require a _margin condition_ :

(28) $\text{dist}(\text{supp }a_{1},\mathbb{R}^{n+1}\backslash
X)\geq\frac{1}{4A_{3}}.$

This will become useful for variable-coefficient decoupling in Section 7 and
can always be achieved by finite decomposition and re-centering (cf. [2]). An
amplitude $a$ is said to be reduced, if it is of product form, satisfies (28)
with $A_{3}=1$ and uniform bounds on the derivatives (27).

######  Definition 3.1.

We say that $(\phi,a)$ are of type $(A_{1},A_{2},A_{3})$, if

  * •

$\phi$ satisfies $C1^{\prime\prime})$, $C2^{\prime\prime})$, $D1)$, $D2)$, and
$a$ satisfies (28),

  * •

$a$ is of product form and satisfies (27).

Moreover, $(\phi,a)$ is said to be reduced if $\phi$ is a reduced phase
function and $a$ is a reduced amplitude.

The following observation will be useful at a later point:

######  Lemma 3.2.

Suppose that $\phi$ is a reduced phase function. Then we have

(29)
$\|\partial_{x_{n}}\partial^{\beta}_{\omega}\phi\|_{L^{\infty}(X\times\Xi)}\lesssim_{N,c_{\text{cone}}}1\text{
for }1\leq|\beta|\leq N.$

###### Proof.

For $\beta=(\beta^{\prime},\beta_{n-1})\in\mathbb{N}_{0}^{n-1}$ with
$|\beta^{\prime}|\geq 3$ this is covered by $D1)$ and for $|\beta|=1$ this is
$C2^{\prime\prime})$. So it remains to show (29) for $|\beta|=2$, and
$|\beta|\geq 3$ with $|\beta^{\prime}|\leq 2$: We obtain by homogeneity

$\partial_{x_{n}}\phi(x;\omega^{\prime},\omega_{n-1})=\omega_{n-1}\partial_{x_{n}}\phi(x;\omega^{\prime}/\omega_{n-1},1).$

From this follows

(30)
$\partial^{2}_{x_{n}\omega_{n-1}}\phi(x;\omega^{\prime},\omega_{n-1})=\partial_{x_{n}}\phi(x;\omega^{\prime}/\omega_{n-1},1)-\partial^{2}_{x_{n}\omega^{\prime}}\phi(x;\omega^{\prime}/\omega_{n-1},1)\cdot\frac{\omega^{\prime}}{\omega_{n-1}}.$

And moreover,

(31)
$\partial^{3}_{x_{n},\,\omega_{n-1},\,\omega_{n-1}}\phi=\frac{1}{\omega_{n-1}}\langle\partial^{3}_{x_{n},\,\omega^{\prime},\,\omega^{\prime}}\phi(x;\omega^{\prime}/\omega_{n-1},1)\frac{\omega^{\prime}}{\omega_{n-1}},\frac{\omega^{\prime}}{\omega_{n-1}}\rangle.$

Hence, by $C2^{\prime\prime})$ this is uniformly bounded.

For $|\beta|=2$ and $|\beta^{\prime}|=1$, the claim follows from taking an
additional derivative in (30) and estimates (26) and $C1^{\prime\prime})$.

For $3\leq|\beta|\leq N$ and $|\beta^{\prime}|\leq 1$, the claim is evident
from taking derivatives in (31), and the estimates $C2^{\prime\prime})$ and
$D1)$. ∎

###  3.2. Geometric consequences

Let $\phi$ be a reduced phase function in the above sense. We shall see how
the corresponding hypersurfaces $\Sigma_{x}$ parametrized by
$\omega\mapsto\partial_{x}\phi(x;\omega)$ resemble the ones from $\phi_{*}$.
To see this, recall that $\Psi:U\rightarrow\Xi$ satisfies
$\partial_{x^{\prime}}\phi(x;\Psi(x;\omega))=\omega$. Hence, $\Sigma_{x}$ is
the graph of the function
$h_{x}(\omega)=\partial_{x_{n}}\phi(x;\Psi(x;\omega))$ over the fibre $U_{x}$.
Each $h_{x}$ is a perturbation of the translation invariant case in the
following sense:

######  Lemma 3.3.

The following estimate holds for all $\omega\in U_{x}$:

(32)
$\|\partial^{2}_{\omega^{\prime}\omega^{\prime}}h_{x}(\omega)-I_{n-1}/\omega_{n-1}\|_{L^{\infty}}\lesssim
c_{cone}.$

Here $c_{cone}>0$ denotes the constant from the definition of a reduced phase
function.

###### Proof.

This is a consequence of properties of $\Psi$. Firstly, we record that
$\Psi(x;e_{n-1})=1$. By the implicit function theorem and non-degeneracy of
$\phi$, we find

$\partial_{\omega}\Psi(x;\omega)=\partial^{2}_{x^{\prime}\omega}\phi(x;\Psi(x;\omega))^{-1}.$

Hence,

$\|\partial_{\omega}\Psi(x;\omega)-I_{n-1}\|_{L^{\infty}}=O(c_{cone}).$

As a consequence of this identity (and choosing $c_{cone}$ to be sufficiently
small),

$|\Psi(x;\omega)-\Psi(x;\omega^{\prime})|\sim|\omega-\omega^{\prime}|\text{
for all }\omega,\omega^{\prime}\in U_{x}$

with implicit constant only depending on $n$.
Additionally, if $1\leq k\leq n-1$, then by twice differentiating the identity

$\partial_{x_{k}}\phi(x;\Psi(x;\omega))=\omega_{k}$

in the $\omega$-variables, it follows that

$\|\partial^{2}_{\omega\omega}\Psi_{k}(x;\omega)\|_{L^{\infty}}=O(c_{cone}).$

By the previous estimate, (32) follows from $C2^{\prime\prime})$. ∎

By similar means, we infer estimates for the generalized Gauss map associated
with $T^{\lambda}$. To give the results, let

$X^{\lambda}=\\{x\in\mathbb{R}^{n}\;|\;\frac{x}{\lambda}\in X\\}$

denote the $\lambda$-dilate of $X$, so that $a^{\lambda}$ is supported in
$X^{\lambda}\times\Xi$.

######  Lemma 3.4.

For all $x,\bar{x}\in X^{\lambda}$ and $\omega,\bar{\omega}\in\Xi$, the
estimates

(33)
$\begin{split}\angle(G^{\lambda}(x;\omega),G^{\lambda}(x;\bar{\omega}))&\sim|\frac{\omega}{|\omega|}-\frac{\bar{\omega}}{|\bar{\omega}|}|\sim\angle(\omega,\bar{\omega}),\\\
\angle(G^{\lambda}(x;\omega),G^{\lambda}(\bar{x};\omega))&\lesssim\lambda^{-1}|x-\bar{x}|\end{split}$

hold true.

This will be helpful to understand the wave packet analysis in the following
sections.

###  3.3. Wave packet decomposition

We carry out the wave packet decomposition with respect to some spatial
parameter $1\ll R\ll\lambda$. For this purpose, we follow [14] and use that
the construction only depends on the non-degeneracy condition $C1)$. We do not
use the usual wave packet decomposition for the cone as e.g. in [26], but
adapt the parabolic case, as previously done by Lee [21]. The reason is that
in Section 6 we would sort the smaller cone tubes into larger tubes anyway. It
appears that the present choice of wave packet decomposition allows us to
transfer arguments from [14] to the homogeneous setting more directly. In the
following we introduce notations from [14].

Cover $A^{n-1}$ by finitely overlapping balls of radius $R^{-1/2}$, and let
$\psi_{\theta}$ be a smooth partition of unity adapted to this cover. These
$\theta$ will frequently be referred to as $R^{-1/2}$-balls. For a ball
$\theta$, cover $\mathbb{R}^{n-1}$ with finitely overlapping balls of size
$R^{\frac{1+\delta}{2}}\times\ldots\times R^{\frac{1+\delta}{2}}$ with center
$v\in R^{\frac{1+\delta}{2}}\mathbb{Z}^{n-1}$. Let $\eta_{v}=\eta(\cdot-v)$
denote a bump function adapted to $B(v,R^{\frac{1+\delta}{2}})$ such that

$\sum_{v\in\mathbb{Z}^{n-1}}\eta_{v}=1$

with $\hat{\eta}_{v}$ essentially supported in
$B(0,CR^{-\frac{1+\delta}{2}})$. This is possible by the Poisson summation
formula.

Let $\mathbb{T}$ denote the collection of all pairs $(\theta,v)$. Then, for
$f:\mathbb{R}^{n-1}\to\mathbb{C}$ with support in $A^{n-1}$ and sufficiently
regular, we find

$f=\sum_{(\theta,v)\in\mathbb{T}}(\eta_{v}(\psi_{\theta}f)\check{\;})\hat{\;}=\sum_{(\theta,v)\in\mathbb{T}}\hat{\eta}_{v}*(\psi_{\theta}f).$

For each $R^{-1/2}$-ball $\theta$, let $\omega_{\theta}$ denote its centre.
Choose a real-valued smooth function $\tilde{\psi}$ so that
$\tilde{\psi}_{\theta}$ is supported in $\theta$, and
$\tilde{\psi}_{\theta}(\omega)=1$ whenever $\omega$ belongs to a
$cR^{-1/2}$-neighbourhood of the support of $\psi_{\theta}$ for some small
$c>0$. Finally, define

$f_{\theta,v}=\tilde{\psi}_{\theta}\cdot[\hat{\eta}_{v}*(\psi_{\theta}f)].$

The function $\hat{\eta}_{v}$ is rapidly decaying outside
$B(0,CR^{-\frac{1+\delta}{2}})$ and, consequently,

$\|f_{\theta,v}-(\hat{\eta}_{v}*(\psi_{\theta}f))\|_{L^{\infty}(\mathbb{R}^{n-1})}\leq\text{RapDec}(R)\|f\|_{L^{2}(A^{n-1})}.$

The functions $f_{\theta,v}$ are almost orthogonal: if
$\mathbb{S}\subseteq\mathbb{T}$, then

$\|\sum_{(\theta,v)\in\mathbb{S}}f_{\theta,v}\|^{2}_{L^{2}(\mathbb{R}^{n-1})}\sim\sum_{(\theta,v)\in\mathbb{S}}\|f_{\theta,v}\|^{2}_{L^{2}(\mathbb{R}^{n-1})}.$

Let $T^{\lambda}$ be an oscillatory integral operator with reduced phase
$\phi$ satisfying $C1^{\prime})$ and amplitude $a$ supported in $X\times\Xi$.
For $(\theta,v)\in\mathbb{T}$ define the curve
$\gamma^{1}_{\theta,v}:I^{1}_{\theta,v}\to\mathbb{R}^{n-1}$ by setting
$\gamma^{1}_{\theta,v}(t)=\Phi(v,t;\omega_{\theta})$, where $\Phi$ is the
function introduced in Subsection 3.1 and

$I^{1}_{\theta,v}=\\{x_{n}\in
T\,|\,\partial_{\omega}\phi(x^{\prime},x_{n};\omega_{\theta})=v\text{ for some
}x^{\prime}\in X^{\prime}\\}.$

Hence,
$\partial_{\omega}\phi(\gamma^{1}_{\theta,v}(x_{n}),x_{n};\omega_{\theta})=v$
for all $x_{n}\in I^{1}_{\theta,v}$. For the rescaled curve

$\gamma^{\lambda}_{\theta,v}(t)=\lambda\gamma^{1}_{\theta,v/\lambda}(t/\lambda),$

we find

$\partial_{\omega}\phi^{\lambda}(\gamma^{\lambda}_{\theta,v}(x_{n}),x_{n};\omega_{\theta})=v\text{
for all }t\in
I^{\lambda}_{\theta,v}=\\{t\in\mathbb{R}\,:\,\frac{t}{\lambda}\in
I^{1}_{\theta,v}\\}.$

Let $\Gamma^{\lambda}_{\theta,v}:I^{\lambda}_{\theta,v}\to\mathbb{R}^{n}$
denote the graph mapping
$\Gamma^{\lambda}_{\theta,v}(x_{n})=(\gamma^{\lambda}_{\theta,v}(x_{n}),x_{n})$;
for the sake of brevity, the image of this mapping is denoted by
$\Gamma^{\lambda}_{\theta,v}$, too.

######  Lemma 3.5 ([14, Lemma 5.2]).

The tangent space
$T_{\Gamma^{\lambda}_{\theta,v}(x_{n})}\Gamma^{\lambda}_{\theta,v}$ lies in
the direction of the unit vector
$G^{\lambda}(\Gamma^{\lambda}_{\theta,v}(x_{n});\omega_{\theta})$ for all
$x_{n}\in I^{\lambda}_{\theta,v}$.

We consider curved tubes

$T_{\theta,v}=\\{(x^{\prime},x_{n})\in B(0,R)\,:\,x_{n}\in
I^{\lambda}_{\theta,v}\text{ and }x^{\prime}\in
B(\gamma^{\lambda}_{\theta,v}(x_{n}),R^{\frac{1}{2}+\delta})\\}.$

We refer to the curve $\Gamma^{\lambda}_{\theta,v}$ as the core of
$T_{\theta,v}$. Since $\phi$ is of reduced form, we find by the diffeomorphism
property of $\Phi$ (writing
$x^{\prime}=\Phi^{-1}_{x_{n},\omega_{\theta}}\circ\Phi_{x_{n},\omega_{\theta}}(x^{\prime})$)

$|x^{\prime}-\gamma^{\lambda}_{\theta,v}|\sim|\partial_{\omega}\phi^{\lambda}(x;\omega_{\theta})-v|,$

for all $x=(x^{\prime},x_{n})\in X_{\lambda}$ with $x_{n}\in
I^{\lambda}_{\theta,v}$ uniformly in $\lambda$. This has the following
consequence:

######  Lemma 3.6 ([14, Lemma 5.4]).

If $1\ll R\ll\lambda$ and $x\in B(0,R)\backslash T_{\theta,v}$, then

$|T^{\lambda}f_{\theta,v}(x)|\leq(1+R^{-1/2}|\partial_{\omega}\phi^{\lambda}(x;\omega_{\theta})-v|)^{-(n+1)}\text{RapDec}(R)\|f\|_{L^{2}(A^{n-1})}.$

###  3.4. $L^{2}$-$L^{2}$-estimate

We recall the following generalization of Parseval’s theorem, only depending
on non-degeneracy $C1^{\prime})$ of the phase function (cf. [31, Section
2.1]):

######  Lemma 3.7 ([14, Lemma 5.5]).

If $1\leq R\leq\lambda$ and $B_{R}$ is any ball of radius $R$, then

(34) $\|T^{\lambda}f\|_{L^{2}(B_{R})}\lesssim R^{1/2}\|f\|_{L^{2}(A^{n-1})}.$

This is based on the following estimate:

######  Lemma 3.8 ([14, Lemma 5.6]).

For any fixed $x_{n}\in\mathbb{R}$, we find the estimate

(35)
$\|T^{\lambda}f\|_{L^{2}(\mathbb{R}^{n-1}\times\\{x_{n}\\})}\lesssim\|f\|_{L^{2}(A^{n-1})}.$

###  3.5. $k$-broad norms

Here we recall basic properties of the $k$-broad norms. Although the naming is
misleading as $k$-broad norms are, strictly speaking, no norms, the properties
are similar enough to make the following arguments work. We shall also see
that $U\mapsto\|T^{\lambda}f\|^{p}_{BL^{p}_{k,A}(U)}$ behaves as a measure.

