# Bochner-Riesz means for the Hermite and
special Hermite expansions

Sanghyuk Lee  and  Jaehyeon Ryu  Department of Mathematical Sciences and RIM,
Seoul National University, Seoul 08826, Republic of Korea<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

We consider the Bochner-Riesz means for the Hermite and special Hermite
expansions and study their $L^{p}$ boundedness with the sharp summability
index in a local setting. In two dimensions we establish the boundedness on
the optimal range of $p$ and extend the previously known range in higher
dimensions. Furthermore, we prove a new lower bound on the $L^{p}$ summability
index for the Hermite Bochner-Riesz means in $\mathbb{R}^{d}$, $d\geq 2.$ This
invalidates the conventional conjecture which was expected to be true.

###### Key words and phrases:

Bochner-Riesz means, Hermite and special Hermite functions

###### 2010 Mathematics Subject Classification:

42B99 (primary); 42C10 (secondary)

##  1\. Introduction

Let $\mathcal{H}$ denote the Hermite operator

$-\Delta+|x|^{2}=-\sum_{i=1}^{d}\partial_{i}^{2}+x_{i}^{2},\quad
x=(x_{1},\cdots,x_{d}),\quad d\geq 1$

which is non-negative and selfadjoint with respect to the Lebesgue measure on
$\mathbb{R}^{d}$. The spectrum of the operator $\mathcal{H}$ is given by the
set $2\mathbb{N}_{0}+d$. Here $\mathbb{N}_{0}$ denotes the set of nonnegative
integers. For each $k\in\mathbb{N}_{0}$, the Hermite polynomial $H_{k}(t)$ on
$\mathbb{R}$ is given by Rodrigues’ formula
$H_{k}(t)=(-1)^{k}e^{t^{2}}{d^{k}\over dt^{k}}\big{(}e^{-t^{2}}\big{)}$, and
the $L^{2}$ normalized Hermite functions
$h_{k}(t):=(2^{k}k!\sqrt{\pi})^{-1/2}H_{k}(t)e^{-t^{2}/2}$,
$k\in\mathbb{N}_{0}$ form an orthonormal basis of $L^{2}(\mathbb{R})$. In
higher dimensions the $d$-dimensional Hermite functions are given by the
tensor products of $h_{k}$:

$\Phi_{\alpha}(x)=\prod_{i=1}^{d}h_{\alpha_{i}}(x_{i}),\quad\alpha=(\alpha_{1},\cdots,\alpha_{d})\
{\in\mathbb{N}_{0}^{d}.}$

The Hermite operator and functions respectively represent the Hamiltonian and
quantum states of the particle for the quantum harmonic oscillator. The
functions $\Phi_{\alpha}$ can also be interpreted as basis functions for the
bosonic Fock space via the Bargmann transform. For a detailed discussion
regarding the matters, we refer the reader to [8]. The Hermite operator also
appears in the representation theory of the Heisenberg group $\mathbb{H}^{d}$
(see for example [32]).

The set $\\{\Phi_{\alpha}\\}_{\alpha\in\mathbb{N}_{0}^{d}}$ forms a complete
orthonormal system in $L^{2}(\mathbb{R}^{d})$ and the functions
$\Phi_{\alpha}$ are eigenfunctions for the Hermite operator with eigenvalue
$2|\alpha|+d$ where $|\alpha|=\sum_{i=1}^{d}\alpha_{i}$. Thus, for every $f\in
L^{2}(\mathbb{R}^{d})$ we have the Hermite expansion

$f=\sum_{\alpha\in\mathbb{N}_{0}^{d}}\langle
f,\Phi_{\alpha}\rangle\Phi_{\alpha}=\sum_{\lambda\in
2\mathbb{N}_{0}+d}\Pi_{\lambda}^{\mathcal{H}}f,$

where $\Pi_{\lambda}^{\mathcal{H}}$ denotes the Hermite spectral projection
given by

$\Pi_{\lambda}^{\mathcal{H}}f=\sum_{2|\alpha|+d=\lambda}\langle
f,\Phi_{\alpha}\rangle\Phi_{\alpha}.$

###  1.1. Bochner-Riesz means for the Hermite expansion

The Hermite expansion is convergent in $L^{2}(\mathbb{R}^{d})$ space, but when
$d\geq 2$ the expansion $\sum_{\lambda\leq N}\Pi_{\lambda}^{\mathcal{H}}f$
does not converge to $f$ as $N\to\infty$ in $L^{p}(\mathbb{R}^{d})$ unless
$p=2$. This can be shown making use of the transplantation theorem due to
Kenig-Stanton-Tomas [16] and Fefferman’s counterexample for $L^{p}$
boundedness of the ball multiplier [7] (also see [30, Theorem 3.1.2]). Thus we
are naturally led to consider the Bochner-Riesz mean:

$S_{\lambda}^{\delta}(\mathcal{H})f(x):=\Big{(}1-\frac{\mathcal{H}}{\lambda}\Big{)}_{+}^{\delta}f(x):=\sum_{\lambda^{\prime}\in
2\mathbb{N}_{0}+d}\Big{(}1-\frac{\lambda^{\prime}}{\lambda}\Big{)}_{+}^{\delta}\Pi_{\lambda^{\prime}}^{\mathcal{H}}f.$

The summability exponent $\delta$ mitigates the influence of new summands
$\Pi_{\lambda^{\prime}}^{\mathcal{H}}f$ which enter into the summation as
$\lambda$ increases. So, the operator $S_{\lambda}^{\delta}(\mathcal{H})$ has
more favorable behavior in perspective of $L^{p}$ summability as $\delta$
becomes larger. The classical Bochner-Riesz problem is to determine the
optimal summability order $\delta$ for which $S_{\lambda}^{\delta}(-\Delta)f$
converges to $f$ in ${L^{p}}$ for a given $p\in[1,\infty]$. When $d=2$, the
problem was settled by Carleson-Sjölin [4]. In higher dimensions progress has
been made, however the problem is still left open. See [25, 28, 19] and also
see [9, 33] for most recent results and references therein.

In this paper we are concerned with ${L^{p}}$ convergence of the Hermite
Bochner-Riesz means, that is to say, the problem of determining the optimal
$\delta$ for which $S_{\lambda}^{\delta}(\mathcal{H})f$ converges to $f$ in
$L^{p}$. By the uniform boundedness principle, this problem is equivalent to
that of characterizing the optimal $\delta$ for which the estimate

(1.1) $\displaystyle\|S_{\lambda}^{\delta}(\mathcal{H})\|_{p}\leq C$

holds with a uniform constant $C$ where $\|T\|_{p}:=\sup_{\|f\|_{p}\leq
1}\|Tf\|_{p}$.

When $d=1$, the problem is almost completely settled except some endpoint
cases. Askey and Wainger [1] proved that (1.1) holds with $\delta=0$ if and
only if $4/3<p<4$. When $p\leq 4/3$ or $p\geq 4$, combining this with the
result due to Thangavelu [29], one can show that
$S_{\lambda}^{\delta}(\mathcal{H})$ is uniformly bounded on
$L^{p}(\mathbb{R}^{d})$ if
$\delta>\max\\{\frac{2}{3}|\frac{1}{p}-\frac{1}{2}|-\frac{1}{6},0\\}$. On the
other hand, Thangavelu [29] showed that (1.1) fails to hold if
$\delta<\max\\{\frac{2}{3}|\frac{1}{p}-\frac{1}{2}|-\frac{1}{6},0\\}$.
However, it looks that the estimate (1.1) (or its weaker variants) with
$\delta=\frac{2}{3}|\frac{1}{p}-\frac{1}{2}|-\frac{1}{6}$ still remains open
when $p<4/3$ or $p>4$.

In higher dimensions, unlike one dimension, only partial results are known. By
the transplantation theorem due to Kenig, Stanton, and Tomas [16], the bound
(1.1) implies that the classical Bochner-Riesz means
$S_{\lambda}^{\delta}(-\Delta)$ is uniformly bounded on
${L^{p}(\mathbb{R}^{d})}$ (see Proposition 4.1 and its proof). Thus, by the
well known necessary condition for ${L^{p}}$ boundedness of
$S_{\lambda}^{\delta}(-\Delta)$ (see for example [10, 7]) we have

(1.2)
$\delta>\delta(d,p):=\max\Big{\\{}d\,\Big{|}\frac{1}{p}-\frac{1}{2}\Big{|}-\frac{1}{2},0\Big{\\}},\quad
p\neq 2$

if the uniform bound (1.1) holds. This naturally leads to the following
conjecture.

######  Conjecture 1.1.

Let $p\in[1,\infty]\setminus\\{2\\}$. The uniform estimate (1.1) holds if and
only if (1.2) holds.

Karadzhov [15] verified Conjecture 1.1 for $\max(p,p^{\prime})\geq 2d/(d-2)$.
His result was based on the optimal $L^{2}$–$L^{p}$ spectral projection
estimate

$\|\Pi_{\lambda}^{\mathcal{H}}\|_{2\to p}\leq
C\lambda^{-\frac{1}{2}+\frac{d}{2}(\frac{1}{2}-\frac{1}{p})},\quad{2d}/(d-2)\leq
p\leq\infty.$

The estimate plays the role of $L^{2}$–$L^{p}$ restriction estimate for the
sphere in Stein’s argument [7] which deduces the sharp ${L^{p}}$ bound on
$S_{\lambda}^{\delta}(-\Delta)$ from the $L^{2}$ restriction estimates.
However, as shown by Koch and Tataru [18], the range of $p$ where the above
estimate is valid can not be extended any further. We refer the reader to [18,
13] and references therein for more about the Hermite spectral projection
operator. This means the approach in [15] relying on the $L^{2}$–$L^{p}$
spectral estimate is no longer viable when one tries to prove $L^{p}$
boundedness with $\delta$ satisfying (1.2) when
$\max(p,p^{\prime})\in(2,{2d}/(d-2))$.

#### Local ${L^{p}}$ estimate for $S_{\lambda}^{\delta}(\mathcal{H})$

Meanwhile, the kernel of $S_{\lambda}^{\delta}(\mathcal{H})$ is expressed on a
critical region as an Airy type integral and such phenomenon does not occur in
the case of the classical Bochner-Riesz operator
$S_{\lambda}^{\delta}(-\Delta)$. Taking this into account, Thangavelu [31]
speculated that Conjecture 1.1 may fails 111That is to say, the summability
index for (1.1) may be bigger than that for the classical Bochner-Riesz means
$S_{\lambda}^{\delta}(-\Delta)$ when $d\geq 2$. when
$\max(p,p^{\prime})\in(2d/(d-2),2).$ Instead of the global estimate (1.1) he
considered a local variant of (1.1). To be specific, let us consider the
estimate

(1.3)
$\displaystyle\|\chi_{E}S_{\lambda}^{\delta}(\mathcal{H})\chi_{F}\|_{p}\leq C$

with a constant $C$ independent of $\lambda$ where $E,F$ are measurable
subsets of $\mathbb{R}^{d}$. It was shown by Thangavelu [31] that (1.3) holds
with a compact set $E$ and $F=\mathbb{R}^{d}$ for $\frac{2(d+1)}{d-1}\leq
p\leq\infty$ when (1.2) holds. The result is clearly sharp in that the
estimates fail if $\delta<\delta(d,p)$ because of the aforementioned
transplantation [16]. In analogy to Karadzhov’s approach, a form of local
$L^{2}$–$L^{2(d+1)/(d-1)}$ estimate for $\Pi_{\lambda}^{\mathcal{H}}$ was
utilized. As was shown in [13], the local spectral projection estimate does
not extend for $p<{2(d+1)}/{(d-1)}$, so we can not expect any progress using
$L^{2}$–$L^{p}$ estimate for $\Pi_{\lambda}^{\mathcal{H}}$.

We shall show that Conjecture 1.1 is generally not true when
$\max(p,p^{\prime})\in(2d/(d-2),2).$ In fact, on a certain range of $p$ we
obtain a new lower bound on the summability index $\delta$ (see Proposition
4.1) for the uniform bound (1.1). This invalidates Conjecture 1.1. Thus, in
order to prove boundedness for $\delta>\delta(d,p)$ one has to consider a
weaker alternative as was done [31]. It would be interesting to determine
whether (1.1) holds up to the new lower bound but for the present the problem
seems to be beyond reach. Instead, we first look into $L^{p}$ convergence of
$S_{\lambda}^{\delta}(\mathcal{H})$ in a local setting to make progress on the
current state regarding $L^{p}$ boundedness of the Hermite Bochner-Riesz
means.

As far as the authors are aware, concerning on $L^{p}$ boundedness of the
Hermite Bochner-Riesz means no further progress has been made beyond
Thangavelu’s result ([31]) until now. In this paper, we extend the range of
$p$ for which (1.3) holds under a suitable condition on $E$ and $F$. Even if
(1.3) is a weaker variant of the global estimate, the local estimate is still
strong enough to imply the sharp $L^{p}$ bound on
$S_{\lambda}^{\delta}(-\Delta)$. More precisely, if the estimate (1.3) holds
with $E,F=B(0,\epsilon)$ for any $\epsilon>0$, from the transplantation
theorem ([16]) we see that the Bochner-Riesz operator
$(\lambda+\Delta)_{+}^{\delta}$ is uniformly bounded on $L^{p}$.

To state our first result, we introduce some notations. Let us set

$\displaystyle p_{0}(d)=\begin{cases}\ 2\cdot\dfrac{3d+2}{3d-2}&\text{if
}d\equiv 0\,(\text{mod }2),\\\\[8.61108pt] \ 2\cdot\dfrac{3d+1}{3d-3}&\text{if
}d\equiv 1\,(\text{mod }2),\end{cases}$

and

$\mathcal{D}(x,y):=1+(\langle
x,y\rangle)^{2}-|x|^{2}-|y|^{2},\quad(x,y)\in\mathbb{R}^{d}\times\mathbb{R}^{d}.$

The following is our first result.

######  Theorem 1.2.

Suppose that $E,F\subset\mathbb{R}^{d}$ are compact sets such that $E\times
F\subset\mathfrak{D}(c_{0}):=\\{(x,y)\in\mathbb{R}^{d}:|x|,\,|y|\leq
1-c_{0},\,\mathcal{D}(x,y)>c_{0}^{2}\\}$ for some $0<c_{0}<1$. Then there is a
constant $C$ independent of $\lambda$ such that

(1.4)
$\|\chi_{E_{\lambda}}S_{\lambda}^{\delta}(\mathcal{H})\chi_{F_{\lambda}}\|_{p}\leq
C,$

provided that $p>p_{0}(d)$ and $\delta>\delta(d,p)$ where $E_{\lambda}$,
$F_{\lambda}$ denote the dilated set $\sqrt{\lambda}E$, $\sqrt{\lambda}F$,
respectively.

When $d=2$, Theorem 1.2 establishes the estimate (1.4) on the optimal range
$p$, that is to say, (1.4) holds if and only if $\delta>\delta(d,p)$, $p\neq
2$. From Theorem (1.2), we have the following $L^{p}$ convergence result.

######  Corollary 1.3.

Suppose that $p>p_{0}(d)$ and $\delta>\delta(d,p)$. Then for any compact set
$K\subset\mathbb{R}^{d}$ and compactly supported
$f\in{L^{p}(\mathbb{R}^{d})}$, we have

$\lim_{\lambda\to\infty}\int_{K}|S_{\lambda}^{\delta}(\mathcal{H})f(x)-f(x)|^{p}dx=0.$

###  1.2. Bochner-Riesz means for the special Hermite expansion

Now we consider the twisted Laplacian which is closely related to the Hermite
operator. The twisted Laplacian $\mathcal{L}$ on
$\mathbb{C}^{d}\cong\mathbb{R}^{2d}$ which is defined by

$\mathcal{L}=-\sum_{j=1}^{d}\Big{(}\big{(}\frac{\partial}{\partial
x_{j}}-\frac{1}{2}iy_{j}\big{)}^{2}+\big{(}\frac{\partial}{\partial{y_{j}}}+\frac{1}{2}ix_{j}\big{)}^{2}\Big{)},\quad
x,y\in\mathbb{R}^{d}$

has the same discrete spectrum $2\mathbb{N}_{0}+d$ as $\mathcal{H}$. The
associated eigenfunctions are the special Hermite functions
$\Phi_{\alpha,\beta}$ which are given by the Fourier-Wigner transform of the
Hermite functions. Indeed, for any multi-index
$\alpha,\beta\in\mathbb{N}_{0}^{d}$,

$\Phi_{\alpha,\beta}(z):=(2\pi)^{-\frac{d}{2}}\int_{\mathbb{R}^{d}}e^{i\langle
x,\xi\rangle}\Phi_{\alpha}(\xi-\frac{1}{2}y)\Phi_{\beta}(\xi+\frac{1}{2}y)d\xi,\quad
z=x+iy.$

Then it follows that
$\mathcal{L}\Phi_{\alpha,\beta}=(2|\beta|+d)\Phi_{\alpha,\beta}$, thus
$\Phi_{\alpha,\beta}$ is an eigenfunction of $\mathcal{L}$ with the eigenvalue
$2|\beta|+d$ and the eigenspaces of $\mathcal{L}$ are infinite dimensional.
Additionally, $\Phi_{\alpha,\beta}$ satisfies
$(-\Delta_{z}+\frac{1}{4}|z|^{2})\Phi_{\alpha,\beta}=(|\alpha|+|\beta|+d)\Phi_{\alpha,\beta}$,
which means $\Phi_{\alpha,\beta}$ is an eigenfunction of the Hermite operator
$-\Delta_{z}+\frac{1}{4}|z|^{2}$. This is the reason that
$\Phi_{\alpha,\beta}$ is called the special Hermite function.

For $\lambda\in 2\mathbb{N}_{0}+d$, by $\Pi_{\lambda}^{\mathcal{L}}$ we denote
the projection to the eigenspace of $\mathcal{L}$ with the eigenvalue
$\lambda$, i.e.,

(1.5)
$\Pi_{\lambda}^{\mathcal{L}}f(z):=\sum_{\beta:2|\beta|+d=\lambda}\sum_{\alpha\in\mathbb{N}_{0}^{d}}\langle
f,\Phi_{\alpha,\beta}\rangle\Phi_{\alpha,\beta}(z),\quad f\in
L^{2}(\mathbb{C}^{d}).$

Since $\\{\Phi_{\alpha,\beta}\\}_{\alpha,\beta}$ is a orthonormal basis of
$L^{2}(\mathbb{C}^{d})$, so one can expand $f$ into the series of special
Hermite functions. In fact, we have

$f(z)=\sum_{\lambda\in 2\mathbb{N}_{0}+d}\Pi_{\lambda}^{\mathcal{L}}f(z),\quad
f\in L^{2}(\mathbb{C}^{d}).$

As seen before in the case of Hermite expansion, by the transplantation in
[16] and Fefferman’s counter example [7]) this series fails to converge in
$L^{p}$ unless $p=2$. So, we need to consider the Bochner-Riesz mean for the
special Hermite expansion which is defined by

$S_{\lambda}^{\delta}(\mathcal{L})f(z)=\sum_{\mu\in
2\mathbb{N}_{0}+d}\Big{(}1-\frac{\mu}{\lambda}\Big{)}_{+}^{\delta}\Pi_{\lambda}^{\mathcal{L}}f(z).$

By the uniform boundedness principle, the $L^{p}$ convergence of
$S_{\lambda}^{\delta}(\mathcal{L})f$ for all $f\in L^{p}$ is equivalent to the
uniform estimate

(1.6) $\displaystyle\|S_{\lambda}^{\delta}(\mathcal{L})f\|_{p}\leq
C\|f\|_{p}.$

The problem has been studied by many authors. By the transplantation theorem
in [16] we see the estimate (1.6) implies $L^{p}$ boundedness of the classical
Bochner-Riesz operator in $\mathbb{R}^{2d}$. Thus (1.6) holds only if
$\delta>\delta(2d,p)$, $p\neq 2$. It seems to be plausible to conjecture that
the uniform estimate (1.6) holds if $\delta>\delta(2d,p)$. In [30], Thangavelu
verified the conjecture for $\frac{2d}{d-1}\leq p\leq\infty$. Later, the range
was extended to $\frac{2(3d+1)}{3d-2}<p\leq\infty$ by Ratnakumar, Rawat, and
Thangavelu [21]. Further progress was made by Thangavelu [31] who showed the
local estimate $\|\chi_{E}S_{\lambda}^{\delta}(\mathcal{L})\|_{p}\leq C$ holds
provided that $\delta>\delta(2d,p)$ and $\frac{2(2d+1)}{2d-1}<p\leq\infty$.
The corresponding global estimate was later established by Stempak and
Zienkiewicz [26], i.e., they showed that the estimate (1.6) holds for
$\frac{2(2d+1)}{2d-1}<p\leq\infty$ and $\delta>\delta(2d,p)$. The common key
ingredient of the previous results is the $L^{2}$–$L^{p}$ projection estimate
of the form

(1.7) $\displaystyle\|\Pi_{\lambda}^{\mathcal{L}}\|_{2\to p}\leq
C\lambda^{\frac{d-1}{2}-\frac{d}{p}}$

with $C$ independent of $\lambda$ which was combined with Stein’s argument
[7]. The projection estimate (1.7) was shown by Stempak and Zienkiewicz [26]
for $\frac{2(2d+1)}{2d-1}<p\leq\infty$. Later, Koch and Ricci [17] proved that
(1.7) holds if and only if $\frac{2(2d+1)}{2d-1}\leq p\leq\infty$ (also, see
[14] for $L^{p}$-$L^{q}$ estimates for $\Pi_{\lambda}^{\mathcal{L}}$). So,
further improvement is no longer possible via the estimate (1.7) when
$\frac{2(2d+1)}{2d-1}>p$.

#### Local $L^{p}$ convergence of $S_{\lambda}^{\delta}(\mathcal{L})$

Currently, no result with the sharp summability exponent $\delta(2d,p)$ is
known when $\frac{2(2d+1)}{2d-1}>p$. Following the approach in [31], we
consider a local variant of (1.6) and prove new estimate with the sharp
summability exponent outside the aforementioned range of $p$. The following is
our result regarding the Bochner-Riesz means for the special Hermite
expansion.

######  Theorem 1.4.

Let $0<c_{0}<2$, $p>p_{0}(2d)$, and $\delta>\delta(2d,p)$. Suppose that
$E,F\subset\mathbb{R}^{2d}$ are compact sets satisfying $|z-z^{\prime}|\leq
2-c_{0}$ for all $z\in E$, $z^{\prime}\in F$. Then there exists a constant $C$
independent of $\lambda$ such that

(1.8)
$\displaystyle\|\chi_{E_{\lambda}}S_{\lambda}^{\delta}(\mathcal{L})\chi_{F_{\lambda}}\|_{p}\leq
C,$

where $E_{\lambda}$, $F_{\lambda}$ denote the dilated set $\sqrt{\lambda}E$,
$\sqrt{\lambda}F$, respectively.

######  Corollary 1.5.

Let $p,\delta$ be as in Theorem 1.4. Then for any compact set
$K\subset\mathbb{R}^{2d}$ and compactly supported
$f\in{L^{p}(\mathbb{C}^{d})}$ we have

(1.9)
$\displaystyle\lim_{\lambda\to\infty}\int_{B}|S_{\lambda}^{\delta}(\mathcal{L})f(x)-f(x)|^{p}dx=0.$

In the case of $d=1$, from this result we have a complete characterization of
$(p,\delta)$ for which the local convergence (1.9) holds. The assumption that
$|z-z^{\prime}|\leq 2-c_{0}$ for all $z\in E$, $z^{\prime}\in F$ is made for
technical reason and we don’t know whether the condition is necessary for the
estimate (1.8) with the sharp summability index. The assumption can be
regarded as a counterpart of the assumption on $\mathcal{D}(x,y)$ in Theorem
1.2. If $|z-z^{\prime}|\geq 2+c_{0}$ for all $z\in E$, $z^{\prime}\in F$, the
kernel of the operator
$\chi_{E_{\lambda}}S_{\lambda}^{\delta}(\mathcal{L})\chi_{F_{\lambda}}$
rapidly decays (See (3.11)). Thus the assumption can be relaxed so that
$||z-z^{\prime}|-2|\geq c_{0}$ for all $z\in E$, $z^{\prime}\in F$. In order
to prove the global bound we need to understand the behavior of the kernel of
$S_{\lambda}^{\delta}(\mathcal{L})(z,z^{\prime})$ when $|z-z^{\prime}|$ is
close to $2$.

