# Linear restriction estimates for Schrödinger equation on metric cones

Junyong Zhang  Department of Mathematics, Beijing Institute of Technology,
Beijing 100081, China and Department of Mathematics, Australian National
University, Canberra, ACT 0200, Australia<EMAIL_ADDRESS>
###### Abstract.

In this paper, we study some modified linear restriction estimates of the
dynamics generated by Schrödinger operator on metric cone $M$, where the
metric cone $M$ is of the form $M=(0,\infty)_{r}\times\Sigma$ with the cross
section $\Sigma$ being a compact $(n-1)$-dimensional Riemannian manifold
$(\Sigma,h)$ and the equipped metric is $g=\mathrm{d}r^{2}+r^{2}h$. Assuming
the initial data possesses additional regularity in angular variable
$\theta\in\Sigma$, we show some linear restriction estimates for the
solutions. As applications, we obtain global-in-time Strichartz estimates for
radial initial data and show small initial data scattering theory for the
mass-critical nonlinear Schrödinger equation on two-dimensional metric cones.

Key Words: Linear restriction estimate, Metric cone, Strichartz estimates
AMS Classification: 42B37, 35Q40, 47J35.

##  1\. Introduction and Statement of Main Result

We study some restriction estimates for the solution of Schrödinger equations
on the setting of metric cone. The metric cone $M$ is of the form
$M=(0,\infty)_{r}\times\Sigma$, where $(\Sigma,h)$ is a compact
$(n-1)$-dimensional Riemannian manifold and the metric of $M$ is
$g=\mathrm{d}r^{2}+r^{2}h$. More precisely, we consider solutions $u:\R\times
M\rightarrow\C$ to the initial problem (IVP) for the Schrödinger equation on
$M$,

(1.1) $i\partial_{t}u(t,z)+Hu(t,z)=0,\quad
u(t,z)|_{t=0}=u_{0}(z),\quad(t,z)\in\R\times M.$

Here, we use the operator $H=-\Delta_{g}+q(\theta)/r^{2}$ where $\Delta_{g}$
denotes the Friedrichs extension of Laplace-Beltrami from the domain
$C_{c}^{\infty}(M^{\circ})$, compactly supported smooth functions on the
interior of the metric cone, and we write $q(\theta)$ for a smooth function on
$\Sigma$ such that $-\Delta_{h}+q(\theta)$ is positive on $L^{2}(\Sigma)$. The
Euclidean space $\R^{n}$ is the simplest example of a metric cone; its cross
section is $(\mathbb{S}^{n-1},\mathrm{d}\theta^{2})$. We note that the general
metric cones have a dilation symmetry analogous to that of Euclidean space but
no other symmetries in general.

There is a large amount of literature focused on the restriction theory on the
Euclidean space, we refer the readers to [1, 23, 32, 34, 35, 36, 38]. Shao
[24, 25] proved the cone and parabolic restriction conjectures hold true for
the spatial rotation invariant functions which are supported on the cone or
parabola. Motivated by [24, 25], Miao, Zheng and the author[18, 19] utilized
the spherical harmonics expansion and analyzed the asymptotic behavior of the
Bessel function to generalize Shao’s results by establishing restriction
estimates with some angular regularity loss. Based on [18], Miao, Zheng and
the author [20] proved a scale of Strichartz estimates (extending the
admissible restriction) for wave equation with an inverse square potential
when the initial data had additional angular regularity.

We are interested in the restriction estimate for the solution of Schrödinger
equations on the metric cone. Cones were studied from the problem of wave
diffraction from a cone point; see [29, 7, 8]. The Laplacian defined on cones
has been studied by Cheeger and Taylor [4, 5]. Other aspects on the metric
cone also have been studied; for example the heat kernel and Riesz transform
kernel were studied in [11, 15]. There has been a lot of interest in the study
of the Schrödinger propagator on the smooth asymptotically conic Riemannian
manifolds. We refer the reader to Hassell, Tao and Wunsch [12, 13] and
Mizutani [16]. In particular, Guillarmou, Hassell and Sikora [9] showed a
estimate of the spectral measure to obtain a Stein-Tomas restriction theorem
in this asymptotically conic setting. The restriction problem is much more
than the Stein-Tomas type restriction estimates. We recall that a
asymptotically conic manifold $X$, outside some compact set, is isometric to a
conical space $M=\R_{+}\times\Sigma$, where $\Sigma$ is a compact
$(n-1)$-dimensional manifold with or without boundary. By analogy with
Euclidean space, we call $r\in\R_{+}$ the radial variable and
$\theta\in\Sigma$ the angular variable. Then $(r,\theta)$ are polar
coordinates on $M$, and we can write the metric as $g=\mathrm{d}r^{2}+r^{2}h$
with the Riemannian metric $h$ on $\Sigma$. We refer the reader to [17, 14]
for more details on the scattering manifolds. Most arguments applying to
metric cones can be recognized as an ingredient of the analysis on
asymptotically conic manifolds. The problems on metric cones appear as model
problems when dealing with similar questions on asymptotically conic
manifolds. We however will prove much more restriction estimates than [9] by
assuming the initial data having additional “angular” regularity. As
applications, we show a global-in-time Strichartz estimate for the Schrödinger
equation on the metric cone for radial initial data. For two-dimensional
metric cone, Ford [6] proved the full range of global-in-time Strichartz
estimates. We remark that the Strichartz estimates established in [12, 13, 16]
for scattering manifolds are local in time.

As pointed out in [9], the Laplacian on the scattering manifolds gives rise to
a family of Poisson operators $P(\lambda)$ defined for $\lambda>0$. The
corresponding extension-restriction problem is to consider the boundedness of
$P(\lambda)$: $L^{p}(\partial M)\rightarrow L^{q}(M)$. Its norm is in terms of
the frequency $\lambda$. The restriction conjecture on the ball and the
parabolic surface with dimension $n$ says that $1\leq p<{2(n+1)}/{n}$ and
${(n+2)}/q\leq{n}/{p^{\prime}}$ is a necessary and sufficient condition. It is
very hard to show the sufficient part when $p$ is close to ${2(n+1)}/{n}$ and
the problem still remains open.

In this paper, we follow the argument in [19, 20] to show modified restriction
estimates with some loss of angular regularity for the solution of Schrödinger
equation on conic manifold when $p$ is close to ${2(n+1)}/{n}$. Since we do
not know how to construct an approximate “global” parametrix for the
propagator $e^{itH}$, we have to write the propagator as a linear combination
of products of the Hankel transform of the radial part and eigenfunctions of
$-\Delta_{h}+q(\theta)$, the Laplace-Beltrami operator on $\Sigma$. Though
this expression may cause some loss of angular regularity, it gives a global
in time expression of the solution. Compared with our previous work [18, 20]
for wave equation, we need to exploit effectively the oscillation of the
multiplier $e^{it\rho^{2}}$ which has much more oscillation than the wave
multiplier $e^{it\rho}$ at high frequency. The Bessel function $J_{\nu}(r)$
appears in the Hankel transform, and the decay property of the Bessel function
plays a key role in our argument. Since $J_{\nu}(r)$ decays more slowly than
$r^{-1/2}$ when $1\ll r\sim\nu$, we overcome this difficulty by exploiting the
oscillations both in $e^{it\rho^{2}}$ and the Bessel function $J_{\nu}(r\rho)$
in proving a localized estimate for $q=\infty$; see Proposition 3.1 below.
However the strategy breaks down for the other general $q$, for example $q=4$.
We need develop the advantage of the parabolic curvature. To do this, we use a
bilinear argument which is in spirit of Carleson-Sjölin argument or
equivalently the $TT^{*}$ method. In the process of using bilinear argument,
we have to divide into two cases $1\ll\nu\sim r\ll\nu^{2}$ and $\nu^{2}\ll r$.
In the former, the low decay of Bessel function leads to a loss of angular
regularity. The latter will be treated by using a complete asymptotic formula
for the Bessel function in [27, 37]. The quantity $\nu^{2}$ is chosen to
balance the two things: the smallest loss of angular regularity and the
absolutely convergent of the series of the coefficients in the complete
asymptotic formula. In the proof of the case $q=4$, we additionally require a
Whitney-type decomposition argument because of the failure of Hardy-
Littlewood-Sobolev inequality.

To state our main result, we need some notation. Let

(1.2)
$\chi_{\infty}=\Big{\\{}\nu:\nu=\sqrt{\lambda+(1/4)(n-2)^{2}},\quad\lambda~{}\text{is
eigenvalue of}-\widetilde{\Delta}_{h}:=-\Delta_{h}+q(\theta)\Big{\\}},$

and let $d(\nu)$ be the multiplicity of
$\lambda_{\nu}=\nu^{2}-\frac{1}{4}(n-2)^{2}$ as eigenvalue of
$-\widetilde{\Delta}_{h}$ and $\\{\varphi_{\nu,\ell}\\}_{1\leq\ell\leq
d(\nu)}$ the associated eigenfunctions of $-\widetilde{\Delta}_{h}$. We then
have the decomposition of $f\in L^{2}(M)$

(1.3)
$f(z)=f(r,\theta)=\sum\limits_{\nu\in\chi_{\infty}}\sum\limits_{\ell=1}^{d(\nu)}a_{\nu,\ell}(r)\varphi_{\nu,\ell}(\theta).$

For more details, we refer to Section 2. We now define the “distorted” Fourier
transform of the Schwartz function $f$ by

(1.4)
$\mathcal{F}_{H}(f)(\rho,\omega)=\sum_{\nu\in\chi_{\infty}}\sum_{\ell=1}^{d(\nu)}\varphi_{\nu,\ell}(\omega)\int_{0}^{\infty}(r\rho)^{-\frac{n-2}{2}}J_{\nu}(r\rho)a_{\nu,\ell}(r)r^{n-1}\mathrm{d}r,$

where $\omega\in\Sigma$ and $J_{\nu}(r)$ is the Bessel function of order
$\nu$. We remark that when $\Sigma=\mathbb{S}^{n-1}$, $\varphi_{\nu,\ell}$ is
the spherical harmonics function $Y_{k,\ell}(\theta)\in
L^{2}(\mathbb{S}^{n-1})$ of order $k$ and $\nu=k+(n-2)/2$, then the
“distorted” Fourier transform defined above, up to some constant, is same as
the classical Fourier transform by [28, Theorem 3.10].

Our main theorem is stated as:

######  Theorem 1.1.

Let $n\geq 2$ and $M$ be an $n$-dimensional metric cone, and let $u$ be the
solution of the equation (1.1). Suppose
$q=\frac{p^{\prime}(n+2)}{n}>\frac{2(n+1)}{n}$ and $p\geq 1$. Then there
exists a constant $C$ only depending on $p,q,n$, and $M$ such that

$1).$ if $u_{0}(z)=f(r)$ is a radial Schwartz function111This is in order to
avoid needless technicalities, but our estimates will not depend on any of the
Schwartz semi-norms of the $u_{0}$ and so can be extended to rougher initial
data., then

(1.5) $\|u(t,z)\|_{L^{q}_{t,z}(\R\times M)}\leq
C_{p,q,n,M}\|\mathcal{F}_{H}(u_{0})\|_{L^{p}(M)};$

$2).$ and if $u_{0}$ is any Schwartz function (not necessarily radial) and
$p\geq 2$, then

(1.6) $\|u(t,z)\|_{L^{q}_{t,z}(\R\times M)}\leq
C_{p,q,n,M}\|\mathcal{F}_{H}{\big{(}(1-\widetilde{\Delta}_{h})^{s}u_{0}\big{)}}\|_{L^{p}(M)},$

where $s=\frac{(q-2)(n-1)}{4q}+\frac{1}{qn}$.

Remarks:

$\mathrm{i}).$ We are interested in the estimate (1.6) with $p=2$, which gives
a global-in-time Strichartz-type estimate with $s$-loss of angular regularity

$\|u(t,z)\|_{L^{{2(n+2)}/{n}}_{t,z}(\R\times M)}\leq
C\|(1-\widetilde{\Delta}_{h})^{s}u_{0}\|_{L^{2}(M)},\quad
s=\frac{(q-2)(n-1)}{4q}+\frac{1}{qn}.$

By (1.5), we obtain a global in time Strichartz estimates for radial initial
data.

$\mathrm{ii}).$ Let $N$ be a dyadic number, if the initial data $u_{0}$ is
radial such that the support of
$\mathcal{F}_{H}(u_{0})\subset\\{\rho:N\leq\rho\leq 2N\\}$, by interpolating
(3.1) and (3.4) in $q$ and summing in $R$, we can obtain the Strichartz
estimate

(1.7) $\|u(t,z)\|_{L^{q}_{t,z}(\R\times M)}\leq
CN^{\frac{n}{2}-\frac{n+2}{q}}\|u_{0}\|_{L^{2}(M)}\quad\text{for}~{}q>{2(2n+1)}/{(2n-1)}.$

The Strichartz estimates in [13, 16] also imply (1.7) holds locally in time,
but for $q\geq 2(n+2)/n$.

$\mathrm{iii}).$ The assumption on the positivity of the operator
$-\widetilde{\Delta}_{h}$ can be satisfied when $q(\theta)\geq 0$. It would be
possible to generalize the result to $-\widetilde{\Delta}_{h}+(n-2)^{2}/4>0$
allowing some negative potential, which includes the special Schrödinger
equation on $\R^{n}$ with a inverse-square potential $a/|z|^{2}$ when
$a>-(n-2)^{2}/4$. In that case, the relationship between $q$ and $p$ should
depend on the square root of the smallest eigenvalue of the operator
$-\widetilde{\Delta}_{h}+(n-2)^{2}/4$.

$\mathrm{iv}).$ In a future work, we hope to use the resolvent and spectral
measure arguments in [10, 9] to show the restriction estimate for $p=2$
without a loss of angular regularity.

As pointed out in the paper [13], the Strichartz estimates established by
Hassell, Tao and Wunsch are not strong enough to obtain a scattering theory
for the nonlinear Schrödinger equations on the scattering manifold. Ford [6]
proved the global-in-time Strichartz estimates for two-dimensional metric cone
$C(\mathbb{S}^{1}_{\rho})$. From Ford’s Strichartz estimates, one can conclude
the global existence and scattering for the mass critical Schrödinger equation
on $2$-dimension metric cone with small initial data. As applications of (1.5)
with $p=2$, we reprove the same result for the mass critical Schrödinger
equation on $2$-dimension metric cone with small radial initial data. We do
this because that one can generalize the result to higher dimension as long as
one could develop a fractional Liebniz rule for Sobolev spaces on cones.
Consider the initial value problem

(1.8) $\begin{cases}i\partial_{t}u-Hu=\gamma|u|^{2}u,\qquad(t,z)\in\R\times
M,\\\ u(t,z)|_{t=0}=u_{0}(z),\qquad\qquad z\in M.\end{cases}$

Indeed by duality, the Strichartz estimate (1.5) implies the inhomogeneous
Strichartz estimate

(1.9)
$\Big{\|}\int_{0}^{t}e^{-i(t-s)H}f(z,s)\mathrm{d}s\Big{\|}_{L^{q}_{t,z}(\R\times
M)}\lesssim\|f\|_{L^{q^{\prime}}_{t,z}(\R\times M)},\quad\text{with}\quad
q=2(n+2)/n.$

And then we can apply the arguments of Cazenave and Weissler [3] or Tao [31]
with Euclidean space replaced by the conic manifold $M$ to show:

######  Corollary 1.1 (Scattering theory for NLS).

Let $M$ be $2$-dimension manifold as in Theorem 1.1 and $\gamma=\pm 1$. Let
$u_{0}\in L^{2}(M)$ be radial such that $\|u_{0}\|_{L^{2}(M)}\leq\epsilon$
with small constant $\epsilon$, then NLS (1.8) is global well-posed in
$L^{2}(M)$ and the solution $u$ is scattering and moreover $u\in
L^{4}_{t,z}(\R\times M)$.