######  Lemma 3.9 (Finite (sub-)additivity, [14, Lemma 6.1]).

Let $U_{1},U_{2}\subseteq\mathbb{R}^{n}$ and $U=U_{1}\cup U_{2}$. If $1\leq
p<\infty$ and $A$ is a non-negative integer, then

(36)
$\|T^{\lambda}f\|^{p}_{BL^{p}_{k,A}(U)}\leq\|T^{\lambda}f\|^{p}_{BL^{p}_{k,A}(U_{1})}+\|T^{\lambda}f\|^{p}_{BL^{p}_{k,A}(U_{2})}$

holds for all integrable $f:A^{n-1}\to\mathbb{C}$.

Secondly, we have the following variant of the triangle inequality:

######  Lemma 3.10 (Triangle inequality, [14, Lemma 6.2]).

If $U\subseteq\mathbb{R}^{n}$, $1\leq p<\infty$ and $A=A_{1}+A_{2}$ for
$A_{1}$ and $A_{2}$ non-negative integers, then

(37)
$\|T^{\lambda}(f_{1}+f_{2})\|_{BL^{p}_{k,A}(U)}\lesssim\|T^{\lambda}f_{1}\|_{BL^{p}_{k,A_{1}}(U)}+\|T^{\lambda}f_{2}\|_{BL^{p}_{k,A_{2}}(U)}$

holds for all integrable $f_{1},f_{2}:A^{n-1}\to\mathbb{C}$.

We further have the following variant of Hölder’s inequality:

######  Lemma 3.11 (Logarithmic convexity, [14, Lemma 6.3]).

Suppose that $U\subseteq\mathbb{R}^{n}$, $1\leq p,p_{1},p_{2}<\infty$ and
$0\leq\alpha_{1},\alpha_{2}\leq 1$ satisfy $\alpha_{1}+\alpha_{2}=1$ and

$\frac{1}{p}=\frac{\alpha_{1}}{p_{1}}+\frac{\alpha_{2}}{p_{2}}.$

If $A=A_{1}+A_{2}$ for $A_{1}$, $A_{2}$ non-negative integers, then

$\|T^{\lambda}f\|_{BL^{p}_{k,A}(U)}\lesssim\|T^{\lambda}f\|^{\alpha_{1}}_{BL^{p_{1}}_{k,A_{1}}(U)}\|T^{\lambda}f\|^{\alpha_{2}}_{BL^{p_{2}}_{k,A_{2}}(U)}.$

Later on, we shall only consider $A\gg 1$, which allows us to use Hölder’s and
Minkowski’s inequality for $k$-broad norms.

###  3.6. Overview of parameters

For the reader’s convenience, we provide an overview of the parameters to be
used in Sections 3-7:

  * •

$0<\varepsilon\ll 1$ denotes the parameter, for which we aim to prove the
estimates in Theorem 1.1 and 1.2,

  * •

$m$ will denote the dimension of the algebraic variety, which is used for the
polynomial partitioning (see Section 4),

  * •

we have the hierarchy of parameters:

$\delta\ll\delta_{n}\ll\ldots\delta_{1}\ll\varepsilon$

with $\delta_{i}=\delta_{i}(\varepsilon)$ and $\delta=\delta(\varepsilon)$.
The parameters $\delta_{i}$ will quantify tangentiality of wave packets with
respect to a variety of dimension $i$ and will be specified in the iteration
in Section 6.

  * •

$\delta$ is the parameter from wave packet decomposition and eventually,
$\delta=\delta(\varepsilon)$, and $N=N(\delta)=N(\varepsilon)$ the parameter
$N$ for reduced data will be chosen only depending on $\varepsilon$.

  * •

In Section 7 we need the derivative bounds for reduced phase functions up to
$N$ for a possibly larger value of $N$ depending on a parameter
$\delta_{1}=\delta_{1}(\varepsilon)$ used in a narrow decoupling estimate.

##  4\. Polynomial partitioning

A key tool in the proof will be polynomial partitioning following previous
work by Guth [12, 13] (see also Guth–Katz [15]) and in the variable
coefficient case Guth–Hickman–Iliopoulou [14]. The idea is to divide the ball
$B_{R}$ by the zero set of a polynomial into cells, which equidistribute the
broad norm. Either $\mu_{T^{\lambda}f}$ will be concentrated in the cells or
at the wall, i.e., an appropriate neighbourhood of the zero locus of the
polynomial. Both cases will be handled by induction. We recall some facts from
[14], which we will use in the following.

###  4.1. Tools from algebraic geometry

Given a polynomial $P$ in $\mathbb{R}^{n}$, its zero set is denoted by $Z(P)$.
To make the varieties $Z(P_{1},\ldots,P_{n-m})$ smooth $m$-dimensional
manifolds, we consider transverse complete intersections:

######  Definition 4.1.

Let $m\in\mathbb{N}$, $m\leq n$, and let $P_{1},\ldots,P_{n-m}$ be polynomials
on $\mathbb{R}^{n}$ whose common zero set is denoted by
$Z(P_{1},\ldots,P_{n-m})$. The variety $Z(P_{1},\ldots,P_{n-m})$ is called a
transverse complete intersection if

$\nabla P_{1}(x)\wedge\ldots\wedge\nabla P_{n-m}(x)\neq 0\qquad\forall x\in
Z(P_{1},\ldots,P_{n-m}).$

The degree of the transverse complete intersection $\overline{\deg}Z$ is
defined as
$\max_{j=1,\ldots,n-m}\deg P_{j}$.

We have the following partitioning argument:

######  Theorem 4.2 ([14, Theorem 7.3]).

Suppose that $W\geq 0$ is a non-zero $L^{1}$-function on $\mathbb{R}^{n}$.
Then, for any degree $D\in\mathbb{N}$, there exists a non-zero polynomial $P$
of degree $\deg P\lesssim D$ such that the following holds:

  * •

The set $Z(P)$ is a finite union of $\lesssim\log D$ transverse complete
intersections.

  * •

If $(O_{i})_{i\in\mathcal{I}}$ denotes the set of connected components of
$\mathbb{R}^{n}\backslash Z(P)$, then $\\#\mathcal{I}\lesssim D^{n}$ and

(38) $\int_{O_{i}}W\sim D^{-n}\int_{\mathbb{R}^{n}}W\text{ for all
}i\in\mathcal{I}.$

The connected components are called _cells_.

We further need the following lemma on transverse intersections of tubes with
varieties:

######  Lemma 4.3 ([13, Lemma 5.7]).

Let $T$ be a cylinder of radius $r$ with central line $\ell$ and suppose that
$Z=Z(P_{1},\ldots,P_{n-m})\subseteq\mathbb{R}^{n}$ is a transverse complete
intersection, where the polynomials $P_{j}$ have degree at most $D$. For
$\alpha>0$, let

$Z_{>\alpha}=\\{z\in Z:\angle(T_{z}Z,\ell)>\alpha\\}.$

Then $Z_{>\alpha}\cap T$ is contained in a union of $\lesssim D^{n}$ balls of
radius $\lesssim r\alpha^{-1}$.

For the application, we are interested in $r=R^{(1+\delta)/2}$, as this will
be the radius of the (thin) tubes and $\alpha=R^{-\frac{1}{2}+\delta}$.

###  4.2. Polynomial approximation

However, with smooth core curves, Lemma 4.3 is not applicable directly. We
approximate the core curves by polynomials such that algebraic methods can
still be applied to the curved tubes. We follow [14, Section 7.2]. Let
$\varepsilon>0$ be a small parameter and let $N=N_{\varepsilon}:=\lceil
1/(2\varepsilon)\rceil\in\mathbb{N}$. Suppose that
$\Gamma:(-1,1)\to\mathbb{R}^{n}$ is a smooth curve with

$\|\Gamma\|_{C^{N+1}(-1,1)}=\max_{0\leq k\leq
N+1}\sup_{|t|<1}|\Gamma^{(k)}(t)|\lesssim 1.$

After the reductions of Section 3.1, we find the following estimates:

######  Lemma 4.4.

The curves $\Gamma^{1}_{\theta,v}$ satisfy

$|(\Gamma^{1}_{\theta,v})^{\prime}(t)|\sim 1\text{ for all }t\in
I^{1}_{\theta,v},$

and

$\sup_{t\in I^{1}_{\theta,v}}|(\Gamma^{1}_{\theta,v})^{(k)}(t)|\lesssim
c_{par}\text{ for }2\leq k\leq N.$

The proof from [14, Lemma 7.4] applies verbatim because of bounds (26) and
$D2)$ from Subsection 3.1 although the phase functions are from different
classes.

We denote by $[\Gamma]_{\varepsilon}:\mathbb{R}\to\mathbb{R}^{n}$ the
polynomial curve given by the degree-$N$ Taylor approximation of $\Gamma$
around zero. Observe that

$\|[\Gamma]_{\varepsilon}\|_{C^{\infty}(-2,2)}\leq
e^{2}\|\Gamma\|_{C^{N}(-1,1)}\lesssim 1.$

Furthermore, for $\lambda\gg 1$, noting that $\lambda^{-\varepsilon
N}\leq\lambda^{-1/2}$, Taylor’s theorem yields

$|\Gamma^{(i)}(t)-[\Gamma]^{(i)}_{\varepsilon}(t)|\lesssim_{\varepsilon}\lambda^{-\frac{1}{2}}|t|^{1-i}\text{
for all }|t|\lesssim_{\varepsilon}\lambda^{-\varepsilon}\text{ and }i=0,1.$

Letting $\Gamma^{\lambda}:(-\lambda,\lambda)\to\mathbb{R}^{n}$ denote the
rescaled curve $\Gamma^{\lambda}(t)=\lambda\Gamma(t/\lambda)$, the above
inequalities imply that

(39)
$\|[\Gamma^{\lambda}]^{\prime}_{\varepsilon}\|_{C^{\infty}(-2\lambda,2\lambda)}\lesssim
1\text{ and
}\|[\Gamma^{\lambda}]^{\prime\prime}_{\varepsilon}\|_{C^{\infty}(-2\lambda,2\lambda)}\lesssim\lambda^{-1},$

and

$|(\Gamma^{\lambda})^{(i)}(t)-([\Gamma^{\lambda}]_{\varepsilon})^{(i)}(t)|\lesssim_{\varepsilon}\lambda^{-\frac{1}{2}}|t|^{1-i}\text{
for all }|t|\lesssim_{\varepsilon}\lambda^{1-\varepsilon}\text{ and }i=0,1.$

As a consequence of
$|(\Gamma^{\lambda})^{\prime}(t)|\sim|[\Gamma^{\lambda}]^{\prime}_{\varepsilon}(t)|\sim
1$, the tangent spaces to the curves $\Gamma^{\lambda}$ and
$[\Gamma^{\lambda}]_{\varepsilon}$ have a small angular separation, i.e.,

(40)
$\angle(T_{\Gamma^{\lambda}(t)}\Gamma^{\lambda},T_{[\Gamma^{\lambda}]_{\varepsilon}(t)}[\Gamma^{\lambda}]_{\varepsilon})\lesssim_{\varepsilon}\lambda^{-\frac{1}{2}}\text{
for all }|t|\lesssim_{\varepsilon}\lambda^{1-\varepsilon}.$

###  4.3. Transverse interactions between curved tubes and varieties

We have the following generalization of Lemma 4.3:

######  Lemma 4.5 ([14, Lemma 7.5]).

Let $n\geq 2$, $1\leq m\leq n$ and
$Z=Z(P_{1},\ldots,P_{n-m})\subseteq\mathbb{R}^{n}$ be a transverse complete
intersection. Suppose that $\Gamma:\mathbb{R}\to\mathbb{R}^{n}$ is a
polynomial graph satisfying

(41) $\|\Gamma^{\prime}\|_{L^{\infty}(-2\lambda,2\lambda)}\lesssim 1\text{ and
}\|\Gamma^{\prime\prime}\|_{L^{\infty}(-2\lambda,2\lambda)}\leq\delta$

for some $\lambda,\delta>0$. There exists a dimensional constant $\bar{C}>0$
such that, for all $\alpha>0$ and $0<r<\lambda$ satisfying
$\alpha\geq\bar{C}\delta r$, the set $Z_{>\alpha,r,\Gamma}\cap B(0,\lambda)$
is contained in a union of

$O((\overline{\deg}Z\cdot\deg\Gamma)^{n})$

balls of radius $r/\alpha$.

This will be used when proving the $k$-broad estimate via polynomial
partitioning.

##  5\. Transverse equidistribution estimates

###  5.1. Outline of the argument.

In this section transverse equidistribution estimates for wave packets
tangential to varieties will be examined. Functions having wave packets
tangential to a variety arise in the _algebraic case_ when applying polynomial
partitioning in the main induction argument. Contrary to [26] or [10],
however, we stick to the wave packet decomposition used in [14]. We make the
following definition (see [14, 13]):

######  Definition 5.1.

Let $Z=Z(P_{1},\ldots,P_{n-m})$ be a transverse complete intersection. A wave
packet $(\theta,v)$ is said to be $R^{-\frac{1}{2}+\delta_{m}}$-tangent to $Z$
in $B(0,R)$ if

(42) $T_{\theta,v}\cap B_{R}\subseteq N_{R^{\frac{1}{2}+\delta_{m}}}(Z)$

and

(43)
$\angle(G^{\lambda}(x;\omega_{\theta}),T_{z}Z)\leq\bar{c}_{tang}R^{-\frac{1}{2}+\delta_{m}}$

for any $x\in T_{\theta,v}$ and $z\in Z\cap B(0,2R)$ with
$|x-z|\leq\bar{C}_{tang}R^{\frac{1}{2}+\delta_{m}}$.

We want to study functions concentrated on the collection of wave packets

$\mathbb{T}_{Z}=\\{(\theta,v)\in\mathbb{T}:T_{\theta,v}\text{ in
}R^{-\frac{1}{2}+\delta_{m}}-\text{tangent to }Z\text{ in }B(0,R)\\}.$

Precisely, we make the following definition:

######  Definition 5.2.

If $\mathbb{S}\subseteq\mathbb{T}$, then $f$ is said to be concentrated on
wave packets from $\mathbb{S}$ if

$f=\sum_{(\theta,v)\in\mathbb{S}}f_{\theta,v}+\text{RapDec}(R)\|f\|_{L^{2}}.$

Let $B\subseteq\mathbb{R}^{n}$ be a ball of radius
$R^{\frac{1}{2}+\delta_{m}}$ with centre $\bar{x}\in B(0,R)$. We study
$\eta_{B}\cdot T^{\lambda}g$, where $\eta_{B}$ is a suitable choice of
Schwartz function adapted to $B$. A stationary phase argument yields that the
Fourier transform of $\eta_{B}\cdot T^{\lambda}g_{\theta,v}$ is concentrated
near the surface $\Sigma=\\{\Sigma(\omega)\,:\omega\in A^{n-1}\\}$, where
$\Sigma(\omega)=\partial_{x}\phi^{\lambda}(\bar{x};\omega)$. This leads to the
refined set of wave packets

$\mathbb{T}_{Z,B}=\\{(\theta,v)\in\mathbb{T}_{Z}:T_{\theta,v}\cap
B\neq\emptyset\\}.$

For $(\theta,v)\in\mathbb{T}_{Z,B}$, the direction
$G^{\lambda}(\bar{x};\omega_{\theta})$ of $T_{\theta,v}$ must make a small
angle with each of the tangent spaces $T_{z}Z$ for all $z\in Z\cap B$. This
constrains $\Sigma(\omega_{\theta})$ to lie in a small neighbourhood of some
typically $m$-dimensional manifold $S_{\xi}$. But in the homogeneous case,
$S_{\xi}$ might only be one-dimensional, or “close” to a one-dimensional
manifold. This will be quantified below. We refer to this as _case of narrow
space-time frequencies_. This case does not contribute in the broad norm, for
which reason it is referred to as _narrow_. The role of space-time frequencies
is emphasized, as in the analysis we are rather considering sectors on the
generalized cone (and not the projection to $\mathbb{R}^{n-1}$).