#### Our approach

The proofs of Theorem 1.4 and 1.2 follow a similar strategy which is inspired
by the recent works by Jeong and the authors [13, 14] on the spectral
projection operators $\Pi_{\lambda}^{\mathcal{H}}$ and
$\Pi_{\lambda}^{\mathcal{L}}$. First, we obtain a explicit expression for the
kernels of the operators $S^{\delta}_{\lambda}(\mathcal{H})$,
$S^{\delta}_{\lambda}(\mathcal{L})$ using the Schrödinger propagators
$e^{-it\mathcal{H}}$, $e^{-it\mathcal{L}}$ (see (2.1), (3.1)) of which kernel
representation is well known. Secondly, combining the expression of kernel
with the method of stationary phase we obtain asymptotic expansions of the
kernels. This reduces the matter to obtaining the sharp estimate for the
oscillatory integral operators which satisfy the Carleson-Sjölin condition [4,
11, 24]. In two dimensions this allows us to obtain the optimal results.
However, in higher dimensions the Carleson-Sjölin condition alone is not
enough, as was shown by Bourgain [3], to give the sharp bound for
$p<2(d+1)/(d-1)$ (or $p<2(2d+1)/(2d-1)$). However, an additional ellipticity
assumption on the second fundamental form of the phase allows to get the bound
on improved range. This observation was first made by one of the author [20].
Thirdly, we show that the phases in the asymptotic expansion satisfy the
ellipticity condition (see Lemma 3.10 and 2.13). We combine this with the
results regarding the oscillatory integral operator [20, 9]. We refer the
reader forward to Section 2.5 for more regarding the oscillatory integral
operators.

#### Notation

Throughout the rest of the paper, we identify $\mathbb{C}^{d}$ with
$\mathbb{R}^{2d}$.

  1. $\bullet$

For given $A,B>0$, we write $B\lesssim A$ if there is a constant $C>0$ such
that $B\leq CA$. Here, if $C$ has to be taken to be small enough, we use the
notation $B\ll A$ to mean that $A$ is sufficiently larger than $B$.
Furthermore, $A\sim B$ denotes that $A\lesssim B$ and $B\lesssim A$.

  2. $\bullet$

$B_{d}(x,r)=\\{y\in\mathbb{R}^{d}:|y-x|<r\\}$.

  3. $\bullet$

For an operator $T$ we denote by $T(x,y)$ (or $T(z,z^{\prime})$) the kernel of
$T$.

  4. $\bullet$

$\partial_{x}:=(\partial_{x_{1}},\dots,\partial_{x_{d}})^{\intercal}$,
$\partial_{x}^{\intercal}:=(\partial_{x_{1}},\dots,\partial_{x_{d}})$, so
$\partial_{x}\partial_{y}^{\intercal}=(\partial_{x_{i}}\partial_{y_{j}})_{1\leq
i,j\leq d}$. In particular, if $a(x)=(a_{1}(x),\dots,a_{d}(x))$ is a
$\mathbb{C}^{d}$-valued differentiable function on $\mathbb{R}^{d}$, then we
have $\partial_{x}^{\intercal}a(x)=(\partial_{x_{j}}a_{i}(x))_{1\leq i,j\leq
d}$.

  5. $\bullet$

By $\mathbf{I}_{d}$ we denote the $d\times d$ identity matrix. If the value of
$d$ is clear from the context, we simply denote $\mathbf{I}_{d}$ by
$\mathbf{I}$.

  6. $\bullet$

For $1\leq i\leq d$, $\mathbf{e}_{i}$ denotes the $i$-th standard basis in
$\mathbb{R}^{d}$.

  7. $\bullet$

For $S\subset\mathbb{R}^{d}$ and a constant $a>0$, we denote $aS=\\{ax:x\in
S\\}$.

##  2\. Bochner-Riesz means for the Hermite expansion:
Proof of Theorem 1.2

In this section we prove Theorem 1.2. We begin by obtaining an explicit
expression of the kernel of the Bochner-Riesz means. For the purpose we make
use of Mehler’s formula for the Schrödinger propagator.

###  2.1. Decomposition of $S_{\lambda}^{\delta}(\mathcal{H})$

We start by considering the Hermite-Schrödinger propagator
$e^{-it\mathcal{H}}$ which is the solution to the Cauchy problem
$(i\partial_{t}-\mathcal{H})u=0$ and $u(0,x)=f(x)$. The propagator
$e^{-it\mathcal{H}}$ can be expressed by the spectral projection operators
$\Pi_{\lambda}^{\mathcal{H}}$:

$e^{-it\mathcal{H}}=\sum_{\lambda\in
2\mathbb{N}_{0}+d}e^{-it\lambda}\Pi_{\lambda}^{\mathcal{H}}.$

On the other hand, by virtue of Mehler’s formula we have an explicit
expression of the kernel of $e^{-it\mathcal{H}}$. It is well known that

(2.1) $\displaystyle e^{-it\mathcal{H}}f(x)=(2\pi i\sin
2t)^{-\frac{d}{2}}e^{i\pi d/4}\int
e^{\frac{i}{2}((|x|^{2}+|y|^{2})\cot{2t}-2\langle x,y\rangle\csc{2t})}f(y)dy$

for $f\in\mathcal{S}(\mathbb{R}^{d})$. For example, see [30] and also see [22]
for a detailed discussion regarding derivation of (2.1).

To study $L^{p}$ boundedness of the Bochner-Riesz means we need to properly
decompose the operator $S_{\lambda}^{\delta}(\mathcal{H})$. Let $\psi\in
C^{\infty}_{c}([\frac{1}{4},1])$ be a smooth bump function such that
$\sum_{j\in\mathbb{Z}}\psi(2^{j}t)=1$ for all $t>0$. Then, setting
$\psi^{\delta}(t):=t^{\delta}\psi(t)$, we denote

$\displaystyle\quad\psi_{j}(t)$ $\displaystyle=\psi^{\delta}(2^{-j}t),\quad
j\geq 1,$ $\displaystyle\psi_{0}(t)$ $\displaystyle=t^{\delta}_{+}\sum_{j\geq
0}\psi(2^{j}t),$

so that $t^{\delta}=\sum_{1\leq 2^{j}\leq 4\lambda}2^{\delta j}\psi_{j}(t)$
for $0\leq t\leq\lambda$. For any bounded continuous function $m$ on
$\mathbb{R}$, we define an operator $m(\mathcal{H})$ by setting

$m(\mathcal{H})=\sum_{\lambda\in
2\mathbb{N}_{0}+d}m(\lambda)\Pi_{\lambda}^{\mathcal{H}}.$

Then, since
$S_{\lambda}^{\delta}(\mathcal{H})=\lambda^{-\delta}\big{(}\lambda-\mathcal{H}\big{)}_{+}^{\delta}$
and since the above summation is taken over the set $2\mathbb{N}_{0}+d$, we
can write

(2.2)
$S_{\lambda}^{\delta}(\mathcal{H})=\lambda^{-\delta}\big{(}\lambda-\mathcal{H}\big{)}_{+}^{\delta}=\lambda^{-\delta}\sum_{1\leq
2^{j}\leq 4\lambda}2^{\delta j}\psi_{j}(\lambda-\mathcal{H}).$

Though $\psi_{0}$ is not smooth, we may assume $\psi_{0}$ is smooth replacing
it with a suitable smooth function because $\mathcal{H}$ has the spectrum
$2\mathbb{N}_{0}+d$. Thus, the proof of Theorem 1.2 is reduced to obtaining
the sharp $L^{p}$ estimate for each operator $\psi_{j}(\lambda-A)$. That is to
say, by (2.2) Theorem 1.2 follows if we show

(2.3)
$\|\chi_{E_{\lambda}}\psi_{j}(\lambda-\mathcal{H})\chi_{F_{\lambda}}\|_{p}\lesssim(\lambda
2^{-j})^{\delta(d,p)},\quad 1\leq 2^{j}\leq 4\lambda$

for $p>p_{0}(d)$.

We now relate the operators $\psi_{j}(\lambda-\mathcal{H})$ to the propagator
$e^{it\mathcal{H}}$ via Fourier inversion. In fact, for
$\eta\in\mathcal{S}(\mathbb{R})$ we have

${\mathchoice{{\ooalign{\hbox{\raise 6.25pt\hbox{\scalebox{0.9}[-1.2]{\lower
6.25pt\hbox{$\displaystyle\widehat{\vrule width=0.0pt,height=4.30554pt\vrule
height=0.0pt,width=4.9653pt}$}}}}\cr\hbox{$\displaystyle\eta$}}}}{{\ooalign{\hbox{\raise
6.25pt\hbox{\scalebox{0.9}[-1.2]{\lower 6.25pt\hbox{$\textstyle\widehat{\vrule
width=0.0pt,height=4.30554pt\vrule
height=0.0pt,width=4.9653pt}$}}}}\cr\hbox{$\textstyle\eta$}}}}{{\ooalign{\hbox{\raise
5.81944pt\hbox{\scalebox{0.9}[-1.2]{\lower
5.81944pt\hbox{$\scriptstyle\widehat{\vrule width=0.0pt,height=3.01389pt\vrule
height=0.0pt,width=3.47571pt}$}}}}\cr\hbox{$\scriptstyle\eta$}}}}{{\ooalign{\hbox{\raise
5.53241pt\hbox{\scalebox{0.9}[-1.2]{\lower
5.53241pt\hbox{$\scriptscriptstyle\widehat{\vrule
width=0.0pt,height=2.15277pt\vrule
height=0.0pt,width=2.48265pt}$}}}}\cr\hbox{$\scriptscriptstyle\eta$}}}}}(\lambda-\mathcal{H})=\frac{1}{2\pi}\int\eta(t)e^{i{t}(\lambda-\mathcal{H})}dt.$

Combining this with (2.1) and changing variables $t\to t/2$, we get

${\mathchoice{{\ooalign{\hbox{\raise 6.25pt\hbox{\scalebox{0.9}[-1.2]{\lower
6.25pt\hbox{$\displaystyle\widehat{\vrule width=0.0pt,height=4.30554pt\vrule
height=0.0pt,width=4.9653pt}$}}}}\cr\hbox{$\displaystyle\eta$}}}}{{\ooalign{\hbox{\raise
6.25pt\hbox{\scalebox{0.9}[-1.2]{\lower 6.25pt\hbox{$\textstyle\widehat{\vrule
width=0.0pt,height=4.30554pt\vrule
height=0.0pt,width=4.9653pt}$}}}}\cr\hbox{$\textstyle\eta$}}}}{{\ooalign{\hbox{\raise
5.81944pt\hbox{\scalebox{0.9}[-1.2]{\lower
5.81944pt\hbox{$\scriptstyle\widehat{\vrule width=0.0pt,height=3.01389pt\vrule
height=0.0pt,width=3.47571pt}$}}}}\cr\hbox{$\scriptstyle\eta$}}}}{{\ooalign{\hbox{\raise
5.53241pt\hbox{\scalebox{0.9}[-1.2]{\lower
5.53241pt\hbox{$\scriptscriptstyle\widehat{\vrule
width=0.0pt,height=2.15277pt\vrule
height=0.0pt,width=2.48265pt}$}}}}\cr\hbox{$\scriptscriptstyle\eta$}}}}}(\lambda-\mathcal{H})(x,y)=C_{d}\int\eta(t/2)(\sin
t)^{-\frac{d}{2}}e^{i\phi_{\lambda}(t,x,y)}dt,$

where

$\phi_{\lambda}(t,x,y):=\frac{\lambda t}{2}+\frac{|x|^{2}+|y|^{2}}{2}\cot
t-\langle x,y\rangle\csc t.$

Instead of dealing with ${\mathchoice{{\ooalign{\hbox{\raise
6.25pt\hbox{\scalebox{0.9}[-1.2]{\lower
6.25pt\hbox{$\displaystyle\widehat{\vrule width=0.0pt,height=4.30554pt\vrule
height=0.0pt,width=4.9653pt}$}}}}\cr\hbox{$\displaystyle\eta$}}}}{{\ooalign{\hbox{\raise
6.25pt\hbox{\scalebox{0.9}[-1.2]{\lower 6.25pt\hbox{$\textstyle\widehat{\vrule
width=0.0pt,height=4.30554pt\vrule
height=0.0pt,width=4.9653pt}$}}}}\cr\hbox{$\textstyle\eta$}}}}{{\ooalign{\hbox{\raise
5.81944pt\hbox{\scalebox{0.9}[-1.2]{\lower
5.81944pt\hbox{$\scriptstyle\widehat{\vrule width=0.0pt,height=3.01389pt\vrule
height=0.0pt,width=3.47571pt}$}}}}\cr\hbox{$\scriptstyle\eta$}}}}{{\ooalign{\hbox{\raise
5.53241pt\hbox{\scalebox{0.9}[-1.2]{\lower
5.53241pt\hbox{$\scriptscriptstyle\widehat{\vrule
width=0.0pt,height=2.15277pt\vrule
height=0.0pt,width=2.48265pt}$}}}}\cr\hbox{$\scriptscriptstyle\eta$}}}}}(\lambda-\mathcal{H})$
it is more convenient to work with the rescaled operator. So, for
$\zeta\in\mathcal{S}(\mathbb{R})$ we define an operator
$[\zeta]^{\mathcal{H}}_{\lambda}$ whose kernel is given by

(2.12) $\displaystyle[\zeta]^{\mathcal{H}}_{\lambda}(x,y)$
$\displaystyle:={\mathchoice{{\ooalign{\hbox{\raise
7.31482pt\hbox{\scalebox{0.9}[-1.2]{\lower
7.31482pt\hbox{$\displaystyle\widehat{\vrule width=0.0pt,height=7.5pt\vrule
height=0.0pt,width=22.15279pt}$}}}}\cr\hbox{$\displaystyle\zeta(2\cdot)$}}}}{{\ooalign{\hbox{\raise
7.31482pt\hbox{\scalebox{0.9}[-1.2]{\lower
7.31482pt\hbox{$\textstyle\widehat{\vrule width=0.0pt,height=7.5pt\vrule
height=0.0pt,width=22.15279pt}$}}}}\cr\hbox{$\textstyle\zeta(2\cdot)$}}}}{{\ooalign{\hbox{\raise
6.56482pt\hbox{\scalebox{0.9}[-1.2]{\lower
6.56482pt\hbox{$\scriptstyle\widehat{\vrule width=0.0pt,height=5.25pt\vrule
height=0.0pt,width=15.50694pt}$}}}}\cr\hbox{$\scriptstyle\zeta(2\cdot)$}}}}{{\ooalign{\hbox{\raise
6.06482pt\hbox{\scalebox{0.9}[-1.2]{\lower
6.06482pt\hbox{$\scriptscriptstyle\widehat{\vrule
width=0.0pt,height=3.75pt\vrule
height=0.0pt,width=11.07635pt}$}}}}\cr\hbox{$\scriptscriptstyle\zeta(2\cdot)$}}}}}(\lambda-\mathcal{H})(\sqrt{\lambda}x,\sqrt{\lambda}y)$
(2.13) $\displaystyle\,=C_{d}\int\zeta(t)(\sin
t)^{-\frac{d}{2}}e^{i\lambda\mathcal{P}_{\\!\,\mathcal{H}}(t,x,y)}dt,$

where $C_{d}$ is a constant depending on $d$ and

(2.14)
$\mathcal{P}_{\\!\,\mathcal{H}}(t,x,y):=\frac{t}{2}+\frac{(|x|^{2}+|y|^{2})\cos
t}{2\sin t}-\frac{\langle x,y\rangle}{\sin t}.$

By scaling, the estimate (2.3) is now equivalent to

(2.15)
$\|\chi_{E}[\widehat{\psi_{j}}(\cdot/2)]^{\mathcal{H}}_{\lambda}\chi_{F}\|_{p}\lesssim\lambda^{-\frac{d}{2}}(\lambda
2^{-j})^{\delta(d,p)},\quad 1\leq 2^{j}\leq 4\lambda.$

We occasionally use the following lemma which is a simple consequence of the
formula (2.13) for the kernel.

######  Lemma 2.1.

Let $\zeta\in C_{c}^{\infty}((0,\infty))$. Then, for any measurable sets
$E,F\subset\mathbb{R}^{d}$,

(2.16)
$\displaystyle\|\chi_{E}[\zeta]^{\mathcal{H}}_{\lambda}\,\chi_{F}\|_{p}$
$\displaystyle=\|\chi_{E}[\zeta(-\,\cdot)]^{\mathcal{H}}_{\lambda}\,\chi_{F}\|_{p},$
(2.17)
$\displaystyle\|\chi_{E}[\zeta]^{\mathcal{H}}_{\lambda}\,\chi_{F}\|_{p}$
$\displaystyle=\|\chi_{E}[\zeta(\cdot+\pi)]^{\mathcal{H}}_{\lambda}\,\chi_{(-F)}\|_{p}.$

###### Proof.

We observe that $\mathcal{P}_{\mathcal{H}}(-t,x,y)=-\phi(t,x,y)$ and
$\mathcal{P}_{\mathcal{H}}(t+\pi,x,y)=\frac{\pi}{2}+\mathcal{P}_{\mathcal{H}}(t,x,-y).$
Using this, (2.13) and changing of variables, we have

(2.18) $\displaystyle[\zeta(-\cdot)]^{\mathcal{H}}_{\lambda}(x,y)$
$\displaystyle=C_{1}\overline{[\zeta]^{\mathcal{H}}_{\lambda}(x,y)},$ (2.19)
$\displaystyle[\zeta(\cdot+\pi)]^{\mathcal{H}}_{\lambda}(x,y)$
$\displaystyle=C_{2}[\zeta]^{\mathcal{H}}_{\lambda}(x,-y),$

where $C_{1},C_{2}\in\mathbb{C}$ such that $|C_{1}|=|C_{2}|=1$. Thus, (2.18),
(2.19) give (2.16), (2.17), respectively. ∎

In the expression (2.13) there are singularities at $t={k\pi},$
$k\in\mathbb{Z}$, so we need to make further decomposition by breaking
$\widehat{\psi_{j}}(\cdot/2)$. To do so, we set

(2.20) $\displaystyle\nu^{l}(t)$ $\displaystyle=\sum_{\pm}\psi(\pm
2^{l}t)+\psi(2^{l}(\pi\pm t)),\quad l\geq 1,$ $\displaystyle\nu^{0}(t)$
$\displaystyle=\chi_{(-\pi,\pi)}(t)-\sum_{l=1}^{\infty}\nu^{l}(t),$

so that $\operatorname{supp}\nu^{l}\subset(-\pi,\pi)$ and
$\sum_{l=0}^{\infty}\nu^{l}(t-2k\pi)=1$ a.e. on $(-\pi,\pi)$. We also set

${\nu_{j}^{l,k}}(t)=\widehat{\psi_{j}}(t/2){\nu^{l}(t-2k\pi),}$

so we have $\sum_{k=-\infty}^{\infty}\sum_{l=0}^{\infty}{\nu^{l}(t-2k\pi)=1}$
a.e. Thus it follows

(2.21)
$[\widehat{\psi_{j}}(\cdot/2)]^{\mathcal{H}}_{\lambda}=\sum_{k=-\infty}^{\infty}\sum_{l=0}^{\infty}[\nu_{j}^{l,k}]^{\mathcal{H}}_{\lambda}.$

Now the proof of (2.15) (and Theorem 1.2) is essentially reduced to showing
the following.

######  Theorem 2.2.

Let $0<\rho<\pi-2^{-5}$ and $\lambda\in 2\mathbb{N}_{0}+d$. Let $\eta_{\rho}$
be a smooth function such that
$\operatorname{supp}\eta_{\rho}\subset[2^{-2}\rho,\rho]$ and
$|\eta_{\rho}^{(n)}(t)|\leq C_{n}\rho^{-n}$, $n\in\mathbb{N}_{0}$. Suppose
that the sets $E,F$ satisfy the condition in Theorem 1.2, i.e., $E\times
F\subset\mathfrak{D}(c_{0})$. Then, for $p>p_{0}(d)$ we have

(2.22)
$\displaystyle\|\chi_{E}[\eta_{\rho}]^{\mathcal{H}}_{\lambda}\chi_{F}\|_{p}\lesssim\begin{cases}\
\lambda^{-\frac{d}{2}}\rho,&\rho\leq\lambda^{-1},\\\ \
\lambda^{-\frac{d}{2}}\lambda^{\delta(d,p)}\rho^{\delta(d,p)+1},&\rho>\lambda^{-1}.\end{cases}$

Once we have Theorem 2.2, the proof of (2.15) is rather straightforward.
Assuming Theorem 2.2 for the moment, we prove (2.15).

###### Proof of (2.15).

Recalling (2.21) we need to obtain bounds on
$\|\chi_{E}[{\nu_{j}^{l,k}}]^{\mathcal{H}}_{\lambda}\chi_{F}\|_{p}$. By (2.17)
we need only to consider, instead of $\nu_{j}^{l,k}$, the multipliers
operators given by $\widehat{\psi_{j}}(k\pi+t/2)\nu^{l}(t)$. Then by (2.17)
and (2.16) the matter reduces to dealing with the multipliers given by

$\nu_{\pm}^{l,k}:=\widehat{\psi_{j}}(k\pi\pm
t/2)\psi(2^{l}t),\quad\nu_{\pm\pi}^{l,k}:=\widehat{\psi_{j}}(k\pi\pm(t-\pi)/2)\psi(2^{l}t),$

which are supported in $[2^{-2-l},2^{-l}]$. One can easily see that the
estimates

$\displaystyle\bigg{|}\frac{d^{n}}{dt^{n}}\nu_{\theta}^{l,k}(t)\bigg{|}\leq
C2^{j}\max(2^{nj},2^{nl})(1+2^{j-l})^{-N}(1+2^{j}|k|)^{-N},\quad\theta\in\\{\pm,\pm\pi\\},$

hold for any $N\in\mathbb{N}$ with a constant $C=C(N)$. Recalling $1\leq
2^{j}\leq 4\lambda$, after normalizing the functions $\nu_{\theta}^{l,k}$, we
apply Theorem 2.2 to
$\|\chi_{E}[{\nu_{\theta}^{l,k}}]^{\mathcal{H}}_{\lambda}\chi_{F}\|_{p}$.
Thus, for any $N$ we get

$\|\chi_{E}[\nu_{j}^{l,k}]^{\mathcal{H}}_{\lambda}\chi_{F}\|_{p}\lesssim\begin{cases}\lambda^{-\frac{d}{2}}2^{-l}2^{j}(1+2^{j}|k|)^{-N},&4\lambda<2^{l},\\\
\lambda^{-\frac{d}{2}}\lambda^{\delta(d,p)}\,2^{-l(\delta(d,p)+1)}\,2^{j}(1+2^{j}|k|)^{-N},&\,2^{j}\leq
2^{l}\leq 4\lambda,\\\
\lambda^{-\frac{d}{2}}\lambda^{\delta(d,p)}\,2^{-j\delta(d,p)}\,2^{-(j-l)N}(1+2^{j}|k|)^{-N},&\
2^{l}<2^{j}.\end{cases}$

Using (2.21) and taking summation over $l,k$ give (2.15). In fact, taking $N$
large enough, it is sufficient to consider the case $k=0$. ∎

###  2.2. Further decomposition and reduction

From now on we extensively utilize the expression (2.13). To obtain estimates
from the oscillatory integral we need to collect some properties of the phase
function $\mathcal{P}_{\mathcal{H}}$.