Remarks: For higher dimensions $n\geq 2$, one could show the small scattering
theory in $H^{s}(M)$ when $s\geq\text{max}(0,\frac{n}{2}-\frac{2}{\kappa-1})$
for the nonlinear Schrödinger equation (1.8) with nonlinearity
$|u|^{\kappa-1}u,(\kappa>1)$. This would require one to develop a fractional
Liebniz rule for Sobolev spaces on these manifolds.

Now we introduce some notation. We use $A\lesssim B$ to denote $A\leq CB$ for
some large constant C which may vary from line to line and depend on various
parameters, and similarly we use $A\ll B$ to denote $A\leq C^{-1}B$. We employ
$A\sim B$ when $A\lesssim B\lesssim A$. If the constant $C$ depends on a
special parameter other than the above, we shall denote it explicitly by
subscripts. For instance, $C_{\epsilon}$ should be understood as a positive
constant not only depending on $p,q,n$, and $M$, but also on $\epsilon$.
Throughout this paper, pairs of conjugate indices are written as
$p,p^{\prime}$, where $\frac{1}{p}+\frac{1}{p^{\prime}}=1$ with $1\leq
p\leq\infty$. We use $L^{p}_{\mu(r)}(\R_{+})$ to denote the usual $L^{p}$
space with the measure $\mathrm{d}\mu(r)=r^{n-1}\mathrm{d}r$.

This paper is organized as follows: In Section 2, we use the Hankel transform
and Bessel function to give the expression of the solution. Section 3 is
devoted to proving the key localized estimates of Hankel transforms. In the
final section, we use the estimates established in Section 3 to show Theorem
1.1.

Acknowledgments: The author would like to express his great gratitude to A.
Hassell for his helpful discussions and comments. He also would like to thank
the anonymous referee for careful reading the manuscript and for giving useful
comments. The author was partly supported by the Fundamental Research
Foundation of Beijing Institute of Technology (20111742015) and Beijing
Natural Science Foundation 1144014).

##  2\. Preliminary

In this section, we introduce a orthogonal decomposition of $L^{2}(\Sigma)$
associated with the eigenfunctions of $-\Delta_{h}+q(\theta)$. We provide some
standard facts about the Hankel transform and the Bessel functions. We
conclude this section by writing the solution of (1.1) as a linear combination
of products of radial functions and the eigenfunctions of
$-\Delta_{h}+q(\theta)$.

###  2.1. Orthogonal decomposition of $L^{2}(\Sigma)$

In this subsection, we decompose $L^{2}(\Sigma)$ into the subspaces spanned by
the eigenfunctions of $-\Delta_{h}+q(\theta)$ associated with its eigenvalues.
We consider the operator

(2.1) $H=-\Delta_{g}+\frac{q(\theta)}{r^{2}},$

on the metric cone $M=(0,\infty)_{r}\times\Sigma$. Here
$(r,\theta)\in\R_{+}\times\Sigma$ are some polar coordinates, $q(\theta)$ is a
real continuous function and the metric $g$ in coordinates
$(r,\theta)\in\R_{+}\times\Sigma$ is a metric of the form

$g=\mathrm{d}r^{2}+r^{2}h(\theta,\mathrm{d}\theta).$

The Riemannian metric $h$ on $\Sigma$ is independent of $r$. If $\Sigma$ has a
boundary, the Dirichlet condition will be used for $H$. Let $\Delta_{h}$ be
the Laplace-Beltrami operator on $(\Sigma,h)$. We will assume that

$-\Delta_{h}+q(\theta)\geq 0$

on $L^{2}(\Sigma)$, that is, for any $f\in L^{2}(\Sigma)$, we have

$\left\langle\big{(}-\Delta_{h}+q(\theta)\big{)}f,f\right\rangle_{L^{2}(\Sigma)}\geq
0.$

Then $H\geq 0$ in $L^{2}(M;\mathrm{d}g(z))$ with
$\mathrm{d}g(z)=\sqrt{|g|}\mathrm{d}z$. We modify $\chi_{\infty}$ by

(2.2)
$\chi_{\infty}=\Big{\\{}\nu:\nu=\sqrt{\lambda+(1/4)(n-2)^{2}};~{}\lambda~{}\text{is
eigenvalue of}-\widetilde{\Delta}_{h}:=-\Delta_{h}+q(\theta)\Big{\\}},$

and let

(2.3) $\chi_{K}=\chi_{\infty}\cap[0,K],\quad K\in\N.$

For $\nu\in\chi_{\infty}$, let $d(\nu)$ be the multiplicity of
$\lambda_{\nu}=\nu^{2}-\frac{1}{4}(n-2)^{2}$ as eigenvalue of
$-\Delta_{h}+q(\theta)$ and $\\{\varphi_{\nu,\ell}(\theta)\\}_{1\leq\ell\leq
d(\nu)}$ the eigenfunctions of $-\Delta_{h}+q(\theta)$, that is

(2.4)
$(-\Delta_{h}+q(\theta))\varphi_{\nu,\ell}=\lambda_{\nu}\varphi_{\nu,\ell},\quad\langle\varphi_{\nu,\ell},\varphi_{\nu,\ell^{\prime}}\rangle_{L^{2}(\Sigma)}=\delta_{\ell,\ell^{\prime}}.$

We remark that $\lambda_{\nu}\geq 0$ hence $\nu\geq(n-2)/2$. Define

$\mathcal{H}^{\nu}=\text{span}\\{\varphi_{\nu,1},\ldots,\varphi_{\nu,d(\nu)}\\},$

then we have the orthogonal decomposition

$L^{2}(\Sigma)=\bigoplus_{\nu\in\chi_{\infty}}\mathcal{H}^{\nu}.$

Let $\pi_{\nu}$ denote the orthogonal projection:

$\pi_{\nu}f=\sum_{\ell=1}^{d(\nu)}\varphi_{\nu,\ell}(\theta)\int_{\Sigma}f(r,\omega)\varphi_{\nu,\ell}(\omega)\mathrm{d}\sigma_{h},\quad
f\in L^{2}(M),$

where $\mathrm{d}\sigma_{h}$ is the measure on $\Sigma$ under the metric $h$.
For any $f\in L^{2}(M)$, we have the expansion formula

(2.5)
$f(z)=\sum_{\nu\in\chi_{\infty}}\pi_{\nu}f=\sum_{\nu\in\chi_{\infty}}\sum_{\ell=1}^{d(\nu)}a_{\nu,\ell}(r)\varphi_{\nu,\ell}(\theta)$

where
$a_{\nu,\ell}(r)=\int_{\Sigma}f(r,\theta)\varphi_{\nu,\ell}(\theta)\mathrm{d}\sigma_{h}$.
By orthogonality, it gives

(2.6)
$\|f(z)\|^{2}_{L^{2}(\Sigma)}=\sum_{\nu\in\chi_{\infty}}\sum_{\ell=1}^{d(\nu)}|a_{\nu,\ell}(r)|^{2}.$

We write $H$ on the cone expressed in polar coordinates as

(2.7)
$H=-\partial^{2}_{r}-\frac{n-1}{r}\partial_{r}+\frac{1}{r^{2}}\big{(}-\Delta_{h}+q(\theta)\big{)}$

and set

(2.8)
$\begin{split}A_{\nu}:=-\partial_{r}^{2}-\frac{n-1}{r}\partial_{r}+\frac{\nu^{2}-\left(\frac{n-2}{2}\right)^{2}}{r^{2}}\end{split}$

in $L^{2}_{\mu(r)}(\R_{+})$. In particular, taking $q(\theta)=a\geq 0$, we
also can consider the equation (1.1) perturbed by an inverse square potential.

###  2.2. The Bessel function and Hankel transform

For our purpose, we recall that the Bessel function $J_{\nu}(r)$ of order
$\nu$ is defined by

$J_{\nu}(r)=\frac{(r/2)^{\nu}}{\Gamma\left(\nu+\frac{1}{2}\right)\Gamma(1/2)}\int_{-1}^{1}e^{isr}(1-s^{2})^{(2\nu-1)/2}\mathrm{d}s,$

where $\nu>-\frac{1}{2}$ and $r>0$. A simple computation gives the rough
estimates

(2.9)
$|J_{\nu}(r)|\leq\frac{Cr^{\nu}}{2^{\nu}\Gamma\left(\nu+\frac{1}{2}\right)\Gamma(1/2)}\left(1+\frac{1}{\nu+1/2}\right),$

where $C$ is an absolute constant and the estimate will be mainly used when
$r\lesssim 1$. Another well known asymptotic expansion about the Bessel
function is

$J_{\nu}(r)=r^{-1/2}\sqrt{\frac{2}{\pi}}\cos(r-\frac{\nu\pi}{2}-\frac{\pi}{4})+O_{\nu}(r^{-3/2}),\quad\text{as}~{}r\rightarrow\infty$

but with a constant depending on $\nu$ (see [28]). As pointed out in [27], if
one seeks a uniform bound for large $r$ and $k$, then the best one can do is
$|J_{\nu}(r)|\leq Cr^{-\frac{1}{3}}$. To investigate the behavior of
asymptotic on $k$ and $r$, we recall Schläfli’s integral representation [37]
of the Bessel function: for $r\in\R^{+}$ and $\nu>-1/2$

(2.10)
$\begin{split}J_{\nu}(r)&=\frac{1}{2\pi}\int_{-\pi}^{\pi}e^{ir\sin\theta-i\nu\theta}\mathrm{d}\theta-\frac{\sin({\nu}\pi)}{\pi}\int_{0}^{\infty}e^{-(r\sinh
s+\nu s)}\mathrm{d}s\\\ &:=\tilde{J}_{\nu}(r)-E_{\nu}(r).\end{split}$

We remark that $E_{\nu}(r)=0$ when $\nu\in\Z^{+}$. A simple computation gives
that for $r>0$

(2.11)
$|E_{\nu}(r)|=\Big{|}\frac{\sin(\nu\pi)}{\pi}\int_{0}^{\infty}e^{-(r\sinh
s+\nu s)}\mathrm{d}s\Big{|}\leq C(r+\nu)^{-1}.$

Next, we recall the properties of Bessel function $J_{\nu}(r)$ in [27], and we
refer the readers to [19] for the detail proof.

######  Lemma 2.1 (Asymptotics of the Bessel function).

Assume $\nu\gg 1$. Let $J_{\nu}(r)$ be the Bessel function of order $\nu$
defined as above. Then there exist a large constant $C$ and a small constant
$c$ independent of $\nu$ and $r$ such that:

$\bullet$ when $r\leq\frac{\nu}{2}$

(2.12) $\begin{split}|J_{\nu}(r)|\leq Ce^{-c(\nu+r)};\end{split}$

$\bullet$ when $\frac{\nu}{2}\leq r\leq 2\nu$

(2.13) $\begin{split}|J_{\nu}(r)|\leq
C\nu^{-\frac{1}{3}}(\nu^{-\frac{1}{3}}|r-\nu|+1)^{-\frac{1}{4}};\end{split}$

$\bullet$ when $r\geq 2\nu$

(2.14) $\begin{split}J_{\nu}(r)=r^{-\frac{1}{2}}\sum_{\pm}a_{\pm}(r,\nu)e^{\pm
ir}+E(r,\nu),\end{split}$

where $|a_{\pm}(r,\nu)|\leq C$ and $|E(r,\nu)|\leq Cr^{-1}$.

Let $f\in L^{2}(M)$, we define the Hankel transform of order $\nu$ by

(2.15)
$(\mathcal{H}_{\nu}f)(\rho,\theta)=\int_{0}^{\infty}(r\rho)^{-\frac{n-2}{2}}J_{\nu}(r\rho)f(r,\theta)r^{n-1}\mathrm{d}r.$

As in [2, 21], we have the following properties of the Hankel transform. We
also refer the readers to M. Taylor [30, Chapter 9].

######  Lemma 2.2.

Let $\mathcal{H}_{\nu}$ and $A_{\nu}$ be defined as above. Then

$(\rm{i})$ $\mathcal{H}_{\nu}=\mathcal{H}_{\nu}^{-1}$,

$(\rm{ii})$ $\mathcal{H}_{\nu}$ is self-adjoint, i.e.
$\mathcal{H}_{\nu}=\mathcal{H}_{\nu}^{*}$,

$(\rm{iii})$ $\mathcal{H}_{\nu}$ is an $L^{2}$ isometry, i.e.
$\|\mathcal{H}_{\nu}\phi\|_{L^{2}(M)}=\|\phi\|_{L^{2}(M)}$,

$(\rm{iv})$
$\mathcal{H}_{\nu}(A_{\nu}\phi)(\rho,\theta)=\rho^{2}(\mathcal{H}_{\nu}\phi)(\rho,\theta)$,
for $\phi\in L^{2}$.

###  2.3. The expression of the solution.

Consider the following Cauchy problem:

(2.16) $\begin{cases}i\partial_{t}u+Hu=0,\\\ u(0,z)=u_{0}(z).\end{cases}$

By (2.5), we have the expansion

$u_{0}(z)=\sum_{\nu\in\chi_{\infty}}\sum_{\ell=1}^{d(\nu)}a_{\nu,\ell}(r)\varphi_{\nu,\ell}(\theta).$

Let us consider the equation (2.16) in polar coordinates $(r,\theta)$. Write
$v(t,r,\theta)=u(t,z)$ and $g(r,\theta)=u_{0}(z)$. Then $v(t,r,\theta)$
satisfies that

(2.17)
$\begin{cases}i\partial_{t}v-\partial_{rr}v-\frac{n-1}{r}\partial_{r}v-\frac{1}{r^{2}}\Delta_{h}v+\frac{q(\theta)}{r^{2}}v=0\\\
v(0,r,\theta)=g(r,\theta),\end{cases}$

where

$g(r,\theta)=\sum_{\nu\in\chi_{\infty}}\sum_{\ell=1}^{d(\nu)}a_{\nu,\ell}(r)\varphi_{\nu,\ell}(\theta).$

Using separation of variables, we can write $v$ as a linear combination of
products of functions and eigenfunctions

(2.18)
$v(t,r,\theta)=\sum_{\nu\in\chi_{\infty}}\sum_{\ell=1}^{d(\nu)}v_{\nu,\ell}(t,r)\varphi_{\nu,\ell}(\theta),$

where $v_{\nu,\ell}$ is given by

$\begin{cases}i\partial_{t}v_{\nu,\ell}-\partial_{rr}v_{\nu,\ell}-\frac{n-1}{r}\partial_{r}v_{\nu,\ell}+\frac{\lambda_{\nu}}{r^{2}}v_{\nu,\ell}=0,\\\
v_{\nu,\ell}(0,r)=a_{\nu,\ell}(r)\end{cases}$

for each $\nu\in\chi_{\infty}$ and $1\leq\ell\leq d(\nu)$. Recall $A_{\nu}$
defined in (2.8), then it reduces to consider

(2.19) $\begin{cases}i\partial_{t}v_{\nu,\ell}+A_{\nu}v_{\nu,\ell}=0,\\\
v_{\nu,\ell}(0,r)=a_{\nu,\ell}(r).\end{cases}$

Applying the Hankel transform to the equation (2.19), we have by $(\rm{iv})$
in Lemma 2.2

(2.20)
$\begin{cases}i\partial_{t}\tilde{v}_{\nu,\ell}+\rho^{2}\tilde{v}_{\nu,\ell}=0\\\
\tilde{v}_{\nu,\ell}(0,\rho)=b_{\nu,\ell}(\rho),\end{cases}$

where

(2.21)
$\tilde{v}_{\nu,\ell}(t,\rho)=(\mathcal{H}_{\nu}v_{\nu,\ell})(t,\rho),\quad
b_{\nu,\ell}(\rho)=(\mathcal{H}_{\nu}a_{\nu,\ell})(\rho).$

Solving this ODE and inverting the Hankel transform, we obtain

$\begin{split}v_{\nu,\ell}(t,r)&=\int_{0}^{\infty}(r\rho)^{-\frac{n-2}{2}}J_{\nu}(r\rho)\tilde{v}_{\nu,\ell}(t,\rho)\rho^{n-1}\mathrm{d}\rho\\\
&=\int_{0}^{\infty}(r\rho)^{-\frac{n-2}{2}}J_{\nu}(r\rho)e^{it\rho^{2}}b_{\nu,\ell}(\rho)\rho^{n-1}\mathrm{d}\rho.\end{split}$

Therefore we get

(2.22) $\begin{split}&u(t,z)=e^{itH}u_{0}=v(t,r,\theta)\\\
&=\sum_{\nu\in\chi_{\infty}}\sum_{\ell=1}^{d(\nu)}\varphi_{\nu,\ell}(\theta)\int_{0}^{\infty}(r\rho)^{-\frac{n-2}{2}}J_{\nu}(r\rho)e^{it\rho^{2}}b_{\nu,\ell}(\rho)\rho^{n-1}\mathrm{d}\rho\\\
&=\sum_{\nu\in\chi_{\infty}}\sum_{\ell=1}^{d(\nu)}\varphi_{\nu,\ell}(\theta)\mathcal{H}_{\nu}\big{[}e^{it\rho^{2}}b_{\nu,\ell}(\rho)\big{]}(r).\end{split}$

##  3\. Localized estimates of Hankel transforms

To prove Theorem 1.1, we need the following linear localized estimates. As
mentioned in the introduction, we need develop the decay of the Bessel
function and explore the oscillation both in $e^{it\rho^{2}}$ and the Bessel
function to prove these localized estimates. Since these estimates take the
same form for radial case and general case, we use the notation $\chi_{K}$ for
finite $K$ or $K=\infty$ to treat the cases together in the following proof.