To linearize $S_{\xi}$, if it is not a “narrow”, essentially one-dimensional
set, let $R^{\frac{1}{2}}<\rho\ll R$ and for the remainder of this section,
let $\tau\subseteq A^{n-1}$ denote a sector of aperture
$O(\rho^{-\frac{1}{2}+\delta_{m}})$. We define

$\mathbb{T}_{Z,B,\tau}=\\{(\theta,v)\in\mathbb{T}_{Z}:\theta\cap\tau\neq\emptyset\text{
and }T_{\theta,v}\cap B\neq\emptyset\\}.$

We recall the constant-coefficient case. Suppose that $Z$ is an
$m$-dimensional affine plane so that $T_{z}Z=V$ for all $z\in Z$, where
$V\parallel Z$.

The extension operator for the cone has the unnormalized Gauss map
$G_{0}(\omega)=\big{(}-\frac{\omega}{|\omega|},1\big{)}$. Let
$V^{+}=\\{\omega\in\mathbb{R}^{n-1}\backslash\\{0\\}:G_{0}(\omega)\in V\\}$.
By a crucial observation due to Ou–Wang [26], if $V^{+}$ is tangent to
$\mathcal{C}=\\{(\omega,\frac{\omega}{|\omega|}):\omega\in A^{n-1}\\}$ up to
an angle $R^{-\delta_{m}}$, then
$N_{R^{-\frac{1}{2}+\delta_{m}}}V^{+}\cap\mathcal{C}$ is a
$O(R^{-\delta_{m}})$-neighbourhood of $O(1)$ radial lines. In the easier case
of the extension operator of the paraboloid (see [13]), this _narrow_ case
does not happen. As an example in case of the cone, we can consider $V=\langle
e_{1}+e_{n},e_{i_{1}},\ldots,e_{i_{m-1}}\rangle\subseteq\mathbb{R}^{n}$ with
$e_{i_{j}}\neq e_{i_{k}}$ for $j\neq k$ and $i_{k}\in\\{2,\ldots,n-1\\}$. Here
$e_{i}$ denotes the $i$th unit vector. We have $\dim(V)=m$, but clearly

$V^{+}=\\{\omega\in\mathbb{R}^{n-1}\backslash\\{0\\}:(\frac{-\omega}{|\omega|},1)\in
V\\}=\\{-\mu e_{1}:\mu>0\\}$

is always one-dimensional.

In the variable coefficient case, we see that if $V^{+}$ is tangent to
$\mathcal{C}_{x}=\\{\partial_{x}\phi^{\lambda}(\bar{x};\omega)\\}$ up to an
angle $R^{-\delta_{m}}$, then
$N_{R^{-\frac{1}{2}+\delta_{m}}}V^{+}\cap\mathcal{C}_{x}$ is a
$O(R^{-\delta_{m}})$-neighbourhood of $O(1)$ radial lines. Hence, the case of
narrow space-time frequencies does not contribute to the $k$-broad norm.
Otherwise, we refer to this as _case of broad space-time frequencies_ , and we
shall see that we find quantitative transversality to hold. We can deduce
transverse equidistribution estimates similar to the paraboloid case (or its
variable coefficient counterpart). In the constant coefficient case, but for
arbitrary cones, this was recently investigated by Gao _et al._ in [10]. We
shall see how the arguments extend to the variable coefficient case.

In this section we aim to prove the following estimate for $g$ concentrated on
wave packets tangential to $Z$ in the case of broad space-time frequencies:

$\int_{B\cap N_{\rho^{\frac{1}{2}+\delta_{m}}}(Z)}|T^{\lambda}g|^{2}\lesssim
R^{\frac{1}{2}+O(\delta_{m})}\big{(}\frac{\rho}{R}\big{)}^{\frac{n-m}{2}}\|g\|^{2}_{L^{2}}.$

The precise statement with additional assumptions on $g$ is provided in Lemma
5.8.

###  5.2. Geometric preliminaries: narrow and broad space-time frequencies

In this subsection we quantify the above cases of narrow and broad space-time
frequencies. There are two linearizations involved. The transverse complete
intersection is linearized and below $V$ plays the role of the tangent space
of $Z$. Moreover, the phase function is linearized such that we can apply
arguments from the constant-coefficient case due to Ou–Wang [26] and Gao _et
al._ [10].

We consider an $m$-dimensional linear subspace $V\subseteq\mathbb{R}^{n}$. Let
$A=(a_{i,j})_{i,j}\in\mathbb{R}^{(n-m)\times n}$ be a matrix of maximal rank
such that

$V=\\{x\in\mathbb{R}^{n}:Ax=0\\}.$

We consider the intersection of $V$ with the hyperplane $\\{x_{n}=0\\}$
perceived as subspace of $\mathbb{R}^{n-1}$:

$V^{-}=\\{x^{\prime}\in\mathbb{R}^{n-1}:A(x^{\prime},0)=0\\}.$

With $V$ playing the role of the tangent space of $Z$, which is close to the
values of the generalized Gauss map, which are not contained in the hyperplane
$\\{x_{n}=0\\}$, we suppose that $\dim V^{-}=\dim V-1$.

The linearization is easier to carry out after changing to graph
parametrization in $u$-frequencies via $\Psi^{\lambda}$, which we recall is
implicitly defined by

$\partial_{x^{\prime}}\phi^{\lambda}(\bar{x};\Psi^{\lambda}(\bar{x};u))=u.$

We use short-hand notation $\Psi(u)=\Psi^{\lambda}(\bar{x};u)$. Note that
$\Psi$ is $1$-homogeneous like $\partial_{x^{\prime}}\phi^{\lambda}$ and the
identity mapping because

$\partial_{x^{\prime}}\phi^{\lambda}(\bar{x};\Psi(\mu u))=\mu
u=\mu\partial_{x^{\prime}}\phi^{\lambda}(\bar{x};\Psi(u))=\partial_{x^{\prime}}\phi^{\lambda}(\bar{x};\mu\Psi(u)).$

By substituting
$\tilde{\phi}(u)=h_{\bar{x}}(u)=\partial_{x_{n}}\phi^{\lambda}(\bar{x};\Psi(u))$
the arguments due to Gao _et al._ [10] apply. We define a set

$L=\\{u\in B_{n-1}(0,2)\backslash
B_{n-1}(0,1/2):A(-\nabla_{u}\tilde{\phi}(u),1)=0\\}.$

The set $\\{(u,\tilde{\phi}(u)):u\in L\\}$ describes the points on the
generalized cone, which have a normal in $V$. The _case of narrow space-time
frequencies_ gives a negligible contribution to the broad norm:

######  Lemma 5.3 ([10, Lemma 4.5]).

Let $\eta\in\mathbb{S}^{n-2}\subseteq\mathbb{R}^{n-1}$. If $\eta\in L$ and
$\angle(\eta,V^{-})>\frac{\pi}{2}-K^{-2}$, then $L$ is contained in the set
$\\{\xi\in\mathbb{R}^{n-1}\backslash\\{0\\}:\angle(\xi,\eta)\lesssim
K^{-2}\\}$.

######  Remark 5.4.

It is important to note that, contrary to the case of broad space-time
frequencies analyzed below, the lemma does not hinge on a stronger
localization of $\eta$. For later purposes, note that balls
$B(\bar{x};R^{\frac{1}{2}+\delta_{m}})$, for which Lemma 5.3 applies, do not
contribute in the $k$-broad norm.

We turn to the more involved case of _broad space-time frequencies_ : In
general $\\{(u,\tilde{\phi}(u)):u\in L\\}$ does not lie in an affine subspace
because $L$ is not an affine subspace. We start by linearizing $L$ at
$\eta\in\mathbb{S}^{n-2}$. By taking the orthogonal complement of a suitable
extension of the tangent space of $L$ at $\eta$ we shall construct $W$, which
is quantitatively transverse to $V$. Let $\eta\in L\cap\mathbb{S}^{n-2}$ with
$\angle(\eta,V^{-})<\frac{\pi}{2}-K^{-2}$. Let

$L_{i}=\\{u\in
A^{n-1}\,:\,\sum_{j=1}^{n-1}a_{i,j}\partial_{j}\tilde{\phi}(\eta)-a_{i,n}=0\\},\quad
i=1,\ldots,n-m$

such that

$L=\bigcap_{i=1}^{n-m}L_{i}.$

We compute the normal $n^{(i)}$ of $L_{i}$ at $\eta$ to be

$n^{(i)}_{k}=\sum_{j=1}^{n-1}a_{i,j}\partial^{2}_{jk}\tilde{\phi}(\eta),\quad
n^{(i)}=\partial^{2}_{\eta\eta}\tilde{\phi}(\eta)\alpha_{i}\text{ with
}\alpha_{i}=(a_{i,1},\ldots,a_{i,n-1}),\;i=1,\ldots,n-m.$

The tangent space of $L_{i}$ at $\eta$ is given by

$T_{\eta}L_{i}=(n^{(i)})^{\perp}=\\{\xi^{\prime}\in\mathbb{R}^{n-1}:(n^{(i)},\xi^{\prime})=0\\}=\\{\xi^{\prime}\in\mathbb{R}^{n-1}\,:\,(\partial^{2}_{\eta\eta}\tilde{\phi}(\eta)\alpha_{i},\xi^{\prime})=0\\}.$

We argue that the normals are linearly independent: Let $\tilde{V}=\langle
n^{(1)},\ldots,n^{(n-m)}\rangle$.

######  Lemma 5.5.

$\tilde{V}\subseteq\mathbb{R}^{n-1}$ is a subspace of dimension $n-m$.

###### Proof.

It is key to observe that
$\partial^{2}_{\eta\eta}\tilde{\phi}(\eta)\alpha_{i}$, $i=1,\ldots,n-m$ are
linearly independent due to the angle condition
$\angle(\eta,V^{-})<\frac{\pi}{2}-K^{-2}$. Indeed, the Hessian is degenerate
only in the direction of $\eta$, but $\alpha_{i}$ is orthogonal to $V^{-}$,
and $A$ has maximal rank. ∎

Therefore, the tangent space of $L$ is given by the intersection of
$T_{\eta}L_{i}$:

$T_{\eta}L=\gamma_{1}^{\perp}\cap\ldots\cap\gamma_{n-m}^{\perp}=\langle\gamma_{1},\ldots,\gamma_{n-m}\rangle^{\perp}.$

We have $T_{\eta}L=\bar{V}^{-}=\tilde{V}^{\perp}$ is the orthogonal complement
of $\tilde{V}$ in $\mathbb{R}^{n-1}$ such that

$\mathbb{R}^{n-1}=\tilde{V}\oplus\bar{V}^{-}.$

Let $\bar{V}=\langle\bar{V}^{-},e_{n}\rangle$ be the linear subspace spanned
by $\bar{V}^{-}$ and $e_{n}$. We let $W=\bar{V}^{\perp}$ be the orthogonal
complement of $\bar{V}$ in $\mathbb{R}^{n}$:

$\mathbb{R}^{n}=\bar{V}\oplus W.$

As pointed out in [10], all the linear subspaces depend on the choice of
$\eta$. We have the following quantitative transversality:

######  Lemma 5.6 ([10, Lemma 4.6]).

Let $\eta\in\mathbb{S}^{n-2}\cap L$. If
$\angle(\eta,V^{-})\leq\frac{\pi}{2}-K^{-2}$, then $W$ and $V$ are transverse
in the sense that $\angle(V,W)\gtrsim K^{-4}$.

###  5.3. Verifying the transverse equidistribution estimate: broad space-time
frequencies

We turn to the key equidistribution estimate in case of broad space-time
frequencies.

######  Remark 5.7.

We can suppose that $Z$ in $2B$ is $K$-flat, i.e., for any $z,z^{\prime}\in
Z\cap 2B$ we have

$\angle(T_{z}Z,T_{z^{\prime}}Z)\lesssim K^{-5}$

with $K\lesssim R^{\delta}\ll\rho^{\delta_{m}}$. If there is a wave packet
with $\angle(G^{\lambda}(x;\omega_{\theta}),T_{z}Z)\lesssim
R^{-\frac{1}{2}+\delta_{m}}$ for all $z\in Z\cap 2B$, it follows from the
triangle inequality for small angles that

$\angle(T_{z}Z,T_{z^{\prime}}Z)\lesssim R^{-\frac{1}{2}+\delta_{m}}.$

If there is no wave packet such that the above holds, then there is nothing to
show.

######  Lemma 5.8.

Let $Z$ be a transverse complete intersection with $\dim Z=m$,
$\overline{\text{deg}}\;Z\lesssim_{\varepsilon}1$,
$B=B(\bar{x},R^{\frac{1}{2}+\delta_{m}})$ a ball of radius
$R^{\frac{1}{2}+\delta_{m}}$, and let $g$ be concentrated on wave packets in
$\mathbb{T}_{Z,B,\tau}$. Suppose that with the notations of Subsection 5.2,
with $\tilde{\phi}=h_{\bar{x}}$, and for some
$\eta\in\Psi^{-1}(\tau)\cap\mathbb{S}^{n-2}$ we are in the situation of Lemma
5.6, that is in the case of broad space-time frequencies. Then, for any
$R^{\frac{1}{2}}\ll\rho\leq R$,

(44) $\int_{B\cap
N_{\rho^{\frac{1}{2}+\delta_{m}}}(Z)}|T^{\lambda}g|^{2}\lesssim
R^{\frac{1}{2}+O(\delta_{m})}\big{(}\frac{\rho}{R}\big{)}^{\frac{n-m}{2}}\|g\|^{2}_{L^{2}}.$

The proof relies on quantifying the uncertainty principle from [14, Subsection
8.2]. Let $G:\mathbb{R}^{n}\to\mathbb{C}$ be such that
$\text{supp}(\hat{G})\subseteq B(c,r)$. Then we have essentially for
$0<\rho<r^{-1}$:

(45) $\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{B(x_{0},\rho)}|G|^{2}\lesssim\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{B(x_{0},r^{-1})}|G|^{2}.$

As a first step in the proof of Lemma 5.8, we consider wave packets tangential
to linear subspaces: In the following transverse equidistribution estimates
are considered with respect to some fixed linear subspace
$V\subseteq\mathbb{R}^{n}$. Recall that $B$ is a ball of radius
$R^{\frac{1}{2}+\delta_{m}}$ with centre $\bar{x}\in\mathbb{R}^{n}$, and
define

$\mathbb{T}_{V,B}=\\{(\theta,v):\angle(G^{\lambda}(\bar{x};\omega_{\theta}),V)\lesssim
R^{-\frac{1}{2}+\delta_{m}}\text{ and }T_{\theta,v}\cap B\neq\emptyset\\}.$

Recall that $R^{\frac{1}{2}}<\rho<R$ and, for $\tau\subseteq\mathbb{R}^{n-1}$
a sector of aperture $O(\rho^{-\frac{1}{2}+\delta_{m}})$ centred around a
point in $A^{n-1}$, define

$\mathbb{T}_{V,B,\tau}=\\{(\theta,v)\in\mathbb{T}_{V,B}:\theta\cap\big{(}\frac{\tau}{10}\big{)}\neq\emptyset\\}.$

######  Lemma 5.9.

If $V\subseteq\mathbb{R}^{n}$ is a linear subspace, then there exists a linear
subspace $W$ with the following properties:

  * (1)

$\dim V+\dim W=n$;

  * (2)

$V$ and $W$ are quantitatively transverse: we have $\angle(V,W)\gtrsim
K^{-4}$;

  * (3)

if $g$ is concentrated on wave packets from $\mathbb{T}_{V,B,\tau}$, and there
is $\eta\in\Psi^{-1}(\tau)$ such that for $\tilde{\phi}=h_{\bar{x}}$ the
assumptions of Lemma 5.6 are valid, $\Pi$ is any plane parallel to $W$ and
$x_{0}\in\Pi\cap B$, then the inequality

$\int_{\Pi\cap
B(x_{0},\rho^{\frac{1}{2}+\delta_{m}})}|T^{\lambda}g|^{2}\lesssim_{\delta}R^{O(\delta_{m})}\big{(}\frac{\rho}{R}\big{)}^{\frac{\dim
W}{2}}\|g\|^{2\delta/(1+\delta)}_{L^{2}}\big{(}\int_{\Pi\cap
2B}|T^{\lambda}g|^{2}\big{)}^{\frac{1}{1+\delta}}$

holds, up to inclusion of $\text{RapDec}(R)\|g\|_{L^{2}}$ on the right-hand
side.