Using (2.14), a computation gives

(2.23) $\displaystyle\partial_{t}\mathcal{P}_{\mathcal{H}}(t,x,y)$
$\displaystyle=-\frac{\cos^{2}t-2\langle x,y\rangle\cos
t+|x|^{2}+|y|^{2}-1}{2\sin^{2}t}.$

Since $\mathcal{D}(x,y)\geq c_{0}^{2}$ for $(x,y)\in\mathfrak{D}(c_{0})$, we
can factor $\cos^{2}t-2\langle x,y\rangle\cos t+|x|^{2}+|y|^{2}-1=(\cos
t-\langle x,y\rangle-\mathcal{D}(x,y))(\cos t-\langle
x,y\rangle+\mathcal{D}(x,y))$. One can easily see $-1<\langle
x,y\rangle\pm\sqrt{\mathcal{D}(x,y)}<1$ for $(x,y)\in\mathfrak{D}(c_{0})$
because $|\langle x,y\rangle|<1$ if $(x,y)\in\mathfrak{D}(c_{0})$. Thus we can
define smooth functions $S_{c}(x,y),$ $S_{*}(x,y)\in(0,\pi)$ by setting

(2.24) $\displaystyle\cos S_{c}(x,y)$ $\displaystyle=\langle
x,y\rangle+\sqrt{\mathcal{D}(x,y)},\quad(x,y)\in\mathfrak{D}(c_{0}),$ (2.25)
$\displaystyle\cos S_{*}(x,y)$ $\displaystyle=\langle
x,y\rangle-\sqrt{\mathcal{D}(x,y)},\quad(x,y)\in\mathfrak{D}(c_{0}).$

So, we may also write

(2.26) $\displaystyle\partial_{t}\mathcal{P}_{\mathcal{H}}(t,x,y)=-\frac{(\cos
t-\cos S_{c}(x,y))(\cos t-\cos S_{*}(x,y))}{2\sin^{2}t}\,.$

Also, since $\sin^{2}S_{c}(x,y)=(1-\langle
x,y\rangle-\sqrt{\mathcal{D}(x,y)})(1+\langle
x,y\rangle+\sqrt{\mathcal{D}(x,y)})$ and $(1-\langle
x,y\rangle)^{2}-\mathcal{D}(x,y)=|x-y|^{2}$, we have

(2.27) $\displaystyle\sin S_{c}(x,y)=|x-y|\bigg{(}\frac{1-\langle
x,y\rangle+\sqrt{\mathcal{D}(x,y)}}{1+\langle
x,y\rangle+\sqrt{\mathcal{D}(x,y)}}\bigg{)}^{\frac{1}{2}}.$

Note that $1\pm\langle x,y\rangle+\sqrt{\mathcal{D}(x,y)}\gtrsim c_{0}$ for
$(x,y)\in\mathfrak{D}(c_{0})$. Thus we have $S_{c}(x,y)\sim|x-y|$. A similar
computation shows $S_{\ast}(x,y)\sim|x+y|$.

Since the integral in (2.13) has slower decay near the critical points, the
kernel $[\eta_{\rho}]^{\mathcal{H}}_{\lambda}$ becomes singular on the sets
$\\{x=y\\}$, $\\{x=-y\\}$, as $S_{c}(x,y)$, $S_{\ast}(x,y)\to 0$ while
$S_{c}(x,y)$, $S_{\ast}(x,y)\in\operatorname{supp}\eta_{\rho}$. We note that

(2.28) $|S_{c}(x,y)-S_{*}(x,y)|\geq 2c_{0}$

for any $(x,y)\in E\times F$. This follows by the mean value theorem and the
fact $\cos S_{*}(x,y)-\cos S_{c}(x,y)=2\sqrt{\mathcal{D}(x,y)}\geq 2c_{0}$.

As a first step of the proof of (2.22), we decompose $E\times F$ into smaller
balls to localize the values of $S_{c}(x,y)$, $S_{*}(x,y)$. We choose a
constant $c>0$ small enough so that

(2.29) $\displaystyle\begin{aligned} |\cos S_{c}(x_{1},y_{1})-\cos
S_{c}(x_{2},y_{2})|\leq c_{0}/10,\\\ |\cos S_{*}(x_{1},y_{1})-\cos
S_{*}(x_{2},y_{2})|\leq c_{0}/10,\end{aligned}$

for $(x_{1},y_{1})$, $(x_{2},y_{2})\in\mathfrak{D}(c_{0})$ if
$|(x_{1},y_{1})-(x_{2},y_{2})|\leq 4cc_{0}$. Since $E$ and $F$ are compact, so
there are finite collections of balls $\\{B(x_{j},cc_{0})\\}_{j=1}^{N}$,
$\\{B(y_{j^{\prime}},cc_{0})\\}_{j^{\prime}=1}^{N^{\prime}}$ which cover $E$,
$F$, respectively. Let $\\{\varphi_{j}\\}_{1\leq j\leq N}$,
$\\{\varphi^{\prime}\\}_{1\leq j^{\prime}\leq N}$ be collections of smooth
functions such that $\sum_{j=1}^{N}\varphi_{j}=1$ on $E$ and
$\operatorname{supp}\varphi_{j}\subset B(x_{j},2cc_{0})$ and
$\sum_{j^{\prime}=1}^{N^{\prime}}\varphi^{\prime}_{j^{\prime}}=1$ on $F$ and
$\operatorname{supp}\varphi^{\prime}_{j^{\prime}}\subset
B(y_{j^{\prime}},2cc_{0})$. In order to prove (2.22) it suffices to prove the
corresponding estimates for
$\varphi_{j}[\eta_{\rho}]_{\lambda}^{\mathcal{H}}\varphi^{\prime}_{j^{\prime}}$
with the same bound.

If $\rho\sim 1$, then the support of $\eta_{\rho}$ may contain both critical
points $S_{c}(x,y)$ and $S_{*}(x,y)$ for some
$(x,y)\in\operatorname{supp}{\varphi_{j}}\times\operatorname{supp}{\varphi^{\prime}_{j^{\prime}}}$.
However, if we use the decomposition in the above we can exclude the case
$S_{*}(x,y)\in\operatorname{supp}{\eta_{\rho}}$ breaking $\eta_{\rho}$ into
smooth functions supported in small intervals. In fact, combining (2.29) with
the symmetric properties (2.16), (2.17) and the separation condition (2.28),
we may assume that

$\operatorname{dist}(S_{*}(x,y),\operatorname{supp}\eta_{\rho})\geq c_{0}/2$

for all
$(x,y)\in\operatorname{supp}{\varphi_{j}}\times\operatorname{supp}{\varphi^{\prime}_{j^{\prime}}}$.

To see this, we fix $j,j^{\prime}$ and decompose $\eta_{\rho}$ into
$O(\rho/c_{0})$ many cutoff functions $\eta_{\rho,k}$ which are supported in
finitely overlapping intervals of length $c_{0}/2$. Then, from the
construction ((2.29)) we have either
$\operatorname{dist}(S_{*}(x,y),\operatorname{supp}\eta_{\rho,k})\geq c_{0}/2$
for all $(x,y)\in B(x_{j},2cc_{0})\times B(y_{j^{\prime}},2cc_{0})$ or
$\operatorname{dist}(S_{*}(x,y),\operatorname{supp}\eta_{\rho,k})\leq c_{0}$
for all $(x,y)\in B(x_{j},2cc_{0})\times B(y_{j^{\prime}},2cc_{0})$. The
former case is acceptable. For the latter case, by (2.28) it follows that
$\operatorname{dist}(S_{c}(x,y),\operatorname{supp}\eta_{\rho,k})\geq c_{0}/2$
for all $(x,y)\in B(x_{j},2cc_{0})\times B(y_{j^{\prime}},2cc_{0})$. We note
that $S_{c}(x,-y)+S_{*}(x,y)=\pi$ for any $x,y$ since $\cos S_{*}(x,y)=-\cos
S_{c}(x,-y)$. Thus
$\operatorname{dist}(\eta_{\rho,k}(\pi-\cdot),S_{\ast}(x,-y))\geq c_{0}/2$ for
all $(x,y)\in B(x_{j},2cc_{0})\times B(y_{j^{\prime}},2cc_{0})$. From (2.16)
and (2.17) we note that

$\|\chi_{B(x_{j},2cc_{0})}[\eta_{\rho,j}]^{\mathcal{H}}_{\lambda}\chi_{B(y_{j^{\prime}},2cc_{0})}\|_{p}=\|\chi_{B(x_{j},2cc_{0})}[\eta_{\rho,k,\pi}]^{\mathcal{H}}_{\lambda}\chi_{(-B(y_{j^{\prime}},2cc_{0}))}\|_{p},$

where $\eta_{\rho,k,\pi}(t):=\eta_{\rho,k}(\pi-t)$. Clearly,
$B(x_{j},2cc_{0})\times(-B(y_{j^{\prime}},2cc_{0}))\subset\mathfrak{D}(c_{0})$
and we have $\operatorname{dist}(\eta_{\rho,k,\pi},S_{\ast}(x,y))\geq c_{0}/2$
for all $(x,y)\in B(x_{j},2cc_{0})\times(-B(y_{j^{\prime}},2cc_{0}))$ as
desired.

Now, the proof of (2.22) reduces to showing, for $p>p_{0}(d)$,

(2.30)
$\|\varphi_{j}[\eta_{\rho}]^{\mathcal{H}}_{\lambda}{\varphi^{\prime}_{j^{\prime}}}\|_{p}\lesssim\begin{cases}\lambda^{-\frac{d}{2}}\rho,&\rho\leq\lambda^{-1},\\\
\lambda^{-\frac{d}{2}+\delta(d,p)}\rho^{\delta(d,p)+1},&\rho>\lambda^{-1}\end{cases}$

for each $j,j^{\prime}$. To show this we break the kernel dyadically away from
the diagonal $\\{x=y\\}$:

(2.31)
$(\varphi_{j}[\eta_{\rho}]^{\mathcal{H}}_{\lambda}{\varphi^{\prime}_{j^{\prime}}})(x,y)=\sum_{l}\mathcal{T}^{l}(x,y):=\sum_{l}[\eta_{\rho}]^{\mathcal{H}}_{\lambda}(x,y)\psi(2^{l}|x-y|)\varphi_{j}(x){\varphi^{\prime}_{j^{\prime}}}(y).$

For each $l$ we further decompose $\varphi_{j}$,
${\varphi^{\prime}_{j^{\prime}}}$ into smooth functions which are supported in
finitely overlapping balls of radius $c2^{-l}$. Since $|x-y|\sim 2^{-l}$, by
this additional decomposition it now suffices to consider the operator

$\chi_{l}[\eta_{\rho}]^{\mathcal{H}}_{\lambda}\chi_{l}^{\prime}$

while $\chi_{l},\chi_{l}^{\prime}$ satisfy the following:

(2.32)
$\displaystyle\operatorname{supp}\chi_{l}\times\operatorname{supp}\chi_{l}^{\prime}\subset\mathfrak{D}(c_{0}/2),$
(2.33) $\displaystyle\operatorname{supp}\chi_{l},\
\operatorname{supp}\chi_{l}^{\prime}\subset B(x_{l},2^{2-l}),\ \text{ for some
}$ $\displaystyle x_{0}\in\mathfrak{D}(c_{0}),$ (2.34) $\displaystyle
2^{-l-2}\leq\operatorname{dist}(\operatorname{supp}\chi_{l},\operatorname{supp}\chi_{l}^{\prime})\leq
2^{-l},$ (2.35) $\displaystyle|\partial_{x}^{\alpha}\chi_{l}|,\
|\partial_{x}^{\alpha}\chi_{l}^{\prime}|\leq C_{\alpha}2^{|\alpha|l}.$

The following is clear.

######  Lemma 2.3.

Let $\chi_{l},\chi_{l}^{\prime}$ be smooth functions satisfying (2.32)–(2.35).
Suppose that
$\|\chi_{l}[\eta_{\rho}]^{\mathcal{H}}_{\lambda}\chi_{l}^{\prime}\|_{p}\leq B$
holds, then we have $\|\mathcal{T}^{l}\|_{p}\leq CB$ for a constant $C$.

###  2.3. Estimates for
$\chi_{l}[\eta_{\rho}]^{\mathcal{H}}_{\lambda}\chi_{l}^{\prime}$

Therefore, we only need to obtain estimate for
$\chi_{l}[\eta_{\rho}]^{\mathcal{H}}_{\lambda}\chi_{l}^{\prime}$. To do this,
we consider the kernel of $[\eta_{\rho}]^{\mathcal{H}}_{\lambda}$ which is
given by the integral in (2.13). So, the decay property of the kernel of
$[\eta_{\rho}]^{\mathcal{H}}_{\lambda}$ largely depends on whether the support
of $\eta_{\rho}$ contains the critical point $S_{c}(x,y)$ or not. As be seen
below, it is easy to handle
$\chi_{l}[\eta_{\rho}]^{\mathcal{H}}_{\lambda}\chi_{l}^{\prime}$ when
$\operatorname{dist}(S_{c}(x,y),\operatorname{supp}(\eta_{\rho}))\geq c>0$.

From (2.27) and (2.34) we have

(2.36) $S_{c}(x,y)\sim
2^{-l},\quad(x,y)\in\operatorname{supp}\chi_{l}\times\operatorname{supp}\chi_{l}^{\prime}.$

We also occasionally make use of the following elementary lemmas.

######  Lemma 2.4.

Let $1\leq p\leq\infty$ and let $E,F$ be measurable subsets of
$\mathbb{R}^{d}$. Suppose that the kernel $T(x,y)$ of $T$ satisfies
$|T(x,y)|\leq B$ for $(x,y)\in E\times F$. Then $\|\chi_{E}T\chi_{F}\|_{p}\leq
B|E|^{\frac{1}{p}}|F|^{\frac{1}{p^{\prime}}}.$

######  Lemma 2.5 ([13, Lemma 2.8]).

Let $\phi\in C^{\infty}(I)$ and $A$ be a smooth bump function on $\mathbb{R}$
supported in an interval $I$ of length $0<\rho<2^{2}$. Suppose that
$|\phi^{\prime}(t)|\geq L$, $|\phi^{(n)}(t)|\leq CL\rho^{1-n}$, and
$|A^{(n)}(t)|\leq C\rho^{-n}$ for any $n\in\mathbb{N}_{0}$ and $t\in I$. Then,
for $\lambda\geq 1$ and $N=0,1,2,\dots$,

$\bigg{|}\int A(t)e^{i\lambda\phi(t)}dt\bigg{|}\leq C_{N}\rho(1+\lambda\rho
L)^{-N}.$

Lemma 2.4 follows from interpolation between the trivial estimates
$\|\chi_{E}T\chi_{F}\|_{\infty}\leq B|F|$ and $\|\chi_{E}T\chi_{F}\|_{1}\leq
B|E|$. Lemma 2.5 can be shown by routine integration by parts, also see [13,
Lemma 2.8] for a proof based on a scaling argument.

In order to show (2.30) we separately handle the cases $\rho\leq\lambda^{-1}$
and $\rho>\lambda^{-1}$. The former case is easier to show.

######  Lemma 2.6.

Let $1\leq p\leq\infty$. If $\rho\leq\lambda^{-1}$, then

(2.37)
$\|\chi_{l}[\eta_{\rho}]^{\mathcal{H}}_{\lambda}\chi_{l}^{\prime}\|_{p}\lesssim\begin{cases}\lambda^{-N}\rho^{1-\frac{d}{2}+N}2^{(2N-d)l},&\quad
2^{-l}\gg\lambda^{-\frac{1}{2}}\rho^{\frac{1}{2}},\\\
\rho^{1-\frac{d}{2}}2^{-dl},&\quad
2^{-l}\lesssim\lambda^{-\frac{1}{2}}\rho^{\frac{1}{2}}.\end{cases}$

###### Proof.

The bound (2.37) follows from estimates for the kernel of
$\chi_{l}[\eta_{\rho}]^{\mathcal{H}}_{\lambda}\chi_{l}^{\prime}$, so we assume
$(x,y)\in\operatorname{supp}\chi_{l}\times\operatorname{supp}\chi_{l}^{\prime}$
and $t\in\operatorname{supp}\eta_{\rho}$. We first deal with the bound (2.37)
in the case $2^{-l}\gg\lambda^{-\frac{1}{2}}\rho^{\frac{1}{2}}$.

From (2.36) we have $1-\cos S_{c}(x,y)\sim 2^{-2l}$. Thus it follows $|\cos
t-\cos{S_{c}(x,y)}|\geq|1-\cos S_{c}(x,y)|-|1-\cos t|\gtrsim 2^{-2l}$ because
$2^{-l}\gg\rho$. Using this and (2.26), we get the lower bound

(2.38) $\displaystyle|\partial_{t}\mathcal{P}_{\mathcal{H}}(t,x,y)|\sim
2^{-2l}\rho^{-2}$

since $t\sim\rho$. We now obtain bounds on
$\partial_{t}^{n}\mathcal{P}_{\mathcal{H}}$, $n\geq 2$. From (2.26) we note

(2.39)
$\displaystyle\\!\\!\partial^{n}_{t}\mathcal{P}_{\mathcal{H}}(t,x,y)=-\frac{1}{2}\\!\\!\\!\\!\sum_{n_{1}+n_{2}+n_{3}=n-1}\\!\\!\\!\\!\\!\\!\\!\partial^{n_{1}}_{t}\\!(\cos
t-\cos S_{c})\partial^{n_{2}}_{t}\\!(\cos t-\cos
S_{*})\partial^{n_{3}}_{t}(\sin t)^{-2}.$

From now on, we occasionally denote $S_{c}(x,y)$, $S_{*}(x,y)$ by $S_{c}$,
$S_{*}$, respectively, concealing $x,y$ for simplicity. The absolute value of
the summand $\partial^{n_{1}}_{t}(\cos t-\cos S_{c})\partial^{n_{2}}_{t}(\cos
t-\cos S_{*})\partial^{n_{3}}_{t}(\sin t)^{-2}$ is bounded by $C\rho^{1-n}$ if
$n_{3}\leq n-2$, or $C2^{-2l}\rho^{-n-1}$ if $n_{3}=n-1$. Since
$2^{-l}\gg\rho$, we thus obtain
$|\partial^{n}_{t}\mathcal{P}_{\mathcal{H}}(t,x,y)|\lesssim
2^{-2l}\rho^{-n-1}$. Also, by a computation it is clear that

(2.40) $\displaystyle\quad|\partial_{t}^{n}(\eta_{\rho}(t)(\sin
t)^{-\frac{d}{2}})|\lesssim\rho^{-\frac{d}{2}-n},\quad n\in\mathbb{N}_{0},$

Putting these estimates together and using Lemma 2.5, we get

$|(\chi_{l}[\eta_{\rho}]^{\mathcal{H}}_{\lambda}\chi_{l}^{\prime})(x,y)|\lesssim\lambda^{-N}\rho^{1-\frac{d}{2}+N}2^{2Nl},\quad
N\in\mathbb{N}.$

Then by (2.33) and Lemma 2.4 we get the desired estimate (2.37) when
$2^{-l}\gg\lambda^{-\frac{1}{2}}\rho^{\frac{1}{2}}$.

Now we turn to the case
$2^{-l}\lesssim\lambda^{-\frac{1}{2}}\rho^{\frac{1}{2}}$. In this case, we
shall make use of the following well known estimate (see [30, p.70]):

(2.41)
$\displaystyle|\Pi_{\lambda}^{\mathcal{H}}(x,y)|\lesssim\lambda^{\frac{d}{2}-1},\quad\lambda\in
2\mathbb{N}_{0}+d,$

for any $x,y\in\mathbb{R}^{d}$. From (2.12) we note that
$[\eta_{\rho}]^{\mathcal{H}}_{\lambda}(\lambda^{-1/2}x,\lambda^{-1/2}y)$ can
be expressed as $\sum_{\lambda^{\prime}\in
2\mathbb{N}_{0}+d}\int\eta_{\rho}(2t)e^{it(\lambda-\lambda^{\prime})}dt\,\Pi_{\lambda^{\prime}}(\mathcal{H})(x,y)$.
Thus we have

$\displaystyle[\eta_{\rho}]^{\mathcal{H}}_{\lambda}(\lambda^{-1/2}x,\lambda^{-1/2}y)=2^{-1}\sum_{\lambda^{\prime}\in
2\mathbb{N}_{0}+d}{\mathchoice{{\ooalign{\hbox{\raise
6.25pt\hbox{\scalebox{0.9}[-1.2]{\lower
6.25pt\hbox{$\displaystyle\widehat{\vrule width=0.0pt,height=4.30554pt\vrule
height=0.0pt,width=7.86058pt}$}}}}\cr\hbox{$\displaystyle\eta_{\rho}$}}}}{{\ooalign{\hbox{\raise
6.25pt\hbox{\scalebox{0.9}[-1.2]{\lower 6.25pt\hbox{$\textstyle\widehat{\vrule
width=0.0pt,height=4.30554pt\vrule
height=0.0pt,width=7.86058pt}$}}}}\cr\hbox{$\textstyle\eta_{\rho}$}}}}{{\ooalign{\hbox{\raise
5.81944pt\hbox{\scalebox{0.9}[-1.2]{\lower
5.81944pt\hbox{$\scriptstyle\widehat{\vrule width=0.0pt,height=3.01389pt\vrule
height=0.0pt,width=5.54376pt}$}}}}\cr\hbox{$\scriptstyle\eta_{\rho}$}}}}{{\ooalign{\hbox{\raise
5.53241pt\hbox{\scalebox{0.9}[-1.2]{\lower
5.53241pt\hbox{$\scriptscriptstyle\widehat{\vrule
width=0.0pt,height=2.15277pt\vrule
height=0.0pt,width=4.5507pt}$}}}}\cr\hbox{$\scriptscriptstyle\eta_{\rho}$}}}}}((\lambda-\lambda^{\prime})/2)\Pi_{\lambda^{\prime}}^{\mathcal{H}}(x,y).$

Since $|{\mathchoice{{\ooalign{\hbox{\raise
6.25pt\hbox{\scalebox{0.9}[-1.2]{\lower
6.25pt\hbox{$\displaystyle\widehat{\vrule width=0.0pt,height=4.30554pt\vrule
height=0.0pt,width=7.86058pt}$}}}}\cr\hbox{$\displaystyle\eta_{\rho}$}}}}{{\ooalign{\hbox{\raise
6.25pt\hbox{\scalebox{0.9}[-1.2]{\lower 6.25pt\hbox{$\textstyle\widehat{\vrule
width=0.0pt,height=4.30554pt\vrule
height=0.0pt,width=7.86058pt}$}}}}\cr\hbox{$\textstyle\eta_{\rho}$}}}}{{\ooalign{\hbox{\raise
5.81944pt\hbox{\scalebox{0.9}[-1.2]{\lower
5.81944pt\hbox{$\scriptstyle\widehat{\vrule width=0.0pt,height=3.01389pt\vrule
height=0.0pt,width=5.54376pt}$}}}}\cr\hbox{$\scriptstyle\eta_{\rho}$}}}}{{\ooalign{\hbox{\raise
5.53241pt\hbox{\scalebox{0.9}[-1.2]{\lower
5.53241pt\hbox{$\scriptscriptstyle\widehat{\vrule
width=0.0pt,height=2.15277pt\vrule
height=0.0pt,width=4.5507pt}$}}}}\cr\hbox{$\scriptscriptstyle\eta_{\rho}$}}}}}(\tau)|\lesssim\rho(1+\rho|\tau|)^{-N}$
for any $N\in\mathbb{N}$, we get

$\big{|}[\eta_{\rho}]^{\mathcal{H}}_{\lambda}(\lambda^{-1/2}x,\lambda^{-1/2}y)\big{|}\lesssim\sum_{\lambda^{\prime}\in
2\mathbb{N}_{0}+d}\rho(1+\rho|\lambda-\lambda^{\prime}|)^{-N}{(\lambda^{\prime})}^{\frac{d}{2}-1}\lesssim\rho^{1-\frac{d}{2}}$

for $x,y\in\mathbb{R}^{d}$. Thus,
$|[\eta_{\rho}]^{\mathcal{H}}_{\lambda}(x,y)|\lesssim\rho^{1-\frac{d}{2}}.$
Applying Lemma 2.4 with (2.33), we get the second estimate in (2.37). ∎

Estimates for $\chi_{l}[\eta_{\rho}]^{\mathcal{H}}_{\lambda}\chi_{l}^{\prime}$
when $\rho>\lambda^{-1}$. We distinguish the three cases

$2^{-l}\ll\rho,\quad 2^{-l}\gg\rho,\quad 2^{-l}\sim\rho.$

Unlike the first and second cases, the support of $\eta_{\rho}$ may contain
$S_{c}(x,y)$ in the third case. So handling the case is the main part of the
proof (see Proposition 2.8). We first deals with the two easier cases.

######  Lemma 2.7.

Let $1\leq p\leq\infty$. If $\rho>\lambda^{-1}$, then

(2.42)
$\|\chi_{l}[\eta_{\rho}]^{\mathcal{H}}_{\lambda}\chi_{l}^{\prime}\|_{p}\lesssim\begin{cases}\lambda^{-N}\rho^{1-\frac{d}{2}-N}2^{-dl},&\quad
2^{-l}\ll\rho,\\\ \lambda^{-N}\rho^{1-\frac{d}{2}+N}2^{(2N-d)l},&\quad
2^{-l}\gg\rho.\end{cases}$

###### Proof.

As before we show the bound (2.42) by obtaining estimates for the kernel of
$\chi_{l}[\eta_{\rho}]^{\mathcal{H}}_{\lambda}\chi_{l}^{\prime}$. Throughout
the proof we assume
$(x,y)\in\operatorname{supp}\chi_{l}\times\operatorname{supp}\chi_{l}^{\prime}$
and $t\in\operatorname{supp}\eta_{\rho}$.