######  Proposition 3.1.

Let $\beta\in C_{c}^{\infty}(\R)$ supported in $I:=[1,2]$ and $R>0$ be a
dyadic number. Then the following linear restriction estimates hold:

$\bullet$ for $q=2$,

(3.1)
$\begin{split}\big{\|}\sum_{\nu\in\chi_{K}}\sum_{\ell=1}^{d(\nu)}\varphi_{\nu,\ell}(\theta)&\mathcal{H}_{\nu}\big{[}e^{it\rho^{2}}b_{\nu,\ell}(\rho)\beta(\rho)\big{]}(r)\big{\|}_{L^{2}_{t}(\R;L^{2}_{\mu(r)}([R,2R];L^{2}_{\theta}(\Sigma)))}\\\
&\lesssim\min\left\\{R^{\frac{1}{2}},R^{\frac{n}{2}}\right\\}\Big{\|}\Big{(}\sum_{\nu\in\chi_{K}}\sum_{\ell=1}^{d(\nu)}|b_{\nu,\ell}(\rho)|^{2}\Big{)}^{\frac{1}{2}}\beta(\rho)\Big{\|}_{L^{2}_{\mu(\rho)}};\end{split}$

$\bullet$ for $q=\infty$,

(3.2)
$\begin{split}\big{\|}\sum_{\nu\in\chi_{K}}\sum_{\ell=1}^{d(\nu)}\varphi_{\nu,\ell}(\theta)&\mathcal{H}_{\nu}\big{[}e^{it\rho^{2}}b_{\nu,\ell}(\rho)\beta(\rho)\big{]}(r)\big{\|}_{L^{\infty}_{t}(\R;L^{\infty}_{\mu(r)}([R,2R];L^{2}_{\theta}(\Sigma)))}\\\
&\lesssim\min\left\\{R^{-\frac{n-1}{2}},1\right\\}\Big{\|}\Big{(}\sum_{\nu\in\chi_{K}}\sum_{\ell=1}^{d(\nu)}(1+\nu)^{\frac{1}{3}}|b_{\nu,\ell}(\rho)|^{2}\Big{)}^{\frac{1}{2}}\beta(\rho)\Big{\|}_{L^{1}_{\mu(\rho)}};\end{split}$

and

(3.3)
$\begin{split}\big{\|}\sum_{\nu\in\chi_{K}}\sum_{\ell=1}^{d(\nu)}\varphi_{\nu,\ell}(\theta)&\mathcal{H}_{\nu}\big{[}e^{it\rho^{2}}b_{\nu,\ell}(\rho)\beta(\rho)\big{]}(r)\big{\|}_{L^{\infty}_{t}(\R;L^{\infty}_{\mu(r)}([R,2R];L^{2}_{\theta}(\Sigma)))}\\\
&\lesssim\min\left\\{R^{-\frac{n-1}{2}},1\right\\}\Big{\|}\Big{(}\sum_{\nu\in\chi_{K}}\sum_{\ell=1}^{d(\nu)}|b_{\nu,\ell}(\rho)|^{2}\Big{)}^{\frac{1}{2}}\beta(\rho)\Big{\|}_{L^{2}_{\mu(\rho)}};\end{split}$

$\bullet$ for $q=3p^{\prime}$ and $2\leq p<4$,

(3.4)
$\begin{split}\big{\|}\sum_{\nu\in\chi_{K}}\sum_{\ell=1}^{d(\nu)}&\varphi_{\nu,\ell}(\theta)\mathcal{H}_{\nu}\big{[}e^{it\rho^{2}}b_{\nu,\ell}(\rho)\beta(\rho)\big{]}(r)\big{\|}_{L^{q}_{t}(\R;L^{q}_{\mu(r)}([R,2R];L^{2}_{\theta}(\Sigma)))}\\\
&\lesssim\min\left\\{R^{(n-1)(\frac{1}{q}-\frac{1}{2})},R^{\frac{n}{q}}\right\\}\Big{\|}\Big{(}\sum_{\nu\in\chi_{K}}\sum_{\ell=1}^{d(\nu)}(1+\nu)^{\frac{4}{q}}|b_{\nu,\ell}(\rho)|^{2}\Big{)}^{\frac{1}{2}}\beta(\rho)\Big{\|}_{L^{p}_{\mu(\rho)}};\end{split}$

and $1\leq p<2$

(3.5)
$\begin{split}\big{\|}\sum_{\nu\in\chi_{K}}\sum_{\ell=1}^{d(\nu)}&\varphi_{\nu,\ell}(\theta)\mathcal{H}_{\nu}\big{[}e^{it\rho^{2}}b_{\nu,\ell}(\rho)\beta(\rho)\big{]}(r)\big{\|}_{L^{q}_{t}(\R;L^{q}_{\mu(r)}([R,2R];L^{2}_{\theta}(\Sigma)))}\\\
&\lesssim\min\left\\{R^{(n-1)(\frac{1}{q}-\frac{1}{2})},R^{\frac{n}{q}}\right\\}\Big{\|}\Big{(}\sum_{\nu\in\chi_{K}}\sum_{\ell=1}^{d(\nu)}(1+\nu)^{\frac{2}{q}+\frac{1}{3}}|b_{\nu,\ell}(\rho)|^{2}\Big{)}^{\frac{1}{2}}\beta(\rho)\Big{\|}_{L^{p}_{\mu(\rho)}};\end{split}$

$\bullet$ for $q=4$ and $\forall\epsilon>0$

(3.6)
$\begin{split}\big{\|}\sum_{\nu\in\chi_{K}}\sum_{\ell=1}^{d(\nu)}&\varphi_{\nu,\ell}(\theta)\mathcal{H}_{\nu}\big{[}e^{it\rho^{2}}b_{\nu,\ell}(\rho)\beta(\rho)\big{]}(r)\big{\|}_{L^{4}_{t}(\R;L^{4}_{\mu(r)}([R,2R];L^{2}_{\theta}(\Sigma)))}\\\
&\lesssim\min\left\\{R^{-\frac{n-1}{4}+\epsilon},R^{\frac{n}{4}}\right\\}\Big{\|}\Big{(}\sum_{\nu\in\chi_{K}}\sum_{\ell=1}^{d(\nu)}(1+\nu)|b_{\nu,\ell}(\rho)|^{2}\Big{)}^{\frac{1}{2}}\beta(\rho)\Big{\|}_{L^{4}_{\mu(\rho)}}.\end{split}$

######  Remark 3.1.

The estimates above are essentially established by breaking things into
$R\lesssim 1$ and $R\gg 1$ due to the different asymptotic behavior of Bessel
function on each regime.

######  Remark 3.2.

The implicit constant is independent of $K$, which allows us to sum over all
of $\chi_{\infty}$ in next section. In other words, we can replace $\chi_{K}$
by $\chi_{\infty}$ in the above estimates. When the initial data is radial
Schwartz function, $K$ is finite hence the sum over $\ell$ and $\nu$
converges. If the initial data is a Schwartz function (not necessary radial),
$K$ may be infinite, and however the summation also converges due to the
Schwartz property. More precisely, since the initial data is Schwartz,
$b_{\nu,\ell}$ decays likely $(1+\nu)^{-N}$ for any $N>0$. On the other hand,
we note $d(\nu)\sim\nu^{n-2}$ hence the sum converges.

######  Remark 3.3.

The loss of angular regularity in (3.5) is much more than (3.4). We only use
(3.5) to conclude (1.5). By the radial assumption, one has that $K$ is finite
hence the loss of angular regularity is trivial.

The rest of this section is devoted to proving this Proposition. We first note
that by orthogonality of the angular eigenfunctions $\varphi_{\nu,\ell}$

(3.7)
$\begin{split}\big{\|}\sum_{\nu\in\chi_{K}}\sum_{\ell=1}^{d(\nu)}\varphi_{\nu,\ell}(\theta)&\mathcal{H}_{\nu}\big{[}e^{it\rho^{2}}b_{\nu,\ell}(\rho)\beta(\rho)\big{]}(r)\big{\|}_{L^{2}_{\theta}(\Sigma)}\\\
&=\left\\{\sum_{\nu\in\chi_{K}}\sum_{\ell=1}^{d(\nu)}\Big{|}\mathcal{H}_{\nu}\big{[}e^{it\rho^{2}}b_{\nu,\ell}(\rho)\beta(\rho)\big{]}(r)\Big{|}^{2}\right\\}^{1/2}\\\
&=r^{-\frac{n-2}{2}}\left(\sum_{\nu\in\chi_{K}}\sum_{\ell=1}^{d(\nu)}\big{|}\int_{0}^{\infty}e^{it\rho^{2}}J_{\nu}(r\rho)b_{\nu,\ell}(\rho)\beta(\rho)\rho^{n/2}\mathrm{d}\rho\big{|}^{2}\right)^{\frac{1}{2}}.\end{split}$

Now we prove (3.1)-(3.6) hold for $R\lesssim 1$. To do this, we need the
following Lemma.

######  Lemma 3.1.

Let $b_{\nu,\ell}(\rho)$ and $\beta(\rho)$ be as in Proposition 3.1, then the
following estimate holds for $q\geq 2$ and $R\lesssim 1$

(3.8)
$\begin{split}\Big{\|}r^{-\frac{n-2}{2}}\Big{(}\sum_{\nu\in\chi_{K}}\sum_{\ell=1}^{d(\nu)}\big{|}\int_{0}^{\infty}e^{it\rho^{2}}J_{\nu}(r\rho)&b_{\nu,\ell}(\rho)\beta(\rho)\rho^{\frac{n}{2}}\mathrm{d}\rho\big{|}^{2}\Big{)}^{\frac{1}{2}}\Big{\|}_{L^{q}_{t}(\R;L^{q}_{\mu(r)}[R,2R])}\\\
&\lesssim
R^{\frac{n}{q}}\Big{\|}\Big{(}\sum_{\nu\in\chi_{K}}\sum_{\ell=1}^{d(\nu)}|b_{\nu,\ell}(\rho)|^{2}\Big{)}^{\frac{1}{2}}\beta(\rho)\Big{\|}_{L^{q^{\prime}}_{\mu(\rho)}}.\end{split}$

We postpone the proof for a moment. Notice the $\nu$-weights appearing in
(3.2), (3.4)-(3.6) are larger than $1$, and note $q^{\prime}\leq p$ and
compact support of $\beta$, we use the Hölder inequality and Lemma 3.1 to show
Proposition 3.1 holds for $R\lesssim 1$.

###### Proof of Lemma 3.1.

Since $q\geq 2$, the Minkowski inequality and Fubini’s theorem show that the
left hand side of (3.8) is bounded by

$\begin{split}\left\|r^{-\frac{n-2}{2}}\left(\sum_{\nu\in\chi_{K}}\sum_{\ell=1}^{d(\nu)}\Big{\|}\int_{0}^{\infty}e^{it\rho^{2}}J_{\nu}(r\rho)b_{\nu,\ell}(\rho)\beta(\rho)\rho^{\frac{n-2}{2}}\rho\mathrm{d}\rho\Big{\|}^{2}_{L^{q}_{t}(\R)}\right)^{\frac{1}{2}}\right\|_{L^{q}_{\mu(r)}([R,2R])}.\end{split}$

We write by making variable changes

(3.9)
$\begin{split}\left\|r^{-\frac{n-2}{2}}\left(\sum_{\nu\in\chi_{K}}\sum_{\ell=1}^{d(\nu)}\Big{\|}\int_{0}^{\infty}e^{it\rho}J_{\nu}(r\sqrt{\rho})b_{\nu,\ell}(\sqrt{\rho})\beta(\sqrt{\rho})\rho^{\frac{n-2}{4}}\mathrm{d}\rho\Big{\|}^{2}_{L^{q}_{t}(\R)}\right)^{\frac{1}{2}}\right\|_{L^{q}_{\mu(r)}([R,2R])}.\end{split}$

Hence we use the Hausdorff-Young inequality in $t$ and change variables back
to obtain

$\begin{split}\text{LHS of
}~{}\eqref{3.7}\lesssim\Big{\|}r^{-\frac{n-2}{2}}\Big{(}\sum_{\nu\in\chi_{K}}\sum_{\ell=1}^{d(\nu)}\big{\|}J_{\nu}(r\rho)b_{\nu,\ell}(\rho)\beta(\rho)\rho^{(n-2)/2+1/q^{\prime}}\big{\|}_{L^{q^{\prime}}_{\rho}}^{2}\Big{)}^{\frac{1}{2}}\Big{\|}_{L^{q}_{\mu(r)}([R,2R])}.\end{split}$

Note the compact support of $\beta$, we obtain by (2.9)

$\begin{split}&\text{LHS of}~{}\eqref{3.7}\\\
\lesssim&\left(\int_{R}^{2R}r^{-\frac{(n-2)q}{2}}\Big{(}\sum_{\nu\in\chi_{K}}\sum_{\ell=1}^{d(\nu)}\Big{|}\frac{(4r)^{\nu}}{2^{\nu}\Gamma(\nu+\frac{1}{2})\Gamma(1/2)}\Big{|}^{2}\big{\|}b_{\nu,\ell}(\rho)\beta(\rho)\big{\|}_{L^{q^{\prime}}_{\rho}}^{2}\Big{)}^{\frac{q}{2}}r^{n-1}\mathrm{d}r\right)^{\frac{1}{q}}.\end{split}$

Note the stirling’s formula
$\Gamma\left(\nu+1\right)\sim\sqrt{\nu}(\nu/e)^{\nu}$, we see the coefficient
is bounded independent of $\nu$. On the other hand, we have the factor
$R^{n/q}R^{\sqrt{\lambda_{0}+(n-2)^{2}/4}-(n-2)/2}$ where $\lambda_{0}\geq 0$
is the smallest eigenvalue of $-\Delta_{h}+q(\theta)$. Note compact support of
$\beta$, thus we can adjust the weight in $\rho$ to prove (3.8). ∎

######  Remark 3.4.

It might help to given an example to show how this works. If $h=(d\theta)^{2}$
is the Euclidean metric on the sphere $\mathbb{S}^{n-1}$, $n\geq 2$ and
$q(\theta)=0$, then we have for $H=-\Delta$

$\chi_{\infty}=\left\\{(n-2)/2+k;k\in\mathbb{N}\right\\}.$

One can follow the above argument to show (3.8).