###### Proof.

The proof largely follows [14]. The difference is that we are dealing with the
larger sectors $\tau$ here having an aperture
$O(\rho^{-\frac{1}{2}+\delta_{m}})$ in contrast to balls of radius
$O(\rho^{-\frac{1}{2}+\delta_{m}})$. This will be compensated by the null
direction of the conic surface.

Constructing the subspace $W$: Recall that
$h_{\bar{x}}(u)=\partial_{x_{n}}\phi^{\lambda}(\bar{x};\Psi(u))$ with
$\partial_{x^{\prime}}\phi^{\lambda}(\bar{x};\Psi(u))=u$ such that
$(u,h_{\bar{x}}(u))$ is a graph parametrization of
$\partial_{x}\phi^{\lambda}(\bar{x};\cdot)$ in $u$-frequencies with
$\omega=\Psi(u)$. If $L\cap\Psi^{-1}(\tau)=\emptyset$, then
$\Psi(L)\cap\tau=\emptyset$, but then, by Lemma 3.4 we had
$\mathbb{T}_{V,B,\tau}=\emptyset$ and there is nothing to show.

Suppose $L\cap\Psi^{-1}(\tau)\neq\emptyset$ in the following and fix $\eta\in
L\cap\Psi^{-1}(\tau)$. Note that

$\partial^{2}_{xu_{1}}\phi^{\lambda}(\bar{x};\Psi(u))\wedge\ldots\wedge\partial^{2}_{xu_{n-1}}\phi^{\lambda}(\bar{x};\Psi(u))=G_{0}(\bar{x};\omega)\cdot\det
J\Psi(u).$

Hence, we can construct $W$ around $\eta\in L\cap\Psi^{-1}(\tau)$ as in
Subsection 5.2. $W$ and $V$ are quantitatively transverse as in (2) by Lemma
5.6.

Verifying the transverse equidistribution estimate: Recall that $g$ is
concentrated on wave packets $\mathbb{T}_{V,B,\tau}$, $B$ is a
$R^{\frac{1}{2}+\delta_{m}}$-ball, and $\tau$ is a
$O(\rho^{-\frac{1}{2}+\delta_{m}})$-sector. Let
$\eta_{B}(x)=\eta((x-\bar{x})/R^{\frac{1}{2}+\delta_{m}})$ denote a Schwartz
cutoff, which satisfies $\eta(x)=1$ for $x\in B(0,2)$.

Let $\Sigma(\omega)=\partial_{x}\phi^{\lambda}(\bar{x};\omega)$. If
$\omega\in\text{supp}(g_{\theta,v})$, then
$|\omega-\omega_{\theta}|<R^{-\frac{1}{2}}$, and so
$|\Sigma(\omega)-\xi_{\theta}|\lesssim R^{-\frac{1}{2}}$, where
$\xi_{\theta}=\Sigma(\omega_{\theta})$. By non-stationary phase, we find like
in [14]

$|(\eta_{B}\cdot
T^{\lambda}g_{\theta,v})|_{\Pi}\widehat{\;}(\xi)|\lesssim_{N}R^{O(1)}w_{B(\text{proj}_{W}\xi_{\theta},R^{-\frac{1}{2}})}(\xi)\|g_{\theta,v}\|_{L^{2}}.$

Let

$\displaystyle L=\\{u\in A^{n-1}:(-\nabla h_{\bar{x}}(u),1)\in V\\},\quad
S_{\omega}=\\{\omega\in A^{n-1}:G^{\lambda}(\bar{x};\omega)\in V\\}.$

Let $A_{u}=\eta+T_{\eta}L$ denote the affine variant of the linear subspace
$T_{\eta}L$, and $A_{\xi}=A_{u}\times\mathbb{R}$. Let $V_{\xi}$ be the linear
subspace corresponding to $A_{\xi}$. We have $V_{\xi}^{\perp}=W$.

Next, we shall show that

(46) $\text{dist}(\xi_{\theta},A_{\xi})\lesssim R^{-\frac{1}{2}+\delta_{m}}.$

Once the above is proved, the proof can then be concluded like in [14]. Since
the space-time frequencies of $\eta_{B}\cdot T^{\lambda}g|_{\Pi}$ are
concentrated in a ball of radius $R^{-\frac{1}{2}+\delta_{m}}$, we can then
use a rigorous version of (45) to conclude.

We turn to the proof of (46): Fix $(\theta,v)\in\mathbb{T}_{V,B,\tau}$ and let

$u_{\theta}=(\Sigma(\omega_{\theta})_{1},\ldots,\Sigma(\omega_{\theta})_{n-1}).$

The triangle inequality yields

(47)
$\text{dist}(\xi_{\theta},A_{\xi})=\text{dist}(u_{\theta},A_{u})\leq\text{dist}(u_{\theta},L\cap\Psi^{-1}(\tau))+\sup_{u_{*}\in
L\cap\Psi^{-1}(\tau)}\text{dist}(u_{*},A_{u}).$

Furthermore, by Lemma 3.4, taking advantage of the null direction,

$\text{dist}(u_{\theta},L\cap\Psi^{-1}(\tau))\sim\text{dist}(\omega_{\theta},S_{\omega}\cap\tau)\lesssim\angle(G^{\lambda}(\bar{x};\omega_{\theta}),V)\lesssim
R^{-\frac{1}{2}+\delta_{m}},$

where the last inequality is by the definition of $\mathbb{T}_{V,B,\tau}$.

We turn to the estimate of the second term in (47): Fix $u_{*}\in
L\cap\Psi^{-1}(\tau)$. We note that
$\text{dist}(u_{*},A_{u})=\text{dist}(u_{*},A_{\bar{u}})$ for
$\bar{u}=\frac{|u_{*}|}{|u|}u$ by null direction. Let
$A_{\bar{u}}=u_{0}+V_{u}$ for some linear subspace $V_{u}$. Now we note that
the surface $L\cap\Psi^{-1}(\tau)\cap(|u_{*}|\cdot\mathbb{S}^{n-2})$, provided
$\rho$ is large enough, can be written as a subset of the graph of a function
$\psi:\mathcal{W}\to V_{u}^{\perp}$, where $\mathcal{W}\subseteq V_{u}$ is a
subset around the origin of size $O(\rho^{-\frac{1}{2}+\delta_{m}})$. More
precisely, we may write

$L\cap\Psi^{-1}(\tau)\cap(\|u_{*}\|\cdot\mathbb{S}^{n-2})\subseteq\\{w+\psi(w):w\in\mathcal{W}\\}+u_{0}$

with $\psi(0)=0$ and $\nabla\psi(0)=0$. The estimate now follows from Taylor
expansion:

$|\psi(w)|=O(\rho^{-1+2\delta_{m}})=O(R^{-\frac{1}{2}+\delta_{m}}).$

Here we used the hypothesis $R^{\frac{1}{2}}<\rho\ll R$ (cf. [14]), which is
absent for constant-coefficients. ∎

For the proof of the transverse equidistribution estimate in Lemma 5.8, we
have to extend the estimate from a fixed vector space to a variety. The
argument follows [14, Section 8.4] with the difference that the quantitative
transversality (see [14, Definition 8.11]) mildly depends on the scale. This
is due to the necessity of distinguishing between broad and narrow space-time
frequencies and reflected in Lemma 5.6. The vector space $W$ constructed in
Subsection 5.2 now satisfies

$\angle(v,w)\gtrsim K^{-4}\text{ for all }0\neq v\in T_{z}Z\text{ and }0\neq
w\in W.$

###### Proof of Lemma 5.8.

First, by Lemma 3.4, we have

$|G^{\lambda}(\bar{x};\theta)-G^{\lambda}(x;\theta)|\lesssim|x-\bar{x}|/\lambda\lesssim
R^{-\frac{1}{2}+\delta_{m}}.$

This yields

$\angle(G^{\lambda}(\bar{x};\theta),T_{z}Z)\lesssim
R^{-\frac{1}{2}+\delta_{m}}\text{ for all }z\in Z\cap 2B.$

Letting $V=T_{z}Z$, we have

$\mathbb{T}_{Z,B,\tau}\subseteq\mathbb{T}_{V,B,\tau}.$

We can apply Lemma 5.9 to find a subspace $W$ such that

(48) $\angle(V,W)\gtrsim K^{-4}$

and

(49) $\int_{\Pi\cap
B(x_{0},\rho^{\frac{1}{2}+\delta_{m}})}|T^{\lambda}g|^{2}\lesssim_{\delta}R^{O(\delta_{m})}\big{(}\frac{\rho}{R}\big{)}^{\frac{\dim
W}{2}}\|g\|^{\frac{2\delta}{1+\delta}}_{L^{2}}\big{(}\int_{\Pi\cap
2B}|T^{\lambda}g|^{2}\big{)}^{\frac{1}{1+\delta}}$

for every affine subspace $\Pi$ parallel to $W$. By Remark 5.7, $(T_{z}Z,W)$
is a quantitatively transverse pair for all $z\in Z\cap 2B$. We have by (the
proof of) [14, Lemma 8.13]

$\Pi\cap N_{\rho^{\frac{1}{2}+\delta_{m}}}(Z)\cap B\subseteq
N_{CK^{4}\rho^{\frac{1}{2}+\delta_{m}}}(\Pi\cap Z)\cap 2B.$

Here is a minor change compared to [14] as the size of the neighbourhood of
$\Pi\cap Z$ is now increased by $K^{4}$. However, this factor can be absorbed
into $R^{O(\delta_{m})}$. Since $\Pi\cap Z$ is a transverse complete
intersection of dimension $\dim W+\dim Z-n$, a result due to Wongkew [41]
yields that $\Pi\cap N_{\rho^{\frac{1}{2}+\delta_{m}}}(Z)\cap B$ can be
covered by

$O\big{(}R^{O(\delta_{m})}\big{(}\frac{R}{\rho}\big{)}^{(\dim W+\dim
Z-n)/2}\big{)}=O(R^{O(\delta_{m})})$

balls of radius $\rho^{\frac{1}{2}+\delta_{m}}$ because $K\lesssim
R^{\delta}\ll\rho^{\delta_{m}}$. Now the argument is concluded by integrating
over $\Pi$ parallel to $W$ and Hörmander’s $L^{2}$-bound (cf. Lemma 3.7). For
details we refer to [14, p. 318]. ∎

##  6\. Main inductive argument: Proof of the $k$-broad estimate

The $k$-broad estimate is a consequence of the following claim, which is
amenable to induction on scales and dimension. Let

$\bar{p}(k,n)=2\cdot\frac{n+k}{n+k-2}.$

######  Theorem 6.1.

For $\varepsilon>0$ sufficiently small, there are

$0<\delta\ll\delta_{n}\ll\delta_{n-1}\ll\ldots\ll\delta_{1}\ll\varepsilon$

and large dyadic parameters $\bar{A}_{\varepsilon}$, $\bar{C}_{\varepsilon}$,
$D_{m,\varepsilon}\lesssim_{\varepsilon}1$ and $\theta_{m}<\varepsilon$ such
that the following holds. Suppose $Z=Z(P_{1},\ldots,P_{n-m})$ is a transverse
complete intersection with $\overline{\deg}Z\leq D_{m,\varepsilon}$. For all
$2\leq k\leq n$, $1\leq A\leq\bar{A}_{\varepsilon}$ dyadic and $1\leq K\leq
R\leq\lambda$, the inequality

(50)
$\|T^{\lambda}f\|_{BL^{p}_{k,A}(B(0,R))}\lesssim_{\varepsilon}K^{\bar{C}_{\varepsilon}}R^{\theta_{m}+\delta(\log\bar{A}_{\varepsilon}-\log
A)-e_{k,n}(p)+\frac{1}{2}}\|f\|_{L^{2}(A^{n-1})}$

holds whenever $f$ is concentrated on wave packets from $\mathbb{T}_{Z}$ and

(51) $2\leq p\leq\bar{p}_{0}(k,m)=\begin{cases}\bar{p}(k,m),\quad\text{if
}k<m,\\\ \bar{p}(m,m)+\delta,\quad\text{if }k=m.\end{cases}$

Above,

$e_{k,n}(p)=\frac{1}{2}\big{(}\frac{1}{2}-\frac{1}{p}\big{)}(n+k).$

For future reference we denote by $E_{m,A}(R)$ the constant on the right-hand
side of (50):

(52) $E_{m,A}(R)=C_{m,\varepsilon}K^{\bar{C}_{\varepsilon}}R^{\varepsilon-
c_{n}\delta_{m}+\delta(\log\bar{A}_{\varepsilon}-\log
A)-e_{k,n}(p)+\frac{1}{2}}.$

In the first step we reduce to
$R\lesssim_{\varepsilon}\lambda^{1-\varepsilon}$ by covering $B(0,\lambda)$
with balls of radius $\lambda^{1-\varepsilon}$. The technical details are
provided in [14, Lemma 10.2]. This reduction is necessary for polynomial
approximation of the core curve $\gamma^{\lambda}_{\omega,v}$ uniformly in
$R$.

Next, we set up the induction argument for $1\leq
R\lesssim_{\varepsilon}\lambda^{1-\varepsilon}$. For $\varepsilon>0$
sufficiently small, it is enough to consider $K\leq R^{\delta}$ by choosing
$\bar{C}_{\varepsilon}$ sufficiently large (as the claim then follows from the
trivial $L^{1}$-$L^{\infty}$-estimate and crude summation). We let furthermore

(53)
$\begin{split}D_{m,\varepsilon}=\varepsilon^{-\delta^{-(2n-m)}},\quad\theta_{m}(\varepsilon)=\varepsilon-
c_{n}\delta_{m},\quad\bar{A}_{\varepsilon}=\lceil
e^{\frac{10n}{\delta}}\rceil,\\\
\delta_{i}=\delta_{i}(\varepsilon)=\varepsilon^{2i+1}\text{ for all
}i=1,\ldots,n,\text{ and }\delta=\delta(\varepsilon)\ll\delta_{n}.\end{split}$

The base case is given by $m\leq k-1$, and $A\geq 2^{10}$. For details we
refer to [14, Subsection 10.3].

###  6.1. Inductive step

Let $2\leq k\leq n-1$, $k\leq m\leq n$, and $K\leq R^{\delta}$. Assume, by way
of induction hypothesis, that (50) holds whenever $\text{dim}Z\leq m-1$ or the
radial parameter is at most $\frac{R}{2}$. Fix $\varepsilon>0$,
$1<A\leq\bar{A}_{\varepsilon}$ and a transverse complete intersection
$Z=Z(P_{1},\ldots,P_{n-m})$ with $\overline{\deg}Z\leq D_{m,\varepsilon}$,
where $\bar{A}_{\varepsilon}$ and $D_{m,\varepsilon}$ are as in (53). Let $f$
be concentrated on wave packets from $\mathbb{T}_{Z}$. It suffices to show
(50) for $p=\bar{p}_{0}(k,m)$ by interpolation with the trivial $L^{2}$-bound.
We recall the two cases to be analyzed:

The cellular case: For any transverse complete intersection $Y^{l}\subseteq Z$
of dimension $1\leq l\leq m-1$ and maximum degree at most
$(D_{m,\varepsilon})^{n}$, the inequality

(54)
$\|T^{\lambda}f\|^{p}_{BL^{p}_{k,A}(N_{R^{\frac{1}{2}+\delta_{m}}/4}(Y^{l})\cap
B(0,R))}<c_{alg}\|T^{\lambda}f\|^{p}_{BL^{p}_{k,A}(B(0,R))}$

holds.