We first deal with the case $\rho\gg 2^{-l}$. By (2.34) and (2.27) we have
$1-\cos S_{c}(x,y)\sim 2^{-2l}$, so $|\cos t-\cos{S_{c}(x,y)}|\geq|1-\cos
t|-|1-\cos S_{c}(x,y)|\gtrsim\rho^{2}$ because $\rho\gg 2^{-l}$. Using this
and (2.26) and (2.39), we get
$|\partial_{t}\mathcal{P}_{\mathcal{H}}(t,x,y)|\gtrsim 1$ and

$\displaystyle|\partial^{n}_{t}\mathcal{P}_{\mathcal{H}}(t,x,y)|\lesssim\rho^{1-n},\quad
n\in\mathbb{N}.$

Since $|\partial^{n}_{t}(\eta_{\rho}(t)(\sin
t)^{-\frac{d}{2}})|\lesssim\rho^{-\frac{d}{2}-n}$, by Lemma 2.5 we get
$|(\chi_{l}[\eta_{\rho}]^{\mathcal{H}}_{\lambda}\chi_{l}^{\prime})(x,y)|\lesssim\lambda^{-N}\rho^{1-\frac{d}{2}-N},$
$N\in\mathbb{N}_{0}.$ Applying Lemma 2.4 yields the first case estimate in
(2.42).

The proof of the second case in (2.42) similar to that of the first case of
(2.37). Note that $|\cos t-\cos{S_{c}(x,y)}|\gtrsim 2^{-2l}$ because $\rho\ll
2^{-l}$. Thus, from (2.26) and (2.39), we have
$|\partial_{t}\mathcal{P}_{\mathcal{H}}(t,x,y)|\gtrsim 2^{-2l}\rho^{-2}$ and
$|\partial^{n}_{t}\mathcal{P}_{\mathcal{H}}(t,x,y)|\lesssim
2^{-2l}\rho^{-1-n},$ $n\in\mathbb{N}.$ The rest of proof is identical, so we
omit the detail. ∎

The following is the key estimate which we need to prove Theorem 2.2.

######  Proposition 2.8.

Let $\rho>\lambda^{-1}$ and $\rho\sim 2^{-l}$. Then we have

(2.43)
$\|\chi_{l}[\eta_{\rho}]^{\mathcal{H}}_{\lambda}\chi_{l}^{\prime}\|_{p}\lesssim\lambda^{-\frac{d}{2}+\delta(d,p)}\rho^{1+\delta(d,p)}$

provided that $p_{0}(d)<p\leq\infty$.

Assuming Proposition 2.8, we show the estimate (2.30). We combine Lemma 2.6,
Lemma 2.7 and Proposition 2.8.

###### Proof of (2.30).

Let us first consider the case $\rho\leq\lambda^{-1}$. By (2.31) we see

(2.44)
$\|\varphi_{j}[\eta_{\rho}]^{\mathcal{H}}_{\lambda}\varphi_{j^{\prime}}\|_{p}\leq\sum_{l}\|\mathcal{T}^{l}\|_{p}.$

Splitting the sum
$\sum_{l}=\sum_{2^{-l}\lesssim\lambda^{-\frac{1}{2}}\rho^{\frac{1}{2}}}+\sum_{2^{-l}\gg\lambda^{-\frac{1}{2}}\rho^{\frac{1}{2}}}$,
we combine Lemma 2.6 with a large $N$ and Lemma 2.3. Thus, we see that
$\|\varphi_{j}[\eta_{\rho}]^{\mathcal{H}}_{\lambda}\varphi_{j^{\prime}}\|_{p}$
is bounded by a constant times

$\displaystyle\sum_{2^{-l}\lesssim\lambda^{-\frac{1}{2}}\rho^{\frac{1}{2}}}\rho^{1-\frac{d}{2}}2^{-dl}+\sum_{2^{-l}\gg\lambda^{-\frac{1}{2}}\rho^{\frac{1}{2}}}\lambda^{-N}\rho^{1-\frac{d}{2}+N}2^{(2N-d)l}.$

Thus, we get
$\|\varphi_{j}[\eta_{\rho}]^{\mathcal{H}}_{\lambda}\varphi_{j^{\prime}}\|_{p}\lesssim\
\lambda^{-\frac{d}{2}}\rho$ as desired.

We now consider the case $\rho>\lambda^{-1}$. Similarly, using Lemma 2.7 with
a large $N$ and Proposition 2.8 together with Lemma 2.3, we have

$\displaystyle\big{(}\sum_{2^{-l}\ll\rho}+\sum_{2^{-l}\sim\rho}+\sum_{2^{-l}\gg\rho}\big{)}\|\mathcal{T}^{l}\|_{p}\lesssim(\rho\lambda)^{-N}\rho^{1+\frac{d}{2}}+\lambda^{-\frac{d}{2}+\delta(d,p)}\rho^{1+\delta(d,p)}$

for $p_{0}(d)<p\leq\infty$. By (2.44) it follows
$\|\varphi_{j}[\eta_{\rho}]^{\mathcal{H}}_{\lambda}\varphi_{j^{\prime}}\|_{p}\lesssim\lambda^{-\frac{d}{2}+\delta(d,p)}\rho^{1+\delta(d,p)}$
since $\rho>\lambda^{-1}$. Therefore we get (2.30). ∎

To complete the proof of Theorem 2.2 it now remains to prove Proposition 2.8.
The rest of this section is devoted to the proof of Proposition 2.8.

###  2.4. Asymptotic expansion of the kernel when
$2^{-l}\sim\rho>\lambda^{-1}$

As discussed before, in this case the support of $\eta_{\rho}$ may contain the
critical point $S_{c}(x,y)$.

Additional decomposition of $\chi_{l}$, $\chi^{\prime}_{l}$ and $\eta_{\rho}$.
If we further break the cutoff functions $\chi_{l}$, $\chi^{\prime}_{l}$ and
$\eta_{\rho}$ into finitely many smooth functions, then in addition to
(2.32)–(2.35) we may assume that

(2.45) $\displaystyle\operatorname{supp}\chi_{l}\subset
B(x_{0},\epsilon_{0}\rho),$
$\displaystyle\quad\operatorname{supp}\chi_{l}^{\prime}\subset
B(y_{0},\epsilon_{0}\rho),$ (2.46) $\displaystyle S_{c}(x,y)\in
S_{c}(x_{0},y_{0})+(-\epsilon_{0}\rho,\epsilon_{0}\rho),$
$\displaystyle\quad(x,y)\in\operatorname{supp}\chi_{l}\times\operatorname{supp}\chi_{l}^{\prime},$
(2.47) $\displaystyle\operatorname{supp}\eta_{\rho}\subset S_{c}(x_{0},y_{0}$
$\displaystyle)+(-2\epsilon_{0}\rho,2\epsilon_{0}\rho),$

for a small $\epsilon_{0}>0$ and $(x_{0},y_{0})\in\mathfrak{D}(c_{0})$.
Indeed, just breaking $\chi_{l}$, $\chi^{\prime}_{l}$ into cutoff functions
which are supported in finitely overlapping balls of radius
$c\epsilon_{0}\rho$ with a small $c>0$, trivially we have (2.45) and (2.46)
and this gives rise to only $O(\epsilon_{0}^{-2d})$ many such functions.
However, the third condition is not completely trivial. This can be achieved
by breaking $\chi_{l}[\eta_{\rho}]^{\mathcal{H}}_{\lambda}\chi_{l}^{\prime}$
into major and minor parts. In fact, let $\widetilde{\eta}_{\rho}$ be a smooth
bump function such that ${\operatorname{supp}\widetilde{\eta_{\rho}}}\subset
S_{c}(x_{0},y_{0})+(-3\epsilon_{0}\rho,3\epsilon_{0}\rho)$,
$\widetilde{\eta_{\rho}}(t)=1$ if $t\in
S_{c}(x_{0},y_{0})+(-2\epsilon_{0}\rho,2\epsilon_{0}\rho)$, and
$|\widetilde{\eta_{\rho}}^{(n)}|\leq C\rho^{-n}$ for any $n\in\mathbb{N}_{0}$.
Then we split

$\chi_{l}[\eta_{\rho}]^{\mathcal{H}}_{\lambda}\chi_{l}^{\prime}=\chi_{l}[\eta_{\rho}\widetilde{\eta}_{\rho}]^{\mathcal{H}}_{\lambda}\chi_{l}^{\prime}+\chi_{l}[\eta_{\rho}(1-\widetilde{\eta}_{\rho})]^{\mathcal{H}}_{\lambda}\chi_{l}^{\prime}.$

The second term can be handled in the same manner as before. From (2.46) we
note that $|\cos t-\cos{S_{c}(x,y)}|\gtrsim\rho^{2}$ on the support of
$\eta_{\rho}(1-\widetilde{\eta}_{\rho})$. Thus, using this, (2.26) and (2.39),
we have $|\partial_{t}\mathcal{P}_{\mathcal{H}}(t,x,y)|\gtrsim 1$ and
$|\partial^{n}_{t}\mathcal{P}_{\mathcal{H}}(t,x,y)|\lesssim\rho^{1-n},$
$n\in\mathbb{N}.$ Following the argument in the proofs of Lemma 2.6 and 2.7 we
get
$\|\chi_{l}[\eta_{\rho}(1-\widetilde{\eta}_{\rho})]^{\mathcal{H}}_{\lambda}\chi_{l}^{\prime}\|_{p}\lesssim(\lambda\rho)^{-N}\rho^{1+\frac{d}{2}}$
for $\rho>\lambda^{-1}$. Therefore we may disregard the contribution from
$\chi_{l}[\eta_{\rho}(1-\widetilde{\eta}_{\rho})]^{\mathcal{H}}_{\lambda}\chi_{l}^{\prime}$
and we may therefore assume (2.47).

We obtain an asymptotic development of the kernel
$[\eta_{\rho}]^{\mathcal{H}}_{\lambda}(x,y)$ via the method of stationary
phase. We recall the formula (2.39) with $n=2$. Then, using (2.24) and (2.25)
we have

(2.48) $\displaystyle\begin{aligned}
\partial_{t}^{2}\mathcal{P}_{\mathcal{H}}(S_{c}(x,y),x,y)=\frac{\cos
S_{*}(x,y)-\cos S_{c}(x,y)}{2\sin
S_{c}(x,y)}=\frac{\sqrt{\mathcal{D}(x,y)}}{\sin S_{c}(x,y)}.\end{aligned}$

Since $c_{0}\leq\sqrt{\mathcal{D}(x,y)}\leq 1$, from (2.34) and (2.36) we have

(2.49)
$\displaystyle\partial_{t}^{2}\mathcal{P}_{\mathcal{H}}(S_{c}(x,y),x,y)\sim\rho^{-1},\quad(x,y)\in\operatorname{supp}\chi_{l}\times\operatorname{supp}\chi_{l}^{\prime}.$

One can easily see

(2.50)
$\displaystyle|\partial^{n}_{t}\partial_{x}^{\alpha}\partial_{y}^{\beta}\mathcal{P}_{\\!\,\mathcal{H}}(t,x,y)|\lesssim\rho^{1-n-|\alpha|-|\beta|},$

where
$(t,x,y)\in\operatorname{supp}\eta_{\rho}\times\operatorname{supp}\chi_{l}\times\operatorname{supp}\chi_{l}^{\prime}$.
To show the estimate (2.50), recalling (2.14), we only need to consider the
case $|\alpha|+|\beta|\leq 2$. When $|\alpha|+|\beta|=0$, (2.50) can be shown
using (2.39) similarly as before. The other cases can be handled in a more
straightforward manner. For example, note that
$\partial_{x}\mathcal{P}_{\\!\,\mathcal{H}}(t,x,y)=(\cos tx-y)/\sin t$. Thus
$\partial_{x}\mathcal{P}_{\\!\,\mathcal{H}}(t,x,y)=O(1)$ because $\cos
t\,x-y=(\cos t-1)\,x+x-y=O(\rho)$ and we also have
$\partial_{t}^{n}\partial_{x}\mathcal{P}_{\\!\,\mathcal{H}}(t,x,y)=O(\rho^{-n}).$
The remaining cases can be handled similarly.

We make change of variables so that the associated phase and amplitude
functions have uniformly bounded derivatives in $\rho$. Let us set

$\displaystyle\sigma(t,x,y)$ $\displaystyle=S_{c}(\rho x+x_{0},\rho
y+y_{0})+\rho t,$ $\displaystyle\widetilde{\mathcal{P}}_{h}(t,x,y)$
$\displaystyle=\rho^{-1}\mathcal{P}_{\mathcal{H}}(\sigma(t,x,y),\rho
x+x_{0},\rho y+y_{0}),$ $\displaystyle\widetilde{\chi}_{l}(x)$
$\displaystyle=\chi_{l}(\rho
x+x_{0}),\quad\widetilde{\chi}_{l}^{\prime}(y)=\chi_{l}(\rho y+y_{0}),$
$\displaystyle a(t,x,y)$
$\displaystyle=\rho^{\frac{d}{2}}(\sin\sigma(t,x,y))^{-\frac{d}{2}}\eta_{\rho}(\sigma(t,x,y))\widetilde{\chi}_{l}(x)\widetilde{\chi}_{l}^{\prime}(y).$

As to be seen later, all of these functions have uniformly bounded derivatives
and the support of $a$ is contained in $(-2\epsilon_{0},2\epsilon_{0})\times
B(0,\epsilon_{0})\times B(0,\epsilon_{0})$. Let us set

$I_{\rho}^{\mathcal{H}}(x,y)=\rho^{1-\frac{d}{2}}\int
a(t,x,y)e^{i\lambda\rho\widetilde{\mathcal{P}}_{h}(t,x,y)}dt.$

By the change of variables $t\to\sigma(t,x,y)$, we have

(2.51) $(\chi_{l}[\eta_{\rho}]^{\mathcal{H}}_{\lambda}\chi_{l}^{\prime})(\rho
x+x_{0},\rho y+y_{0})=C_{d}I_{\rho}^{\mathcal{H}}(x,y),$

where $C_{d}$ is a constant depending on $d$. We obtain an asymptotic
expansion of the integral $I_{\rho}^{\mathcal{H}}(x,y)$ using the method of
stationary phase.

To do so, we first note

(2.52)
$|\partial_{t}^{n}\partial_{x}^{\alpha}\partial_{y}^{\beta}\sigma(t,x,y)|\leq
C\rho$

for
$(t,x,y)\in\operatorname{supp}a\times\operatorname{supp}\widetilde{\chi}_{l}\times\operatorname{supp}\widetilde{\chi}_{l}^{\prime}$.
This can be shown using the following.

######  Lemma 2.9.

If
$(x,y)\in\operatorname{supp}\chi_{l}\times\operatorname{supp}\chi_{l}^{\prime},$
then

(2.53)
$\displaystyle|\partial_{x}^{\alpha}\partial_{y}^{\beta}S_{c}(x,y)|\lesssim\rho^{1-|\alpha|-|\beta|}.$

###### Proof.

From (2.27) we see (2.53) is trivially true if $|\alpha|+|\beta|=0$. We may
assume $|\alpha|+|\beta|\geq 1$ and by symmetry we may also assume
$|\alpha|\geq 1$. Since $\partial_{x}S_{c}(x,y)=-(\sin
S_{c}(x,y))^{-1}\partial_{x}\cos S_{c}(x,y)$ we note that

$\partial_{x}^{\alpha}\partial_{y}^{\beta}(\partial_{x}S_{c}(x,y))=-\sum_{\alpha_{1}+\alpha_{2}=\alpha,\,|\alpha_{2}|=1}\partial_{x}^{\alpha_{1}}\partial_{y}^{\beta}(\sin
S_{c}(x,y))^{-1}\partial_{x}^{\alpha_{2}}\cos S_{c}(x,y)+\mathcal{E}$

where $\mathcal{E}$ is a linear combination of the terms

$\partial_{x,y}^{\kappa}(\sin S_{c}(x,y))^{-1}\partial_{x,y}^{\gamma}\cos
S_{c}(x,y)$

with $|\gamma|\geq 2$ and $|\kappa|=|\alpha|+|\beta|-|\gamma|$. On the other
hand, for any multi-indices $\alpha^{\prime},\beta^{\prime}$, using (2.24), we
get $|\partial_{x}^{\alpha^{\prime}}\partial_{y}^{\beta^{\prime}}\cos
S_{c}(x,y)|\leq C_{\alpha^{\prime},\beta^{\prime}}$ because
$\mathcal{D}(x,y)\geq c_{0}^{2}/4$ and from (2.27) it similarly follows that
$\partial_{x}^{\alpha^{\prime}}\partial_{y}^{\beta^{\prime}}(\sin
S_{c}(x,y))^{-1}=O(|x-y|^{-|\alpha^{\prime}|-|\beta^{\prime}|-1}).$ Thus, we
see $|\mathcal{E}|\lesssim|x-y|^{1-|\alpha|-|\beta|}$ and
$|\partial_{x}^{\alpha-1}\partial_{y}^{\beta}((\sin
S_{c}(x,y))^{-1})\partial_{x}\cos
S_{c}(x,y)|\lesssim|x-y|^{1-|\alpha|-|\beta|}$. Hence we get (3.31). ∎

Using (2.52), (2.40), and (2.53), one can easily see

(2.54)
$\displaystyle|\partial_{t}^{n}\partial_{x}^{\alpha}\partial_{y}^{\beta}a(t,x,y)|\leq
C_{n,\alpha,\beta}$

for
$(t,x,y)\in\operatorname{supp}a\times\operatorname{supp}\widetilde{\chi}_{l}\times\operatorname{supp}\widetilde{\chi}_{l}^{\prime}.$
Similarly, combining (2.50) with (2.53), we also have

(2.55)
$\displaystyle|\partial_{t}^{n}\partial_{x}^{\alpha}\partial_{y}^{\beta}\widetilde{\mathcal{P}}_{h}(t,x,y)|\leq
C_{n,\alpha,\beta}$

for
$(t,x,y)\in\operatorname{supp}a\times\operatorname{supp}\widetilde{\chi}_{l}\times\operatorname{supp}\widetilde{\chi}_{l}^{\prime}.$
From (2.49) it follows that

(2.56) $\displaystyle\partial_{t}^{2}\widetilde{\mathcal{P}}_{h}(t,x,y)\sim 1$

for
$(t,x,y)\in\operatorname{supp}a\times\operatorname{supp}\widetilde{\chi}_{l}\times\operatorname{supp}\widetilde{\chi}_{l}^{\prime}.$
Since $\partial_{t}\widetilde{\mathcal{P}}_{h}(0,x,y)=0$, the map
$t\to\widetilde{\mathcal{P}}_{h}(t,x,y)$ has a nondegenerate critical point at
$t=0$. Thus we may now apply the method of stationary phase. In fact, applying
[12, Theorem 7.7.5] together with (2.54), (2.55), and (2.56), we obtain the
following.

######  Lemma 2.10.

For $N\in\mathbb{N}$, we have

(2.57) $\displaystyle I_{\rho}^{\mathcal{H}}(x,y)=$
$\displaystyle\rho^{\frac{2-d}{2}}\sum_{n=0}^{N-1}(\lambda\rho)^{-\frac{1}{2}-n}A_{n}(x,y)e^{i\lambda\rho\widetilde{\mathcal{P}}_{h}(0,x,y)}+E_{N}(x,y),$

where $\operatorname{supp}A_{n},\operatorname{supp}E_{N}\subset
B(0,\epsilon_{0})\times B(0,\epsilon_{0})$ and

$\displaystyle|\partial_{x}^{\alpha}\partial_{y}^{\beta}A_{n}(x,y)|\leq
C_{\alpha,\beta},\quad|E_{N}(x,y)|\leq
C_{N}\rho^{\frac{2-d}{2}}(\lambda\rho)^{-N}$

with $C_{\alpha,\beta}$ and $C_{N}$ independent of $\lambda,\rho$.

In the expansion (2.57), the error term $E_{N}(x,y)$ is negligible if we take
$N$ large enough. Indeed, from Lemma 2.4 we have
$\|\chi_{l}E_{N}\chi_{l}^{\prime}\|_{p}\lesssim\rho^{\frac{2-d}{2}}(\lambda\rho)^{-N}\rho^{d}$.
With a large $N$ it is clear that
$\rho^{\frac{2-d}{2}}(\lambda\rho)^{-N}\rho^{d}\lesssim\lambda^{-\frac{d}{2}+\delta(d,p)}\rho^{1+\delta(d,p)}$.
Thus, to obtain the estimate (2.43) we are led to consider the operators with
the oscillatory kernels
$A_{n}(x,y)e^{i\lambda\rho\widetilde{\mathcal{P}}_{h}(0,x,y)}$. Boundedness of
properties of such an operator are determined by the phase function
$\widetilde{\mathcal{P}}_{h}(0,x,y)$. So, we need to take a close look at it.

Let us define

(2.58) $\Phi_{\mathcal{H}}(x,y)=\mathcal{P}_{\mathcal{H}}(S_{c}(x,y),x,y).$

From (2.14), using (2.65) and (2.25), we see

$\displaystyle\Phi_{\mathcal{H}}(x,y)$
$\displaystyle=\frac{1}{2}\Big{(}S_{c}(x,y)+\frac{(|x|^{2}+|y|^{2})\cos
S_{c}(x,y)-2\langle x,y\rangle}{\sin S_{c}(x,y)}\Big{)}$
$\displaystyle=\frac{1}{2}(S_{c}(x,y)-\cos S_{*}(x,y)\sin S_{c}(x,y)).$

Combining this with (2.27) one can easily see
$\Phi_{\mathcal{H}}(x,y)-2^{-1}|x-y|=O(|x-y|^{2})$ as $|x-y|\to 0$. Thus, this
suggest the phase function $\Phi_{\mathcal{H}}(x,y)$ can be viewed as a small
perturbation of the function $|x-y|$ when $(x,y)$ is contained in a small ball
with center $0$. This is natural in view of the transplantation theorem due to
Kenig, Stanton, and Tomas [16] because the phase function $|x-y|$ arises as a
counterpart of $\Phi_{\mathcal{H}}(x,y)$ for the classical Bochner-Riesz
operator $(1+\Delta)_{+}^{\delta}$.

###  2.5. Carleson-Sjölin type operator

Let $A\in C_{c}^{\infty}(\mathbb{R}^{d}\times\mathbb{R}^{d-1})$ and $\phi\in
C^{\infty}(\operatorname{supp}A)$. Let $T_{\lambda}[\phi,A]$ denote the
operator defined by

$T_{\lambda}[\phi,A]f(x)=\int_{\mathbb{R}^{d-1}}e^{i\lambda\phi(x,\xi)}A(x,\xi)f(\xi)d\xi.$

We assume that the mixed Hessian $\partial_{\xi}\partial^{\intercal}_{x}\phi$
has the maximal rank, i.e.,

(C1)
$\displaystyle\operatorname{rank}(\partial_{\xi}\partial^{\intercal}_{x}\phi(x,\xi))=d-1,\quad(x,\xi)\in\operatorname{supp}A$

This means that the image of $\xi\to\partial_{x}\phi(x_{0},\xi)$ is a smooth
immersed surface in $\mathbb{R}^{d}$. The condition guarantees that, for any
$(x_{0},\xi_{0})\in\operatorname{supp}A$, there is a unique vector
$\nu(x_{0},y_{0})\in\mathbb{S}^{d-1}$ modulo $\pm$ such that

$\partial_{\xi}\langle\partial_{x}\phi(x_{0},\xi),\nu(x_{0},\xi_{0})\rangle\big{|}_{\xi=\xi_{0}}=0.$

We further assume that the parameterized surface
$\xi\to\partial_{x}\phi(x_{0},\xi)$ has nonvanishing Gaussian curvature.
Equivalently, the second fundamental form of the surface parameterized by
$\xi\to\partial_{x}\phi(x_{0},\xi)$ is not singular, i.e.,

$\operatorname{rank}(\partial_{\xi}\partial_{\xi}^{\intercal}\langle\partial_{x}\phi(x_{0},\xi),\nu(x_{0},\xi_{0})\rangle\big{|}_{\xi=\xi_{0}})=d-1,\
\ (x_{0},\xi_{0})\in\operatorname{supp}A.$

The conditions (C1) and (2.5) together are called the Carleson-Sjölin
condition, and if an oscillatory integral operator $T_{\lambda}[\phi,A]$
satisfies both (C1) and (2.5), we say $T_{\lambda}[\phi,A]$ is a Carleson-
Sjölin type operator. Concerning the Carleson-Sjölin type operators, the
estimate of the form

(2.59)
$\displaystyle\|T_{\lambda}[\phi,A]f\|_{L^{q}(\mathbb{R}^{d})}\lesssim\lambda^{-\frac{d}{q}}\|f\|_{L^{p}(\mathbb{R}^{d-1})}$

has been studied by various authors [4, 11, 24, 3, 20, 9]. Hörmander [11]
conjectured that (2.59) holds for $\frac{1}{q}<\frac{d-1}{2d}$ and
$\frac{1}{q}\leq\frac{d-1}{(d+1)p^{\prime}}$ 222The range of $p,q$ is the best
possible one. This follows from the well known necessity condition for the
restriction estimate to the surfaces with nonzero curvature since the estimate
(2.59) implies the adjoint restriction estimate to such surfaces.  if
$T_{\lambda}[\phi,A]$ is a Carleson-Sjölin type operator. This was verified by
Hörmander when $d=2$. He also showed that this range of $p,q$ is optimal for
(2.59).