To prove Proposition 3.1, it suffices to prove the followings estimates: for
$R\gg 1$

$\bullet$ for $q=2$

(3.10)
$\begin{split}\Big{\|}r^{-\frac{n-2}{2}}\Big{(}\sum_{\nu\in\chi_{K}}\sum_{\ell=1}^{d(\nu)}\big{|}\int_{0}^{\infty}e^{it\rho^{2}}J_{\nu}(r\rho)&b_{\nu,\ell}(\rho)\beta(\rho)\rho^{\frac{n}{2}}\mathrm{d}\rho\big{|}^{2}\Big{)}^{\frac{1}{2}}\Big{\|}_{L^{2}_{t}(\R;L^{2}_{\mu(r)}[R,2R])}\\\
&\lesssim
R^{\frac{1}{2}}\big{(}\sum_{\nu\in\chi_{K}}\sum_{\ell=1}^{d(\nu)}\|b_{\nu,\ell}(\rho)\beta(\rho)\|^{2}_{L^{2}_{\mu(\rho)}}\big{)}^{\frac{1}{2}};\end{split}$

$\bullet$ for $q=\infty$

(3.11)
$\begin{split}\Big{\|}r^{-\frac{n-2}{2}}\Big{(}\sum_{\nu\in\chi_{K}}\sum_{\ell=1}^{d(\nu)}\big{|}\int_{0}^{\infty}&e^{it\rho^{2}}J_{\nu}(r\rho)b_{\nu,\ell}(\rho)\beta(\rho)\rho^{\frac{n}{2}}\mathrm{d}\rho\big{|}^{2}\Big{)}^{\frac{1}{2}}\Big{\|}_{L^{\infty}_{t}(\R;L^{\infty}_{\mu(r)}[R,2R])}\\\
&\lesssim
R^{-\frac{n-1}{2}}\Big{\|}\big{(}\sum_{\nu\in\chi_{K}}\sum_{\ell=1}^{d(\nu)}(1+\nu)^{\frac{1}{3}}\big{|}b_{\nu,\ell}(\rho)\big{|}^{2}\big{)}^{\frac{1}{2}}\beta(\rho)\Big{\|}_{L^{1}_{\mu(\rho)}};\end{split}$

and

(3.12)
$\begin{split}\Big{\|}r^{-\frac{n-2}{2}}\Big{(}\sum_{\nu\in\chi_{K}}\sum_{\ell=1}^{d(\nu)}\big{|}\int_{0}^{\infty}e^{it\rho^{2}}J_{\nu}(r\rho)&b_{\nu,\ell}(\rho)\beta(\rho)\rho^{\frac{n}{2}}\mathrm{d}\rho\big{|}^{2}\Big{)}^{\frac{1}{2}}\Big{\|}_{L^{\infty}_{t}(\R;L^{\infty}_{\mu(r)}[R,2R])}\\\
&\lesssim
R^{-\frac{n-1}{2}}\big{(}\sum_{\nu\in\chi_{K}}\sum_{\ell=1}^{d(\nu)}\|b_{\nu,\ell}(\rho)\beta(\rho)\|^{2}_{L^{2}_{\mu(\rho)}}\big{)}^{\frac{1}{2}};\end{split}$

$\bullet$ for $q=3p^{\prime}$ and $2\leq p<4$,

(3.13)
$\begin{split}\Big{\|}r^{-\frac{n-2}{2}}\Big{(}\sum_{\nu\in\chi_{K}}\sum_{\ell=1}^{d(\nu)}\big{|}\int_{0}^{\infty}&e^{it\rho^{2}}J_{\nu}(r\rho)b_{\nu,\ell}(\rho)\beta(\rho)\rho^{\frac{n}{2}}\mathrm{d}\rho\big{|}^{2}\Big{)}^{\frac{1}{2}}\Big{\|}_{L^{q}_{t}(\R;L^{q}_{\mu(r)}[R,2R])}\\\
&\lesssim
R^{(n-1)(\frac{1}{q}-\frac{1}{2})}\Big{\|}\big{(}\sum_{\nu\in\chi_{K}}\sum_{\ell=1}^{d(\nu)}(1+\nu)^{\frac{4}{q}}\big{|}b_{\nu,\ell}(\rho)\big{|}^{2}\big{)}^{\frac{1}{2}}\beta(\rho)\Big{\|}_{L^{p}_{\mu(\rho)}};\end{split}$

and $1\leq p<2$

(3.14)
$\begin{split}\Big{\|}r^{-\frac{n-2}{2}}\Big{(}\sum_{\nu\in\chi_{K}}\sum_{\ell=1}^{d(\nu)}\big{|}\int_{0}^{\infty}&e^{it\rho^{2}}J_{\nu}(r\rho)b_{\nu,\ell}(\rho)\beta(\rho)\rho^{\frac{n}{2}}\mathrm{d}\rho\big{|}^{2}\Big{)}^{\frac{1}{2}}\Big{\|}_{L^{q}_{t}(\R;L^{q}_{\mu(r)}[R,2R])}\\\
&\lesssim
R^{(n-1)(\frac{1}{q}-\frac{1}{2})}\Big{\|}\big{(}\sum_{\nu\in\chi_{K}}\sum_{\ell=1}^{d(\nu)}(1+\nu)^{{\frac{2}{q}+\frac{1}{3}}}\big{|}b_{\nu,\ell}(\rho)\big{|}^{2}\big{)}^{\frac{1}{2}}\beta(\rho)\Big{\|}_{L^{p}_{\mu(\rho)}};\end{split}$

$\bullet$ for $q=4$, $\forall\epsilon>0$

(3.15)
$\begin{split}\Big{\|}r^{-\frac{n-2}{2}}\Big{(}\sum_{\nu\in\chi_{K}}\sum_{\ell=1}^{d(\nu)}\big{|}\int_{0}^{\infty}&e^{it\rho^{2}}J_{\nu}(r\rho)b_{\nu,\ell}(\rho)\beta(\rho)\rho^{\frac{n}{2}}\mathrm{d}\rho\big{|}^{2}\Big{)}^{\frac{1}{2}}\Big{\|}_{L^{4}_{t}(\R;L^{4}_{\mu(r)}[R,2R])}\\\
&\lesssim
R^{-\frac{n-1}{4}+\epsilon}\Big{\|}\big{(}\sum_{\nu\in\chi_{K}}\sum_{\ell=1}^{d(\nu)}(1+\nu)\big{|}b_{\nu,\ell}(\rho)\big{|}^{2}\big{)}^{\frac{1}{2}}\beta(\rho)\Big{\|}_{L^{4}_{\mu(\rho)}}.\end{split}$

Step 1. We first prove (3.10) holds for $R\gg 1$. After changing variables as
(3.9) and canceling some factors $r$, we use the Plancherel theorem in $t$ to
show

(3.16) $\begin{split}\text{LHS of }~{}\eqref{3.8}\lesssim
R^{\frac{1}{2}}\Big{\|}\left(\sum_{\nu\in\chi_{K}}\sum_{\ell=1}^{d(\nu)}\big{\|}J_{\nu}(r\rho)b_{\nu,\ell}(\rho)\beta(\rho)\rho^{(n-1)/2}\big{\|}^{2}_{L^{2}_{\rho}}\right)^{\frac{1}{2}}\Big{\|}_{L^{2}_{r}([R,2R])}.\end{split}$

Along with (3.16), it is easy to verify (3.10), if we could prove

(3.17) $\int_{R}^{2R}|J_{\nu}(r)|^{2}\mathrm{d}r\leq C,\quad R\gg 1,$

where the constant $C$ is independent of $\nu$ and $R$. To prove (3.17), we
write

$\begin{split}\int_{R}^{2R}|J_{\nu}(r)|^{2}\mathrm{d}r=\int_{I_{1}}|J_{\nu}(r)|^{2}\mathrm{d}r+\int_{I_{2}}|J_{\nu}(r)|^{2}\mathrm{d}r+\int_{I_{3}}|J_{\nu}(r)|^{2}\mathrm{d}r\end{split}$

where $I_{1}=[R,2R]\cap[0,\frac{\nu}{2}],I_{2}=[R,2R]\cap[\frac{\nu}{2},2\nu]$
and $I_{3}=[R,2R]\cap[2\nu,\infty]$. By using (2.12) and (2.14) in Lemma 2.1,
we have

(3.18) $\begin{split}\int_{I_{1}}|J_{\nu}(r)|^{2}\mathrm{d}r\leq
C\int_{I_{1}}e^{-cr}\mathrm{d}r\leq Ce^{-cR},\end{split}$

and

(3.19) $\begin{split}\int_{I_{3}}|J_{\nu}(r)|^{2}\mathrm{d}r\leq
C.\end{split}$

On the other hand, one has by (2.13)

$\begin{split}\int_{[\frac{\nu}{2},2\nu]}|J_{\nu}(r)|^{2}\mathrm{d}r&\leq
C\int_{[\frac{\nu}{2},2\nu]}\nu^{-\frac{2}{3}}(1+\nu^{-\frac{1}{3}}|r-\nu|)^{-\frac{1}{2}}\mathrm{d}r\leq
C.\end{split}$

Observing $[R,2R]\cap[\frac{\nu}{2},2\nu]=\emptyset$ unless $R\sim\nu$, we
obtain

(3.20) $\begin{split}\int_{I_{2}}|J_{\nu}(r)|^{2}\mathrm{d}r\leq
C.\end{split}$

This together with (3.18) and (3.19) yields (3.17). Hence we finally prove
(3.10).

Step 2. To prove (3.11) and (3.12) hold for $R\gg 1$, we utilize the
Schläfli’s integral representation of the Bessel function (2.10) to write
${J}_{\nu}(r\rho)={E}_{\nu}(r\rho)+\tilde{J}_{\nu}(r\rho)$. As before using
the Minkowski inequality and the Hausdorff-Young inequality in $t$, we have by
(2.11),

$\begin{split}\Big{\|}r^{-\frac{n-2}{2}}\Big{(}\sum_{\nu\in\chi_{K}}\sum_{\ell=1}^{d(\nu)}\big{|}\int_{0}^{\infty}e^{it\rho^{2}}&{E}_{\nu}(r\rho)b_{\nu,\ell}(\rho)\beta(\rho)\rho^{\frac{n}{2}}\mathrm{d}\rho\big{|}^{2}\Big{)}^{\frac{1}{2}}\Big{\|}_{L^{\infty}_{t}(\R;L^{\infty}_{\mu(r)}([R,2R]))}\\\
&\lesssim
R^{-\frac{n}{2}}\Big{\|}\big{(}\sum_{\nu\in\chi_{K}}\sum_{\ell=1}^{d(\nu)}\big{|}b_{\nu,\ell}(\rho)\big{|}^{2}\big{)}^{\frac{1}{2}}\beta(\rho)\Big{\|}_{L^{1}_{\mu(\rho)}}.\end{split}$

Thus it remains to prove (3.11) and (3.12) replacing ${J}_{\nu}$ by
$\tilde{J}_{\nu}$. We decompose $[-\pi,\pi]$ into three partitions as follows

$[-\pi,\pi]=I_{1}\cup I_{2}\cup I_{3}$

where

(3.21) $I_{1}=\\{\theta:|\theta|\leq\delta\\},\quad
I_{2}=[-\pi,-\frac{\pi}{2}-\delta]\cup[\frac{\pi}{2}+\delta,\pi],\quad
I_{3}=[-\pi,\pi]\setminus(I_{1}\cup I_{2}),$

with $0<\delta\ll 1$. We define

(3.22) $\Phi_{r,\nu}(\theta)=\sin\theta-\nu\theta/r,$

and $\chi_{\delta}(\theta)$ is a smooth function given by

$\chi_{\delta}(\theta)=\begin{cases}1,\quad\theta\in[-\delta,\delta];\\\
0,\quad\theta\not\in[-2\delta,2\delta].\end{cases}$

Then we divide $\tilde{J}_{\nu}(r)$ into three pieces and write

(3.23)
$\begin{split}\tilde{J}_{\nu}(r)&=\frac{1}{2\pi}\int_{-\pi}^{\pi}e^{ir\Phi_{r,\nu}(\theta)}\mathrm{d}\theta\\\
&=\frac{1}{2\pi}\Big{(}\int_{-\pi}^{\pi}e^{ir\Phi_{r,\nu}(\theta)}\chi_{\delta}(\theta)\mathrm{d}\theta+\int_{I_{2}}e^{ir\Phi_{r,\nu}(\theta)}\mathrm{d}\theta+\int_{I_{3}}e^{ir\Phi_{r,\nu}(\theta)}(1-\chi_{\delta}(\theta))\mathrm{d}\theta\Big{)}\\\
&=:\tilde{J}^{1}_{\nu}(r)+\tilde{J}^{2}_{\nu}(r)+\tilde{J}^{3}_{\nu}(r).\end{split}$

When $\theta\in I_{2}$, the function
$\Phi^{\prime}_{r,\nu}(\theta)=\cos\theta-\nu/r$ is monotonic in the intervals
$[-\pi,-\frac{\pi}{2}-\delta]$ and $[\frac{\pi}{2}+\delta,\pi]$ respectively
and satisfies that

$|\Phi^{\prime}_{r,\nu}(\theta)|\geq\nu/r+|\cos\theta|\geq\sin\delta.$

Then by [27, Proposition 2, Chapter VIII], we have the following estimate
uniformly in $\nu$

(3.24)
$\Big{|}\frac{1}{2\pi}\int_{I_{2}}e^{ir\Phi_{r,\nu}(\theta)}\mathrm{d}\theta\Big{|}\leq
c_{\delta}r^{-1}.$

When $\theta\in I_{3}$, then
$|\Phi^{\prime\prime}_{r,\nu}(\theta)|\geq\sin\delta$, we have by [27,
Proposition 2, Chapter VIII]

(3.25)
$\Big{|}\frac{1}{2\pi}\int_{I_{3}}e^{ir\Phi_{r,\nu}(\theta)}(1-\chi_{\delta}(\theta))\mathrm{d}\theta\Big{|}\leq
c_{\delta}r^{-1/2},$

uniformly in $\nu$. Using the similar arguments as above, it follows from
(3.24) and (3.25) that

(3.26)
$\begin{split}\Big{\|}r^{-\frac{n-2}{2}}\Big{(}\sum_{\nu\in\chi_{K}}\sum_{\ell=1}^{d(\nu)}\big{|}\int_{0}^{\infty}e^{it\rho^{2}}&\big{(}{\tilde{J}^{2}}_{\nu}(r\rho)+{\tilde{J}^{3}}_{\nu}(r\rho)\big{)}b_{\nu,\ell}(\rho)\beta(\rho)\rho^{\frac{n}{2}}\mathrm{d}\rho\big{|}^{2}\Big{)}^{\frac{1}{2}}\Big{\|}_{L^{\infty}_{t}(\R;L^{\infty}_{\mu(r)}([R,2R]))}\\\
&\lesssim
R^{-\frac{n-1}{2}}\Big{\|}\big{(}\sum_{\nu\in\chi_{K}}\sum_{\ell=1}^{d(\nu)}\big{|}b_{\nu,\ell}(\rho)\big{|}^{2}\big{)}^{\frac{1}{2}}\beta(\rho)\Big{\|}_{L^{1}_{\mu(\rho)}}.\end{split}$

By using Lemma 2.1, we see $|\tilde{J}^{1}_{\nu}(r)|\lesssim r^{-1/3}$ when
$r\sim\nu$. Then arguing as before, we have

(3.27)
$\begin{split}\Big{\|}r^{-\frac{n-2}{2}}\Big{(}\sum_{\nu\in\chi_{K}}\sum_{\ell=1}^{d(\nu)}\big{|}\int_{0}^{\infty}e^{it\rho^{2}}&{\tilde{J}^{1}}_{\nu}(r\rho)b_{\nu,\ell}(\rho)\beta(\rho)\rho^{\frac{n}{2}}\mathrm{d}\rho\big{|}^{2}\Big{)}^{\frac{1}{2}}\Big{\|}_{L^{\infty}_{t}(\R;L^{\infty}_{\mu(r)}([R,2R]))}\\\
&\lesssim
R^{-\frac{n-1}{2}}\Big{\|}\big{(}\sum_{\nu\in\chi_{K}}\sum_{\ell=1}^{d(\nu)}(1+\nu)^{\frac{1}{3}}\big{|}b_{\nu,\ell}(\rho)\big{|}^{2}\big{)}^{\frac{1}{2}}\beta(\rho)\Big{\|}_{L^{1}_{\mu(\rho)}}.\end{split}$