The algebraic case: There exists a transverse complete intersection
$Y^{l}\subseteq Z$ of dimension $1\leq l\leq m-1$ of maximum degree at most
$(D_{m,\varepsilon})^{n}$ such that

(55)
$\|T^{\lambda}f\|^{p}_{BL^{p}_{k,A}(N_{R^{\frac{1}{2}+\delta_{m}}/4}(Y^{l})\cap
B(0,R))}\geq c_{alg}\|T^{\lambda}f\|^{p}_{BL^{p}_{k,A}(B(0,R))}.$

Here $c_{alg}>0$ depends on $n$ and $\varepsilon$.

####  6.1.1. Cellular case

The cellular case is as usually treated by induction on the radius. Via
polynomial partitioning the $BL^{p}_{k,A}$-norm is equidistributed among the
cells, and the induction closes. This case is handled as in [14, Section
10.5]. We omit the details.

####  6.1.2. Algebraic case

The algebraic case is more involved: $T^{\lambda}f$ can be regarded as
concentrated near a low-dimensional and low degree variety $Y^{l}$ (for an
oversimplification, one can think of a hyperplane). In the _tangential
subcase_ the wave packets from $f$ are also tangential to this variety, then
we can use induction on the dimension to conclude. In the _transverse subcase_
, many wave packets are transverse to $Y^{l}$. Then we can conclude via
transverse equidistribution estimates. For the homogeneous phase functions the
transverse equidistribution estimates are different from [14, Section 10.6],
and we give the details.

Fix a transverse complete intersection $Y^{l}$ of dimension $1\leq l\leq m-1$
of maximum degree $\overline{\deg}Y^{l}\leq(D_{m,\varepsilon})^{n}$, which
satisfies (50). Let $R^{\frac{1}{2}}\ll\rho\ll R$ be such that
$\rho^{\frac{1}{2}+\delta_{l}}=R^{\frac{1}{2}+\delta_{m}}$, and note that

$R\leq R^{2\delta_{l}}\rho\text{ and }\rho\leq R^{-\delta_{l}/2}R.$

Let $\mathcal{B}_{\rho}$ be a finitely overlapping cover of $B(0,R)$ by
$\rho$-balls, and for each $B\in\mathcal{B}_{\rho}$ define

$\mathbb{T}_{B}=\\{(\theta,v)\in\mathbb{T}:T_{\theta,v}\cap
N_{R^{\frac{1}{2}+\delta_{m}}/4}(Y^{l})\cap B\neq\emptyset\\}$

and

$f_{B}:=\sum_{(\theta,v)\in\mathbb{T}_{B}}f_{\theta,v}.$

We have by the triangle inequality for broad norms

$\|T^{\lambda}f\|^{p}_{BL^{p}_{k,A}(B(0,R))}\lesssim\sum_{B\in\mathcal{B}_{\rho}}\|T^{\lambda}f_{B}\|^{p}_{BL^{p}_{k,A}(N_{R^{\frac{1}{2}+\delta_{m}}/4}(Y^{l})\cap
B)}$

up to $\text{RapDec}(R)\|f\|^{p}_{L^{2}}$ on the right-hand side by the rapid
decay of the wave packets away from the tubes.

For $B=B(y,\rho)\in\mathcal{B}_{\rho}$, let $\mathbb{T}_{B,tang}$ denote the
set of all $(\theta,v)\in\mathbb{T}_{B}$ with the property that, whenever
$x\in T_{\theta,v}$ and $z\in Y^{l}\cap B(y,2\rho)$ satisfy $|x-z|\leq
2\bar{C}_{tang}\rho^{\frac{1}{2}+\delta_{l}}$, it follows that

$\angle(G^{\lambda}(x;\omega_{\theta}),T_{z}Y^{l})\leq\frac{1}{2}\bar{c}_{tang}\rho^{-\frac{1}{2}+\delta_{l}},$

where $\bar{C}_{tang}$ and $\bar{c}_{tang}$ are the constants appearing in the
definition of tangency. Furthermore, let
$\mathbb{T}_{B,trans}=\mathbb{T}_{B}\backslash\mathbb{T}_{B,tang}$ and define

$f_{B,tang}=\sum_{(\theta,v)\in\mathbb{T}_{B,tang}}f_{\theta,v}\text{ and
}f_{B,trans}=\sum_{(\theta,v)\in\mathbb{T}_{B,trans}}f_{\theta,v}.$

It follows that $f_{B}=f_{B,tang}+f_{B,trans}$ and, by the triangle inequality
for broad norms, we conclude that

$\|T^{\lambda}f\|^{p}_{BL^{p}_{k,A}(B(0,R))}\lesssim\sum_{B\in\mathcal{B}_{\rho}}\|T^{\lambda}f_{B,tang}\|^{p}_{BL^{p}_{k,A/2}(B)}+\sum_{B\in\mathcal{B}_{\rho}}\|T^{\lambda}f_{B,trans}\|^{p}_{BL^{p}_{k,A/2}(B)}.$

Either the tangential or the transverse contribution to the above sum
dominates, and each case is treated separately.

Tangential subcase: Suppose that the tangential term dominates and we have

(56)
$\|T^{\lambda}f\|^{p}_{BL^{p}_{k,A}(B(0,R))}\lesssim\sum_{B\in\mathcal{B}_{\rho}}\|T^{\lambda}f_{B,tang}\|^{p}_{BL^{p}_{k,A/2}(B)}.$

This case can be handled by applying the dimensional induction hypothesis. We
refer to [14, pp. 345-346] for the details.

Transverse sub-case: In this case, we have

$\|T^{\lambda}f\|^{p}_{BL^{p}_{k,A}(B(0,R))}\lesssim\sum_{B\in\mathcal{B}_{\rho}}\|T^{\lambda}f_{B,trans}\|^{p}_{BL^{p}_{k,A/2}(B)}.$

The strategy in the transverse case is to use induction on radius to show that
for some $\overline{c}_{\varepsilon}>0$ one has (redenoting $f_{j}$ for
$f_{B_{j},trans}$)

(57)
$\|T^{\lambda}f_{j}\|_{BL^{p}_{k,A/2}(B)}\leq\overline{c}_{\varepsilon}E_{m,A}(R)\|f_{j}\|_{L^{2}(A^{n-1})}$

for all $B_{j}\in\mathcal{B}_{\rho}$.

Provided $\overline{c}_{\varepsilon}>0$ is chosen sufficiently small,
depending only on $n$ and $\varepsilon$, the proof is concluded by almost
orthogonality of $f_{B,\text{trans}}$ (see [14, Eq. (10.23)]):

$\|T^{\lambda}f\|_{BL^{p}_{k,A/2}(B(0,R))}\lesssim_{\varepsilon}\bar{c}_{\varepsilon}E_{m,A}(R)\|f\|^{1-\frac{2}{p}}_{L^{2}}\big{(}\sum_{B\in\mathcal{B}_{\rho}}\|f_{B,trans}\|^{2}_{L^{2}}\big{)}^{\frac{1}{p}}\leq
E_{m,A}(R)\|f\|_{L^{2}}.$

This estimate relies on Lemma 4.5.

The main obstacle to (57) is that we cannot expect $f_{j}$ to satisfy the
hypothesis of Theorem 6.1 at scale $\rho$. The remedy is to break $f_{j}$ into
pieces $f_{j,b}$, which are $\rho^{\frac{1}{2}+\delta_{m}}$-tangent to a
translated variety of $Z+b$. To obtain a favorable estimate when reassembling
the pieces, we use transverse equidistribution estimates. This still follows
[14]. Recall however, as argued in Section 5, that after restricting to a ball
of size $R^{\frac{1}{2}+\delta_{m}}$, the transverse equidistribution
estimates do not always hold true. But if these do not hold true, then the
contribution can actually be neglected. The first step is to separate the
essentially and non-essentially contributing balls.

Separating essentially and non-essentially contributing
$R^{\frac{1}{2}+\delta_{m}}$-balls: We cover $B_{j}$ by finitely overlapping
$R^{\frac{1}{2}+\delta_{m}}$-balls $B_{j,k}$. Let
$(\theta,v)\in\mathbb{T}_{Z,B_{j,k}}$ and $x\in T_{\theta,v}\cap
N_{R^{\frac{1}{2}+\delta_{m}}}(Z)\cap B_{j,k}$, $z\in Z$ with
$|x-z|\leq\bar{C}_{tang}R^{\frac{1}{2}+\delta_{m}}$. By definition of
tangency, we have

$\angle(G^{\lambda}(x;\omega_{\theta}),T_{z}Z)\lesssim
R^{-\frac{1}{2}+\delta_{m}}.$

Let $V=T_{z}Z$. By Lemma 3.4, we have

$\angle(G^{\lambda}(\bar{x};\omega_{\theta}),V)\lesssim
R^{-\frac{1}{2}+\delta_{m}}.$

Now we consider the linearization
$\tilde{\phi}(u)=\partial_{x_{n}}\phi^{\lambda}(\bar{x};\Psi^{\lambda}(u))$
around $\bar{x}$, the centre of $B_{j,k}$. We consider like in Section 5.2 a
matrix $A\in\mathbb{R}^{(n-m)\times n}$ of maximal rank such that

$V=\\{x\in\mathbb{R}^{n}:Ax=0\\},\quad L=\\{u\in B_{n-1}(0,2)\backslash
B_{n-1}(0,1/2):A(-\nabla_{u}\tilde{\phi}(u),1)=0\\},$

where $L$ denotes the set of $u$-frequencies with normal in $V$.

We apply the dichotomy of Section 5.2:

  * Case I:

$L$ is contained in $O(1)$ sectors of size $1\times K^{-2}\times\ldots\times
K^{-2}$ by Lemma 5.3. This is referred to as Case I. In Subsection 5.2 this
was called the case of narrow space-time frequencies. Note that if $L$ is
contained in $O(1)$ $1\times K^{-2}\times\ldots\times K^{-2}$-sectors, then so
is $\bigcup\theta$ with
$\angle(G^{\lambda}(\bar{x};\omega_{\theta}),V)\lesssim
R^{-\frac{1}{2}+\delta_{m}}$ by Lemma 3.4. Consequently, by $2\leq k\leq m$,
Case I-balls can be neglected in the $k$-broad norm (see (58) below).

  * Case II:

This was referred to as case of broad space-time frequencies. We consider the
further refinement $\mathbb{T}_{V,B_{j,k},\tau}$ with $\tau$ a
$\rho^{-\frac{1}{2}}$-sector. By Lemma 5.6, there is a quantitatively
transverse subspace $W$ with $\bar{V}\oplus W=\mathbb{R}^{n}$ and

$\angle(V,W)\gtrsim K^{-4}.$

$\bar{V}$ denotes the extension of a tangent space of $L$ from Subsection 5.2.

We let $X_{I}$ and $X_{II}$ denote the union of balls $B_{j,k}$ from Cases I
and II.

Sorting into medium tubes: Redenote $g=f_{j,trans}$. Next, we use the sorting
into medium tubes as in [14, Section 9]111The arguments in [14, Section 9]
clearly only depend on the non-degeneracy condition $C1)$.. The idea is to
carry out a wave packet decomposition at smaller scale $\rho\ll R$ on the
$\rho$-ball. This requires us to mildly modify the phase function, amplitude
function, and input function $\phi,a,g\to\tilde{\phi},\tilde{a},\tilde{g}$ to
center the $\rho$-ball at the origin. The wave packets at scale $\rho$ are
denoted by $\tilde{\mathbb{T}}$ with indices $\tilde{\theta}$ (a
$\rho^{-\frac{1}{2}}$-cap) and
$\tilde{v}\in\rho^{\frac{1+\delta}{2}}\mathbb{Z}^{n-1}$. We can break the
$R$-wave packet $(\theta,v)\in\mathbb{T}$ into smaller scale $\rho$-wave
packets essentially contributing to $(\theta,v)$. These $\rho$-wave packets
are denoted by $\tilde{\mathbb{T}}_{\theta,v}$ (see [14, Lemma 9.1]).

We use the following facts from [14]: Firstly, we recall how the tangency
properties are inherited. The $\rho$-wave packets obtained from a wave packet
$R^{-\frac{1}{2}+\delta_{m}}$-tangential to a variety $Z$ are
$\rho^{-\frac{1}{2}+\delta_{m}}$-tangential to a translate $Z-b$ (cf. [14,
Proposition 9.2]).

There is a sorting
$\big{(}\mathcal{T}_{\tilde{\theta},w}\big{)}_{\tilde{\theta},w}$,
$\mathcal{T}_{\tilde{\theta},w}\subseteq\mathbb{T}$ with $\tilde{\theta}$ a
$\rho^{-\frac{1}{2}}$-cap and $w\in R^{\frac{1+\delta}{2}}\mathbb{Z}^{n-1}$,
which partitions $\mathbb{T}$ (see [14, Section 9.3]). The corresponding
_medium tubes_ of length $\rho$ and width $R^{\frac{1}{2}+\delta}$ are given
by

$T_{\tilde{\theta},w}=\big{(}\bigcup_{(\theta,v)\in\mathcal{T}_{\tilde{\theta},w}}T_{\theta,v}\big{)}\cap
B(y,\rho).$

We have the almost orthogonality:

$\|g\|_{2}^{2}\sim\sum_{(\tilde{\theta},w)\in\mathcal{T}}\|g_{\tilde{\theta},w}\|_{2}^{2},$

and we have that for every $(\theta,v)\in\mathcal{T}_{\tilde{\theta},w}$ it
holds

$\text{dist}_{H}(T_{\theta,v}\cap B(y,\rho),T_{\tilde{\theta},w})\lesssim
R^{\frac{1}{2}+\delta}.$

We can also consider the $\rho$-wave packet decomposition of
$(g_{\tilde{\theta},w})\tilde{\;}$. This is concentrated on scale $\rho$-wave
packets belonging to
$\bigcup_{(\theta,v)\in\mathcal{T}_{\tilde{\theta},w}}\tilde{\mathbb{T}}_{\theta,v}$.
We obtain a covering
$(\tilde{\mathcal{T}}_{\tilde{\theta},w})_{\tilde{\theta},w}$ of
$\tilde{\mathbb{T}}$ by almost disjoint sets. Hence, the sorting into medium
tubes is suitable to switch between $R$ and $\rho$-wave packets.