In higher dimensions, Stein [24] obtained (2.59) for $\frac{2(d+1)}{d-1}\leq
q\leq\infty$ and $\frac{1}{q}\leq\frac{d-1}{(d+1)p^{\prime}}$ but Bourgain [3]
essentially disproved the Hörmander’s conjecture by constructing a phase
function $\phi$ which satisfies the Carleson-Sjölin condition but the estimate
(2.59) does fails for any $q>\frac{2(d+1)}{d-1}$ if $d$ is odd. Nevertheless,
one of the authors [20] proved that the range of $p,q$ for (2.59) can be
improved to $q>\frac{2(d+2)}{d}$ under the additional condition:

(C3) For $(x_{0},\xi_{0})\in\operatorname{supp}A$, all nonzero eigenvalues of
the matrix
$\displaystyle\partial_{\xi}\partial_{\xi}^{\intercal}\langle\partial_{x}\phi(x_{0},\xi),\nu(x_{0},\xi_{0})\rangle|_{\xi=\xi_{0}}\text{
have the same sign.}$

It was also shown that the range is optimal when $d=3$. Recently, Guth,
Hickman and Iliopoulou [9] obtained the sharp result for (2.59) with $p=q$
under the assumptions (C1) – (C3) for $d\geq 4$. The main ingredients of their
result are multilinear estimates due to Bennett, Carbery and Tao [2] and the
method of polynomial partitioning. For our purpose, we summarize the
previously known results when $p=q$ as follows ([4, 11, 20, 9]).

######  Theorem 2.11.

Suppose that $d\geq 2$ and $\phi$ satisfies the conditions (C1) – (C3). Then
we have (2.59) whenever $p=q>p_{0}(d)$.

Furthermore, the range of $p$ is sharp up to the endpoint in that there exist
counterexamples of $T_{\lambda}[\phi,A]$ whose phase function $\phi$ satisfies
the conditions C1) – C3) such that (2.59) with $p=q$ fails whenever
$p<p_{0}(d)$ (see [9]).

###  2.6. The phase function $\Phi_{\mathcal{H}}$

In this section we investigate the phase function $\Phi_{\mathcal{H}}$ and its
curvature condition. We begin with considering the vectors
$\mathbf{a}(x,y),\mathbf{b}(x,y)$ which are given by

(2.60) $\displaystyle\mathbf{a}(x,y)$ $\displaystyle:=\cos S_{c}(x,y)x-y,$
(2.61) $\displaystyle\mathbf{b}(x,y)$ $\displaystyle:=x-\cos S_{c}(x,y)y.$

The following show how the vectors $\mathbf{a}(x,y),\mathbf{b}(x,y)$, and
$x,y$ are related.

######  Lemma 2.12.

Let $\mathcal{D}(x,y)>0$. Then, we have the following:

(2.62) $\displaystyle|\mathbf{a}(x,y)|^{2}$
$\displaystyle=(1-|x|^{2})\sin^{2}S_{c}(x,y),$ (2.63)
$\displaystyle|\mathbf{b}(x,y)|^{2}$
$\displaystyle=(1-|y|^{2})\sin^{2}S_{c}(x,y),$ (2.64)
$\displaystyle\langle\mathbf{a}(x,y),\mathbf{b}(x,y)\rangle$
$\displaystyle=\sqrt{\mathcal{D}(x,y)}\sin^{2}S_{c}(x,y).$

In particular, (2.62) shows that the map
$y\to\partial_{x}\Phi_{\mathcal{H}}(x,y)$ can not have the maximal rank $d$.
To see this, note that

$\partial_{x}\Phi_{\mathcal{H}}(x,y)=\partial_{x}\mathcal{P}_{\mathcal{H}}(S_{c}(x,y),x,y)=\frac{\cos
S_{c}(x,y)x-y}{\sin S_{c}(x,y)}.$

For the first equality we use
$\partial_{t}\mathcal{P}_{\mathcal{H}}(S_{c}(x,y),x,y)=0$. Thus, by (2.62) we
have $|\partial_{x}\Phi_{\mathcal{H}}(x,y)|=\sqrt{1-|x|^{2}}$. So, the image
of $y\to\partial_{x}\Phi_{\mathcal{H}}(x,y)$ is contained in the sphere of
radius $\sqrt{1-|x|^{2}}$.

###### Proof of Lemma 2.12.

From (2.23) we note that $-2\langle x,y\rangle\cos
S_{c}+|y|^{2}=1-|x|^{2}-\cos^{2}S_{c}$. Thus, it follows

$\displaystyle|\cos S_{c}\mspace{1.0mu}x-y|^{2}$
$\displaystyle=|x|^{2}\cos^{2}S_{c}-2\langle x,y\rangle\cos
S_{c}+|y|^{2}=(1-|x|^{2})\sin^{2}S_{c}.$

This gives (2.62). Since $\mathbf{b}(x,y)=-\mathbf{a}(y,x)$, (2.63) follows
from (2.62). It remains to show (2.64). Let us set $\mathcal{Q}(t,x,y)=\langle
x,y\rangle\cos^{2}t-(|x|^{2}+|y|^{2})\cos t+\langle x,y\rangle$. Then we note

$\displaystyle\langle\mathbf{a}(x,y),\mathbf{b}(x,y)\rangle=-\mathcal{Q}(S_{c},x,y).$

We also note that
$\partial_{t}^{2}\mathcal{P}_{\mathcal{H}}(t,x,y)=-\frac{\mathcal{Q}(t,x,y)}{\sin^{3}t}.$
Combining this with (2.48) we get

(2.65) $\mathcal{Q}(S_{c}(x,y),x,y)=-\sqrt{\mathcal{D}}\sin^{2}S_{c}$

and hence (2.64). ∎

Let us set

(2.66)
$\mathbf{M}(x,y)=\partial_{z}\partial_{z}^{\intercal}\Big{\langle}\partial_{x}\Phi_{\mathcal{H}}(x,z),\frac{\mathbf{a}(x,y)}{|\mathbf{a}(x,y)|}\Big{\rangle}\Big{|}_{z=y}.$

What follows is crucial in showing that the phase $\Phi_{\mathcal{H}}$
satisfies C1) – C3) after rescaling and freezing a suitable coordinate.

######  Lemma 2.13.

Let
$(x,y)\in\operatorname{supp}\chi_{l}\times\operatorname{supp}\chi_{l}^{\prime}$.
Then, $(i)$ the matrix
$\partial_{y}\partial_{x}^{\intercal}\Phi_{\mathcal{H}}(x,y)$ has rank $d-1$
and

(2.67)
$\partial_{y}\partial_{x}^{\intercal}\Phi_{\mathcal{H}}(x,y)\mathbf{a}(x,y)=0.$

Additionally, $(ii)$ if
$(x,y)\in\operatorname{supp}\chi_{l}\times\operatorname{supp}\chi_{l}^{\prime}$
satisfies

(2.68) $\frac{\mathbf{b}(x,y)}{|\mathbf{b}(x,y)|}=\mathbf{e}_{d},$

then the submatrix
$\widetilde{\mathbf{M}}(x,y):=\\{\mathbf{M}(x,y)_{i,j}\\}_{1\leq i,j\leq d-1}$
of $\mathbf{M}(x,y)$ has negative eigenvalues
$\lambda_{1},\dots,\lambda_{d-1}$ such that

$-\lambda_{i}\sim|x-y|^{-2},\quad 1\leq i\leq d-1.$

From now on, to simplify the notation, we denote by $\mathbf{a},\mathbf{b}$
the vectors $\mathbf{a}(x,y)$, $\mathbf{b}(x,y)$, respectively, and we also
drop the variables $x,y$ from $\mathcal{D}(x,y)$ as long as no ambiguity
arises. From (2.58) a computation shows

(2.69)
$\displaystyle\partial_{y}\partial_{x}^{\intercal}\Phi_{\mathcal{H}}(x,y)$
$\displaystyle=\partial_{y}\partial_{x}^{\intercal}\mathcal{P}_{\mathcal{H}}(S_{c},x,y)+\partial_{y}\partial_{t}\mathcal{P}_{\mathcal{H}}(S_{c},x,y)\partial_{x}^{\intercal}S_{c}$
$\displaystyle\qquad+\partial_{y}S_{c}\partial_{x}^{\intercal}\partial_{t}\mathcal{P}_{\mathcal{H}}(S_{c},x,y)+(\partial_{y}S_{c}\partial_{x}^{\intercal}S_{c})\partial_{t}^{2}\mathcal{P}_{\mathcal{H}}(S_{c},x,y).$

Here we use $\partial_{t}\mathcal{P}_{\mathcal{H}}(S_{c},x,y)=0$. Using (2.14)
it is easy to show

$\displaystyle\partial_{y}\partial_{x}^{\intercal}\mathcal{P}_{\mathcal{H}}(S_{c},x,y)$
$\displaystyle=-\frac{\mathbf{I}}{\sin S_{c}},$
$\displaystyle\partial_{x}^{\intercal}\partial_{t}\mathcal{P}_{\mathcal{H}}(S_{c},x,y)$
$\displaystyle=-\frac{\mathbf{b}^{\intercal}}{\sin^{2}S_{c}},$
$\displaystyle\partial_{y}\partial_{t}\mathcal{P}_{\mathcal{H}}(S_{c},x,y)$
$\displaystyle=\frac{\mathbf{a}}{\sin^{2}S_{c}}.$

Since $\partial_{x}^{\intercal}S_{c}=-{\partial_{x}^{\intercal}(\cos
S_{c})}/{\sin S_{c}}$ and $\partial_{y}S_{c}=-{\partial_{y}(\cos S_{c})}/{\sin
S_{c}}$, using (2.24) we also have

(2.70)
$\displaystyle\partial_{x}^{\intercal}S_{c}=\frac{\mathbf{b}^{\intercal}}{\sin
S_{c}\sqrt{\mathcal{D}}},\quad\partial_{y}S_{c}=-\frac{\mathbf{a}}{\sin
S_{c}\sqrt{\mathcal{D}}}.$

Thus, putting these identities and (2.48) into (2.69), we have

(2.71)
$\displaystyle\partial_{y}\partial_{x}^{\intercal}\Phi_{\mathcal{H}}(x,y)=-\frac{\mathbf{I}_{d}}{\sin
S_{c}}+\frac{\mathbf{a}\mathbf{b}^{\intercal}}{\sin^{3}S_{c}\sqrt{\mathcal{D}}}=\frac{1}{\sin^{3}S_{c}\sqrt{\mathcal{D}}}\big{(}\mathbf{a}\mathbf{b}^{\intercal}-\mathbf{a}^{\intercal}\mathbf{b}\,\mathbf{I}_{d}\big{)}.$

The second equality follows from (2.64). This allows us to obtain an explicit
expression for the matrix $\mathbf{M}(x,y).$

######  Lemma 2.14.

Let $(x,y)\in\mathfrak{D}(c_{0})$ and
$\omega(x,y)=\sqrt{(1-|x|^{2})\mathcal{D}(x,y)}\sin^{4}S_{c}(x,y)$. Then we
have

(2.72)
$\mathbf{M}(x,y)=\frac{\mathbf{a}^{\intercal}\mathbf{b}\,\mathbf{I}-\mathbf{a}\mathbf{b}^{\intercal}}{\omega(x,y)\mathbf{a}^{\intercal}\mathbf{b}}\Big{(}{\mathbf{b}\mathbf{a}^{\intercal}-\cos
S_{c}\mathbf{a}\mathbf{a}^{\intercal}-\mathbf{a}^{\intercal}\mathbf{b}\,\mathbf{I}}\Big{)}.$

###### Proof.

From (2.66) and (2.71), using (2.62), we see

$\displaystyle\mathbf{M}(x,y)$ $\displaystyle=\frac{-1}{\sqrt{1-|x|^{2}}\sin
S_{c}(x,y)}\partial_{z}^{\intercal}\bigg{(}\frac{\mathcal{G}(x,y,z)}{\sin^{3}S_{c}(x,z)\sqrt{\mathcal{D}(x,z)}}\bigg{)}\bigg{|}_{z=y},$

where

$\mathcal{G}(x,y,z)=\langle\mathbf{a}(x,z),\mathbf{b}(x,z)\rangle\mathbf{a}(x,y)-\langle\mathbf{a}(x,y),\mathbf{b}(x,z)\rangle\mathbf{a}(x,z).$

Since $\mathcal{G}(x,y,y)=0$, it follows that

(2.73)
$\displaystyle\mathbf{M}(x,y)=-\frac{\partial_{z}^{\intercal}\mathcal{G}(x,y,z)|_{z=y}}{\omega(x,y)}.$

Via a straightforward calculation we have

$\displaystyle\partial_{z}^{\intercal}(\langle\mathbf{a}(x,z),\mathbf{b}(x,z)\rangle\mathbf{a}(x,y))=\mathbf{a}(x,y)\mathbf{a}(x,z)^{\intercal}\partial_{z}^{\intercal}\mathbf{b}(x,z)+\mathbf{a}(x,y)\mathbf{b}(x,z)^{\intercal}\partial_{z}^{\intercal}\mathbf{a}(x,z),$
$\displaystyle\partial_{z}^{\intercal}(\langle\mathbf{a}(x,y),\mathbf{b}(x,z)\rangle\mathbf{a}(x,z))=\langle\mathbf{a}(x,y),\mathbf{b}(x,z)\rangle\partial_{z}^{\intercal}\mathbf{a}(x,z)+\mathbf{a}(x,z)\mathbf{a}(x,y)^{\intercal}\partial_{z}^{\intercal}\mathbf{b}(x,z).$

Thus we have

$\displaystyle\partial_{z}^{\intercal}\mathcal{G}(x,y,z)|_{z=y}=(\mathbf{a}\mathbf{b}^{\intercal}-\mathbf{a}^{\intercal}\mathbf{b}\,\mathbf{I})\,\partial_{y}^{\intercal}\mathbf{a}.$

Differentiating both sides of the equations (2.60) and (2.24), one can easily
see
$\partial_{y}^{\intercal}\mathbf{a}(x,y)=\mathcal{D}^{-\frac{1}{2}}x\mathbf{a}^{\intercal}-\mathbf{I}$.
Combining this with an identity $x=\frac{\mathbf{b}-\cos
S_{c}\mathbf{a}}{\sin^{2}S_{c}}$ gives

$\partial_{y}^{\intercal}\mathbf{a}(x,y)=\frac{(\mathbf{b}-\cos
S_{c}\mathbf{a})\mathbf{a}^{\intercal}}{\sin^{2}S_{c}\sqrt{\mathcal{D}}}-\mathbf{I}=\frac{\mathbf{b}\mathbf{a}^{\intercal}-\cos
S_{c}\mathbf{a}\mathbf{a}^{\intercal}-\mathbf{a}^{\intercal}\mathbf{b}\,\mathbf{I}}{\mathbf{a}^{\intercal}\mathbf{b}}.$

For the last inequality we use (2.64). Hence, combining the above identities
with (2.73) we obtain (2.72). ∎

We are now ready to prove Lemma 2.13.

###### Proof of Lemma 2.13.

We begin by noting from Lemma 2.12 that
$|\mathbf{a}|,|\mathbf{b}|,|\mathbf{a}^{\intercal}\mathbf{b}|\gtrsim\rho$
since
$(x,y)\in\operatorname{supp}\chi_{l}\times\operatorname{supp}\chi_{l}^{\prime}\subset\mathfrak{D}(c_{0}/2)$.
Using (2.71) we have

$\displaystyle\partial_{y}\partial_{x}^{\intercal}\Phi_{\mathcal{H}}v=\begin{cases}\
\qquad\quad 0,&v=\mathbf{a},\\\ \
\dfrac{\langle\mathbf{b},v\rangle\mathbf{a}-\langle\mathbf{a},\mathbf{b}\rangle
v}{\sin^{3}S_{c}\sqrt{\mathcal{D}}},&v\perp\text{span}\\{\mathbf{a}\\}.\end{cases}$

This proves $(i)$ in Lemma 2.13 because
$\langle\mathbf{b},v\rangle\mathbf{a}-\langle\mathbf{a},\mathbf{b}\rangle
v\neq 0$ by (2.64).

We now verify the next assertion $(ii)$. Let us set

$\mathbf{G}(x,y)=(\mathbf{a}^{\intercal}\mathbf{b}\,\mathbf{I}-\mathbf{a}\mathbf{b}^{\intercal})({\mathbf{b}\mathbf{a}^{\intercal}-\cos
S_{c}\mathbf{a}\mathbf{a}^{\intercal}-\mathbf{a}^{\intercal}\mathbf{b}\,\mathbf{I}}).$

Then we note that $\mathbf{b}^{\intercal}\mathbf{G}(x,y)=0$ and
$\mathbf{G}(x,y)\mathbf{b}=(\mathbf{a}^{\intercal}\mathbf{b}\,\mathbf{I}-\mathbf{a}\mathbf{b}^{\intercal})(-\cos
S_{c}\,\mathbf{a})=0.$ Since
$\mathbf{b}(x,y)=|\mathbf{b}(x,y)|\mathbf{e}_{d}$, it follows that

$\displaystyle\mathbf{G}(x,y)=\begin{pmatrix}\widetilde{\mathbf{G}}(x,y)&0\\\
0&0\end{pmatrix}$

where $\widetilde{\mathbf{G}}(x,y)$ is a $d-1\times d-1$ submatrix of
$\mathbf{G}(x,y)$. Also, we observe that

$\displaystyle\mathbf{G}(x,y)v=\begin{cases}-{|\mathbf{a}|^{2}|\mathbf{b}|^{2}}v,&\text{if
}v=\mathbf{a}-\dfrac{\mathbf{a}^{\intercal}\mathbf{b}}{|\mathbf{b}|^{2}}\mathbf{b},\text{}\\\
-(\mathbf{a}^{\intercal}\mathbf{b})^{2}\,v,&\text{if
}v\perp\text{span}\\{\mathbf{a},\mathbf{b}\\}.\end{cases}$

A computation gives
$\mathbf{G}(x,y)\mathbf{a}=-|\mathbf{a}|^{2}|\mathbf{b}|^{2}(\mathbf{a}-\frac{\mathbf{a}^{\intercal}\mathbf{b}}{|\mathbf{b}|^{2}}\mathbf{b})$.
Thus, the first follows since $\mathbf{G}(x,y)\mathbf{b}=0$ and the second is
easy to see. Thus, $\widetilde{\mathbf{G}}(x,y)$ has two eigenvalues
${|\mathbf{a}|^{2}|\mathbf{b}|^{2}}$, $(\mathbf{a}^{\intercal}\mathbf{b})^{2}$
of multiplicity $1$, $d-2$, respectively. Therefore, from (2.73) we see that
$\widetilde{\mathbf{M}}(x,y)$ has eigenvalues
$\lambda_{1},\lambda_{2}=\dots=\lambda_{d-1},$ where

$\displaystyle-\lambda_{1}=\frac{|\mathbf{a}|^{2}|\mathbf{b}|^{2}}{\mathbf{a}^{\intercal}\mathbf{b}\sqrt{(1-|x|^{2})\mathcal{D}}\sin^{4}S_{c}},\quad-\lambda_{2}=\frac{\mathbf{a}^{\intercal}\mathbf{b}}{\sqrt{(1-|x|^{2})\mathcal{D}}\sin^{4}S_{c}}.$

Using (2.62)–(2.64), and (2.27), we see

$\displaystyle|\lambda_{1}|$
$\displaystyle=\frac{\sqrt{1-|x|^{2}}(1-|y|^{2})}{\sin^{2}S_{c}(x,y)\mathcal{D}(x,y)}\sim\frac{1}{|x-y|^{2}},$
$\displaystyle|\lambda_{2}|$
$\displaystyle=\frac{1}{\sqrt{(1-|x|^{2})}\sin^{2}S_{c}(x,y)}\sim\frac{1}{|x-y|^{2}}$

because $(x,y)\in\mathfrak{D}(c_{0}/2)$. Hence $\widetilde{\mathbf{M}}(x,y)$
satisfies $(ii)$. This completes the proof. ∎

###  2.7. Proof of Proposition 2.8

We note that $\mathbf{a}(\mathbf{R}x,\mathbf{R}y)=\mathbf{R}\mathbf{a}(x,y)$,
$\mathbf{b}(\mathbf{R}x,\mathbf{R}y)=\mathbf{R}\mathbf{b}(x,y)$ for any
rotation matrix $\mathbf{R}$ (see (2.60) and (2.61)) because $S_{c}(x,y)$ is
invariant under the simultaneous rotation, i.e.,
$S_{c}(x,y)=S_{c}(\mathbf{R}x,\mathbf{R}y).$ It is clear that
$\mathcal{P}_{h}(t,x,y)=\mathcal{P}_{h}(t,\mathbf{R}x,\mathbf{R}y)$. Thus,
from (2.13) we see
$[\eta_{\rho}]^{\mathcal{H}}_{\lambda}(\mathbf{R}x,\mathbf{R}y)=[\eta_{\rho}]^{\mathcal{H}}_{\lambda}(x,y)$,
so we have

$\|\chi_{l}[\eta_{\rho}]^{\mathcal{H}}_{\lambda}\chi_{l}^{\prime}\|_{p}=\|\chi_{l}(\mathbf{R}\cdot)[\eta_{\rho}]^{\mathcal{H}}_{\lambda}\chi_{l}^{\prime}(\mathbf{R}\cdot)\|_{p}$

for any rotation $\mathbf{R}$. In order to show (2.43), performing
simultaneous rotation for $x,y$, we may assume

(2.74)
$\frac{\mathbf{b}(x_{0},y_{0})}{|\mathbf{b}(x_{0},y_{0})|}=\mathbf{e}_{d},$

after replacing $\mathbf{R}x_{0},\mathbf{R}y_{0}$ with $x_{0},y_{0}$.