We here obtain more decay $r^{-\frac{1}{6}}$ from the loss of the angular
regularity $\nu^{1/6}$ when $r\sim\nu$. Therefore we prove (3.11). To prove
(3.12) concerning $\tilde{J}^{1}_{\nu}(\rho r)$ without loss of angular
regularity, we need to use effectively the oscillation of $e^{it\rho^{2}}$. We
write Fourier series of $b_{\nu,\ell}(\rho)$ as

$\begin{split}b_{\nu,\ell}(\rho)=\sum_{j}b_{\nu,\ell}^{j}e^{i\rho^{2}j}\quad\text{with}\quad
b_{\nu,\ell}^{j}=\frac{1}{4\pi}\int_{0}^{16}e^{-i\rho^{2}j}b_{\nu,\ell}(\rho)\rho\mathrm{d}\rho.\end{split}$

By the Plancherel theorem and the orthogonality, we remark that

(3.28)
$\begin{split}\big{(}\sum_{\nu\in\chi_{K}}\sum_{\ell=1}^{d(\nu)}\|b_{\nu,\ell}(\rho)\rho^{\frac{1}{2}}\|^{2}_{L^{2}_{\mu(\rho)}(I)}\big{)}^{\frac{1}{2}}\cong\big{(}\sum_{\nu\in\chi_{K}}\sum_{\ell=1}^{d(\nu)}\sum_{j}|b_{\nu,\ell}^{j}|^{2}\big{)}^{\frac{1}{2}}.\end{split}$

Thus it suffices to prove

(3.29)
$\begin{split}\Big{\|}r^{-\frac{n-2}{2}}\Big{(}\sum_{\nu\in\chi_{K}}\sum_{\ell=1}^{d(\nu)}\big{|}\int_{0}^{\infty}e^{it\rho^{2}}&{\tilde{J}^{1}}_{\nu}(r\rho)\sum_{j}b_{\nu,\ell}^{j}e^{i\rho^{2}j}\beta(\rho)\rho^{\frac{n}{2}}\mathrm{d}\rho\big{|}^{2}\Big{)}^{\frac{1}{2}}\Big{\|}_{L^{\infty}_{t}(\R;L^{\infty}_{\mu(r)}([R,2R]))}\\\
&\lesssim
R^{-\frac{n-1}{2}}\Big{\|}\big{(}\sum_{\nu\in\chi_{K}}\sum_{\ell=1}^{d(\nu)}\big{|}b_{\nu,\ell}(\rho)\big{|}^{2}\big{)}^{\frac{1}{2}}\beta(\rho)\Big{\|}_{L^{2}_{\mu(\rho)}}.\end{split}$

For simplicity, we define

(3.30)
$\begin{split}\psi_{t+\frac{j}{4}}^{\nu}(r)=\int_{0}^{\infty}e^{i(t+\frac{j}{4})\rho^{2}}\int_{\R}e^{i\rho
r\sin\theta-i\nu\theta}\chi_{\delta}(\theta)\mathrm{d}\theta\beta(\rho)\rho^{\frac{n}{2}}\mathrm{d}\rho.\end{split}$

Let $m=t+\frac{j}{4}$, then we write

(3.31) $\begin{split}\psi_{m}^{\nu}(r)&=\int_{\R^{2}}e^{i\rho(r\sin\theta+\rho
m)}e^{-i\nu\theta}\beta(\rho)\mathrm{d}\rho\chi_{\delta}(\theta)\mathrm{d}\theta.\end{split}$

For our purpose, we need to investigate the asymptotic behavior of the
function $\psi_{m}^{\nu}(r)$. To this end, we consider the following two
cases. Write the phase function

$\begin{split}\Phi_{r,m,\nu}(\rho,\theta)=m\rho^{2}+\rho
r\sin\theta-\nu\theta.\end{split}$

$\bullet$ Subcase $(a)$: $4R\leq|m|$. Since $R\geq 1$, then $|m|\geq 4$. Note
that $\rho\in[1/2,4]$, then the derivative of the phase function in $\rho$
satisfies

$\begin{split}|\partial_{\rho}\Phi_{r,m,\nu}(\rho,\theta)|=|r\sin\theta+2m\rho|\geq|m|-r|\sin\theta|\geq|m|/100,\end{split}$

by making use of $r\leq 2R\leq|m|$ and $|\theta|\leq 2\delta$. Integrating by
part in $\rho$ gives that

(3.32) $\begin{split}|\psi_{m}^{\nu}(r)|\leq
C_{\delta,N}(1+|m|)^{-N}.\end{split}$

Hence keeping in mind $m=t+\frac{j}{4}$, we have

$\begin{split}&\Big{\|}r^{-\frac{n-2}{2}}\Big{(}\sum_{\nu\in\chi_{K}}\sum_{\ell=1}^{d(\nu)}\big{|}\sum_{\\{j:4R\leq|t+\frac{j}{4}|\\}}b_{\nu,\ell}^{j}\psi_{t+\frac{j}{4}}^{\nu}(r)\big{|}^{2}\Big{)}^{\frac{1}{2}}\Big{\|}_{L^{\infty}_{t}(\R;L^{\infty}_{\mu(r)}([R,2R]))}\\\
&\leq
C_{\delta,N}R^{-N}\Big{\|}r^{-\frac{n-2}{2}}\Big{(}\sum_{\nu\in\chi_{K}}\sum_{\ell=1}^{d(\nu)}\Big{|}\sum_{\\{j:4R\leq|t+\frac{j}{4}|\\}}|b_{\nu,\ell}^{j}|\left(1+|t+\frac{j}{4}|\right)^{-N}\Big{|}^{2}\Big{)}^{\frac{1}{2}}\Big{\|}_{L^{\infty}_{t}(\R;L^{\infty}_{\mu(r)}([R,2R]))}.\end{split}$

By the Cauchy-Schwarz inequality and choosing $N$ large enough, the above is
bounded by

(3.33)
$\begin{split}&C_{\delta,N}R^{-N}\Big{\|}r^{-\frac{n-2}{2}}\Big{(}\sum_{\nu\in\chi_{K}}\sum_{\ell=1}^{d(\nu)}\sum_{j}|b_{\nu,\ell}^{j}|^{2}(1+|t+\frac{j}{4}|)^{-N}\Big{)}^{\frac{1}{2}}\Big{\|}_{L^{\infty}_{t}(\R;L^{\infty}_{\mu(r)}([R,2R]))}\\\
&\leq
C_{\delta,N}R^{-N}\Big{(}\sum_{\nu\in\chi_{K}}\sum_{\ell=1}^{d(\nu)}\sum_{j}|b_{\nu,\ell}^{j}|^{2}\Big{)}^{\frac{1}{2}}\lesssim
R^{-N}\Big{\|}\big{(}\sum_{\nu\in\chi_{K}}\sum_{\ell=1}^{d(\nu)}\big{|}b_{\nu,\ell}(\rho)\big{|}^{2}\big{)}^{\frac{1}{2}}\beta(\rho)\Big{\|}_{L^{2}_{\mu(\rho)}(I)}.\end{split}$

$\bullet$ Subcase $(b)$: $|m|<4R$. We recall that

$\begin{split}\psi_{m}^{\nu}(r)&=\frac{1}{2\pi}\int_{\R^{2}}e^{ir\tilde{\Phi}_{r,m,\nu}(\rho,\theta)}\beta(\rho)\chi_{\delta}(\theta)\mathrm{d}\rho\mathrm{d}\theta,\end{split}$

where $\tilde{\Phi}_{r,m,\nu}(\rho,\theta)=\Phi_{r,m,\nu}(\rho,\theta)/r$.
Then a direct computation yields

(3.34)
$\begin{split}\nabla_{\rho,\theta}\tilde{\Phi}_{r,m,\nu}=\Big{(}-2m\rho/r+\sin\theta,\rho\cos\theta-\nu/r\Big{)}\end{split}$

and

(3.35)
$\begin{split}\frac{\partial^{2}\tilde{\Phi}_{r,m,\nu}}{\partial(\rho,\theta)^{2}}=\begin{pmatrix}-2m/r,&\cos\theta\\\
\cos\theta,&\rho\sin\theta\end{pmatrix}.\end{split}$

Since $|\theta|<\delta\ll 1$ and $|m|<4R$, there exists a small constant $c>0$
which is independent of $r,m,\nu$ such that

$\begin{split}\Big{|}\det\Big{(}\frac{\partial^{2}\tilde{\Phi}_{r,m,\nu}}{\partial(\rho,\theta)^{2}}\Big{)}\Big{|}=\left|\frac{2m}{r}\rho\sin\theta-\cos^{2}\theta\right|\geq\cos^{2}\theta-4m|\sin\theta|/r\geq
c.\end{split}$

Then the modified phase function $\tilde{\Phi}_{r,m,\nu}(\rho,\theta)$ is non-
degenerate, the standard stationary phase argument gives that there exists a
constant $C>0$ which is independent of $r,m,\nu$ such that

(3.36) $\begin{split}|\psi_{m}^{\nu}(r)|\leq Cr^{-1}.\end{split}$

For fixed $t,R$, we define $A=\\{j\in\Z:|t+\frac{j}{4}|\leq 4R\\}$. It is easy
to see $\sharp A$ is $O(R)$. Thus it follows from (3.36) and the Cauchy-
Schwarz inequality that

(3.37)
$\begin{split}&\Big{\|}r^{-\frac{n-2}{2}}\Big{(}\sum_{\nu\in\chi_{K}}\sum_{\ell=1}^{d(\nu)}\big{|}\sum_{j\in
A}b_{\nu,\ell}^{j}\psi_{t+\frac{j}{4}}^{\nu}(r)\big{|}^{2}\Big{)}^{\frac{1}{2}}\Big{\|}_{L^{\infty}_{t}(\R;L^{\infty}_{\mu(r)}([R,2R]))}\\\
&\leq
C_{\delta,N}R^{-\frac{n-1}{2}}\Big{(}\sum_{\nu\in\chi_{K}}\sum_{\ell=1}^{d(\nu)}\sum_{j}|b_{\nu,\ell}^{j}|^{2}\Big{)}^{\frac{1}{2}}\lesssim
R^{-\frac{n-1}{2}}\Big{\|}\big{(}\sum_{\nu\in\chi_{K}}\sum_{\ell=1}^{d(\nu)}\big{|}b_{\nu,\ell}(\rho)\big{|}^{2}\big{)}^{\frac{1}{2}}\beta(\rho)\Big{\|}_{L^{2}_{\mu(\rho)}}.\end{split}$

Together with (3.33), this gives (3.29). Thus it proves (3.12).

Step 3. We prove (3.13) and (3.15), i.e. the case $q=3p^{\prime}$ and $2\leq
p\leq 4$. The (3.14) follows from the interpolation of (3.13) and (3.11). To
do so, we need to use the bilinear argument to explore the oscillation both in
$e^{it\rho^{2}}$ and the Bessel function $J_{\nu}(r\rho)$. For our purpose, we
have to use the complete asymptotic formula for the Bessel function [27, 37]
and verify the sum of the coefficient is absolutely convergent when
$\nu^{2}\ll r$. On the other hand the Hardy-Littlewood-Sobolev inequality
fails at $q=4$, we require the Whitney-type decomposition to overcome this
difficulty.

To prove (3.13) and (3.15), it suffices to prove: for $q=3p^{\prime}$ and
$2\leq p\leq 4$

(3.38)
$\begin{split}\Big{\|}\Big{(}\sum_{\nu\in\chi_{K}}\sum_{\ell=1}^{d(\nu)}\big{|}\int_{0}^{\infty}e^{it\rho^{2}}&J_{\nu}(r\rho)b_{\nu,\ell}(\rho)\beta(\rho)\rho^{\frac{n}{2}}\mathrm{d}\rho\big{|}^{2}\Big{)}^{\frac{1}{2}}\Big{\|}_{L^{q}_{t,r}(\R\times[R,2R])}\\\
&\lesssim
R^{-\frac{1}{2}+\epsilon_{q}}\Big{\|}\Big{(}\sum_{\nu\in\chi_{K}}\sum_{\ell=1}^{d(\nu)}(1+\nu)^{\frac{4}{q}}\big{|}b_{\nu,\ell}(\rho)\big{|}^{2}\Big{)}^{\frac{1}{2}}\beta(\rho)\Big{\|}_{L^{p}_{\mu(\rho)}(I)},\end{split}$

where $\epsilon_{q}=\epsilon$ if $q=4$ otherwise $\epsilon_{q}=0$.

$\bullet$ Case 1: $\nu\in\Omega_{1}:=\\{\nu\in\chi_{K}:R\ll\nu\\}$.

By the Minkowski inequality, (2.8) and the Hausdorff-Young inequality in $t$,
it shows that

(3.39)
$\begin{split}\Big{\|}\Big{(}\sum_{\nu\in\Omega_{1}}\sum_{\ell=1}^{d(\nu)}&\big{|}\int_{0}^{\infty}e^{it\rho^{2}}J_{\nu}(r\rho)b_{\nu,\ell}(\rho)\beta(\rho)\rho^{\frac{n}{2}}\mathrm{d}\rho\big{|}^{2}\Big{)}^{\frac{1}{2}}\Big{\|}_{L^{q}_{t,r}(\R\times[R,2R])}\\\
&\lesssim\Big{(}\sum_{\nu\in\Omega_{1}}\sum_{\ell=1}^{d(\nu)}\big{\|}J_{\nu}(r\rho)b_{\nu,\ell}(\rho)\rho^{\frac{n}{2}-{\frac{1}{q}}}\beta(\rho)\big{\|}_{L^{q^{\prime}}_{\rho}L^{q}_{r}([R,2R])}^{2}\Big{)}^{\frac{1}{2}}\\\
&\lesssim\Big{(}\sum_{\nu\in\chi_{K}}\sum_{\ell=1}^{d(\nu)}\big{\|}e^{-cr}b_{\nu,\ell}(\rho)\rho^{\frac{n}{2}-{\frac{1}{q}}}\beta(\rho)\big{\|}_{L^{q^{\prime}}_{\rho}L^{q}_{r}([R,2R])}^{2}\Big{)}^{\frac{1}{2}}\\\
&\lesssim
Ce^{-cR}\Big{\|}\big{(}\sum_{\nu\in\chi_{K}}\sum_{\ell=1}^{d(\nu)}\big{|}b_{\nu,\ell}(\rho)\big{|}^{2}\big{)}^{\frac{1}{2}}\beta(\rho)\Big{\|}_{L^{p}_{\mu(\rho)}(I)}.\end{split}$

$\bullet$ Case 2: $\nu\in\Omega_{2}:=\\{\nu\in\chi_{K}:\nu\lesssim
R\lesssim\nu^{2}\\}$.