We define the essential part like in [26, p. 3582]:

$g_{ess}=\sum_{(\tilde{\theta},w)\in\mathcal{T}_{ess}}g_{\tilde{\theta},w}=g-\sum_{(\tilde{\theta},w)\in\mathcal{T}_{tail}}g_{\tilde{\theta},w},$

where

$\begin{split}\mathcal{T}_{ess}&=\\{(\tilde{\theta},w):\,\exists(\theta,v)\in\mathcal{T}_{\tilde{\theta},w}:T_{\theta,v}\cap
X_{II}\neq\emptyset\\},\\\
\mathcal{T}_{tail}&=\\{(\tilde{\theta},w):\,\forall(\theta,v)\in\mathcal{T}_{\tilde{\theta},w}:T_{\theta,v}\cap
X_{II}=\emptyset\\}.\end{split}$

Like in [26], we infer that

(58)
$\|T^{\lambda}g\|_{BL^{p}_{k,A/2}(B_{j})}\leq\|T^{\lambda}g_{ess}\|_{BL^{p}_{k,A/4}(B_{j})}+\text{RapDec}(R)\|f\|_{L^{2}}.$

Breaking the essential contribution into smaller neighbourhoods of translates:
As in [14], we choose a set of translates $\mathcal{B}$, so that we can write

(59) $\|T^{\lambda}g_{ess}\|_{BL^{p}_{k,A/4}(B_{j})}\lesssim(\log
R)^{2}\big{(}\sum_{b\in\mathcal{B}}\|T^{\lambda}g_{ess,b}\|^{p}_{BL^{p}_{k,A/4}(B_{j})}\big{)}^{\frac{1}{p}},$

where each piece $g_{ess,b}$ is defined so that it is concentrated on scale
$\rho$ wave packets, which are tangential to some translate $Z-y+b$ of $Z$.
This construction is provided in [14, Lemma 10.5]. The set of translates
$\mathcal{B}$ are chosen so that effectively

  * (i)

$N_{R^{\frac{1}{2}+\delta_{m}}}(Z)\subseteq\bigcup_{b\in\mathcal{B}}N_{\rho^{\frac{1}{2}+\delta_{m}}}(Z-y+b)$,

  * (ii)

$N_{\rho^{\frac{1}{2}+\delta_{m}}}(Z-y+b)$ are essentially disjoint.

Technically, this decomposition also involves to “tune” the wave packets of
$g_{ess,b}$ to a collection of balls $\mathcal{B}^{\prime}\subseteq
B_{K^{2}}$. This quantifies (i) and (ii) and is necessary due to the
definition of the broad norm. In the following we take the resulting almost
orthogonality (61) for granted (see [14, Lemma 9.6]).

By modifying $T^{\lambda}$ and $g_{ess,b}$, we can center the ball $B_{j}$ at
the origin:

$\|T^{\lambda}g_{ess,b}\|_{BL^{p}_{k,A/4}(B_{j})}\lesssim(\log
R)^{2}\big{(}\sum_{b\in\mathcal{B}}\|\tilde{T}^{\lambda}(g_{ess,b})\tilde{\;}\|^{p}_{BL^{p}_{k,A/4}(B(0,\rho))}\big{)}^{\frac{1}{p}}.$

Now we can apply the radial induction hypothesis:

$\big{(}\sum_{b\in\mathcal{B}}\|\tilde{T}^{\lambda}(g_{ess,b})\tilde{\;}\|^{p}_{BL^{p}_{k,A/4}(B(0,\rho))}\big{)}^{\frac{1}{p}}\leq
E_{m,A/4}(\rho)\big{(}\sum_{b\in\mathcal{B}}\|g_{ess,b}\|^{p}_{L^{2}}\big{)}^{\frac{1}{p}}.$

Once we have proved

(60)
$\big{(}\sum_{b\in\mathcal{B}}\|\tilde{g}_{ess,b}\|^{p}_{L^{2}}\big{)}^{\frac{1}{p}}\lesssim
R^{O(\delta_{m})}\big{(}\frac{\rho}{R}\big{)}^{(n-m)\big{(}\frac{1}{4}-\frac{1}{2p}\big{)}}\|g_{ess}\|_{2},$

the computation to conclude the proof is like in [14]. (60) is proved via
interpolation between $p=2$ and $p=\infty$.

Reassembling the set of translates: For $p=2$ we can argue like in [14] based
on the almost orthogonality of the $\rho$-wave packets contributing to
$(g_{ess,b})\tilde{\;}$ (here we also use the essential disjointness of the
translates in (ii) above):

(61)
$\big{(}\sum_{b\in\mathcal{B}}\|g_{ess,b}\|^{2}_{L^{2}}\big{)}^{\frac{1}{2}}\lesssim\|g_{ess}\|_{2}.$

We turn to the proof of (60) for $p=\infty$, which requires an additional
argument compared to [14]. The crucial ingredient remains transverse
equidistribution. By almost orthogonality of
$(\tilde{\theta},w)\in\mathcal{T}$ and the definition of $\mathcal{T}_{ess}$
we have

$\|g_{ess,b}\|^{2}_{L^{2}}\sim\sum_{(\tilde{\theta},w)\in\mathcal{T}_{ess}}\|(g_{ess,\tilde{\theta},w})_{b}\tilde{\;}\|^{2}_{L^{2}}.$

By construction of $g_{ess,\tilde{\theta},w}$ there is
$(\theta,v)\in\mathcal{T}_{\tilde{\theta},w}$ such that $T_{\theta,v}$
intersects $X_{II}$. Let $B=B(\bar{x};R^{\frac{1}{2}+\delta_{m}})$ denote the
corresponding ball in $X_{II}$. Since the Hausdorff distance between
$T_{\theta_{1},v_{1}}$ for further
$(\theta_{1},v_{1})\in\mathcal{T}_{\tilde{\theta},w}$ is $\lesssim
R^{\frac{1}{2}+\delta}$, we can apply the reverse Hörmander estimate from [14]
at scale $\rho$ with $r\sim R^{\frac{1}{2}+\delta_{m}}$ to find that

(62) $\|(g_{ess,\tilde{\theta},w})_{b}\tilde{\;}\|_{L^{2}}^{2}\lesssim
R^{-\frac{1}{2}-\delta_{m}}\|\tilde{T}_{*}^{\lambda}(g_{ess,\tilde{\theta},w})_{b}\tilde{\;}\|_{L^{2}(B(\bar{x}-y;10R^{\frac{1}{2}+\delta_{m}}))}^{2}.$

The reverse Hörmander estimate [14, Lemma 9.5] relies on every $\rho$-wave
packet of $(g_{ess,\tilde{\theta},w})_{b}\tilde{\;}$ intersecting the
$R^{\frac{1}{2}+\delta_{m}}$-ball $B$. The proof follows from Fourier series
expansion and the constant-coefficient analog. This yields a slightly
different oscillatory integral operator $\tilde{T}_{*}^{\lambda}$, whose data
however inherits the properties of $(\phi,a)$. We refer to [14].

Next, we can apply Lemma 5.8 to find that222This holds up to
$\text{RapDec}(R)$. Note the shift and localization to
$N_{\rho^{\frac{1}{2}+\delta_{m}}}(Z+b)$ on the left-hand side, which is
explained in [14, Equation (9.10)].

(63)
$\|T_{*}^{\lambda}g_{ess,b,\tilde{\theta},w}\|^{2}_{L^{2}(B(\bar{x};10R^{\frac{1}{2}+\delta_{m}})\cap
N_{\rho^{\frac{1}{2}+\delta_{m}}}(Z+b))}\lesssim
R^{\frac{1}{2}+O(\delta_{m})}\big{(}\frac{\rho}{R}\big{)}^{\frac{n-m}{2}}\|(g_{ess,\tilde{\theta},w})_{b}\tilde{\;}\|^{2}_{L^{2}}.$

Now we can use the almost orthogonality induced by
$\tilde{\mathcal{T}}_{\tilde{\theta},w}$ to find after taking (62) and (63)
together:

$\begin{split}\sum_{(\tilde{\theta},w)\in\mathcal{T}_{ess}}\|(g_{ess,\tilde{\theta},w})_{b}\tilde{\;}\|_{L^{2}}^{2}&\lesssim
R^{O(\delta_{m})}\big{(}\frac{\rho}{R}\big{)}^{\frac{n-m}{2}}\sum_{(\tilde{\theta},w)\in\mathcal{T}_{ess}}\|(g_{ess,\tilde{\theta},w})_{b}\tilde{\;}\|_{L^{2}}^{2}\\\
&\lesssim
R^{O(\delta_{m})}\big{(}\frac{\rho}{R}\big{)}^{\frac{n-m}{2}}\|g_{ess,b}\|_{L^{2}}^{2}\\\
&\lesssim
R^{O(\delta_{m})}\big{(}\frac{\rho}{R}\big{)}^{\frac{n-m}{2}}\|g_{ess}\|_{L^{2}}^{2}.\end{split}$

This finishes the proof of (60), and we can argue that induction closes like
in [14].

$\hfill\Box$

##  7\. From $k$-broad to linear estimates

In this section we deduce the linear estimates from the $k$-broad estimates by
applying the Bourgain–Guth argument [7]. We show the following proposition:

######  Proposition 7.1.

Suppose that for all $K\geq 1$ and all $\varepsilon>0$ any oscillatory
integral operator $T^{\lambda}$ built from reduced data $(\phi,a)$ obeys the
$k$-broad inequality

(64)
$\|T^{\lambda}f\|_{BL^{p}_{k,A}(B(0,R))}\lesssim_{\varepsilon}K^{C_{\varepsilon}}R^{\varepsilon}\|f\|_{L^{p}(A^{n-1})}$

for some fixed $k$, $A=A_{\varepsilon}$, $p$, $C_{\varepsilon}$, and all
$R\geq 1$. If

(65) $p(k,n)\leq p\leq\frac{2n}{n-2},\qquad
p(k,n)=\begin{cases}2\cdot\frac{n-1}{n-2}\quad\text{ if }2\leq k\leq 3,\\\
2\cdot\frac{2n-k+1}{2n-k-1}\quad\text{ if }k>3,\end{cases}$

then any oscillatory integral operator with $C1)$ and $C2^{+})$ phase $\phi$
and amplitude $a$ satisfies

(66)
$\|T^{\lambda}f\|_{L^{p}(\mathbb{R}^{n})}\lesssim_{\phi,\varepsilon,a}\lambda^{\varepsilon}\|f\|_{L^{p}(A^{n-1})}.$

From this proposition Theorem 1.1 is immediate by choosing $k=\frac{n+1}{2}$
for $n$ odd and $k=\frac{n}{2}+1$ for $n$ even as $\max(p(k,n),\bar{p}(k,n))$
gives the lower bound for $p$ in Theorem 1.1.

###  7.1. Outline of the argument

For the proof we use induction on scales: $Q_{p,\delta}(R)$ will denote the
infimum over all constants $C$ for which the estimate

$\|T^{\lambda}f\|_{L^{p}(B(0,r))}\leq C\|f\|_{L^{p}(A^{n-1})}$

holds for $1\leq r\leq R$ and all oscillatory integral operators built from a
suitable class of phase functions. This class is supposed to be invariant
under rescaling and amenable to narrow decoupling, which is explained below.

With this definition, it remains to prove that for $p$ as in Proposition 7.1

$Q_{p,\delta}(R)\lesssim_{\varepsilon}R^{\varepsilon}$

for all $\varepsilon>0$ and $1\leq R\leq\lambda$.
To this end, we decompose $B(0,R)$ into finitely overlapping balls $B_{K^{2}}$
of radius $K^{2}$ and estimate $\|T^{\lambda}f\|_{L^{p}(B_{K^{2}})}$. $f$ is
decomposed into “broad” and “narrow” term. The narrow term is of the form

(67) $\sum_{\begin{subarray}{c}\tau\in V_{a}\\\ \text{for some
}a\end{subarray}}f_{\tau},$

consisting of contributions to $f$ from sectors for which
$G^{\lambda}(\bar{x};\tau)$ makes a small angle with some member of a family
of $(k-1)$-planes. Here $\bar{x}$ denotes the centre of $B_{K^{2}}$. The broad
term consists of contributions to $f$ from the remaining sectors. One may
choose the planes $V_{1},\ldots,V_{A}$ so that the broad term can be bounded
by the $k$-broad inequality. Thus, $f$ of the form (67) has to be analyzed.
This is accomplished by narrow $\ell^{p}$-decoupling and rescaling.

###  7.2. Narrow decoupling and the induction quantity

In this subsection we shall find an estimate for $V\subseteq\mathbb{R}^{n}$ an
$m$-dimensional linear subspace:

$\big{\|}\sum_{\tau\in
V}T^{\lambda}g_{\tau}\big{\|}_{L^{p}(B_{K^{2}})}\lesssim_{\delta}\max(1,K^{(m-2)\big{(}\frac{1}{2}-\frac{1}{p}\big{)}})K^{\delta}\big{(}\sum_{\tau\in
V}\|T^{\lambda}g_{\tau}\|^{p}_{L^{p}(w_{B_{K^{2}}})}\big{)}^{\frac{1}{p}}+l.o.t..$

The additional assumptions to be imposed for the above estimate to hold will
motivate our choice of induction quantity below.

In the translation-invariant case, e.g., with $\mathcal{E}$ as in (1), this
follows from the $\ell^{2}$-decoupling

$\big{\|}\sum_{\tau}\mathcal{E}g_{\tau}\big{\|}_{L^{p}(B_{K^{2}})}\lesssim_{\delta}K^{\delta}\big{(}\sum_{\tau\in
V}\|\mathcal{E}g_{\tau}\|^{2}_{L^{p}(w_{B_{K^{2}}})}\big{)}^{\frac{1}{2}}$

for $2\leq p\leq\frac{2n}{n-2}$ and by counting the sectors $\tau$ such that
$\angle(G(\tau),V)\leq K^{-2}$. This is carried out in [26]; see also [17,
Lemma 2.2]. It appears that the argument from [17, 26] does not suffice to
count the sectors already for general translation-invariant phases under a
convexity assumption, for which reason an additional technical assumption is
introduced.

Let $\phi:\mathbb{R}^{n-1}\backslash\\{0\\}\to\mathbb{R}$ be a $1$-homogeneous
and smooth function with $\text{supp}(\phi)\subseteq\Xi$.333Recall that
$\Xi\subseteq A^{n-1}$ denotes a sector with small aperture $c\ll 1$ centred
at $e_{n-1}$. By Taylor expansion we find

(68)
$\begin{split}\phi(\omega^{\prime},\omega_{n-1})&=\omega_{n-1}\phi(\omega^{\prime}/\omega_{n-1},1)\\\
&=\omega_{n-1}\phi(e_{n-1})+\partial_{\omega^{\prime}}\phi(e_{n-1})\omega^{\prime}+\frac{\langle\partial^{2}_{\omega^{\prime}\omega^{\prime}}\phi(e_{n-1})\omega^{\prime},\omega^{\prime}\rangle}{2\omega_{n-1}}+K^{-4}E(\omega),\end{split}$

with $E(\omega)$ $1$-homogeneous. By comparison with Taylor’s formula we have

$K^{-4}E(\omega)=\sum_{|\alpha|=3}\frac{3}{\alpha!}\int_{0}^{1}(1-s)^{2}(\partial_{\omega^{\prime}}^{\alpha}\phi)(\frac{s\omega^{\prime}}{\omega_{n-1}},1)ds\frac{(\omega^{\prime})^{\alpha}}{\omega_{n-1}^{2}}.$

Gao _et al._ [10] showed that if $E$ is bounded in $C^{N}$ one can recover the
narrow decoupling of general homogeneous phases satisfying a convexity
condition in the constant coefficient case. This leads to the following notion
of $K$-flatness.

######  Definition 7.2.

Let $\phi$ be like above. We say that $\phi$ is $K$-flat if $E$ like in (68)
satisfies

$\|\partial^{\alpha}E\|_{L^{\infty}(\Xi)}\leq C_{flat}$

for $0\leq|\alpha|\leq N_{0}$.

In this definition $N_{0}$ and $C_{flat}$ are universal constants, which allow
us to extend the arguments of [17, 26] for the circular cone to $K$-flat phase
functions as shown in [10].

To apply narrow decoupling for the variable coefficient operator on a small
$K^{2}$-ball with $K^{2}\lesssim\lambda^{\frac{1}{2}-\varepsilon}$, we
approximate the variable coefficient phase with a constant coefficient phase.
Beltran–Hickman–Sogge [2] worked out that this is possible by Taylor
expansion.