By (2.51) and (2.57) the matter is reduced to obtaining estimates for the
operator

$\mathcal{T}f(x)=\int
A(x,y)e^{i\lambda\rho\widetilde{\mathcal{P}}_{h}(0,x,y)}f(y)dy,$

where $A\in C_{c}^{\infty}(B(0,\epsilon_{0})\times B(0,\epsilon_{0}))$. Since
$\|\chi_{l}[\eta_{\rho}]^{\mathcal{H}}_{\lambda}\chi_{l}^{\prime}\|_{p}=\rho^{d}\|(\chi_{l}[\eta_{\rho}]^{\mathcal{H}}_{\lambda}\chi_{l}^{\prime})(\rho\,\cdot+x_{0},\rho\,\cdot+y_{0})\|_{p}$,
taking a large enough $N$ in (2.57) and using Lemma 2.10, the estimate (2.43)
follows if we show

(2.75) $\|\mathcal{T}f\|_{p}\leq C(\lambda\rho)^{-\frac{d}{p}}\|f\|_{p}$

for $p_{0}(d)<p\leq\infty$. We note that

(2.76) $\widetilde{\mathcal{P}}_{h}(0,x,y)=\rho^{-1}\Phi_{\mathcal{H}}(\rho
x+x_{0},\rho y+y_{0}).$

Let us write $y=(\xi,y_{d})\in\mathbb{R}^{d-1}\times\mathbb{R}$ and set

$\phi_{y_{d}}(x,\xi)=\widetilde{\mathcal{P}}_{h}(0,x,\xi,y_{d}).$

Then by (2.76) and (2.67) it follows that

$\partial_{\zeta}\Big{\langle}\partial_{x}\phi_{y_{d}}(x,\zeta),\frac{\mathbf{a}_{\rho}(x,\xi,y_{d})}{|\mathbf{a}_{\rho}(x,\xi,y_{d})|}\Big{\rangle}\Big{|}_{\zeta=\xi}=0,$

for ${(x,\xi,y_{d})}\in B(0,\epsilon_{0})\times B(0,\epsilon_{0})$ where
$\mathbf{a}_{\rho}(x,y)=\mathbf{a}(\rho x+x_{0},\rho y+y_{0})$. From (2.66) we
also see

$\partial_{\xi}\partial_{\xi}^{\intercal}\Big{\langle}\partial_{x}\phi_{(y_{0})_{d}}(x,\xi),\frac{\mathbf{a}(x_{0},y_{0})}{|\mathbf{a}(x_{0},y_{0})|}\Big{\rangle}\Big{|}_{(x,\xi)=(x_{0},\xi_{0})}=\rho^{2}\widetilde{\mathbf{M}}(x_{0},y_{0}).$

Recalling (2.74), by Lemma 2.13 we see the matrix has negative eigenvalues
$-\lambda_{1},\dots,-\lambda_{d-1}$ while $\lambda_{i}\sim 1$,
$i=1,\dots,d-1$. We now note that

$|\partial_{x}^{\alpha}\partial_{y}^{\beta}\widetilde{\mathcal{P}}_{h}(0,x,y)|\lesssim
1,\quad\Big{|}\partial_{x}^{\alpha}\partial_{y}^{\beta}\Big{(}\frac{\mathbf{a}_{\rho}(x,y)}{|\mathbf{a}_{\rho}(x,y)|}\Big{)}\Big{|}\lesssim
1.$

The former follows from (2.55) and Lemma 2.9. The latter can be shown
similarly. Therefore, taking small enough $\epsilon_{0}>0$, by continuity we
see that the matrix

$\partial_{\zeta}\partial_{\zeta}^{\intercal}\Big{\langle}\partial_{x}\phi_{y_{d}}(x,\zeta),\frac{\mathbf{a}_{\rho}(x,\xi,y_{d})}{|\mathbf{a}_{\rho}(x,\xi,y_{d})|}\Big{\rangle}\Big{|}_{\zeta=\xi}$

has negative eigenvalues $-\lambda_{1},\dots,-\lambda_{d-1}$ with
$\lambda_{i}\sim 1$, $i=1,\dots,d-1$ for $(x,\xi,y_{d})\in
B(0,\epsilon_{0})\times B(0,\epsilon_{0})$. Therefore the phase function
$\phi_{y_{d}}$ satisfies the elliptic Carleson-Sjölin condition, i.e.,
(C1)–(C3). We now set

$\mathcal{T}_{y_{d}}g(x)=\int
A(x,\xi,y_{d})e^{i\lambda\rho\widetilde{\mathcal{P}}_{h}(0,x,\xi,y_{d})}g(\xi)d\xi.$

Thus we can use Theorem 2.11 to obtain

$\|T_{y_{d}}g\|_{p}\leq C(\lambda\rho)^{-\frac{d}{p}}\|g\|_{p}.$

Since $\mathcal{T}f=\int\mathcal{T}_{y_{d}}f(\cdot,y_{d})dy_{d}$, by
Minkowski’s inequality followed by Hölder’s inequality gives (2.75). This
completes the proof. ∎

##  3\. Bochner-Riesz means for the special Hermite expansion:
Proof of Theorem 1.4

In this section we consider the Bochner-Riesz means for the special Hermite
expansion. Basically, we follow the same strategy for the Hermite expansion.
We begin with noting that the expression (1.5) can be simplified by making use
of the twisted convolution which is defined as follows:

$f\times g(z)=\int_{\mathbb{R}^{2d}}f(z-w)g(w)e^{\frac{i}{2}\langle
z,\mathbf{S}z^{\prime}\rangle}dw,$

where $\mathbf{S}$ is a skew-symmetric $2d\times 2d$ matrix given by

$\mathbf{S}=\begin{pmatrix}0&-\mathbf{I}_{d}\\\
\,\,\mathbf{I}_{d}&\,0\end{pmatrix}.$

By the argument using the Weyl transform, it can be shown that
$\Pi_{\lambda}^{\mathcal{L}}f=f\times\varsigma_{k}$, where
$k=\frac{\lambda-d}{2}$ and
$\varsigma_{k}(z)=(2\pi)^{-d}L_{k}^{d-1}(\frac{1}{2}|z|^{2})e^{-\frac{1}{4}|z|^{2}}$,
and $L_{k}^{\alpha}$ is the $k$-th Laguerre polynomial of type $\alpha$. Thus
the Schrödinger propagator $e^{-itL}f$ is given by

$e^{-itL}f=\sum_{k=0}^{\infty}e^{-it(2k+d)}f\times\varphi_{k}.$

If $f\in\mathcal{S}(\mathbb{R}^{2d})$, by Lemma 3.1 below we see that the sum
in the right-hand side is uniformly and absolutely convergent. Using the
kernel formula for the heat operator $e^{-tL}$ by replacing $t$ with $it$ (see
[23, p.37]),444In fact, one can make it rigorous via analytic continuation
using the operator $e^{-zL}$ with $\Re z>0$.  we have

(3.1) $\displaystyle e^{-itL}f(z)=\frac{1}{(-4\pi i\sin
t)^{d}}\int_{\mathbb{R}^{2d}}e^{i(\frac{|z-z^{\prime}|^{2}}{4}\cot
t+\frac{1}{2}\langle
z,\mathbf{S}z^{\prime}\rangle)}f(z^{\prime})dz^{\prime},\quad
f\in\mathcal{S}(\mathbb{R}^{2d}).$

Here and henceforth, we regard the variables $z,z^{\prime}$ as real variables,
i.e., $z,z^{\prime}\in\mathbb{R}^{2d}$.

######  Lemma 3.1.

Let $f\in\mathcal{S}(\mathbb{R}^{2d})$ and $N\in\mathbb{N}$. Then, there
exists a constant $C=C(N,f,d)$ that satisfies the estimate
$\|f\times\varphi_{k}\|_{\infty}\leq C|k|^{-N}$ for any $k\in\mathbb{N}_{0}$.

###### Proof.

We recall the known identity
$\varphi_{k}(z)=(2\pi)^{-d/2}\sum_{|\alpha|=k}\Phi_{\alpha,\alpha}(z)$ (see
[30, p. 30]). From this, we have

(3.2)
$\displaystyle\|\varphi_{k}\|_{2}=(2\pi)^{-\frac{d}{2}}(|\\{|\alpha|=k\\}|)^{\frac{1}{2}}\sim\lambda^{\frac{d-1}{2}},$

where $\lambda=2k+d$. Clearly we have
$f\times\mathcal{L}^{N}\varphi_{k}=\mathcal{L}^{N}(f\times\varphi_{k})=\lambda^{N}f\times\varphi_{k}$
and, on the other hand, by routine integration by parts we also have
$f\times\mathcal{L}g=(Lf)\times g$ for a second order differential operator
$L$ whose coefficients are $O(|z|^{2})$. Thus we obtain
$f\times\varphi_{k}=\lambda^{-N}(f\times\mathcal{L}^{N}\varphi_{k})=\lambda^{-N}\big{(}L^{N}f\times\varphi_{k}\big{)}$.
By Hölder’s inequality and (3.2) we get
$|f\times\varphi_{k}(z)|\leq\lambda^{-N}\|L^{N}f\|_{2}\|\varphi_{k}\|_{2}\leq
C_{N}\lambda^{-N+\frac{d-1}{2}}$ for any $N$ since
$f\in\mathcal{S}(\mathbb{C}^{d})$. ∎

Using the same notation as in Section 2, we decompose (c.f. (2.2))

(3.3)
$S_{\lambda}^{\delta}(\mathcal{L})=\Big{(}1-\frac{\mathcal{L}}{\lambda}\Big{)}_{+}^{\delta}=\lambda^{-\delta}\sum_{1\leq
2^{j}\leq 4\lambda}2^{\delta j}\psi_{j}(\lambda-\mathcal{L}).$

As before, Theorem 1.4 follows if we show

(3.4)
$\|\chi_{E_{\lambda}}\psi_{j}(\lambda-\mathcal{L})\chi_{F_{\lambda}}\|_{p}\lesssim(\lambda
2^{-j})^{\delta(2d,p)}$

for $p>p_{0}(d)$. As before, for any bounded continuous function $m$ on
$\mathbb{R}$, by $m(\mathcal{L})$ we denote the operator defined by
$m(\mathcal{L})=\sum_{\lambda\in
2\mathbb{N}_{0}+d}m(\lambda)\Pi_{\lambda}^{\mathcal{L}}.$

For $\eta\in\mathcal{S}(\mathbb{R})$ we define the scaled operator
$[\eta]^{\mathcal{L}}_{\lambda}$ of which kernel is given by

(3.5) $[\eta]^{\mathcal{L}}_{\lambda}(z,z^{\prime})=\int\eta(t)(\sin
t)^{-\frac{d}{2}}e^{i\lambda\mathcal{P}_{\mathcal{L}}(t,z,z^{\prime})}dt,$

where

(3.6)
$\displaystyle\mathcal{P}_{\\!\mathcal{L}}(t,z,z^{\prime}):=t+\frac{|z-z^{\prime}|^{2}\cos
t}{4\sin t}+\frac{\langle z,\mathbf{S}z^{\prime}\rangle}{2}.$

The explicit kernel form (3.5) gives the following periodic and symmetric
property of $[\eta]^{\mathcal{L}}_{\lambda}$.

######  Lemma 3.2.

Let $\eta\in C_{c}^{\infty}((0,\infty))$. Let $\mathbf{L}$ denote a $2d\times
2d$ rotation matrix

$\mathbf{L}=\frac{1}{\sqrt{2}}\begin{pmatrix}\mathbf{I}_{d}&-\mathbf{I}_{d}\\\
\mathbf{I}_{d}&\ \ \mathbf{I}_{d}\end{pmatrix}.$

Then we have

(3.7)
$\displaystyle\|\chi_{E}[\eta]^{\mathcal{L}}_{\lambda}\chi_{F}\|_{p}=\|\chi_{\mathbf{L}(E)}[\eta(-\,\cdot)]^{\mathcal{L}}_{\lambda}\chi_{\mathbf{L}(F)}\|_{p},$
(3.8)
$\displaystyle\|\chi_{E}[\eta]^{\mathcal{L}}_{\lambda}\chi_{F}\|_{p}=\|\chi_{E}[\eta(\cdot+k\pi)]^{\mathcal{L}}_{\lambda}\chi_{F}\|_{p},\quad
k\in\mathbb{Z}.$

###### Proof of Lemma 3.2.

The first identity (3.7) follows from

$[\eta(-\,\cdot)]^{\mathcal{L}}_{\lambda}(z,z^{\prime})=c\,\overline{[\eta]^{\mathcal{L}}_{\lambda}(\mathbf{L}z,\mathbf{L}z^{\prime})},\quad
z,z^{\prime}\in\mathbb{R}^{2d},$

where $c$ is a constant such that $|c|=1$. This can be shown changing
variables $t\to-t$ and using the fact that
$\mathbf{S}=-\mathbf{L}^{\intercal}\mathbf{S}\mathbf{L}$. For (3.8), the
change of variable $t\to t+k\pi$ gives
$[\eta(\cdot+k\pi)]^{\mathcal{L}}_{\lambda}(z,z^{\prime})=c[\eta]^{\mathcal{L}}_{\lambda}(z,z^{\prime})$
with $|c|=1$. Hence we get (3.8). ∎

###  3.1. Reductions

Similarly as before, we observe that

$\psi_{j}(\lambda-\mathcal{L})(x,y)=[\widehat{\psi}_{j}]^{\mathcal{L}}_{\lambda}(\sqrt{\lambda}x,{\sqrt{\lambda}y}).$

Thus, by scaling and (3.1) we see the estimate (3.4) is equivalent to

(3.9)
$\|\chi_{E}[\widehat{\psi}_{j}]^{\mathcal{L}}_{\lambda}\chi_{F}\|_{p}\lesssim\lambda^{-d}(\lambda
2^{-j})^{\delta(2d,p)}.$

To prove the estimate (3.9), we proceed in the similar manner as in the
Hermite case. We decompose $[\widehat{\psi}_{j}]^{\mathcal{L}}_{\lambda}$
using the cutoff functions $\nu_{j}^{l,k}$ in Section 2 (see (2.20)). Thus we
have
$[\widehat{\psi_{j}}]^{\mathcal{L}}_{\lambda}=\sum_{k=-\infty}^{\infty}\sum_{l=0}^{\infty}[\psi_{j}^{l,k}]^{\mathcal{L}}_{\lambda}$.
If we combine this with the symmetric and periodic properties in Lemma 3.2,
the proof of (3.9) is basically reduced to showing the following. (See _Proof
of ( 2.15)._)

######  Theorem 3.3.

Let $0<\rho<\pi-\epsilon_{0}$ for some $\epsilon_{0}>0$. Let $\eta_{\rho}$ be
a smooth function such that
$\operatorname{supp}\eta_{\rho}\subset[2^{-2}\rho,\rho]$ and
$|\eta_{\rho}^{(n)}(t)|\leq C_{n}\rho^{-n}$ for $t>0$, $n\in\mathbb{N}_{0}$.
Then for any $\lambda\in 2\mathbb{N}_{0}+d$ and the sets $E,F$ satisfying the
condition in Theorem 1.4, we have

(3.10)
$\displaystyle\|\chi_{E}[\eta_{\rho}]^{\mathcal{L}}_{\lambda}\chi_{F}\|_{p}\lesssim\begin{cases}\
\lambda^{-d}\rho,&\rho\leq\lambda^{-1},\\\ \
\lambda^{-d}\lambda^{\delta(2d,p)}\rho^{\delta(2d,p)+1},&\rho\geq\lambda^{-1}\end{cases}$

whenever $p>p_{0}(2d)$.

In order to show Theorem 3.3 we make a simple observation. From the assumption
on $E$, $F$ in Theorem 1.4, there exists $z_{0}\in\mathbb{R}^{2d}$ such that
$E,F\subset B(z_{0},4)$. Using (3.5) it is easy to see that

$[\eta]_{\lambda}^{\mathcal{L}}(z+z_{0},z^{\prime}+z_{0})=[\eta]_{\lambda}^{\mathcal{L}}(z,z^{\prime})e^{i\frac{\lambda}{2}(\langle
z_{0},\mathbf{S}z^{\prime}\rangle+\langle z,\mathbf{S}z_{0}\rangle)}.$

Clearly the oscillating term $e^{i\frac{\lambda}{2}(\langle
z_{0},\mathbf{S}z^{\prime}\rangle+\langle z,\mathbf{S}z_{0}\rangle)}$ does not
affect the $L^{p}$ norm of
$\chi_{E}[\eta_{\rho}]^{\mathcal{L}}_{\lambda}\chi_{F}$, so we may assume that
$E,F\subset B(0,4)$ after making a change of variable
$(z,z^{\prime})\to(z+z_{0},z^{\prime}+z_{0})$.

We now recall (3.5) and (3.6). To obtain estimates for the kernel of
$\chi_{E}[\eta_{\rho}]^{\mathcal{L}}_{\lambda}\chi_{F}$ we consider the phase
function $\mathcal{P}_{\\!\mathcal{L}}(t,z,z^{\prime})$. A computation gives

(3.11)
$\displaystyle\partial_{t}\mathcal{P}_{\\!\mathcal{L}}(t,z,z^{\prime})=1-\frac{|z-z^{\prime}|^{2}}{4\sin^{2}t},\quad
t\in(0,\pi).$

Since $|z-z^{\prime}|\leq 2-c_{0}$,
$\mathcal{P}_{\\!\mathcal{L}}(t,z,z^{\prime})$ has two critical points
$\mathfrak{S}_{c}(z,z^{\prime})$ and $\mathfrak{S}_{c}^{*}(z,z^{\prime})$ such
that $\mathfrak{S}_{c}(z,z^{\prime})\in(0,\pi/2-2\epsilon_{0})$ and
$\mathfrak{S}_{c}^{*}(z,z^{\prime})\in(\pi/2+2\epsilon_{0},\pi)$ for a small
$\epsilon_{0}>0$ and

(3.12)
$\displaystyle\sin\mathfrak{S}_{c}(z,z^{\prime})=\sin\mathfrak{S}_{c}^{*}(z,z^{\prime})=\frac{|z-z^{\prime}|}{2}.$

So we have
$\pi-\mathfrak{S}_{c}^{*}(z,z^{\prime})=\mathfrak{S}_{c}(z,z^{\prime})$.

As in the Hermite case which we have handled, we may further assume that

(3.13)
$\operatorname{dist}(\mathfrak{S}_{c}^{*}(z,z^{\prime}),\operatorname{supp}{\eta_{\rho}})\geq\epsilon_{0},\quad(z,z^{\prime})\in
E\times F.$

To justify this, let us define $\eta_{1},\eta_{2},\eta_{3}\in
C_{c}^{\infty}([0,\pi])$ such that
$\operatorname{supp}\eta_{1}\subset[0,\pi/2-\epsilon_{0})$,
$\operatorname{supp}\eta_{2}\subset(\pi/2-3\epsilon_{0}/2,\pi/2+3\epsilon_{0}/2)$,
$\operatorname{supp}\eta_{3}\subset(\pi/2+\epsilon_{0},\pi]$ and
$\eta_{1}+\eta_{2}+\eta_{3}=1$ on $[0,\pi]$. Then we split
$[\eta_{\rho}]^{\mathcal{L}}_{\lambda}$ into the sum

$[\eta_{\rho}\eta_{1}]^{\mathcal{L}}_{\lambda}+[\eta_{\rho}\eta_{2}]^{\mathcal{L}}_{\lambda}+[\eta_{\rho}\eta_{3}]^{\mathcal{L}}_{\lambda}.$

Since
$\operatorname{supp}{(\eta_{\rho}\eta_{1})}\subset[0,\pi/2-\epsilon_{0})$, so
the desired assumption is valid in this case. Also, from (3.11) it follows
that
$|\partial_{t}\mathcal{P}_{\mathcal{L}}(t,z,z^{\prime})|\gtrsim\epsilon_{0}^{3}$
on $\operatorname{supp}{(\eta_{\rho}\eta_{2})}$. So, applying Lemma 2.5 yields
that
$|(\chi_{E}[\eta_{\rho}\eta_{2}]^{\mathcal{L}}_{\lambda}\chi_{F})(z,z^{\prime})|\lesssim\lambda^{-N}$
for any $(z,z^{\prime})\in E\times F$. Hence we may disregard the contribution
from $[\eta_{\rho}\eta_{2}]^{\mathcal{L}}_{\lambda}.$ For the remaining term
$[\eta_{\rho}\eta_{3}]^{\mathcal{L}}_{\lambda}$, by utilizing (3.7), (3.8) we
note

$\|\chi_{E}[\eta_{\rho}\eta_{3}]^{\mathcal{L}}_{\lambda}\chi_{F}\|_{p}=\|\chi_{\mathbf{L}(E)}[(\eta_{\rho}\eta_{3})(\pi-\cdot)]^{\mathcal{L}}_{\lambda}\chi_{\mathbf{L}(F)}\|_{p}$

where $\mathbf{L}$ is the rotation matrix in Lemma 3.2. Obviously
$\mathbf{L}(E)$, $\mathbf{L}(F)$ satisfy the assumption of Theorem 3.3 and
$\operatorname{supp}((\eta_{\rho}\eta_{3})(\pi-\cdot))\subset[0,\pi/2-\epsilon_{0})$.
Hence it is sufficient to show the estimate (3.10) under the additional
assumption (3.13).

In order to show (3.10), by additional decomposition and the standard argument
as in Section 2 it is sufficient to show

(3.14)
$\|\varphi[\eta_{\rho}]^{\mathcal{L}}_{\lambda}\varphi^{\prime}\|_{p}\lesssim\begin{cases}\lambda^{-d}\rho,&\rho\leq\lambda^{-1},\\\
\lambda^{-d+\delta(2d,p)}\rho^{\delta(2d,p)+1},&\rho>\lambda^{-1},\end{cases}$

where $\varphi$, $\varphi^{\prime}$ are smooth functions such that
$\operatorname{supp}\varphi\times\operatorname{supp}\varphi^{\prime}\subset\\{(z,z^{\prime}):|z|,|z^{\prime}|<4,|z-z^{\prime}|<2-c_{0}\\}$.
For the purpose we decompose the integral kernel in dyadic manner:

(3.15)
$(\varphi[\eta_{\rho}]^{\mathcal{L}}_{\lambda}\varphi^{\prime})(z,z^{\prime})=\sum_{l}\mathfrak{T}^{l}(z,z^{\prime}):=\sum_{l}[\eta_{\rho}]^{\mathcal{L}}_{\lambda}(z,z^{\prime})\psi(2^{l}|z-z^{\prime}|)\varphi(z)\varphi^{\prime}(z^{\prime}).$

The matter now is reduced to showing the estimate for $\mathfrak{T}^{l}$. For
the purpose it is sufficient to consider the operator which is given by the
kernel

$(\chi_{l}[\eta_{\rho}]^{\mathcal{L}}_{\lambda}\chi_{l}^{\prime})(z,z^{\prime})=[\eta_{\rho}]^{\mathcal{L}}_{\lambda}(z,z^{\prime})\chi_{l}(z)\chi_{l}^{\prime}(z^{\prime}),$

where $\chi_{l},\chi_{l}^{\prime}$ satisfy (2.33)–(2.35) and
$\operatorname{supp}\chi_{l}\times\operatorname{supp}\chi_{l}^{\prime}\subset\\{(z,z^{\prime}):|z|,|z^{\prime}|<4,|z-z^{\prime}|<2-c_{0}\\}$.
Thus, by (3.12) we have

(3.16) $\mathfrak{S}_{c}(z,z^{\prime})\sim
2^{-l},\quad(z,z^{\prime})\in\operatorname{supp}\chi_{l}\times\operatorname{supp}\chi_{l}^{\prime}.$

#### Estimate for
$\chi_{l}[\eta_{\rho}]^{\mathcal{L}}_{\lambda}\chi_{l}^{\prime}$ when
$\rho\leq\lambda^{-1}$

We first handle the case $\rho\leq\lambda^{-1}$.

######  Lemma 3.4.

Let $1\leq p\leq\infty$. If $\rho\leq\lambda^{-1}$, then for any $N\geq 0$ we
have

(3.17)
$\|\chi_{l}[\eta_{\rho}]^{\mathcal{L}}_{\lambda}\chi_{l}^{\prime}\|_{p}\lesssim\begin{cases}\lambda^{-N}\rho^{1-d+N}2^{(2N-2d)l},&\quad
2^{-l}\gg\lambda^{-\frac{1}{2}}\rho^{\frac{1}{2}},\\\
\rho^{1-d}2^{-2dl},&\quad
2^{-l}\lesssim\lambda^{-\frac{1}{2}}\rho^{\frac{1}{2}}.\end{cases}$

###### Proof.

We assume
$(t,z,z^{\prime})\in\operatorname{supp}\eta_{\rho}\times\operatorname{supp}\chi_{l}\times\operatorname{supp}\chi_{l}^{\prime}$.
We first consider the case
$2^{-l}\gg\lambda^{-\frac{1}{2}}\rho^{\frac{1}{2}}$. Since $2^{-l}\gg\rho$, by
(3.16) it follows that $0<\sin t\ll|z-z^{\prime}|$. From (3.11), we have

(3.18)
$\displaystyle|\partial_{t}\mathcal{P}_{\\!\mathcal{L}}(t,z,z^{\prime})|\sim\frac{|z-z^{\prime}|^{2}}{4\sin^{2}t}\sim
2^{-2l}\rho^{-2}.$

Also a simple calculation shows

(3.19)
$\displaystyle|\partial_{t}^{n}\mathcal{P}_{\\!\mathcal{L}}(t,z,z^{\prime})|\lesssim\frac{|z-z^{\prime}|^{2}}{(\sin
t)^{n+1}}\lesssim 2^{-2l}\rho^{-n-1},\quad n\geq 2,$ (3.20)
$\displaystyle|\partial_{t}^{n}(\eta_{\rho}(t)(\sin
t)^{-d})|\lesssim\rho^{-d-n},\quad n\geq 1.$

Hence Lemma 2.5 gives
$|(\chi_{l}[\eta_{\rho}]^{\mathcal{L}}_{\lambda}\chi_{l}^{\prime})(z,z^{\prime})|\lesssim_{N}\lambda^{-N}\rho^{1-d+N}2^{2Nl}$.
Using Lemma 2.4, we get
$\|\chi_{l}[\eta_{\rho}]^{\mathcal{L}}_{\lambda}\chi_{l}^{\prime}\|_{p}\lesssim_{N}\lambda^{-N}\rho^{1-d+N}2^{2(N-d)l}$
for any $1\leq p\leq\infty$.