By (3.10), we have by canceling some $r$-weights

(3.40)
$\begin{split}\Big{\|}\Big{(}\sum_{\nu\in\Omega_{2}}\sum_{\ell=1}^{d(\nu)}\big{|}\int_{0}^{\infty}e^{it\rho^{2}}&J_{\nu}(r\rho)b_{\nu,\ell}(\rho)\beta(\rho)\rho^{\frac{n}{2}}\mathrm{d}\rho\big{|}^{2}\Big{)}^{\frac{1}{2}}\Big{\|}_{L^{2}_{t,r}(\R\times[R,2R])}\\\
&\lesssim
R^{-\frac{1}{2}}\Big{\|}\Big{(}\sum_{\nu\in\chi_{K}}\sum_{\ell=1}^{d(\nu)}(1+\nu)^{2}\big{|}b_{\nu,\ell}(\rho)\big{|}^{2}\Big{)}^{\frac{1}{2}}\Big{\|}_{L^{2}_{\mu(\rho)}(I)}.\end{split}$

On the other hand, we obtain by (3.12)

$\begin{split}\Big{\|}\Big{(}\sum_{\nu\in\Omega_{2}}\sum_{\ell=1}^{d(\nu)}\big{|}\int_{0}^{\infty}e^{it\rho^{2}}J_{\nu}(r\rho)b_{\nu,\ell}(\rho)&\beta(\rho)\rho^{\frac{n}{2}}\mathrm{d}\rho\big{|}^{2}\Big{)}^{\frac{1}{2}}\Big{\|}_{L^{\infty}_{t,r}(\R\times[R,2R])}\\\
&\lesssim
R^{-\frac{1}{2}}\Big{\|}\big{(}\sum_{\nu\in\chi_{K}}\sum_{\ell=1}^{d(\nu)}\big{|}b_{\nu,\ell}(\rho)\big{|}^{2}\big{)}^{\frac{1}{2}}\beta(\rho)\Big{\|}_{L^{2}_{\mu(\rho)}(I)}.\end{split}$

Interpolating this with (3.40), we have

(3.41)
$\begin{split}\Big{\|}\Big{(}\sum_{\nu\in\Omega_{2}}\sum_{\ell=1}^{d(\nu)}\big{|}\int_{0}^{\infty}e^{it\rho^{2}}&J_{\nu}(r\rho)b_{\nu,\ell}(\rho)\beta(\rho)\rho^{\frac{n}{2}}\mathrm{d}\rho\big{|}^{2}\Big{)}^{\frac{1}{2}}\Big{\|}_{L^{q}_{t,r}(\R\times[R,2R])}\\\
&\lesssim
R^{-\frac{1}{2}}\Big{\|}\Big{(}\sum_{\nu\in\chi_{K}}\sum_{\ell=1}^{d(\nu)}(1+\nu)^{\frac{4}{q}}\big{|}b_{\nu,\ell}(\rho)\big{|}^{2}\Big{)}^{\frac{1}{2}}\Big{\|}_{L^{2}_{\mu(\rho)}(I)}.\end{split}$

$\bullet$ Case 3: $\nu\in\Omega_{3}:=\\{\nu\in\chi_{K}:\nu^{2}\ll R\\}$.

To prove (3.38) in this case, since the $\nu$-weight is large than $1$, it
suffices to show

(3.42)
$\begin{split}\Big{\|}\sum_{\nu\in\Omega_{3}}\sum_{\ell=1}^{d(\nu)}\big{|}\int_{I}e^{it\rho^{2}}b_{\nu,\ell}(\rho)&J_{\nu}(\rho
r)\beta(\rho)\rho^{\frac{n}{2}}{\mathrm{d}\rho}\big{|}^{2}\Big{\|}_{L^{\frac{q}{2}}_{t,r}(\R\times[R,2R])}\\\
&\lesssim
R^{-1+\epsilon_{q}}\Big{\|}\big{(}\sum_{\nu\in\chi_{K}}\sum_{\ell=1}^{d(\nu)}\big{|}b_{\nu,\ell}(\rho)\big{|}^{2}\big{)}^{\frac{1}{2}}\beta(\rho)\Big{\|}^{2}_{L^{p}_{\mu(\rho)}(I)}.\end{split}$

To this end, let $\widetilde{\beta}(\rho)=\beta(\rho)\rho^{\frac{n}{2}}$, we
rewrite

$\begin{split}&\sum_{\nu\in\Omega_{3}}\sum_{\ell=1}^{d(\nu)}\big{|}\int_{I}e^{it\rho^{2}}b_{\nu,\ell}(\rho)J_{\nu}(\rho
r)\beta(\rho)\rho^{\frac{n}{2}}{\mathrm{d}\rho}\big{|}^{2}\\\
&\qquad=\sum_{\nu\in\Omega_{3}}\sum_{\ell=1}^{d(\nu)}\int_{I}e^{it\rho_{1}^{2}}b_{\nu,\ell}(\rho_{1})J_{\nu}(\rho_{1}r)\widetilde{\beta}(\rho_{1}){\mathrm{d}\rho_{1}}\\\
&\qquad\qquad\qquad\qquad\qquad\qquad\int_{I}e^{-it\rho_{2}^{2}}\overline{b_{\nu,\ell}(\rho_{2})J_{\nu}(\rho_{2}r)}\widetilde{\beta}(\rho_{2}){\mathrm{d}\rho_{2}}.\end{split}$

$\bullet$ Subcase (a): $q=3p^{\prime}$ with $2\leq p<4$. Before proving
(3.42), we recall a complete asymptotic formula for the Bessel function [27,
37]. When $\nu$ is fixed, the complete asymptotic formula for $J_{\nu}(\rho
r)$, as $r\rightarrow\infty$, is

(3.43) $\begin{split}J_{\nu}(\rho r)\sim&\big{(}\rho
r\big{)}^{-\frac{1}{2}}\cos\big{(}\rho
r-\frac{\nu\pi}{2}-\frac{\pi}{4}\big{)}\sum_{m=0}^{\infty}(\rho
r)^{-2m}a_{m}(\nu)\\\ &+\big{(}\rho r\big{)}^{-\frac{1}{2}}\sin\big{(}\rho
r-\frac{\nu\pi}{2}-\frac{\pi}{4}\big{)}\sum_{m=0}^{\infty}(\rho
r)^{-2m-1}b_{m}(\nu)\end{split}$

where

$\begin{split}a_{m}(\nu)=\frac{(-1)^{m}\Gamma(\nu+\frac{1}{2}+2m)}{2^{2m}(2m)!\cdot\Gamma(\nu+\frac{1}{2}-2m)},~{}~{}b_{m}(\nu)=\frac{(-1)^{m}\Gamma(\nu+\frac{3}{2}+2m)}{2^{(2m+1)}(2m+1)!\cdot\Gamma(\nu-\frac{1}{2}-2m)}.\end{split}$

Now we aim to estimate

$\begin{split}&\sum_{\nu\in\Omega_{3}}\sum_{\ell=1}^{d(\nu)}\big{|}\int_{I}e^{it\rho^{2}}b_{\nu,\ell}(\rho)J_{\nu}(\rho
r)\beta(\rho)\rho^{\frac{n}{2}}{\mathrm{d}\rho}\big{|}^{2}\\\ &\sim
r^{-1}\sum_{\nu\in\Omega_{3}}e^{i\nu\pi}\sum_{\ell=1}^{d(\nu)}\sum_{m_{1}=0}^{\infty}\sum_{m_{2}=0}^{\infty}r^{-2(m_{1}+m_{2})}a_{m_{1}}(\nu)a_{m_{2}}(\nu)\\\
&\quad\int_{I\times
I}e^{it(\rho_{1}^{2}-\rho_{2}^{2})}b_{\nu,\ell}(\rho_{1})\overline{b_{\nu,\ell}(\rho_{2})}\widetilde{\beta}(\rho_{1})\widetilde{\beta}(\rho_{2})e^{-ir(\rho_{1}\pm\rho_{2})}\rho_{1}^{-2m_{1}-\frac{1}{2}}\rho_{2}^{-2m_{2}-\frac{1}{2}}{\mathrm{d}\rho_{1}}{\mathrm{d}\rho_{2}}\\\
&\qquad+\text{similar terms}.\end{split}$

Since the similar terms can be estimated by the same argument, we only
estimate

$\begin{split}&\Big{\|}r^{-1}\sum_{m_{1},m_{2}=0}^{\infty}(2\pi
r)^{-2(m_{1}+m_{2})}\int_{I\times
I}e^{it(\rho_{1}^{2}-\rho_{2}^{2})}e^{-ir(\rho_{1}\pm\rho_{2})}\\\
&\quad\sum_{\nu\in\Omega_{3}}\sum_{\ell=1}^{d(\nu)}e^{i\nu\pi}a_{m_{1}}(\nu)a_{m_{2}}(\nu)b_{\nu,\ell}(\rho_{1})\overline{b_{\nu,\ell}(\rho_{2})}\widetilde{\beta}(\rho_{1})\widetilde{\beta}(\rho_{2})\rho_{1}^{-2m_{1}-\frac{1}{2}}\rho_{2}^{-2m_{2}-\frac{1}{2}}{\mathrm{d}\rho_{1}}{\mathrm{d}\rho_{2}}\Big{\|}_{L^{\frac{q}{2}}_{t,r}(\R\times[R,2R])}.\end{split}$

Let

$s_{1}=\rho_{1}\pm\rho_{2},\quad s_{2}=\rho_{1}^{2}-\rho_{2}^{2}$

and $\Omega\subset\R\times\R$ be the image of $I\times I$ under such change of
variables. Then by changing variables, we need estimate

$\begin{split}\Big{\|}r^{-1}\sum_{m_{1},m_{2}=0}^{\infty}r^{-2(m_{1}+m_{2})}&\Big{(}\int_{\Omega}e^{i(ts_{2}+rs_{1})}\sum_{\nu\in\Omega_{3}}\sum_{\ell=1}^{d(\nu)}a_{m_{1}}(\nu)a_{m_{2}}(\nu)b_{\nu,\ell}(\rho_{1})\overline{b_{\nu,\ell}(\rho_{2})}\\\
&\times\frac{\widetilde{\beta}(\rho_{1})\widetilde{\beta}(\rho_{2})\rho_{1}^{-2m_{1}-\frac{1}{2}}\rho_{2}^{-2m_{2}-\frac{1}{2}}}{|\rho_{1}\pm\rho_{2}|}{\mathrm{d}s_{1}}{\mathrm{d}s_{2}}\Big{)}\Big{\|}_{L^{\frac{q}{2}}_{t,r}(\R\times[R,2R])}.\end{split}$

Since $q>4$, by the Hausdorff-Young inequality, it suffices to show

$\begin{split}&\sum_{m_{1},m_{2}=0}^{\infty}(2\pi
R)^{-2(m_{1}+m_{2})}\Big{\|}\sum_{\nu\in\Omega_{3}}\sum_{\ell=1}^{d(\nu)}a_{m_{1}}(\nu)a_{m_{2}}(\nu)b_{\nu,\ell}(\rho_{1})\overline{b_{\nu,\ell}(\rho_{2})}\\\
&\qquad\qquad\qquad\qquad\qquad\times\frac{\widetilde{\beta}(\rho_{1})\widetilde{\beta}(\rho_{2})\rho_{1}^{-2m_{1}-\frac{1}{2}}\rho_{2}^{-2m_{2}-\frac{1}{2}}}{|\rho_{1}\pm\rho_{2}|}\Big{\|}_{L^{\frac{q}{q-2}}_{s_{1},s_{2}}(\Omega)}\\\
&\qquad\qquad\qquad\qquad\qquad\lesssim\Big{\|}\big{(}\sum_{\nu\in\chi_{K}}\sum_{\ell=1}^{d(\nu)}\big{|}b_{\nu,\ell}(\rho)\big{|}^{2}\big{)}^{\frac{1}{2}}\rho^{\frac{n-1}{p}}\Big{\|}^{2}_{L^{p}_{\mu(\rho)}(I)}.\end{split}$

By changing variables back, it reduces to prove

$\begin{split}\sum_{m_{1},m_{2}=0}^{\infty}(2\pi
R)^{-2(m_{1}+m_{2})}&\Big{\|}\sum_{\nu\in\Omega_{3}}\sum_{\ell=1}^{d(\nu)}a_{m_{1}}(\nu)a_{m_{2}}(\nu)b_{\nu,\ell}(\rho_{1})\overline{b_{\nu,\ell}(\rho_{2})}\frac{\widetilde{\beta}(\rho_{1})\widetilde{\beta}(\rho_{2})}{|\rho_{1}\pm\rho_{2}|^{\frac{2}{q}}}\Big{\|}_{L^{\frac{q}{q-2}}_{\rho_{1},\rho_{2}}(I^{2})}\\\
&\qquad\qquad\qquad\lesssim\Big{\|}\big{(}\sum_{\nu\in\chi_{K}}\sum_{\ell=1}^{d(\nu)}\big{|}b_{\nu,\ell}(\rho)\big{|}^{2}\big{)}^{\frac{1}{2}}\rho^{\frac{n-1}{p}}\Big{\|}^{2}_{L^{p}_{\rho}(I)}.\end{split}$

Recalling

$a_{m}(\nu)=\frac{(-1)^{m}\Gamma(\nu+\frac{1}{2}+2m)}{2^{2m}(2m)!\cdot\Gamma(\nu+\frac{1}{2}-2m)},$

it gives that

$\sup_{\nu\in\Omega_{3}}|a_{m}(\nu)|=\frac{\Gamma(\sqrt{R}+\frac{1}{2}+2m)}{2^{2m}(2m)!\cdot\Gamma(\sqrt{R}+\frac{1}{2}-2m)}.$

On the other hand, we have the uniformly estimate

$\begin{split}\sum_{m=0}^{\infty}(2\pi
R)^{-2m}\frac{\Gamma(\sqrt{R}+\frac{1}{2}+2m)}{2^{2m}(2m)!\cdot\Gamma(\sqrt{R}+\frac{1}{2}-2m)}\leq
C.\end{split}$

Thus it suffices to prove

$\begin{split}\Big{\|}\sum_{\nu\in\chi_{K}}\sum_{\ell=1}^{d(\nu)}|b_{\nu,\ell}(\rho_{1})\overline{b_{\nu,\ell}(\rho_{2})}|\frac{\widetilde{\beta}(\rho_{1})\widetilde{\beta}(\rho_{2})}{|\rho_{1}\pm\rho_{2}|^{2/q}}\Big{\|}_{L^{\frac{q}{q-2}}_{\rho_{1},\rho_{2}}(I^{2})}\lesssim\Big{\|}\big{(}\sum_{\nu\in\chi_{K}}\sum_{\ell=1}^{d(\nu)}\big{|}b_{\nu,\ell}(\rho)\big{|}^{2}\big{)}^{\frac{1}{2}}\rho^{\frac{n-1}{p}}\Big{\|}^{2}_{L^{p}_{\mu(\rho)}(I)}.\end{split}$

Since $p>\frac{q}{q-2}$ and $|\rho_{1}+\rho_{2}|\geq 1$, the case concerning
$|\rho_{1}+\rho_{2}|$ is obvious to be proved. By the Cauchy-Schwarz
inequality, it is enough to prove

(3.44)
$\begin{split}\Big{\|}\int_{I}\Big{(}\sum_{\nu\in\chi_{K}}\sum_{\ell=1}^{d(\nu)}|b_{\nu,\ell}(\rho_{2})|^{2}\big{)}^{1/2}\frac{1}{|\rho_{1}-\rho_{2}|^{2/q}}&\Big{)}^{\frac{q}{q-2}}\mathrm{d}\rho_{2}\Big{\|}^{\frac{q-2}{q}}_{L^{[\frac{q-2}{q}-\frac{1}{p}]^{-1}\frac{q-2}{q}}_{\rho_{1}}(I)}\\\
&\lesssim\Big{\|}\big{(}\sum_{\nu\in\chi_{K}}\sum_{\ell=1}^{d(\nu)}\big{|}b_{\nu,\ell}(\rho)\big{|}^{2}\big{)}^{\frac{1}{2}}\Big{\|}_{L^{p}_{\rho}(I)}.\end{split}$

Since assuming $q=3p^{\prime}>4$, we have

$1+\frac{q}{q-2}(\frac{q-2}{q}-\frac{1}{p})=\frac{q}{q-2}\frac{2}{q}+\frac{1}{p}\frac{q}{q-2}.$

Then (3.44) follows from the Hardy-Littlewood-Sobolve inequality.