We need the following notations: Let $\phi$ be a reduced phase and
$\bar{x}\in\mathbb{R}^{n}$, which will be the centre of the small ball on
which we want to apply decoupling. Recall that
$u\mapsto\partial_{x}\phi^{\lambda}(\bar{x};\Psi^{\lambda}(\bar{x};u))$ is a
graph parametrization of the hypersurface $\Sigma_{\bar{u}}$. We have

$\langle
x,(\partial_{x}\phi^{\lambda})(\bar{x};\Psi^{\lambda}(\bar{x};u))\rangle=\langle
x^{\prime},u\rangle+x_{n}h_{\bar{x}}(u)$

for all $x=(x^{\prime},x_{n})\in\mathbb{R}^{n}$ with
$h_{\bar{x}}(u)=(\partial_{x_{n}}\phi^{\lambda})(\bar{x};\Psi^{\lambda}(\bar{x};u))$.
Recall that we can suppose $a(x;\omega)=a_{1}(x)a_{2}(\omega)$ by Fourier
series expansion. Let $E_{\bar{x}}$ denote the extension operator associated
to $\Sigma_{\bar{x}}$, given by

$E_{\bar{x}}g(x)=\int_{\mathbb{R}^{n-1}}e^{i(\langle
x^{\prime},u\rangle+x_{n}h_{\bar{x}}(u))}a_{\bar{x}}(u)g(u)du\text{ for all
}x\in\mathbb{R}^{n},$

where
$a_{\bar{x}}(u)=a_{2}\circ\Psi^{\lambda}(\bar{x};u)|\det\partial_{u}\Psi^{\lambda}(\bar{x};u)|$.
We recall how $T^{\lambda}$ is approximated by $E_{\bar{x}}$: Let $x\in
B(\bar{x};K^{2})\subseteq B(0,3\lambda/4)$. By change of variables
$\omega=\Psi^{\lambda}(\bar{x};u)$ and a Taylor expansion of $\phi^{\lambda}$
around $\bar{x}$, we have

$T^{\lambda}f(x)=\int_{\mathbb{R}^{n-1}}e^{i(\langle
x-\bar{x},(\partial_{x}\phi^{\lambda})(\bar{x};\Psi^{\lambda}(\bar{x};u))\rangle+\mathcal{E}^{\lambda}_{\bar{x}}(x-\bar{x};u))}a_{1}^{\lambda}(x)a_{\bar{x}}(u)f_{\bar{z}}(u)du$

with
$f_{\bar{x}}=e^{i\phi^{\lambda}(\bar{x};\Psi^{\lambda}(\bar{x};\cdot))}f\circ\Psi^{\lambda}(\bar{x};\cdot)$
and by Taylor expansion

$\mathcal{E}^{\lambda}_{\bar{x}}(v;u)=\frac{1}{\lambda}\int_{0}^{1}(1-r)\langle(\partial_{xx}\phi)((\bar{x}+rv)/\lambda;\Psi^{\lambda}(\bar{x};u))v,v\rangle
dr.$

Let $N\gg 1$ be a large constant to be specified. By the derivative bounds

$\sup_{(v;u)\in
B(0,K^{2})\times\text{supp}a_{\bar{x}}}|\partial^{\beta}_{\omega}\mathcal{E}_{\bar{x}}^{\lambda}(v;u)|\lesssim_{N}1$

and Fourier series expansion, the oscillation of
$\mathcal{E}^{\lambda}_{\bar{x}}$ can be neglected. This yields the following
lemma:

######  Lemma 7.3 ([2, Lemma 2.6]).

Let $T^{\lambda}$ be an oscillatory integral operator built from reduced data
$(\phi,a)$. Let $0<\delta\leq 1/2$, $1\leq
K^{2}\leq\lambda^{\frac{1}{2}-\delta}$ and $\bar{x}/\lambda\in X$ so that
$B(\bar{x};K^{2})\subseteq B(0,3\lambda/4)$.

  * •

Then

(69)
$\|T^{\lambda}f\|_{L^{p}(w_{B(\bar{x};K^{2})})}\lesssim_{N}\|E_{\bar{x}}f_{\bar{x}}\|_{L^{p}(w_{B(0;K^{2})})}+\lambda^{-\frac{\delta
N}{2}}\|f\|_{L^{2}}$

holds provided that $N$ is sufficiently large depending on $n$, $\delta$, and
$p$, and $w_{B_{K^{2}}}=(1+K^{-2}|x-\bar{x}|)^{-N}$ is a rapidly decaying
weight off $B_{K^{2}}$.

  * •

Suppose that $|\bar{x}|\leq\lambda^{1-\delta^{\prime}}$. There exists a family
of operators $\mathbf{T}^{\lambda}$ all with phase $\phi$ and of type
$(1,1,C)$ data (see Definition 3.1) such that

(70)
$\|E_{\bar{x}}f_{\bar{x}}\|_{L^{p}(w_{B(0,K^{2})})}\lesssim_{N}\|T^{\lambda}_{*}f\|_{L^{p}(w_{B(\bar{x};K^{2})})}+\lambda^{-\frac{N\min(\delta,\delta^{\prime})}{2}}\|f\|_{2}$

holds for some $T^{\lambda}_{*}\in\mathbf{T}^{\lambda}$. The family
$\mathbf{T}^{\lambda}$ has cardinality $O_{N}(1)$ and is independent of
$B(\bar{x};K^{2})$.

To apply the narrow decoupling to $E_{\bar{x}}f_{\bar{x}}$, we need the
constant coefficient phase

$h_{\bar{x}}(u)=\partial_{x_{n}}\phi^{\lambda}(\bar{x};\Psi^{\lambda}(\bar{x};u))$

to be $K$-flat.

######  Definition 7.4.

Let $K\gg 1$. We say that a reduced homogeneous phase
$\phi:\mathbb{R}^{n}\times\mathbb{R}^{n-1}\backslash\\{0\\}\to\mathbb{R}$ is
$K$-flat, if all its constant-coefficient approximations $h_{\bar{x}}$ for
$\bar{x}\in X$ are $K$-flat and $\phi$ satisfies

  * •

for $1\leq k\leq n-1$ and
$\beta=(\beta^{\prime},\beta_{n-1})\in\mathbb{N}_{0}^{n-1}$ with $|\beta|\leq
N_{0}+5$ and $|\beta^{\prime}|\geq 2$:

(71)
$\|\partial_{x_{k}}\partial^{\beta}_{\omega}\phi\|_{L^{\infty}(X\times\Xi)}\leq
K^{-4},$

  * •

for $\beta=(\beta^{\prime},\beta_{n-1})\in\mathbb{N}_{0}^{n-1}$ with
$|\beta|\leq N_{0}+5$ and $|\beta^{\prime}|\geq 3$:

(72)
$\|\partial_{x_{n}}\partial^{\beta}_{\omega}\phi\|_{L^{\infty}(X\times\Xi)}\leq
K^{-4}.$

The derivative bounds are used to facilitate induction: They will allow us to
argue that the rescaled phase function becomes “flatter”. The choice of size
of $\beta^{\prime}$ in (71) and (72) will become clear from the formula for
the phase function after rescaling. The size conditions (71) and (72) (but not
the number of derivatives!) are more restrictive than the definition of a
reduced phase function. We remark that with this definition, Proposition 7.5
now follows from the constant-coefficient decoupling and the approximation by
constant-coefficient operators provided by the previous lemma.

Regarding the choice of $N$ in Lemma 7.3: We choose
$N=N(\delta)=N(\varepsilon)\geq N_{0}+5$ (since $\delta=\delta(\varepsilon)$)
such that the error term $\lambda^{-\delta N/2}\|f\|_{2}$ becomes manageable
when we carry out the induction on scales.

######  Proposition 7.5 (Narrow variable coefficient decoupling).

Suppose that $T^{\lambda}$ is an oscillatory integral operator with reduced
$C1)$ and $C2^{+})$ phase $\phi$, which is $K$-flat, and let
$B_{K^{2}}\subseteq B(0,\lambda^{1-\delta})$ with $1\leq
K^{2}\leq\lambda^{\frac{1}{2}-\delta}$, $0<\delta\leq 1/2$. If
$V\subseteq\mathbb{R}^{n}$ is an $m$-dimensional linear subspace, then for
$2\leq p\leq\frac{2n}{n-2}$ and any $\delta>0$ one has

$\begin{split}\big{\|}\sum_{\tau\in
V}T^{\lambda}g_{\tau}\big{\|}_{L^{p}(B_{K^{2}})}&\lesssim_{\delta,N}\max(1,K^{(m-2)\big{(}\frac{1}{2}-\frac{1}{p}\big{)}})K^{\delta}\big{(}\sum_{\tau\in
V}\|T_{*}^{\lambda}g_{\tau}\|_{L^{p}(w_{B_{K^{2}}})}^{p}\big{)}^{\frac{1}{p}}\\\
&\quad+\lambda^{-\frac{\delta N}{2}}\|g\|_{L^{2}}\end{split}$

provided that $N$ is chosen sufficiently large depending on $n$, $\delta$, and
$p$. Here, the sum ranges over sectors $\tau$ for which
$\angle(G^{\lambda}(\bar{x};\tau),V)\leq K^{-2}$, where $\bar{x}$ is the
centre of $B_{K^{2}}$ and $w_{B_{K^{2}}}=(1+K^{-2}|x-\bar{x}|)^{-N}$ is a
rapidly decaying weight off $B_{K^{2}}$. $T^{\lambda}_{*}$ is an oscillatory
integral operator with phase $\phi$ and some amplitude $a_{*}$ chosen from a
family of $O_{N}(1)$ many amplitudes. The amplitudes $a_{*}$ are reduced after
uniform decomposition of the support of $a_{*}$.

###### Proof.

We apply (69) from Lemma 7.3 to approximate $T^{\lambda}$ with $E_{\bar{x}}$:

(73) $\|\sum_{\tau\in
V}T^{\lambda}f_{\tau}\|_{L^{p}(B_{K^{2}})}\lesssim_{N}\|E_{\bar{x}}\sum_{\tau\in
V}f_{\bar{x},\tau}\|_{L^{p}(w_{B_{K^{2}}})}+\lambda^{-\frac{\delta
N}{2}}\|f\|_{2}.$

Since by our assumption on $\phi$, the underlying phase $h_{\bar{x}}$ for
$E_{\bar{x}}$ is $K$-flat. Moreover, the transformed functions
$f_{\bar{x},\tau}$ are disjoint sectors by diffeomorphism property of
$\Psi^{\lambda}(\bar{x};\cdot)$. Hence, we can apply the narrow decoupling for
$K$-flat constant-coefficient phase functions by [10]:

(74) $\|\sum_{\tau\in
V}E_{\bar{x}}f_{\bar{x},\tau}\|_{L^{p}(w_{B_{K^{2}}})}\lesssim_{\delta}K^{\delta}\max(1,K^{(m-2)\big{(}\frac{1}{2}-\frac{1}{p}\big{)}})\big{(}\sum_{\tau\in
V}\|E_{\bar{x}}f_{\bar{x},\tau}\|^{p}_{L^{p}(w_{B_{K^{2}}})}\big{)}^{\frac{1}{p}}.$

We can apply (70) from Lemma 7.3 to find:

(75)
$\|E_{\bar{x}}f_{\bar{x},\tau}\|_{L^{p}(w_{B_{K^{2}}})}\lesssim_{\delta,N}\|T^{\lambda}_{*}f_{\tau}\|_{L^{p}(w_{B_{K^{2}}})}+\lambda^{-\frac{\delta
N}{2}}\|f_{\tau}\|_{L^{2}}.$

Recall that $T^{\lambda}_{*}$ is a variable-coefficient extension operator
from a family $\mathbf{T}^{\lambda}$, which is of size $O_{N}(1)$. The
operators from $\mathbf{T}^{\lambda}$ have phase $\phi$, but possibly
different amplitude $a_{*}$, which is independent of $\bar{x}$, but still
satisfies the bounds

$|\partial_{\xi}^{\alpha}a_{*}|\lesssim_{N}1\text{ for
}\alpha\in\mathbb{N}_{0}^{n-1},\;|\alpha|\leq N.$

Hence, taking the $p$-th power, summing over the sectors, and pigeonholing in
$T^{\lambda}_{*}$ yields

$\big{(}\sum_{\tau}\|E_{\bar{x}}f_{\bar{x},\tau}\|_{L^{p}(w_{B_{K^{2}}})}^{p}\big{)}^{\frac{1}{p}}\lesssim_{\delta,N}\big{(}\sum_{\tau}\|T^{\lambda}_{*}f_{\tau}\|_{L^{p}(w_{B_{K^{2}}})}^{p}\big{)}^{\frac{1}{p}}+\lambda^{-\frac{\delta
N}{2}}\|f\|_{L^{p}}.$

We take (73), (74), and (75) together to find

$\begin{split}\big{\|}\sum_{\tau\in
V}T^{\lambda}f_{\tau}\big{\|}_{L^{p}(B_{K^{2}})}&\lesssim_{N,\delta}K^{\delta}\max(1,K^{(m-2)\big{(}\frac{1}{2}-\frac{1}{p}\big{)}})\big{(}\sum_{\tau}\|T^{\lambda}_{*}f_{\tau}\|^{p}_{L^{p}(w_{B_{K^{2}})})}\big{)}^{\frac{1}{p}}\\\
&\quad+\lambda^{\frac{n}{2}-\frac{\delta N}{2}}\|f\|_{p}.\end{split}$

∎

We choose $N=5n/\delta$, which keeps the error term $\lambda^{-\frac{\delta
N}{2}}\|g\|_{L^{2}}$ manageable even after summing over the balls.

The technicality of dealing with different amplitude functions is handled by
considering an appropriate class of data $(\phi,a)$, for which the induction
on scales is carried out:

######  Definition 7.6.

For $1\leq p\leq\infty$ and $R\geq 1$ let $Q_{p,\delta}(R)$ denote the infimum
over all constants $C$ for which the estimate

$\|T^{\lambda}f\|_{L^{p}(B(0,r))}\leq C\|f\|_{L^{p}(A^{n-1})}$

holds for $1\leq r\leq R\leq\lambda$ and all oscillatory integral operators
$T^{\lambda}$ built from reduced data $(\phi,a)$ with $\lambda^{\delta}$-flat
phase function.

###  7.3. Parabolic rescaling

In this section we shall see how parabolic rescaling flattens the phase
function. Moreover, we observe how for functions supported on a small sector,
we can apply the estimate from the previous definition on the smaller scale.

####  7.3.1. Change of spatial variables

For the parabolic rescaling we also make use of a change of variables on the
spatial side.