We now handle the case
$2^{-l}\lesssim\lambda^{-\frac{1}{2}}\rho^{\frac{1}{2}}.$ We first recall

(3.21)
$\displaystyle\|\Pi_{\lambda}^{\mathcal{L}}\|_{1\to\infty}\lesssim\lambda^{d-1},\quad\lambda\in
2\mathbb{N}_{0}+d.$

We refer the reader to [30, Section 2.6] for the proof of (3.21). Then, by
following the same argument in the proof of Lemma 2.6, we have

$|[\eta_{\rho}]^{\mathcal{L}}_{\lambda}(z,z^{\prime})|\lesssim\sum_{\nu\in
2\mathbb{N}_{0}+d}\rho(1+\rho|\lambda-\nu|)^{-N}\nu^{d-1}\lesssim\rho^{1-d},\quad
z,z^{\prime}\in\mathbb{R}^{2d}.$

This and Lemma 2.4 give the desired estimate. ∎

#### Estimate for
$\chi_{l}[\eta_{\rho}]^{\mathcal{L}}_{\lambda}\chi_{l}^{\prime}$ when
$\rho>\lambda^{-1}$

Now we deal with the case $\rho>\lambda^{-1}$. We separately consider three
sub-cases $2^{-l}\ll\rho,$ $2^{-l}\gg\rho,$ and $2^{-l}\sim\rho.$ The first
two cases are easy to handle.

######  Lemma 3.5.

Let $1\leq p\leq\infty$. If $\rho>\lambda^{-1}$, then

(3.22)
$\|\chi_{l}[\eta_{\rho}]^{\mathcal{L}}_{\lambda}\chi_{l}^{\prime}\|_{p}\lesssim\begin{cases}\lambda^{-N}\rho^{1-\frac{d}{2}-N}2^{-dl},&\quad
2^{-l}\ll\rho,\\\ \lambda^{-N}\rho^{1-\frac{d}{2}+N}2^{(2N-d)l},&\quad
2^{-l}\gg\rho.\end{cases}$

###### Proof.

We assume
$(t,z,z^{\prime})\in\operatorname{supp}\eta_{\rho}\times\operatorname{supp}\chi_{l}\times\operatorname{supp}\chi_{l}^{\prime}$
and we first consider the case $2^{-l}\ll\rho$. The condition $2^{-l}\ll\rho$
implies

$\displaystyle|\partial_{t}\mathcal{P}_{\\!\mathcal{L}}(t,z,z^{\prime})|=1-\frac{|z-z^{\prime}|^{2}}{4\sin^{2}t}\gtrsim
1.$

Thus, the estimates (3.19), (3.20) remains valid. So, from Lemma 2.5 it
follows that
$|\chi_{l}[\eta_{\rho}]^{\mathcal{L}}_{\lambda}\chi_{l}^{\prime}(z,z^{\prime})|\lesssim_{N}\lambda^{-N}\rho^{1-d-N}$
for any $N\in\mathbb{N}$. This estimate combined with Lemma 2.4 yields
$\|\chi_{l}[\eta_{\rho}]^{\mathcal{L}}_{\lambda}\chi_{l}^{\prime}\|_{p}\lesssim\lambda^{-N}\rho^{1-d-N}2^{-2dl}$
for $2\leq p\leq\infty$.

Now we consider the case $2^{-l}\gg\rho$. In this case we have
$0<\rho\ll\mathfrak{S}_{c}(z,z^{\prime})<\pi/2$ because of (3.16), so as in
the previous case the derivatives of $\mathcal{P}_{\mathcal{L}}$ satisfy
(3.18) and (3.19). Also the amplitude function $\eta_{\rho}(t)(\sin t)^{-d}$
satisfies the bound (3.20). With these estimates, we obtain the second case
estimate in (3.22) following the same argument as before. ∎

What follows is the key estimate which we need for the proof of Theorem 3.3.

######  Proposition 3.6.

Let $\rho>\lambda^{-1}$ and $\rho\sim 2^{-l}$. Then we have

(3.23)
$\|\chi_{l}[\eta_{\rho}]^{\mathcal{L}}_{\lambda}\chi_{l}^{\prime}\|_{p}\lesssim\lambda^{-d+\delta(2d,p)}\rho^{1+\delta(2d,p)}$

provided that $p_{0}(2d)\leq p\leq\infty$.

Now, putting Lemma 3.4, Lemma 3.5, and Proposition 3.6 together, we can prove
Theorem 3.3 by establishing (3.14). Let us consider the case
$\rho>\lambda^{-1}$ first. By (3.15) we have

$\|\varphi[\eta_{\rho}]^{\mathcal{L}}_{\lambda}\varphi^{\prime}\|_{p}\leq\sum_{l}\|\mathfrak{T}^{l}\|_{p}.$

Using Lemma 3.5 and Proposition 3.6, by a standard argument (for example,
Lemma 2.3) we have

$\|\mathfrak{T}^{l}\|_{p}\lesssim\begin{cases}\lambda^{-N}\rho^{1-\frac{d}{2}-N}2^{-dl},&\quad
2^{-l}\ll\rho,\\\ \lambda^{-d+\delta(2d,p)}\rho^{1+\delta(2d,p)},&\quad
2^{-l}\sim\rho,\\\ \lambda^{-N}\rho^{1-\frac{d}{2}+N}2^{(2N-d)l},&\quad
2^{-l}\gg\rho.\end{cases}$

Taking summation over $l$ as in the _Proof of ( 2.30)_, we get the desired
bound (3.14). The remaining case $\rho>\lambda^{-1}$ can be handled similarly
using Lemma 3.4, so we omit the detail.

###  3.2. Asymptotic expansion of the kernel when $2^{-l}\sim\rho$

Unlike the previous cases, the support of $\eta_{\rho}$ may contain the
critical point $\mathfrak{S}_{c}(z,z^{\prime})$.

Additional decomposition of $\chi_{l}$, $\chi^{\prime}_{l}$ and $\eta_{\rho}$.
Let $\epsilon_{0}>0$ be a small number. If we further break $\chi_{l}$,
$\chi^{\prime}_{l}$ and $\eta_{\rho}$ into finitely many the cutoff functions,
then in addition to (2.33)–(2.35) we may assume that

(3.24)
$\displaystyle\operatorname{supp}\chi_{l}\times\operatorname{supp}\chi_{l}^{\prime}\subset\\{(z,z^{\prime}):|z|,\
$ $\displaystyle|z^{\prime}|<2^{2},|z-z^{\prime}|<2-2^{-1}c_{0}\\},$ (3.25)
$\displaystyle\operatorname{supp}\chi_{l}\subset B(z_{0},\epsilon_{0}\rho),$
$\displaystyle\quad\operatorname{supp}\chi_{l}^{\prime}\subset
B(z_{0}^{\prime},\epsilon_{0}\rho),$ (3.26)
$\displaystyle\mathfrak{S}_{c}(z,z^{\prime})\in\mathfrak{S}_{c}(z_{0},z_{0}^{\prime})+(-\epsilon_{0}\rho,\epsilon_{0}\rho),$
$\displaystyle\quad(z,z^{\prime})\in\operatorname{supp}\chi_{l}\times\operatorname{supp}\chi_{l}^{\prime},$
(3.27)
$\displaystyle\operatorname{supp}\eta_{\rho}\subset\mathfrak{S}_{c}(z_{0},z_{0}^{\prime}$
$\displaystyle)+(-2\epsilon_{0}\rho,2\epsilon_{0}\rho)$

for some $z_{0}$ and $z_{0}^{\prime}$. The conditions (3.24)–(3.26) can be
achieved splitting $\chi_{l}$, $\chi^{\prime}_{l}$ into cutoff functions which
are supported in balls of radius $c\epsilon_{0}\rho$ with a small $c>0$. For
the last (3.27) we break

$\chi_{l}[\eta_{\rho}]^{\mathcal{L}}_{\lambda}\chi_{l}^{\prime}=\chi_{l}[\eta_{\rho}\widetilde{\eta}_{\rho}]^{\mathcal{L}}_{\lambda}\chi_{l}^{\prime}+\chi_{l}[\eta_{\rho}(1-\widetilde{\eta}_{\rho})]^{\mathcal{L}}_{\lambda}\chi_{l}^{\prime}$

where $\widetilde{\eta}_{\rho}$ is a smooth bump function such that
$\operatorname{supp}\eta_{\rho}\subset\mathfrak{S}_{c}(z_{0},z^{\prime}_{0})+(-3\epsilon_{0}\rho,3\epsilon_{0}\rho)$,
$\widetilde{\eta_{\rho}}(t)=1$ if
$t\in\mathfrak{S}_{c}(z_{0},z^{\prime}_{0})+(-2\epsilon_{0}\rho,2\epsilon_{0}\rho)$,
and $|\widetilde{\eta_{\rho}}^{(n)}|\leq C\rho^{-n}$ for any
$n\in\mathbb{N}_{0}$. By the same argument as before we get

(3.28)
$\|\chi_{l}[\eta_{\rho}(1-\widetilde{\eta}_{\rho})]^{\mathcal{H}}_{\lambda}\chi_{l}^{\prime}\|_{p}\lesssim(\lambda\rho)^{-N}\rho^{1+d}$

for $\rho>\lambda^{-1}.$ Therefore, the contribution from
$\chi_{l}[\eta_{\rho}(1-\widetilde{\eta}_{\rho})]^{\mathcal{L}}_{\lambda}\chi_{l}^{\prime}$
is negligible. Thus, it is sufficient to deal with
$\chi_{l}[\eta_{\rho}\widetilde{\eta}_{\rho}]^{\mathcal{L}}_{\lambda}\chi_{l}^{\prime}$,
so we may assume (3.27). To show (3.28), we note that
$|\partial_{t}\mathcal{P}_{\mathcal{H}}(t,x,y)|\gtrsim 1$ and
$|\partial^{n}_{t}\mathcal{P}_{\mathcal{H}}(t,x,y)|\lesssim\rho^{1-n},$
$n\in\mathbb{N}$ if
$t\in\operatorname{supp}(\eta_{\rho}(1-\widetilde{\eta}_{\rho}))$ and
$(z,z^{\prime})\in\operatorname{supp}\chi_{l}\times\operatorname{supp}\chi_{l}^{\prime}$.
Then, using Lemma 2.5 and Lemma 2.4, we obtain the estimate (3.28).

Since $\mathfrak{S}_{c}(z,z^{\prime})\sim\rho$ for
$(z,z^{\prime})\in\operatorname{supp}\chi_{l}\times\operatorname{supp}\chi_{l}^{\prime}$,
we have

(3.29)
$\partial_{t}^{2}\mathcal{P}_{\\!\mathcal{L}}(\mathfrak{S}_{c}(z,z^{\prime}),z,z^{\prime})=\frac{|z-z^{\prime}|^{2}\cos\mathfrak{S}_{c}(z,z^{\prime})}{2\sin^{3}\mathfrak{S}_{c}(z,z^{\prime})}\sim\rho^{-1}$

for
$(z,z^{\prime})\in\operatorname{supp}\chi_{l}\times\operatorname{supp}\chi_{l}^{\prime}$.
Recalling (3.6) it is easy to see

(3.30)
$\partial_{t}^{n}\partial_{z}^{\alpha}\partial_{z^{\prime}}^{\beta}\mathcal{P}_{\\!\mathcal{L}}(t,z,z^{\prime})=O(\rho^{1-n-|\alpha|-|\beta|}),\quad
n\geq 1$

for
$(t,z,z^{\prime})\in\operatorname{supp}\eta_{\rho}\times\operatorname{supp}\chi_{l}\times\operatorname{supp}\chi_{l}^{\prime}$.
Since $\sin\mathfrak{S}_{c}(z,z^{\prime})=|z-z^{\prime}|/2$ and $\partial
z(\mathfrak{S}_{c}(z,z^{\prime}))=\partial
z(\sin\mathfrak{S}_{c}(z,z^{\prime}))/\cos\mathfrak{S}_{c}(z,z^{\prime})$, we
see that $\partial
z(\mathfrak{S}_{c}(z,z^{\prime}))=(4-|z-z^{\prime}|^{2}))^{-1/2}\partial
z(|z-z^{\prime}|).$ Thus a routine computation gives

(3.31) $\partial
z^{\alpha}\partial_{z^{\prime}}^{\beta}(\mathfrak{S}_{c}(z,z^{\prime}))=O(|z-z^{\prime}|^{1-|\alpha|-|\beta|})=O(\rho^{1-|\alpha|-|\beta|}).$

for
$(z,z^{\prime})\in\operatorname{supp}\chi_{l}\times\operatorname{supp}\chi_{l}^{\prime}$
since $\rho\sim 2^{-l}$.

The derivatives of the phase $\mathcal{P}_{\mathcal{L}}$ and the amplitude
$\eta_{\rho}$ are not uniformly bounded in $\rho$. Following the same approach
in the previous section, we can get around this making change of variables.
Let us set

$\displaystyle\phi(t,z,z^{\prime})$ $\displaystyle=\mathfrak{S}_{c}(\rho
z+z_{0},\rho z^{\prime}+z^{\prime}_{0})+\rho t,$ (3.32)
$\displaystyle\widetilde{\mathcal{P}}_{\\!\mathcal{L}}(t,z,z^{\prime})$
$\displaystyle=\rho^{-1}\mathcal{P}_{\\!\mathcal{L}}(\phi(t,z,z^{\prime}),\rho
z+z_{0},\rho z^{\prime}+z^{\prime}_{0}),$
$\displaystyle\widetilde{\chi}_{l}(z)$ $\displaystyle=\chi_{l}(\rho
z+z_{0}),\quad\widetilde{\chi}_{l}^{\prime}(z^{\prime})=\chi_{l}^{\prime}(\rho
z^{\prime}+z^{\prime}_{0}),$ $\displaystyle a(t,z,z^{\prime})$
$\displaystyle=\rho^{d}(\sin\phi(t,z,z^{\prime}))^{-d}\eta_{\rho}(\phi(t,z,z^{\prime})))\widetilde{\chi}_{l}(z)\widetilde{\chi}_{l}^{\prime}(z^{\prime}).$

Then we also set

$I_{\rho}^{\mathcal{L}}(x,y)=\rho^{1-d}\int
a(t,z,z^{\prime})e^{i\lambda\rho\widetilde{\mathcal{P}}_{\\!\mathcal{L}}(t,z,z^{\prime})}dt.$

By the change of variables $t\to\phi(t,z,z^{\prime})$, we write

(3.33) $(\chi_{l}[\eta_{\rho}]^{\mathcal{L}}_{\lambda}\chi_{l}^{\prime})(\rho
z+z_{0},\rho z^{\prime}+z^{\prime}_{0})=C_{d}I_{\rho}^{\mathcal{L}}(x,y)$

where $C_{d}$ is a constant depending on $d$. Then, the support of $a$ is
contained in the interval $(-2\epsilon_{0},2\epsilon_{0})\times
B(0,\epsilon_{0})\times B(0,\epsilon_{0})$. From (3.31) we first note

(3.34)
$|\partial_{t}^{n}\partial_{z}^{\alpha}\partial_{z}^{\beta}\phi(t,z,z^{\prime})|\leq
C_{n,\alpha,\beta}\,\rho$

for
$(t,z,z^{\prime})\in\operatorname{supp}a\times\operatorname{supp}\widetilde{\chi}_{l}\times\operatorname{supp}\widetilde{\chi}_{l}^{\prime}$.
Using this and (3.30), one can easily see

(3.35)
$\displaystyle|\partial_{t}^{n}\partial_{z}^{\alpha}\partial_{z}^{\beta}a(t,z,z^{\prime})|\leq
C_{n,\alpha,\beta},$ (3.36)
$\displaystyle|\partial_{t}^{n}\partial_{z}^{\alpha}\partial_{z}^{\beta}\widetilde{\mathcal{P}}_{\\!\mathcal{L}}(t,z,z^{\prime})|\leq
C_{n,\alpha,\beta}$

for
$(t,z,z^{\prime})\in\operatorname{supp}a\times\operatorname{supp}\widetilde{\chi}_{l}\times\operatorname{supp}\widetilde{\chi}_{l}^{\prime}$.
From (3.29) we have

(3.37)
$\displaystyle\partial_{t}^{2}\widetilde{\mathcal{P}}_{h}(t,z,z^{\prime})\sim
1,\qquad(t,z,z^{\prime})\in\operatorname{supp}a\times\operatorname{supp}\widetilde{\chi}_{l}\times\operatorname{supp}\widetilde{\chi}_{l}^{\prime}.$

and $\partial_{t}\widetilde{\mathcal{P}}_{h}(0,z,z^{\prime})=0$. As before, we
now apply the method of stationary phase ([12, Theorem 7.7.5]) to the integral
$I_{\rho}^{\mathcal{L}}(x,y)$, and we obtain

######  Lemma 3.7.

Let $N\in\mathbb{N}$. Then, for
$(z,z^{\prime})\in\operatorname{supp}\chi_{l}\times\operatorname{supp}\chi_{l}^{\prime}$,
we have

(3.38) $\displaystyle
I_{\rho}^{\mathcal{L}}(x,y)=\rho^{1-d}\sum_{n=0}^{N-1}(\lambda\rho)^{-\frac{1}{2}-n}A_{n}(z,z^{\prime})\mspace{1.0mu}e^{i\lambda\rho\widetilde{\mathcal{P}}_{\\!\mathcal{L}}(0,z,z^{\prime})}+E_{N}(z,z^{\prime}),$

where $A_{n},E_{N}\in C_{c}^{\infty}(B(0,\epsilon_{0}))$ satisfy
$|\partial_{z}^{\alpha}\partial_{z}^{\beta}A_{n}(z,z^{\prime})|\leq
C_{\alpha,\beta}$ and $\sup_{x,y}|E_{N}(x,y)|\leq
C_{N}\rho^{\frac{2-d}{2}}(\lambda\rho)^{-N}$ with $C_{\alpha,\beta}$ and
$C_{N}$ independent of $\lambda,\rho$.

Let us set

(3.39)
$\Phi_{\\!\mathcal{L}}(z,z^{\prime}):=\mathcal{P}_{\\!\mathcal{L}}(\mathfrak{S}_{c}(z,z^{\prime}),z,z^{\prime}).$

In the next section we investigate the curvature condition of the phase
function $\Phi_{\\!\mathcal{L}}$.

###  3.3. The phase function $\Phi_{\\!\mathcal{L}}(z,z^{\prime})$

For
$(z,z^{\prime})\in\operatorname{supp}\widetilde{\chi}_{l}\times\operatorname{supp}\widetilde{\chi}_{l}^{\prime}$,
we set

$\mathbf{v}(z,z^{\prime}):=z-z^{\prime},\quad$

and

$\mathbf{R}(z,z^{\prime}):=\cos{\mathfrak{S}_{c}(z,z^{\prime})}\mathbf{I}_{2d}-\sin{\mathfrak{S}_{c}(z,z^{\prime})}\mathbf{S}.$

It should be noted that
$\mathbf{R}(z,z^{\prime})\mathbf{R}(z,z^{\prime})^{\intercal}=\mathbf{I}_{2d}$,
so $\mathbf{R}(z,z^{\prime})$ is a rotation matrix. We occasionally denote
$\mathbf{v}=\mathbf{v}(z,z^{\prime})$,
$\mathfrak{S}_{c}=\mathfrak{S}_{c}(z,z^{\prime})$, and
$\mathbf{R}=\mathbf{R}(z,z^{\prime})$ for simplicity.

Since
$\partial_{t}\mathcal{P}_{\mathcal{L}}(\mathfrak{S}_{c},z,z^{\prime})=0$, from
(3.39) we have
$\partial_{z}\Phi_{\\!\mathcal{L}}(z,z^{\prime})=\partial_{z}\mathcal{P}_{\mathcal{L}}(\mathfrak{S}_{c},z,z^{\prime})$.
Thus, using (3.6), we get
$\partial_{z}\Phi_{\\!\mathcal{L}}(z,z^{\prime})=\frac{(z-z^{\prime})\cos\mathfrak{S}_{c}}{2\sin\mathfrak{S}_{c}}+\frac{\mathbf{S}z^{\prime}}{2}$.
Now, by (3.12) we have

$\displaystyle\partial_{z}\Phi_{\\!\mathcal{L}}(z,z^{\prime})=\frac{\mathbf{R}(z,z^{\prime})(z-z^{\prime})}{|z-z^{\prime}|}+\frac{\mathbf{S}z}{2}.$

Thus, we see
$|\partial_{z}\Phi_{\\!\mathcal{L}}(z,z^{\prime})-\mathbf{S}z/2|=1$ for any
$z^{\prime}$. Hence the map
$z^{\prime}\to\partial_{z}\Phi_{\\!\mathcal{L}}(z,z^{\prime})$ has its rank at
most $2d-1$. As we shall see later,
$\operatorname{rank}{(\partial_{z^{\prime}}\partial_{z}^{\intercal}\Phi_{\\!\mathcal{L}})}=2d-1$
for any
$(z,z^{\prime})\in\operatorname{supp}\chi_{l}\times\operatorname{supp}\chi_{l}^{\prime}$.