$\bullet$ Subcase (b): $q=4$ and $p=4$. In this subcase, the Hardy-Littlewood-
Sobolev inequality fails, we cannot use the above argument to prove (3.42). We
need a Whitney-type decomposition to $I$. Performing a Whitney decomposition
to $I$, for each $j\geq 0$, we break up $I$ into $O(2^{j})$ dyadic intervals
$Q_{\bar{k}}^{j}$ of length $2^{-j}$ and also define $Q_{\bar{k}}^{j}\simeq
Q_{\bar{k}^{\prime}}^{j}$ if they are cousins, i.e. $Q_{\bar{k}}^{j}$ and
$Q_{\bar{k}^{\prime}}^{j}$ are not adjacent but have adjacent parents. Then by
(2.17), we can write the above as the following decomposition

$\begin{split}\sum_{\nu\in\Omega_{3}}\sum_{\ell=1}^{d(\nu)}\sum_{j\geq
0}\sum_{\bar{k}}\sum_{\bar{k}^{\prime}:Q_{\bar{k}}^{j}\simeq
Q_{\bar{k}^{\prime}}^{j}}F_{\bar{k}}^{j}~{}\overline{G_{\bar{k}^{\prime}}^{j}},\end{split}$

where

$\begin{split}F_{\bar{k}}^{j}=F_{\bar{k}}^{j}(t,r)=\int_{Q_{\bar{k}}^{j}}e^{it\rho_{1}^{2}}b_{\nu,\ell}(\rho_{1})J_{\nu}(\rho_{1}r)\widetilde{\beta}(\rho_{1}){\mathrm{d}\rho_{1}},\end{split}$

and

$\begin{split}G_{\bar{k}^{\prime}}^{j}=G_{\bar{k}^{\prime}}^{j}(t,r)=\int_{Q_{{\bar{k}}^{\prime}}^{j}}e^{it\rho_{2}^{2}}b_{\nu,\ell}(\rho_{2})J_{\nu}(\rho_{2}r)\widetilde{\beta}(\rho_{2}){\mathrm{d}\rho_{2}}.\end{split}$

Thus by triangle inequality and $\rho\in[1,2]$, it suffices to prove

(3.45) $\begin{split}\sum_{j\geq\log
R}\Big{\|}\sum_{k\in\Omega_{3}}\sum_{\ell=1}^{d(\nu)}\sum_{\bar{k}}\sum_{\bar{k}^{\prime}:Q_{\bar{k}}^{j}\simeq
Q_{\bar{k}^{\prime}}^{j}}&F_{\bar{k}}^{j}~{}\overline{G_{\bar{k}^{\prime}}^{j}}\Big{\|}_{L^{2}_{t,r}(\R\times[R,2R])}\\\
&\lesssim
R^{-1}\Big{\|}\big{(}\sum_{\nu\in\chi_{K}}\sum_{\ell=1}^{d(\nu)}\big{|}b_{\nu,\ell}(\rho)\big{|}^{2}\big{)}^{\frac{1}{2}}\Big{\|}^{2}_{L^{4}_{\rho}(I)},\end{split}$

and

(3.46) $\begin{split}\sum_{j\leq\log
R}\Big{\|}\sum_{k\in\Omega_{3}}\sum_{\ell=1}^{d(\nu)}\sum_{\bar{k}}\sum_{\bar{k}^{\prime}:Q_{\bar{k}}^{j}\simeq
Q_{\bar{k}^{\prime}}^{j}}&F_{\bar{k}}^{j}~{}\overline{G_{\bar{k}^{\prime}}^{j}}\Big{\|}_{L^{2}_{t,r}(\R\times[R,2R])}\\\
&\lesssim
R^{-1+\epsilon}\Big{\|}\big{(}\sum_{\nu\in\chi_{K}}\sum_{\ell=1}^{d(\nu)}\big{|}b_{\nu,\ell}(\rho)\big{|}^{2}\big{)}^{\frac{1}{2}}\Big{\|}^{2}_{L^{4}_{\rho}(I)}.\end{split}$

Firstly, we prove (3.45). To this end, by the Cauchy-Schwarz inequality and
the triangle inequality, it follows

(3.47) $\begin{split}\text{LHS of }~{}\eqref{4.32}\lesssim\sum_{j\geq\log
R}\sum_{\bar{k}}\sum_{\bar{k}^{\prime}:Q_{\bar{k}}^{j}\simeq
Q_{\bar{k}^{\prime}}^{j}}&\Big{\|}\big{(}\sum_{\nu\in\Omega_{3}}\sum_{\ell=1}^{d(\nu)}|F_{\bar{k}}^{j}|^{2}\big{)}^{\frac{1}{2}}\Big{\|}_{L^{2}_{t,r}(\R\times[R,2R])}\\\
&\times\Big{\|}\big{(}\sum_{\nu\in\Omega_{3}}\sum_{\ell=1}^{d(\nu)}|G_{\bar{k}^{\prime}}^{j}|^{2}\big{)}^{\frac{1}{2}}\Big{\|}_{L^{\infty}_{t,r}(\R\times[R,2R])}.\end{split}$

By (2.14), the Minkowski inequality, Hölder’s inequality and the Hausdorff-
Young inequality in $t$, we have by arguing as before

(3.48)
$\begin{split}&\Big{\|}\big{(}\sum_{\nu\in\Omega_{3}}\sum_{\ell=1}^{d(\nu)}|G_{\bar{k}^{\prime}}^{j}|^{2}\big{)}^{\frac{1}{2}}\Big{\|}_{L^{\infty}_{t,r}(\R\times[R,2R])}\\\
&\quad\lesssim\Big{(}\sum_{\nu\in\Omega_{3}}\sum_{\ell=1}^{d(\nu)}\big{\|}\int_{Q_{{\bar{k}}^{\prime}}^{j}}e^{it\rho_{2}^{2}}b_{\nu,\ell}(\rho_{2})J_{\nu}(\rho_{2}r)\widetilde{\beta}(\rho_{2}){\mathrm{d}\rho_{2}}\big{\|}_{L^{\infty}_{t,r}(\R\times[R,2R])}^{2}\Big{)}^{\frac{1}{2}}\\\
&\quad\lesssim
R^{-\frac{1}{2}}\Big{(}\sum_{\nu\in\chi_{K}}\sum_{\ell=1}^{d(\nu)}\big{\|}b_{\nu,\ell}(\rho_{2})\widetilde{\beta}(\rho_{2})\big{\|}_{L^{1}_{\rho_{2}}(Q_{{\bar{k}}^{\prime}}^{j})}^{2}\Big{)}^{\frac{1}{2}}\\\
&\quad\lesssim
R^{-\frac{1}{2}}|Q_{{\bar{k}}^{\prime}}^{j}|^{\frac{1}{2}}\Big{\|}\Big{(}\sum_{\nu\in\chi_{K}}\sum_{\ell=1}^{d(\nu)}|b_{\nu,\ell}(\rho_{2})|^{2}\Big{)}^{\frac{1}{2}}\Big{\|}_{L^{2}_{\rho_{2}}(Q_{{\bar{k}}^{\prime}}^{j})},\end{split}$

where we make use of $\rho_{2}\in Q_{{\bar{k}}^{\prime}}^{j}\subset[1,2]$. On
the other hand, the Hausdorff-Young inequality in $t$ and similar argument as
before imply that

(3.49)
$\begin{split}&\Big{\|}\big{(}\sum_{\nu\in\Omega_{3}}\sum_{\ell=1}^{d(\nu)}|F_{\bar{k}}^{j}|^{2}\big{)}^{\frac{1}{2}}\Big{\|}_{L^{2}_{t,r}(\R\times[R,2R])}\\\
&\quad=\Big{(}\sum_{\nu\in\Omega_{3}}\sum_{\ell=1}^{d(\nu)}\Big{\|}\int_{Q_{\bar{k}}^{j}}e^{it\rho_{1}^{2}}b_{\nu,\ell}(\rho_{1})J_{\nu}(\rho_{1}r)\widetilde{\beta}(\rho_{1}){\mathrm{d}\rho_{1}}\Big{\|}_{L^{2}_{t,r}(\R\times[R,2R])}^{2}\Big{)}^{\frac{1}{2}}\\\
&\quad\lesssim\Big{\|}\Big{(}\sum_{\nu\in\chi_{K}}\sum_{\ell=1}^{d(\nu)}|b_{\nu,\ell}(\rho_{1})|^{2}\Big{)}^{\frac{1}{2}}\Big{\|}_{L^{2}_{\rho_{1}}(Q_{{\bar{k}}}^{j})}.\end{split}$

Together with (3.47) and (3.48), it gives

$\begin{split}\text{RHS of }~{}\eqref{4.34}\lesssim
R^{-\frac{1}{2}}\sum_{j\geq\log
R}2^{-\frac{j}{2}}\sum_{\bar{k}}\sum_{\bar{k}^{\prime}:Q_{\bar{k}}^{j}\simeq
Q_{\bar{k}^{\prime}}^{j}}&\Big{\|}\Big{(}\sum_{\nu\in\chi_{K}}\sum_{\ell=1}^{d(\nu)}|b_{\nu,\ell}(\rho_{1})|^{2}\Big{)}^{\frac{1}{2}}\Big{\|}_{L^{2}_{\rho_{1}}(Q_{{\bar{k}}}^{j})}\\\
&\Big{\|}\Big{(}\sum_{\nu\in\chi_{K}}\sum_{\ell=1}^{d(\nu)}|b_{\nu,\ell}(\rho_{2})|^{2}\Big{)}^{\frac{1}{2}}\Big{\|}_{L^{2}_{\rho_{2}}(Q_{{\bar{k}}^{\prime}}^{j})}.\end{split}$

Recalling the property of the Whitney decomposition that for each fixed
$\bar{k}$, there are only $O(1)$ cousins of $Q_{{\bar{k}}}^{j}$, then we have

$\begin{split}\text{RHS of }~{}\eqref{4.34}\lesssim
R^{-1}\Big{\|}\Big{(}\sum_{\nu\in\chi_{K}}\sum_{\ell=1}^{d(\nu)}|b_{\nu,\ell}(\rho_{2})|^{2}\Big{)}^{\frac{1}{2}}\Big{\|}^{2}_{L^{2}(I)}.\end{split}$

Thus we prove (3.45).

Now we prove (3.46) to complete the proof. Recalling (3.43) and the
definitions of $F_{\bar{k}}^{j}$ and $G_{\bar{k}^{\prime}}^{j}$, now we aim to
estimate

$\begin{split}&\sum_{\nu\in\Omega_{3}}\sum_{\ell=1}^{d(\nu)}\sum_{\bar{k}}\sum_{\bar{k}^{\prime}:Q_{\bar{k}}^{j}\simeq
Q_{\bar{k}^{\prime}}^{j}}F_{\bar{k}}^{j}~{}\overline{G_{\bar{k}^{\prime}}^{j}}\\\
&\sim r^{-1}\sum_{\bar{k}}\sum_{\bar{k}^{\prime}:Q_{\bar{k}}^{j}\simeq
Q_{\bar{k}^{\prime}}^{j}}\sum_{\nu\in\Omega_{3}}e^{i\nu\pi}\sum_{\ell=1}^{d(\nu)}\sum_{m_{1}=0}^{\infty}\sum_{m_{2}=0}^{\infty}r^{-2(m_{1}+m_{2})}a_{m_{1}}(k)a_{m_{2}}(k)\\\
&\quad\int_{Q_{\bar{k}}^{j}\times
Q_{\bar{k}^{\prime}}^{j}}e^{it(\rho_{1}^{2}-\rho_{2}^{2})}b_{\nu,\ell}(\rho_{1})\overline{b_{\nu,\ell}(\rho_{2})}\widetilde{\beta}(\rho_{1})\widetilde{\beta}(\rho_{2})e^{-ir(\rho_{1}\pm\rho_{2})}\rho_{1}^{-2m_{1}-\frac{1}{2}}\rho_{2}^{-2m_{2}-\frac{1}{2}}{\mathrm{d}\rho_{1}}{\mathrm{d}\rho_{2}}\\\
&\qquad+\text{similar terms}.\end{split}$

As before, since the similar terms can be estimated by the same argument, we
only consider

$\begin{split}&\Big{\|}r^{-1}\sum_{m_{1},m_{2}=0}^{\infty}r^{-2(m_{1}+m_{2})}\sum_{\bar{k}}\sum_{\bar{k}^{\prime}:Q_{\bar{k}}^{j}\simeq
Q_{\bar{k}^{\prime}}^{j}}\int_{Q_{\bar{k}}^{j}\times
Q_{\bar{k}^{\prime}}^{j}}e^{it(\rho_{1}^{2}-\rho_{2}^{2})}e^{-ir(\rho_{1}\pm\rho_{2})}\\\
&\quad\sum_{\nu\in\Omega_{3}}e^{i\nu\pi}\sum_{\ell=1}^{d(\nu)}a_{m_{1}}(k)a_{m_{2}}(k)b_{\nu,\ell}(\rho_{1})\overline{b_{\nu,\ell}(\rho_{2})}\widetilde{\beta}(\rho_{1})\widetilde{\beta}(\rho_{2})\rho_{1}^{-2m_{1}-\frac{1}{2}}\rho_{2}^{-2m_{2}-\frac{1}{2}}{\mathrm{d}\rho_{1}}{\mathrm{d}\rho_{2}}\Big{\|}_{L^{2}_{t,r}(\R\times[R,2R])}.\end{split}$

For this purpose, let $s_{1}=\rho_{1}\pm\rho_{2},\quad
s_{2}=\rho_{1}^{2}-\rho_{2}^{2}$ and
$\Omega_{\bar{k},\bar{k}^{\prime}}^{j}\subset\R\times\R$ be the image of
$Q_{\bar{k}}^{j}\times Q_{\bar{k}^{\prime}}^{j}$ under such change of
variables. Then we aim to estimate

$\begin{split}&\sum_{j\leq\log
R}\Big{\|}r^{-1}\sum_{m_{1},m_{2}=0}^{\infty}r^{-2(m_{1}+m_{2})}\sum_{\bar{k}}\sum_{\bar{k}^{\prime}:Q_{\bar{k}}^{j}\simeq
Q_{\bar{k}^{\prime}}^{j}}\Big{(}\int_{\Omega_{\bar{k},\bar{k}^{\prime}}^{j}}e^{i(ts_{2}+rs_{1})}\\\
&\times\sum_{\nu\in\Omega_{3}}e^{i\nu\pi}\sum_{\ell=1}^{d(k)}a_{m_{1}}(k)a_{m_{2}}(k)b_{\nu,\ell}(\rho_{1})\overline{b_{\nu,\ell}(\rho_{2})}\frac{\widetilde{\beta}(\rho_{1})\widetilde{\beta}(\rho_{2})\rho_{1}^{-2m_{1}-\frac{1}{2}}\rho_{2}^{-2m_{2}-\frac{1}{2}}}{|\rho_{1}\pm\rho_{2}|}{\mathrm{d}s_{1}}{\mathrm{d}s_{2}}\Big{)}\Big{\|}_{L^{2}_{t,r}(\R\times[R,2R])}.\end{split}$

To prove (3.42), by the Hausdorff-Young inequality and the quasi-
orthogonality(see [36, Lemma 6.1]), it suffices to establish

$\begin{split}\sum_{j\leq\log
R}\sum_{m_{1},m_{2}=0}^{\infty}R^{-2(m_{1}+m_{2})}&\Big{(}\sum_{\bar{k}}\sum_{\bar{k}^{\prime}:Q_{\bar{k}}^{j}\simeq
Q_{\bar{k}^{\prime}}^{j}}\Big{\|}\sum_{\nu\in\Omega_{3}}\sum_{\ell=1}^{d(\nu)}a_{m_{1}}(k)a_{m_{2}}(k)b_{\nu,\ell}(\rho_{1})\overline{b_{\nu,\ell}(\rho_{2})}\\\
&\qquad\times\frac{\widetilde{\beta}(\rho_{1})\widetilde{\beta}(\rho_{2})\rho_{1}^{-2m_{1}-\frac{1}{2}}\rho_{2}^{-2m_{2}-\frac{1}{2}}}{|\rho_{1}-\rho_{2}|}\Big{\|}^{2}_{L^{2}_{s_{1},s_{2}}(\Omega_{\bar{k},\bar{k}^{\prime}}^{j})}\Big{)}^{\frac{1}{2}}\\\
&\qquad\lesssim
R^{\epsilon}\Big{\|}\big{(}\sum_{\nu\in\chi_{K}}\sum_{\ell=1}^{d(\nu)}\big{|}b_{\nu,\ell}(\rho)\big{|}^{2}\big{)}^{\frac{1}{2}}\Big{\|}^{2}_{L^{4}_{\rho}(I)}.\end{split}$