######  Lemma 7.7 (Change of spatial variables).

Let
$x=(x^{\prime\prime},x_{n-1},x_{n})\in\mathbb{R}^{n-2}\times\mathbb{R}\times\mathbb{R}$
and $\phi$ be a reduced phase function. For $x_{n}\in T$, $\omega\in\Xi$ there
is a smooth mapping $\Upsilon(\cdot,x_{n};\omega)$, which satisfies

$\partial_{\omega^{\prime}}\phi(\Upsilon(x^{\prime},x_{n};\omega),x_{n};\omega)=x^{\prime\prime}\text{
and }\phi(\Upsilon(x;\omega),x_{n};\omega)=x_{n-1}.$

Moreover, we have the uniform derivative bounds:

(76) $|\partial_{x^{\prime}}\Upsilon|\leq C_{\Upsilon}.$

###### Proof.

We shall apply the implicit function theorem. We consider the defining
equations

$F(\Upsilon(x^{\prime},x_{n};\omega),x_{n};\omega)=\begin{pmatrix}\partial_{\omega^{\prime}}\phi(\Upsilon(x^{\prime},x_{n};\omega),x_{n};\omega)\\\
\phi(\Upsilon(x;\omega),x_{n};\omega)\end{pmatrix}=\begin{pmatrix}x^{\prime\prime}\\\
x_{n-1}\end{pmatrix}$

and differentiate:

$\underbrace{\begin{pmatrix}\partial^{2}_{x^{\prime}\omega}\phi(\Upsilon(x^{\prime},x_{n};\omega),x_{n};\omega)\\\
\partial_{x^{\prime}}\phi(\Upsilon(x^{\prime},x_{n};\omega),x_{n};\omega)\end{pmatrix}}_{A}\partial_{x^{\prime}}\Upsilon(x^{\prime},x_{n};\omega)=1_{n-1}.$

By $1$-homogeneity we have

$\partial_{x^{\prime}}\phi(x;\omega)=\sum_{j=1}^{n-1}\omega_{j}\cdot\partial^{2}_{x^{\prime}\omega_{j}}\phi(x;\omega).$

Thus, for each $x_{n}\in T$ and $\omega\in\Xi$, the Jacobian determinant of
the map
$x^{\prime}\mapsto((\partial_{\omega^{\prime}}\phi(x;\omega),\phi(x;\omega))$
is given by $\omega_{n-1}\cdot\det\partial^{2}_{\omega
x^{\prime}}\phi(x;\omega)$ and hence is non-vanishing. In fact, it is
uniformly bounded from above and below for reduced phases.

Hence,
$\partial_{x^{\prime}}F(\Upsilon(x^{\prime},x_{n};\omega),x_{n};\omega)$ is
invertible, and therefore $\Upsilon(\cdot,x_{n};\omega)$ exists for $x_{n}\in
T$, $\omega\in\Xi$. We obtain by Cramer’s rule

$\partial_{x^{\prime}}\Upsilon(x^{\prime},x_{n};\omega)=\begin{pmatrix}\partial^{2}_{x^{\prime}\omega}\phi(\Upsilon(x^{\prime},x_{n};\omega))\\\
\partial_{x^{\prime}}\phi(x^{\prime},x_{n};\omega)\end{pmatrix}^{-1}=\frac{\text{ad}(A)}{\det(A)}.$

By reducedness of $\phi$, we have $\det A\sim 1$. Secondly, the components of
$\partial^{2}_{x^{\prime}\omega}\phi$ are uniformly bounded for a reduced
phase function and by homogeneity,

$\partial_{x^{\prime}}\phi(x^{\prime},x_{n};\omega)=\sum_{i=1}^{n-1}\omega_{i}\partial^{2}_{x^{\prime}\omega_{i}}\phi(x^{\prime},x_{n};\omega).$

This also gives a uniform estimate on $\partial_{x^{\prime}}\phi$ for a
reduced phase function. ∎

####  7.3.2. Phase and amplitude function after rescaling

We carry out the parabolic rescaling $T^{\lambda}g$ for a function $g$
supported in a $\rho^{-1}$-sector. In [2] the phase function was computed. It
was shown how after rescaling we find the bounds for higher order derivatives
introduced in Section 3.1 to hold. For an arbitrary phase function $\phi$,
$\rho$ has to be chosen large enough depending on $\phi$. For phase functions,
which are reduced before rescaling, $\rho$ can be chosen uniform; see also
Lemma 7.8 and its proof. Since we need some expressions from [2] to verify
that the phase is “flattened” in the sense of (71) and (72) upon rescaling,
some details are repeated. Furthermore, the amplitude and its derivatives
satisfy stronger bounds aswell. However, the scale of the amplitude exceeds
the scale of the phase function, but this can be remedied by an argument due
to Guth–Hickman–Iliopoulou [14]; see the end of the proof of Lemma 7.9.

Let $\omega\in B_{n-2}(0,1)$ with $(\omega,1)$ the centre of the
$\rho^{-1}$-sector encasing the support of $g$:

$\text{supp}(g)\subseteq\\{(\xi^{\prime},\xi_{n-1})\in\mathbb{R}^{n-1}:\,1/2\leq\xi_{n-1}\leq
2\text{ and
}\big{|}\frac{\xi^{\prime}}{\xi_{n-1}}-\omega\big{|}\leq\rho^{-1}\\}.$

We perform the change of variables

$(\xi^{\prime},\xi_{n-1})=(\eta_{n-1}\omega+\rho^{-1}\eta^{\prime},\eta_{n-1}),$

after which follows

$T^{\lambda}g(x)=\int_{\mathbb{R}^{n-1}}e^{i\phi^{\lambda}(x;\eta_{n-1}\omega+\rho^{-1}\eta^{\prime},\eta_{n-1})}a^{\lambda}(x;\eta_{n-1}\omega+\rho^{-1}\eta^{\prime},\eta_{n-1})\tilde{g}(\eta)d\eta,$

where
$\tilde{g}(\eta)=\rho^{-(n-2)}g(\eta_{n-1}\omega+\rho^{-1}\eta^{\prime},\eta_{n-1})$
and $\text{supp}(\tilde{g})\subseteq\Xi$. By Taylor expansion and homogeneity
of the phase, we find

$\begin{split}\phi(x;\eta_{n-1}\omega+\rho^{-1}\eta^{\prime},\eta_{n-1})&=\phi(x;\omega,1)\eta_{n-1}+\rho^{-1}\langle\partial_{\omega^{\prime}}\phi(x;\omega,1),\eta^{\prime}\rangle\\\
&\quad+\rho^{-2}\int_{0}^{1}(1-r)\langle\partial^{2}_{\omega^{\prime}\omega^{\prime}}\phi(x;\eta_{n-1}\omega+r\rho^{-1}\eta^{\prime},\eta_{n-1})\eta^{\prime},\eta^{\prime}\rangle
dr.\end{split}$

Let
$\Upsilon_{\omega}(y^{\prime},y_{n})=(\Upsilon(y^{\prime},y_{n};\omega,1),y_{n})$
and
$\Upsilon^{\lambda}_{\omega}(y^{\prime},y_{n})=\lambda\Upsilon_{\omega}(y^{\prime}/\lambda,y_{n}/\lambda)$
and consider anisotropic dilations

$D_{\rho}(y^{\prime\prime},y_{n-1},y_{n})=(\rho
y^{\prime\prime},y_{n-1},\rho^{2}y_{n})\text{ and
}D^{\prime}_{\rho^{-1}}(y^{\prime\prime},y_{n-1})=(\rho^{-1}y^{\prime\prime},\rho^{-2}y_{n-1})$

on $\mathbb{R}^{n}$ and $\mathbb{R}^{n-1}$, respectively. By definition of
$\Upsilon$, we find

(77) $T^{\lambda}g\circ\Upsilon^{\lambda}_{\omega}\circ
D_{\rho}=\tilde{T}^{\lambda/\rho^{2}}\tilde{g}$

where

$\tilde{T}^{\lambda/\rho^{2}}\tilde{g}(y)=\int_{\mathbb{R}^{n-1}}e^{i\tilde{\phi}^{\lambda/\rho^{2}}(y;\eta)}\tilde{a}^{\lambda}(y;\eta)\tilde{g}(\eta)d\eta$

for the phase $\tilde{\phi}(y;\eta)$ given by

$\langle
y^{\prime},\eta\rangle+\int_{0}^{1}(1-r)\langle\partial^{2}_{\xi^{\prime}\xi^{\prime}}\phi(\Upsilon_{\omega}(D^{\prime}_{\rho^{-1}}y^{\prime},y_{n});\eta_{n-1}\omega+r\rho^{-1}\eta^{\prime},\eta_{n-1})\eta^{\prime},\eta^{\prime}\rangle
dr$

and the amplitude
$\tilde{a}(y;\eta)=a(\Upsilon_{\omega}(D^{\prime}_{\rho^{-1}}y^{\prime},y_{n});\eta_{n-1}\omega+\rho^{-1}\eta^{\prime},\eta_{n-1})$.

We make a harmless linear change of variables: Let $L\in GL(n-1;\mathbb{R})$
be such that $Le_{n-1}=e_{n-1}$ and

$\partial^{2}_{\eta^{\prime}\eta^{\prime}}\partial_{y_{n}}\tilde{\phi}_{L}(0,0;e_{n-1})=I_{n-2},$

where

$\tilde{\phi}_{L}(y;\eta)=\tilde{\phi}(L^{-1}y^{\prime},y_{n};L\eta).$

It suffices to analyze $\tilde{T}^{\lambda/\rho^{2}}_{L}\tilde{g}_{L}$ with
$\tilde{T}^{\lambda/\rho^{2}}_{L}$ defined with respect to the data
$(\tilde{\phi}_{L},\tilde{a}_{L})$ for $\tilde{\phi}_{L}$ as above,
$\tilde{a}_{L}(y;\eta)=\tilde{a}(L^{-1}y^{\prime},y_{n};L\eta)$ and
$\tilde{g}_{L}=|\det L|\tilde{g}\circ L$.
To see that $\tilde{\phi}_{L}$ is still a reduced phase, we use the
representations

(78)
$\tilde{\phi}_{L}(y;\eta)=\rho^{2}\phi(\Upsilon_{\omega}(D^{\prime}_{\rho^{-1}}\circ
L^{-1}y^{\prime},y_{n}),y_{n};\eta_{n-1}\omega+\rho^{-1}L^{\prime}\eta^{\prime},\eta_{n-1})$

and

(79) $\langle
y^{\prime},\eta\rangle+\int_{0}^{1}(1-r)\langle\partial^{2}_{\omega^{\prime}\omega^{\prime}}\phi(\Upsilon_{\omega}(D^{\prime}_{\rho^{-1}}\circ
L^{-1}y^{\prime},y_{n});\eta_{n-1}\omega+r\rho^{-1}L^{\prime}\eta^{\prime},\eta_{n-1})L^{\prime}\eta^{\prime},L^{\prime}\eta^{\prime}\rangle
dr,$

where $L^{\prime}$ denotes the $(n-2)\times(n-2)$-submatrix of $L$, containing
the first $n-2$ rows and columns. We can argue like in [2], that
$\tilde{\phi}_{L}$ is again a reduced phase provided that $\phi$ was a reduced
phase. Note that the notion of reduced phase is slightly weaker in [2]. We
have strengthened the second condition in $D1)$ for technical reasons. For
reduced phase functions the components of $L$ are uniformly bounded (see [2]).
Regarding $K$-flatness of the rescaled phase functions, we have the following:

######  Lemma 7.8.

Suppose that $\phi$ is a reduced $K$-flat phase function supported in a sector
of aperture $\sim\rho^{-1}$. Then $\tilde{\phi}_{L}$ is a reduced
$cK\rho^{\frac{1}{4}}$-flat phase function provided that $\rho$ is chosen
large enough.

###### Proof.

We begin with the verification of (71). Let

$C_{m,\rho^{-1}}=\begin{cases}\rho^{-1},\quad m=1,\ldots,n-2,\\\
\rho^{-2},\quad m=n-1.\end{cases}$

From the representation (78) we find for $k=1,\ldots,n-1$

$\begin{split}\partial_{y_{k}}\tilde{\phi}_{L}&=\rho^{2}\sum_{\ell=1}^{n-1}\partial_{y_{\ell}}\phi(\Upsilon_{\omega}(D^{\prime}_{\rho^{-1}}\circ
L^{-1}y^{\prime},y_{n});\eta_{n-1}\omega+\rho^{-1}L^{\prime}\eta^{\prime},\eta_{n-1})\\\
&\qquad\times\frac{\partial}{\partial
y_{k}}\Upsilon^{\ell}(D^{\prime}_{\rho^{-1}}\circ
L^{-1}y^{\prime},y_{n};\omega,1)\\\
&=\rho^{2}\sum_{\ell,m=1}^{n-1}\partial_{y_{\ell}}\phi(\Upsilon_{\omega}(D^{\prime}_{\rho^{-1}}\circ
L^{-1}y^{\prime},y_{n});\eta_{n-1}\omega+\rho^{-1}L\eta^{\prime},\eta_{n-1})\\\
&\qquad\times\frac{\partial\Upsilon^{\ell}}{\partial
y_{m}}(D^{\prime}_{\rho^{-1}}\circ
L^{-1}y^{\prime},y_{n};\omega,1)C_{m,\rho^{-1}}L^{-1}_{mk}.\end{split}$

We take derivatives in $\eta^{\prime}$, which gives factors of $\rho^{-1}$ and
components of $L$: Recall that these are uniformly bounded. We obtain by
$C_{m,\rho^{-1}}\leq\rho^{-1}$ and $|\partial\Upsilon|\leq C_{\Upsilon}$ for
$2\leq|\alpha|\leq N_{0}+5$ by hypothesis:

$\begin{split}|\partial_{y_{k}}\partial^{\alpha}_{\eta^{\prime}}\tilde{\phi}_{L}|&\leq\rho^{2}\rho^{-1}\rho^{-|\alpha|}C_{L}^{|\alpha|+1}\sum_{|\tilde{\alpha}|=|\alpha|}\sum_{\ell=1}^{n-1}|\partial_{y_{\ell}}\partial_{\eta^{\prime}}^{\tilde{\alpha}}\phi|\\\
&\leq\rho^{1-|\alpha|}C_{L}^{|\alpha|+1}C_{\Upsilon}C_{N_{0}}K^{-4}.\end{split}$

Taking derivatives in $\eta_{n-1}$ only changes the result by a constant (use
e.g. homogeneity in $\eta_{n-1}$): For $2\leq|\alpha|\leq N_{0}+5$ and
$0\leq|\beta|\leq N_{0}+5-|\alpha|$ we obtain

$|\partial_{y_{k}}\partial^{\alpha}_{\eta^{\prime}}\partial_{\eta_{n-1}}^{\beta}\tilde{\phi}_{L}|\leq\rho^{1-|\alpha|}C_{L}^{|\alpha|+|\beta|+1}C_{\Upsilon}C_{N_{0}}K^{-4}.$

For $\rho=\rho(N_{0},n,C_{\Upsilon})$ large enough this verifies (71) with
$c\rho^{\frac{1}{4}}K$ instead of $K$.

We turn to the proof of (72). By the chain rule

$\begin{split}\partial_{y_{n}}\tilde{\phi}_{L}(y;\eta)&=\rho^{2}\sum_{k=1}^{n-1}\partial_{y_{k}}\phi(\Upsilon_{\omega}(D^{\prime}_{\rho^{-1}}\circ
L^{-1}y^{\prime},y_{n});\eta_{n-1}\omega+\rho^{-1}L^{\prime}\eta^{\prime},\eta_{n-1})\\\
&\quad\quad\times\frac{\partial\Upsilon^{k}_{\omega}}{\partial
x_{n}}(D^{\prime}_{\rho^{-1}}\circ L^{-1}y^{\prime},y_{n})\\\
&\quad+\rho^{2}\partial_{y_{n}}\phi(\Upsilon_{\omega}(D^{\prime}_{\rho^{-1}}\circ
L^{-1}y^{\prime},y_{n});\eta_{n-1}\omega+\rho^{-1}L^{\prime}\eta^{\prime},\eta_{n-1})\end{split}$

We find when taking derivatives in $\eta^{\prime}$:

$|\partial^{\alpha}_{\eta^{\prime}}\partial_{y_{n}}\tilde{\phi}_{L}(y;\eta)|\leq\rho^{2}C_{L}^{|\alpha|}\rho^{-|\alpha|}C_{\Upsilon}\sum_{k=1}^{n}|\partial_{x_{k}}\partial^{\alpha}_{\eta^{\prime}}\phi|.$

For $|\alpha|\geq 3$ we find by hypothesis

$|\partial^{\alpha}_{\eta^{\prime}}\partial_{y_{n}}\tilde{\phi}_{L}(y;\eta)|\leq\rho^{2-|\alpha|}C_{L}^{|\alpha|}C_{\Upsilon}C_{N_{0},n}K^{-4}.$