We compute the mixed Hessian
$\partial_{z^{\prime}}\partial_{z}^{\intercal}\Phi_{\mathcal{L}}$. Using the
chain rule, we have

(3.40) $\displaystyle\begin{aligned}
\partial_{z^{\prime}}\partial_{z}^{\intercal}\Phi_{\\!\mathcal{L}}&=\partial_{z^{\prime}}\partial_{z}^{\intercal}\mathcal{P}_{\mathcal{L}}(\mathfrak{S}_{c},z,z^{\prime})+\partial_{z^{\prime}}\partial_{t}\mathcal{P}_{\mathcal{L}}(\mathfrak{S}_{c},z,z^{\prime})\cdot\partial_{z}^{\intercal}\mathfrak{S}_{c}\\\
&\qquad+\partial_{z^{\prime}}\mathfrak{S}_{c}\cdot\partial_{z}^{\intercal}\partial_{t}\mathcal{P}_{\mathcal{L}}(\mathfrak{S}_{c},z,z^{\prime})+(\partial_{z^{\prime}}\mathfrak{S}_{c}\cdot\partial_{z}^{\intercal}\mathfrak{S}_{c})\partial_{t}^{2}\mathcal{P}_{\mathcal{L}}(\mathfrak{S}_{c},z,z^{\prime}).\end{aligned}$

Here we use $\
\partial_{t}\mathcal{P}_{\mathcal{L}}(\mathfrak{S}_{c},z,z^{\prime})=0$. From
(3.6), one can easily see

$\displaystyle\partial_{z^{\prime}}\partial_{z}^{\intercal}\mathcal{P}_{\mathcal{L}}(\mathfrak{S}_{c},z,z^{\prime})$
$\displaystyle=-\frac{\cos\mathfrak{S}_{c}}{2\sin\mathfrak{S}_{c}}\,\mathbf{I}-\frac{1}{2}\,\mathbf{S},$
$\displaystyle\partial_{z}^{\intercal}\partial_{t}\mathcal{P}_{\mathcal{L}}(\mathfrak{S}_{c},z,z^{\prime})$
$\displaystyle=-\frac{\mathbf{v}^{\intercal}}{2\sin^{2}\mathfrak{S}_{c}},$
$\displaystyle\partial_{z^{\prime}}\partial_{t}\mathcal{P}_{\mathcal{L}}(\mathfrak{S}_{c},z,z^{\prime})$
$\displaystyle=\frac{\mathbf{v}}{2\sin^{2}\mathfrak{S}_{c}}.$

Since $\sin\mathfrak{S}_{c}=|z-z^{\prime}|/2$, by (3.29) we also have

$\displaystyle\partial_{t}^{2}\mathcal{P}_{\mathcal{L}}(\mathfrak{S}_{c},z,z^{\prime})=\frac{4\cos\mathfrak{S}_{c}}{|z-z^{\prime}|},$

Now we note that
$\partial_{z}^{\intercal}\mathfrak{S}_{c}={\partial_{z}^{\intercal}(\sin\mathfrak{S}_{c})}/{\cos\mathfrak{S}_{c}}$,
$\partial_{z^{\prime}}\mathfrak{S}_{c}={\partial_{z^{\prime}}(\sin\mathfrak{S}_{c})}/{\cos\mathfrak{S}_{c}}$.
Thus, differentiating $\sin\mathfrak{S}_{c}=|z-z^{\prime}|/2$, we obtain

(3.41) $\displaystyle\partial_{z}^{\intercal}\mathfrak{S}_{c}$
$\displaystyle=\frac{\mathbf{v}^{\intercal}}{2\cos\mathfrak{S}_{c}|z-z^{\prime}|},$
$\displaystyle\partial_{z^{\prime}}\mathfrak{S}_{c}$
$\displaystyle=-\frac{\mathbf{v}}{2\cos\mathfrak{S}_{c}|z-z^{\prime}|}.$

Combining the identities together with (3.40), we obtain

(3.42) $\displaystyle\begin{aligned}
\partial_{z^{\prime}}\partial_{z}^{\intercal}\Phi_{\\!\mathcal{L}}=\frac{1}{|z-z^{\prime}|^{3}\cos\mathfrak{S}_{c}}\big{(}\mathbf{v}\mathbf{v}^{\intercal}-\cos^{2}\mathfrak{S}_{c}\mathbf{v}^{\intercal}\mathbf{v}\,\mathbf{I}-\sin\mathfrak{S}_{c}\cos\mathfrak{S}_{c}\mathbf{v}^{\intercal}\mathbf{v}\,\mathbf{S}\big{)}.\end{aligned}$

Since $\mathbf{S}^{2}=-\mathbf{I}$ and
$\mathbf{R}(z,z^{\prime})^{\intercal}=\cos\mathfrak{S}_{c}\mathbf{I}+\sin\mathfrak{S}_{c}\mathbf{S}$,
it is easy to see the matrix
$\mathbf{v}\mathbf{v}^{\intercal}-\cos^{2}\mathfrak{S}_{c}\mathbf{v}^{\intercal}\mathbf{v}\,\mathbf{I}-\sin\mathfrak{S}_{c}\cos\mathfrak{S}_{c}\mathbf{v}^{\intercal}\mathbf{v}\,\mathbf{S}$
can be factored as follows:

$\displaystyle(\cos\mathfrak{S}_{c}\mathbf{v}\mathbf{v}^{\intercal}-\sin\mathfrak{S}_{c}\mathbf{v}\mathbf{v}^{\intercal}\mathbf{S}-\cos\mathfrak{S}_{c}\mathbf{v}^{\intercal}\mathbf{v}\,\mathbf{I})\mathbf{R}(z,z^{\prime})^{\intercal}.$

Now we observe

$\displaystyle(\cos\mathfrak{S}_{c}\mathbf{v}\mathbf{v}^{\intercal}-\cos\mathfrak{S}_{c}\mathbf{v}^{\intercal}\mathbf{v}\,\mathbf{I}-\sin\mathfrak{S}_{c}\mathbf{v}\mathbf{v}^{\intercal}\mathbf{S})v=\begin{cases}\
\ \ \ \ \ \ 0,&v=\mathbf{v},\\\
-\mathbf{v}^{\intercal}\mathbf{v}\,\mathbf{R}(z,z^{\prime})\mathbf{v},&v=\mathbf{S}\mathbf{v},\\\
-\cos\mathfrak{S}_{c}\mathbf{v}^{\intercal}\mathbf{v}\,v,&v\in(\text{span}\\{\mathbf{v},\mathbf{S}\mathbf{v}\\})^{\perp}.\end{cases}$

For the second case we use the fact that $\mathbf{S}$ is skew symmetric, i.e.,
$\mathbf{v}^{\intercal}\mathbf{S}\mathbf{v}=0$. Note that the vectors
$\mathbf{v}^{\intercal}\mathbf{v}\,\mathbf{R}(z,z^{\prime})\mathbf{v}$ and
$\cos\mathfrak{S}_{c}\mathbf{v}^{\intercal}\mathbf{v}\,v$, $v\neq 0$ are
nonzero. From this and (3.42) we see that the mixed Hessian
$\partial_{z^{\prime}}\partial_{z}^{\intercal}\Phi_{\\!\mathcal{L}}$ has rank
$2d-1$ and
$\partial_{z^{\prime}}\partial_{z}^{\intercal}\Phi_{\\!\mathcal{L}}\,\mathbf{R}(z,z^{\prime})\mathbf{v}=0$
for any $z^{\prime}$. So, we obtain the following.

######  Lemma 3.8.

For
$(z,z^{\prime})\in\operatorname{supp}\chi_{l}\times\operatorname{supp}\chi_{l}^{\prime}$
the matrix
$\partial_{z^{\prime}}\partial_{z}^{\intercal}\Phi_{\\!\mathcal{L}}(z,z^{\prime})$
has rank $2d-1$ and its null space is generated by the vector

$\nu(z,z^{\prime})=\mathbf{R}(z,z^{\prime})\frac{\mathbf{v}(z,z^{\prime})}{|\mathbf{v}(z,z^{\prime})|}.$

The vector $\nu(z,z^{\prime})$ is the unique (modulo $\pm$) unit vector such
that
$\partial_{z^{\prime}}\partial_{z}^{\intercal}\Phi_{\\!\mathcal{L}}\,\nu(z,z^{\prime})=0$.
We now consider the matrix
$\partial_{w}\partial^{\intercal}_{w}\langle\partial_{z}\Phi_{\\!\mathcal{L}}(z,w),\nu(z,z^{\prime})\rangle|_{w=z}$
for which we can obtain an explicit expression.

######  Lemma 3.9.

For
$(z,z^{\prime})\in\operatorname{supp}\chi_{l}\times\operatorname{supp}\chi_{l}^{\prime}$
we have

(3.43)
$\displaystyle\partial_{w}\partial^{\intercal}_{w}\langle\partial_{z}\Phi_{\\!\mathcal{L}}(z,w),\nu(z,z^{\prime})\rangle|_{w=z^{\prime}}=\frac{-1}{\cos^{2}\mathfrak{S}_{c}|z-z^{\prime}|^{4}}\,\mathbf{M}(z,z^{\prime}),$

where the matrix $\mathbf{M}(z,z^{\prime})$ is given by

$\mathbf{M}(z,z^{\prime})=\mathbf{v}\mathbf{v}^{\intercal}-2\cos^{2}\mathfrak{S}_{c}\mathbf{v}\mathbf{v}^{\intercal}+\cos^{2}\mathfrak{S}_{c}\mathbf{v}^{\intercal}\mathbf{v}\,\mathbf{I}+\sin\mathfrak{S}_{c}\cos\mathfrak{S}_{c}(\mathbf{v}\mathbf{v}^{\intercal}\,\mathbf{S}-\mathbf{S}\mathbf{v}\mathbf{v}^{\intercal}).$

###### Proof of Lemma 3.9.

For simplicity we set

$\mathbf{v}_{w}(z)=\mathbf{v}(z,w),\quad\mathfrak{S}_{w}(z)=\mathfrak{S}_{c}(z,w).$

Using (3.42) and
$\mathbf{R}(z,z^{\prime})=\cos\mathfrak{S}_{c}\mathbf{I}-\sin\mathfrak{S}_{c}\mathbf{S}$,
after some computation we have

$\partial_{w}\partial_{z}^{\intercal}\Phi_{\\!\mathcal{L}}(z,w)\cdot\nu(z,z^{\prime})=(\cos\mathfrak{S}_{w}|z-w|^{3}|z-z^{\prime}|)^{-1}\mathbf{k}(z,z^{\prime},w),$

where $\mathbf{k}(z,z^{\prime},w)\in\mathbb{R}^{2d}$ is given by

$\displaystyle\mathbf{k}(z,$ $\displaystyle
z^{\prime},w)=\cos\mathfrak{S}_{c}\,\mathbf{v}_{w}\mathbf{v}_{w}^{\intercal}\mathbf{v}-\sin\mathfrak{S}_{c}\,\mathbf{v}_{w}\mathbf{v}_{w}^{\intercal}\mathbf{S}\mathbf{v}$
$\displaystyle-\mathbf{v}_{w}^{\intercal}\mathbf{v}_{w}\big{(}\cos\mathfrak{S}_{w}\cos(\mathfrak{S}_{w}-\mathfrak{S}_{c})\mathbf{v}+\cos\mathfrak{S}_{w}\sin(\mathfrak{S}_{w}-\mathfrak{S}_{c})\mathbf{S}\mathbf{v}\big{)}.$

Clearly $\mathbf{k}(z,z^{\prime},z^{\prime})=0$ for any $z^{\prime}$. Thus, we
have

$\partial_{w}\partial^{\intercal}_{w}\langle\partial_{z}\Phi_{\\!\mathcal{L}}(z,w),\nu(z,z^{\prime})\rangle|_{w=z^{\prime}}=\frac{1}{\cos\mathfrak{S}_{c}|z-z^{\prime}|^{4}}\,\partial_{w}^{\intercal}\mathbf{k}(z,z^{\prime},z^{\prime}).$

Now, (3.43) follows if we express the matrix
$\partial_{w}^{\intercal}\mathbf{k}(z,z^{\prime},z^{\prime})$ in the desired
form, i.e.,

(3.44)
$\partial_{w}^{\intercal}\mathbf{k}(z,z^{\prime},z^{\prime})=-\frac{1}{\cos\mathfrak{S}_{c}}\mathbf{M}(z,z^{\prime}).$

Via a routine computation we write

$\displaystyle\partial_{w}^{\intercal}$
$\displaystyle\mathbf{k}(z,z^{\prime},w)=\cos\mathfrak{S}_{c}\partial_{w}^{\intercal}(\mathbf{v}_{w}\mathbf{v}_{w}^{\intercal}\mathbf{v})-\sin\mathfrak{S}_{c}\partial_{w}^{\intercal}(\mathbf{v}_{w}\mathbf{v}_{w}^{\intercal}\mathbf{S}\mathbf{v})$
$\displaystyle-\cos
S_{w}\cos(S_{w}-\mathfrak{S}_{c})\mathbf{v}\cdot\partial_{w}^{\intercal}(\mathbf{v}_{w}^{\intercal}\mathbf{v}_{w})+\mathbf{v}_{w}^{\intercal}\mathbf{v}_{w}\sin
S_{w}\cos(S_{w}-\mathfrak{S}_{c})\mathbf{v}\cdot\partial_{w}^{\intercal}S_{w}$
$\displaystyle+\mathbf{v}_{w}^{\intercal}\mathbf{v}_{w}\cos
S_{w}\sin(S_{w}-\mathfrak{S}_{c})\mathbf{v}\cdot\partial_{w}^{\intercal}S_{w}-\cos
S_{w}\sin(S_{w}-\mathfrak{S}_{c})\mathbf{S}\mathbf{v}\cdot\partial_{w}^{\intercal}(\mathbf{v}_{w}^{\intercal}\mathbf{v}_{w})$
$\displaystyle+\mathbf{v}_{w}^{\intercal}\mathbf{v}_{w}\sin
S_{w}\sin(S_{w}-\mathfrak{S}_{c})\mathbf{S}\mathbf{v}\cdot\partial_{w}^{\intercal}S_{w}-\mathbf{v}_{w}^{\intercal}\mathbf{v}_{w}\cos
S_{w}\cos(S_{w}-\mathfrak{S}_{c})\mathbf{S}\mathbf{v}\cdot\partial_{w}^{\intercal}S_{w}.$

Here we can discard the $5,6,7$-th terms on the right hand side since they
vanish if we put $w=z^{\prime}$. Also, we have

$\displaystyle\partial_{w}^{\intercal}(\mathbf{v}_{w}\mathbf{v}_{w}^{\intercal}\mathbf{v})$
$\displaystyle=-(\mathbf{v}_{w}\mathbf{v}^{\intercal}+\mathbf{v}_{w}^{\intercal}\mathbf{v}\mathbf{I}),$
$\displaystyle\partial_{w}^{\intercal}(\mathbf{v}_{w}^{\intercal}\mathbf{v}_{w}\mathbf{S}\mathbf{v})$
$\displaystyle=(\mathbf{v}_{w}\mathbf{v}^{\intercal}\,\mathbf{S}-\mathbf{v}_{w}^{\intercal}\mathbf{S}\mathbf{v}\,\mathbf{I}).$

Using these identities, (3.41), and
$4\sin^{2}{\mathfrak{S}_{c}}=\mathbf{v}^{\intercal}\mathbf{v}$ (see (3.12)),
we now get

$\displaystyle\partial_{w}^{\intercal}\mathbf{k}(z,z^{\prime},z^{\prime})$
$\displaystyle=-\cos\mathfrak{S}_{c}(\mathbf{v}\mathbf{v}^{\intercal}+\mathbf{v}^{\intercal}\mathbf{v}\mathbf{I})-\sin\mathfrak{S}_{c}(\mathbf{v}\mathbf{v}^{\intercal}\,\mathbf{S}-\mathbf{v}^{\intercal}\mathbf{S}\mathbf{v}\,\mathbf{I})+2\cos\mathfrak{S}_{c}\,\mathbf{v}\mathbf{v}^{\intercal}$
$\displaystyle\qquad\qquad-(\cos\mathfrak{S}_{c})^{-1}{\sin^{2}{\mathfrak{S}_{c}}}\mathbf{v}\mathbf{v}^{\intercal}+\sin{\mathfrak{S}_{c}}\mathbf{S}\mathbf{v}\mathbf{v}^{\intercal}.$

Since $\mathbf{v}^{\intercal}\mathbf{S}\mathbf{v}=0$, we therefore obtain
(3.44). ∎

######  Lemma 3.10.

Let
$(z,z^{\prime})\in\operatorname{supp}\chi_{l}\times\operatorname{supp}\chi_{l}^{\prime}$
with $2^{-l}\sim\rho$. Define the matrix $\mathbf{B}(z,z^{\prime})$ by setting

$\mathbf{B}(z,z^{\prime})=\left(\mathbf{v}_{1}\dots,\mathbf{v}_{2d-2},\frac{\mathbf{S}\mathbf{v}(z,z^{\prime})}{|\mathbf{v}(z,z^{\prime})|},\,\frac{\mathbf{v}(z,z^{\prime})}{|\mathbf{v}(z,z^{\prime})|}\right),$

where $\\{\mathbf{v}_{i}\\}_{i=1}^{2d-2}$ denotes an orthonormal basis of
$(\mathrm{span}\\{\mathbf{v}(z,z^{\prime}),\mathbf{S}\mathbf{v}(z,z^{\prime})\\})^{\perp}$.
Then the $2d\times 2d$ matrix

(3.45)
$\mathbf{B}^{\intercal}(z,z^{\prime})\mathbf{R}(z,z^{\prime})\mathbf{M}(z,z^{\prime})\mathbf{R}(z,z^{\prime})^{\intercal}\mathbf{B}(z,z^{\prime})$

is a diagonal matrix such that $(2d,2d)$-th entry is zero and the other
diagonal entries $\lambda_{1},\dots,\lambda_{2d-1}$ satisfy
$\lambda_{i}\sim\rho^{2}$, $1\leq i\leq 2d-1.$

###### Proof of Lemma 3.10.

The matrix $\mathbf{M}=\mathbf{M}(z,z^{\prime})$ in Lemma 3.9 can be written
as

$\mathbf{M}=\cos^{2}\mathfrak{S}_{c}(\mathbf{v}^{\intercal}\mathbf{v}\,\mathbf{I}+\mathbf{S}\mathbf{v}\mathbf{v}^{\intercal}\mathbf{S}-\mathbf{v}\mathbf{v}^{\intercal})-\mathbf{R}^{\intercal}\mathbf{S}\mathbf{v}\mathbf{v}^{\intercal}\mathbf{S}\,\mathbf{R}.$

Then, multiplying $\mathbf{R}=\mathbf{R}(z,z^{\prime})$ and
$\mathbf{R}^{\intercal}=\mathbf{R}(z,z^{\prime})^{\intercal}$ to
$\mathbf{M}(z,z^{\prime})$, we have

$\displaystyle\mathbf{R}\mathbf{M}\mathbf{R}^{\intercal}=\cos^{2}\mathfrak{S}_{c}\mathbf{v}^{\intercal}\mathbf{v}\,\mathbf{I}+\cos^{2}\mathfrak{S}_{c}\mathbf{R}(\mathbf{S}\mathbf{v}\mathbf{v}^{\intercal}\mathbf{S}-\mathbf{v}\mathbf{v}^{\intercal})\mathbf{R}^{\intercal}-\mathbf{S}\mathbf{v}\mathbf{v}^{\intercal}\mathbf{S}.$

Using
$\mathbf{R}=\cos\mathfrak{S}_{c}\mathbf{I}_{2d}-\sin\mathfrak{S}_{c}\mathbf{S}$,
one can easily see
$\mathbf{R}(\mathbf{S}\mathbf{v}\mathbf{v}^{\intercal}\mathbf{S}-\mathbf{v}\mathbf{v}^{\intercal})\mathbf{R}^{\intercal}=\mathbf{S}\mathbf{v}\mathbf{v}^{\intercal}\mathbf{S}-\mathbf{v}\mathbf{v}^{\intercal}$.
Thus we have

$\displaystyle\mathbf{R}\mathbf{M}\mathbf{R}^{\intercal}=\cos^{2}\mathfrak{S}_{c}(\mathbf{v}^{\intercal}\mathbf{v}\,\mathbf{I}-\mathbf{v}\mathbf{v}^{\intercal})-\sin^{2}\mathfrak{S}_{c}\,\mathbf{S}\mathbf{v}\mathbf{v}^{\intercal}\mathbf{S}.$

Then it is easy to verify

$\displaystyle\mathbf{R}\mathbf{M}\mathbf{R}^{\intercal}v=\begin{cases}\mathbf{v}^{\intercal}\mathbf{v}\cos^{2}\mathfrak{S}_{c}\,v,&v\in(\text{span}\\{\mathbf{v},\mathbf{S}\mathbf{v}\\})^{\perp},\\\
\mathbf{v}^{\intercal}\mathbf{v}\,v,&v=\mathbf{S}\mathbf{v},\\\ \ \ \
0,&v=\mathbf{v}.\end{cases}$

Therefore, $\mathbf{R}\mathbf{M}\mathbf{R}^{\intercal}$ has the eigenvalues
$\mathbf{v}^{\intercal}\mathbf{v}\cos^{2}\mathfrak{S}_{c}(z,z^{\prime}),\,\mathbf{v}^{\intercal}\mathbf{v},0$
with multiplicity $2d-2,1,1,$ respectively. It is clear that
$\mathbf{v}^{\intercal}\mathbf{v},\,\mathbf{v}^{\intercal}\mathbf{v}\cos^{2}\mathfrak{S}_{c}(z,z^{\prime})\sim|z-z^{\prime}|^{2}\sim\rho^{2}$
because
$(z,z^{\prime})\in\operatorname{supp}\chi_{l}\times\operatorname{supp}\chi_{l}^{\prime}$.
Now the elementary (diagonalization) argument gives (3.45). ∎

###  3.4. Proof of Proposition 3.6

We now prove (3.23). Let us set

$\mathcal{T}_{\\!\mathcal{L}}f(z)=\int
A(z,z^{\prime})e^{i\lambda\rho\widetilde{\mathcal{P}}_{\\!\mathcal{L}}(0,z,z^{\prime})}f(z^{\prime})dz^{\prime},$

where $A\in C_{c}^{\infty}(B(0,\epsilon_{0})\times B(0,\epsilon_{0}))$. Then
we note

$\|\chi_{l}[\eta_{\rho}]^{\mathcal{L}}_{\lambda}\chi_{l}^{\prime}\|_{p}=\rho^{2d}\|(\chi_{l}[\eta_{\rho}]^{\mathcal{L}}_{\lambda}\chi_{l}^{\prime})(\rho\,\cdot+z_{0},\rho\,\cdot+z^{\prime}_{0})\|_{p}.$

If we use (3.33) and (3.38) with a large $N$, for the proof of (3.23) it is
sufficient to show

(3.46) $\|\mathcal{T}_{\\!\mathcal{L}}f\|_{p}\leq
C(\lambda\rho)^{-\frac{2d}{p}}\|f\|_{p}$

for $p_{0}(2d)<p\leq\infty$. By (3.36) and (3.31) it follows that

(3.47)
$|\partial_{z}^{\alpha}\partial_{z}^{\beta}\widetilde{\mathcal{P}}_{\\!\mathcal{L}}(0,z,z^{\prime})|\leq
C_{\alpha,\beta}.$

Recalling
$\widetilde{\mathcal{P}}_{\\!\mathcal{L}}(0,z,z^{\prime})=\rho^{-1}\Phi_{\\!\mathcal{L}}(\rho
z+z_{0},\rho z^{\prime}+z^{\prime}_{0})$ (see (3.39) and (3.32)), we put

$\widetilde{\Phi}(z,z^{\prime})=\rho^{-1}\Phi_{\\!\mathcal{L}}(\rho\mathbf{R}_{0}z+z_{0},\rho\mathbf{R}_{0}z^{\prime}+z^{\prime}_{0}),$

where

$\mathbf{R}_{0}=\mathbf{R}(z_{0},z^{\prime}_{0})^{\intercal}\mathbf{B}(z_{0},z^{\prime}_{0}).$

We define

$\widetilde{\mathcal{T}}_{\\!\mathcal{L}}f(z)=\int
A(\mathbf{R}_{0}z,\mathbf{R}_{0}z^{\prime})e^{i\lambda\rho\widetilde{\Phi}(z,z^{\prime})}f(z^{\prime})dz^{\prime},$

Since
$\widetilde{\mathcal{P}}_{\\!\mathcal{L}}(0,z,z^{\prime})=\rho^{-1}\Phi_{\\!\mathcal{L}}(\rho
z+z_{0},\rho z^{\prime}+z^{\prime}_{0})$, (3.46) is equivalent to

(3.48) $\|\widetilde{\mathcal{T}}_{\\!\mathcal{L}}f\|_{p}\leq
C(\lambda\rho)^{-\frac{2d}{p}}\|f\|_{p}.$

Fixing $z_{2d}\in(-\epsilon_{0},\epsilon_{0})$, let us set

$\widetilde{\mathcal{T}}_{\\!\mathcal{L}}^{z_{2d}}g(z)=\int\widetilde{A}(z,\zeta)e^{i\lambda\rho\widetilde{\Phi}(z,\zeta,z_{2d})}g(\zeta)d\zeta,\quad\zeta\in\mathbb{R}^{2d-1},$

where $\widetilde{A}\in C_{c}^{\infty}(B(0,\epsilon_{0})\times
B(0,\epsilon_{0}))$. By the same argument as before, the estimate (3.46)
follows if we show

(3.49) $\|\widetilde{\mathcal{T}}_{\\!\mathcal{L}}^{z_{2d}}g\|_{p}\leq
C(\lambda\rho)^{-\frac{2d}{p}}\|g\|_{p},\quad
z_{2d}\in(-\epsilon_{0},\epsilon_{0}).$

To show the estimate it is sufficient to show that the function

$\widetilde{\Phi}_{z_{2d}}(z,\zeta):=\widetilde{\Phi}(z,\zeta,z_{2d}),\quad(z,\zeta)\in\mathbb{R}^{d}\times\mathbb{R}^{d-1}$

satisfies the Carleson-Sjölin condition with ellipticity for
${z_{2d}}\in(-\epsilon_{0},\epsilon_{0})$ on the support of $\widetilde{A}$
since the desired estimate (3.49) follows from Theorem 2.11.

Now we note that

$\partial_{z^{\prime}}\partial_{z}^{\intercal}\widetilde{\Phi}(z,z^{\prime})=\rho\mathbf{R}_{0}^{\intercal}\partial_{z^{\prime}}\partial_{z}^{\intercal}\Phi_{\\!\mathcal{L}}(\rho\mathbf{R}_{0}z+z_{0},\rho\mathbf{R}_{0}z^{\prime}+z^{\prime}_{0})\mathbf{R}_{0}.$

By Lemma 3.8 we have

$\partial_{z^{\prime}}\partial_{z}^{\intercal}\widetilde{\Phi}(z,z^{\prime})\mathrm{N}(z,z^{\prime})=0,$

where
$\mathrm{N}(z,z^{\prime})=\mathbf{R}_{0}^{\intercal}\nu(\rho\mathbf{R}_{0}z+z_{0},\rho\mathbf{R}_{0}z^{\prime}+z^{\prime}_{0})$.
Thus, it is clear that

(3.50)