By changing variables back, it reduces to prove

$\begin{split}&\sum_{j\leq\log
R}\sum_{m_{1},m_{2}=0}^{\infty}R^{-2(m_{1}+m_{2})}\\\
&\Big{(}\sum_{\bar{k}}\sum_{\bar{k}^{\prime}:Q_{\bar{k}}^{j}\simeq
Q_{\bar{k}^{\prime}}^{j}}\Big{\|}\sum_{\nu\in\Omega_{3}}\sum_{\ell=1}^{d(\nu)}a_{m_{1}}(k)a_{m_{2}}(k)b_{\nu,\ell}(\rho_{1})\overline{b_{\nu,\ell}(\rho_{2})}\frac{\widetilde{\beta}(\rho_{1})\widetilde{\beta}(\rho_{2})}{|\rho_{1}\pm\rho_{2}|^{\frac{1}{2}}}\Big{\|}^{2}_{L^{2}_{\rho_{1},\rho_{2}}(Q_{\bar{k}}^{j}\times
Q_{\bar{k}^{\prime}}^{j})}\Big{)}^{\frac{1}{2}}\\\
&\qquad\qquad\qquad\qquad\qquad\lesssim
R^{\epsilon}\Big{\|}\big{(}\sum_{\nu\in\chi_{K}}\sum_{\ell=1}^{d(\nu)}\big{|}b_{\nu,\ell}(\rho)\big{|}^{2}\big{)}^{\frac{1}{2}}\Big{\|}^{2}_{L^{4}_{\rho}(I)}.\end{split}$

As before, we also have the uniformly estimate

$\begin{split}\sum_{m=0}^{\infty}R^{-2m}\frac{\Gamma(\sqrt{R}+\frac{1}{2}+2m)}{2^{2m}(2m)!\cdot\Gamma(\sqrt{R}+\frac{1}{2}-2m)}\leq
C.\end{split}$

Thus it suffices to prove

$\begin{split}&\sum_{j\leq\log
R}\Big{(}\sum_{\bar{k}}\sum_{\bar{k}^{\prime}:Q_{\bar{k}}^{j}\simeq
Q_{\bar{k}^{\prime}}^{j}}\Big{\|}\sum_{\nu\in\Omega_{3}}\sum_{\ell=1}^{d(\nu)}|b_{\nu,\ell}(\rho_{1})\overline{b_{\nu,\ell}(\rho_{2})}|\frac{\beta(\rho_{1})\beta(\rho_{2})}{|\rho_{1}\pm\rho_{2}|^{\frac{1}{2}}}\Big{\|}^{2}_{L^{2}_{\rho_{1},\rho_{2}}(Q_{\bar{k}}^{j}\times
Q_{\bar{k}^{\prime}}^{j})}\Big{)}^{\frac{1}{2}}\\\
&\qquad\qquad\qquad\qquad\qquad\lesssim
R^{\epsilon}\Big{\|}\big{(}\sum_{\nu\in\chi_{K}}\sum_{\ell=1}^{d(\nu)}\big{|}b_{\nu,\ell}(\rho)\big{|}^{2}\big{)}^{\frac{1}{2}}\Big{\|}^{2}_{L^{4}_{\rho}(I)}.\end{split}$

By the Cauchy-Schwarz inequality and
$\text{dist}(Q_{\bar{k}}^{j},Q_{\bar{k}^{\prime}}^{j})\geq 2^{-j}$, we need to
prove

$\begin{split}\sum_{j\leq\log
R}2^{\frac{j}{2}}\Big{(}\sum_{\bar{k}}&\sum_{\bar{k}^{\prime}:Q_{\bar{k}}^{j}\simeq
Q_{\bar{k}^{\prime}}^{j}}\Big{\|}\Big{(}\sum_{\nu\in\Omega_{3}}\sum_{\ell=1}^{d(\nu)}|b_{\nu,\ell}(\rho_{1})|^{2}\Big{)}^{\frac{1}{2}}\Big{\|}^{2}_{L^{2}_{\rho_{1}}(Q_{\bar{k}}^{j})}\Big{\|}\Big{(}\sum_{\nu\in\Omega_{3}}\sum_{\ell=1}^{d(\nu)}|b_{\nu,\ell}(\rho_{2})|^{2}\Big{)}^{\frac{1}{2}}\Big{\|}^{2}_{L^{2}_{\rho_{2}}(Q_{\bar{k}^{\prime}}^{j})}\Big{)}^{\frac{1}{2}}\\\
&\qquad\qquad\qquad\qquad\qquad\lesssim
R^{\epsilon}\Big{\|}\big{(}\sum_{\nu\in\chi_{K}}\sum_{\ell=1}^{d(\nu)}\big{|}b_{\nu,\ell}(\rho)\big{|}^{2}\big{)}^{\frac{1}{2}}\Big{\|}^{2}_{L^{4}_{\rho}(I)}.\end{split}$

Since $|Q_{\bar{k}}^{j}|=|Q_{{\bar{k}}^{\prime}}^{j}|=2^{-j}$, by Hölder’s
inequality, we can bound the left hand side by

$\begin{split}\sum_{j\leq\log
R}&\Big{(}\sum_{\bar{k}}\Big{\|}\Big{(}\sum_{\nu\in\chi_{K}}\sum_{\ell=1}^{d(\nu)}|b_{\nu,\ell}(\rho_{1})|^{2}\Big{)}^{\frac{1}{2}}\Big{\|}^{4}_{L^{4}_{\rho_{1}}(Q_{\bar{k}}^{j})}\Big{)}^{\frac{1}{2}}\end{split}$

Moreover it is controlled by

$\begin{split}\log
R\Big{\|}\big{(}\sum_{\nu\in\chi_{K}}\sum_{\ell=1}^{d(\nu)}\big{|}b_{\nu,\ell}(\rho)\big{|}^{2}\big{)}^{\frac{1}{2}}\Big{\|}^{2}_{L^{4}_{\rho}(I)}\lesssim
R^{\epsilon}\Big{\|}\big{(}\sum_{\nu\in\chi_{K}}\sum_{\ell=1}^{d(\nu)}\big{|}b_{\nu,\ell}(\rho)\big{|}^{2}\big{)}^{\frac{1}{2}}\Big{\|}^{2}_{L^{4}_{\rho}(I)}.\end{split}$

Hence it follows (3.46). Therefore it completes the proof of Proposition 3.1.

##  4\. Proof of the Theorem 1.1

In this section, we utilize Proposition 3.1 to prove Theorem 1.1. We only need
prove (1.6). Indeed, when the initial data $u_{0}=f(r)$ is radial Schwartz, so
is the Schwartz solution $u(t)$ by (2.22). We can follow the argument in
proving (1.6) to easily obtain (1.5), since the $L^{q}$-norms on the compact
set $\Sigma$ of a constant function are equivalent for $1\leq q\leq\infty$. We
remark that one need use (3.5) to obtain (1.5) for $1\leq p\leq 2$.

Now we prove (1.6). By the Sobolev embedding
$H^{\alpha}(\Sigma)\hookrightarrow L^{q}(\Sigma)$ with
$\alpha=(n-1)(\frac{1}{2}-\frac{1}{q})$, it suffices to show

(4.1)
$\begin{split}&\|u(t,z)\|_{L^{{q}}_{t}L^{{q}}_{\mu(r)}L^{2}_{\theta}(\R\times\R_{+}\times\Sigma)}\lesssim\|\mathcal{F}_{H}{\big{(}(1-\widetilde{\Delta}_{h})^{\frac{1}{qn}}u_{0}\big{)}}\|_{L^{p}(M)}\end{split}$

holds for the conditions $q>\frac{2(n+1)}{n}$ and
$\frac{n+2}{q}=\frac{n}{p^{\prime}}$ with $p\geq 2$. By (2.22), we have the
dyadic decomposition

(4.2)
$\begin{split}&\|u(t,z)\|_{L^{{q}}_{t}L^{{q}}_{\mu(r)}L^{2}_{\theta}(\R\times\R_{+}\times\Sigma)}\\\
&\lesssim\big{\|}\sum_{\nu\in\chi_{\infty}}\sum_{\ell=1}^{d(\nu)}\varphi_{\nu,\ell}(\theta)\mathcal{H}_{\nu}\big{[}e^{it\rho^{2}}b_{\nu,\ell}(\rho)\big{]}(r)\big{\|}_{L^{q}_{t}(\R;L^{q}_{\mu(r)}(\R_{+};L^{2}_{\theta}(\Sigma)))}\\\
&\lesssim\Big{(}\sum_{R}\big{(}\sum_{N}\big{\|}\sum_{\nu\in\chi_{\infty}}\sum_{\ell=1}^{d(\nu)}\varphi_{\nu,\ell}(\theta)\\\
&\quad\times\int_{0}^{\infty}(r\rho)^{-\frac{n-2}{2}}J_{\nu}(r\rho)e^{it\rho^{2}}b_{\nu,\ell}(\rho)\rho^{n-1}\beta(\frac{\rho}{N})\mathrm{d}\rho\big{\|}_{L^{q}_{t}(\R;L^{q}_{\mu(r)}([R,2R];L^{2}_{\theta}(\Sigma)))}\big{)}^{q}\Big{)}^{\frac{1}{q}}\end{split}$

where $\beta\in C_{c}^{\infty}(\R)$ supported in $[1,2]$ and $R,N>0$ are
dyadic numbers. Define

(4.3)
$\begin{split}G(R,N;q):=\Big{\|}&\Big{(}\sum_{\nu\in\chi_{\infty}}\sum_{\ell=1}^{d(\nu)}\big{|}\int_{0}^{\infty}(r\rho)^{-\frac{n-2}{2}}\\\
&\quad\times
J_{\nu}(r\rho)e^{it\rho^{2}}b_{\nu,\ell}(\rho)\rho^{n-1}\beta(\frac{\rho}{N})\mathrm{d}\rho\big{|}\Big{)}^{\frac{1}{2}}\Big{\|}_{L^{q}_{t}(\R;L^{q}_{\mu(r)}([R,2R]))}.\end{split}$

Now we use Proposition 3.1. As mentioned in remarks after Proposition 3.1, we
can replace $\chi_{K}$ by $\chi_{\infty}$. By scaling argument and (3.1), we
have

(4.4) $\begin{split}&G(R,N;2)\\\
&\lesssim\min\\{(RN)^{\frac{1}{2}},(RN)^{\frac{n}{2}}\\}N^{n-\frac{n+2}{2}-\frac{n}{2}}\Big{\|}\Big{(}\sum_{\nu\in\chi_{\infty}}\sum_{\ell=1}^{d(\nu)}|b_{\nu,\ell}(\rho)|^{2}\Big{)}^{\frac{1}{2}}\beta(\frac{\rho}{N})\Big{\|}_{L^{2}_{\mu(\rho)}}.\end{split}$

On the other hand, for $\bar{q}=3\bar{p}^{\prime}$ and $2\leq\bar{p}<4$, we
have by (3.4)

(4.5)
$\begin{split}G(R,N;\bar{q})\lesssim&\min\big{\\{}(RN)^{(n-1)(\frac{1}{\bar{q}}-\frac{1}{2})},(RN)^{\frac{n}{\bar{q}}}\big{\\}}N^{n-\frac{n+2}{\bar{q}}-\frac{n}{\bar{p}}}\\\
&\times\Big{\|}\Big{(}\sum_{\nu\in\chi_{\infty}}\sum_{\ell=1}^{d(\nu)}(1+\nu)^{\frac{4}{\bar{q}}}|b_{\nu,\ell}(\rho)|^{2}\Big{)}^{\frac{1}{2}}\beta(\frac{\rho}{N})\Big{\|}_{L^{\bar{p}}_{\mu(\rho)}}.\end{split}$

Applying interpolation theorem to (4.4) and (4.5) with index
$\delta=2-\frac{3}{q}-\frac{1}{p}$,

$\frac{1}{q}=\frac{1-\delta}{2}+\frac{\delta}{\bar{q}},\qquad\frac{1}{p}=\frac{1-\delta}{2}+\frac{\delta}{\bar{p}}$

where ${\bar{q}}=3{\bar{p}}^{\prime}$, we hence have for
$\frac{n+2}{q}=\frac{n}{p^{\prime}}$,

(4.6)
$\begin{split}G(R,N;{q})\lesssim&\min\big{\\{}(RN)^{\frac{n}{q}},(RN)^{-\frac{n-1}{2}[1-\frac{2(n+1)}{qn}]}\big{\\}}\\\
\quad\quad&\times\Big{\|}\Big{(}\sum_{\nu\in\chi_{\infty}}\sum_{\ell=1}^{d(\nu)}(1+\nu)^{\frac{4}{qn}}|b_{\nu,\ell}(\rho)|^{2}\Big{)}^{\frac{1}{2}}\beta(\frac{\rho}{N})\Big{\|}_{L^{p}_{\mu(\rho)}}.\end{split}$

Combining (4.2) with (4.6), we have

$\begin{split}\|u(t,z)\|_{L^{{q}}_{t}L^{{q}}_{\mu(r)}L^{2}_{\theta}(\R\times\R_{+}\times\Sigma)}&\lesssim\Big{(}\sum_{R}\Big{(}\sum_{N}\min\big{\\{}(RN)^{\frac{n}{q}},(RN)^{-\frac{n-1}{2}[1-\frac{2(n+1)}{qn}]}\big{\\}}\\\
&\times\Big{\|}\Big{(}\sum_{\nu\in\chi_{\infty}}\sum_{\ell=1}^{d(\nu)}(1+\nu)^{\frac{4}{qn}}|b_{\nu,\ell}(\rho)|^{2}\Big{)}^{\frac{1}{2}}\beta(\frac{\rho}{N})\Big{\|}_{L^{p}_{\mu(\rho)}}\Big{)}^{q}\Big{)}^{\frac{1}{q}}.\end{split}$

Since $q>\frac{2(n+1)}{n}$ and $R,N$ are both dyadic number, we have

$\begin{split}&\sup_{R>0}\sum_{N}\min\left\\{(RN)^{\frac{n}{q}},(RN)^{-\frac{n-1}{2}[1-\frac{2(n+1)}{qn}]}\right\\}<\infty,\\\
&\sup_{N>0}\sum_{R}\min\left\\{(RN)^{\frac{n}{q}},(RN)^{-\frac{n-1}{2}[1-\frac{2(n+1)}{qn}]}\right\\}<\infty.\end{split}$

By using the Schur’s test, for $p$ and $q$ where $q>\frac{2(n+1)}{n}>p\geq 2$,
we have

$\begin{split}&\|u(t,z)\|_{L^{{q}}_{t}L^{{q}}_{\mu(r)}L^{2}_{\theta}(\R\times\R_{+}\times\Sigma)}\\\
&\lesssim\Big{(}\sum_{N}\Big{\|}\Big{(}\sum_{\nu\in\chi_{\infty}}\sum_{\ell=1}^{d(\nu)}(1+\nu)^{\frac{4}{qn}}|b_{\nu,\ell}(\rho)|^{2}\Big{)}^{\frac{1}{2}}\beta(\frac{\rho}{N})\Big{\|}_{L^{p}_{\mu(\rho)}}^{p}\Big{)}^{\frac{1}{q}}\\\
&\lesssim\|\mathcal{F}_{H}{\big{(}(1-\widetilde{\Delta}_{h})^{\frac{1}{qn}}u_{0}\big{)}}\|_{L^{p}(M)}.\end{split}$

Therefore we prove (4.1).

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