# Strichartz estimates and nonlinear wave equation on nontrapping
asymptotically conic manifolds

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.

We prove the global-in-time Strichartz estimates for wave equations on the
nontrapping asymptotically conic manifolds. We obtain estimates for the full
set of wave admissible indices, including the endpoint. The key points are the
properties of the microlocalized spectral measure of Laplacian on this setting
showed in [20] and a Littlewood-Paley squarefunction estimate. As
applications, we prove the global existence and scattering for a family of
nonlinear wave equations on this setting.

Key Words: Strichartz estimate, Asymptotically conic manifold, Spectral
measure, Global existence, Scattering theory
AMS Classification: 35Q40, 35S30, 47J35.

##  1\. Introduction and Statement of Main Results

Let $(M^{\circ},g)$ be a Riemannian manifold of dimension $n\geq 2$, and let
$I\subset\R$ be a time interval. Suppose $u(t,z)$: $I\times
M^{\circ}\rightarrow\mathbb{C}$ to be the solutions of the wave equation

$\partial_{t}^{2}u+\mathrm{H}u=0,\quad
u(0)=u_{0}(z),~{}\partial_{t}u(0)=u_{1}(z)$

where $\mathrm{H}=-\Delta_{g}$ denotes the minus Laplace-Beltrami operator on
$(M^{\circ},g)$. The general homogeneous Strichartz estimates read

$\|u(t,z)\|_{L^{q}_{t}L^{r}_{z}(I\times M^{\circ})}\leq
C\big{(}\|u_{0}\|_{H^{s}(M^{\circ})}+\|u_{1}\|_{H^{s-1}(M^{\circ})}\big{)},$

where $H^{s}$ denotes the $L^{2}$-Sobolev space over $M^{\circ}$, and $2\leq
q,r\leq\infty$ satisfy

$s=n(\frac{1}{2}-\frac{1}{r})-\frac{1}{q},\quad\frac{2}{q}+\frac{n-1}{r}\leq\frac{n-1}{2},\quad(q,r,n)\neq(2,\infty,3).$

In the flat Euclidean space, where $M^{\circ}=\R^{n}$ and
$g_{jk}=\delta_{jk}$, one can take $I=\R$; see Strichartz [30], Ginibre and
Velo [10], Keel and Tao [22], and references therein. In general manifolds,
for instance the compact manifold with or without boundary, most of the
Strichartz estimates are local in time. If $M^{\circ}$ is a compact manifold
without boundary, due to finite speed of propagation one usually works in
coordinate charts and establishes local Strichartz estimates for variable
coefficient wave operators on $\R^{n}$. See for examples [21, 26, 32].
Strichartz estimates also are considered on compact manifold with boundary,
see [6], [2] and references therein. When we consider the noncompact manifold
with nontrapping condition, one can obtain global-in-time Strichartz
estimates. For instance, when $M^{\circ}$ is a exterior manifold in
$\mathbb{R}^{n}$ to a convex obstacle, for metrics $g$ which agree with the
Euclidean metric outside a compact set with nontrapping assumption, the global
Strichartz estimates are obtained by Smith-Sogge [27] for odd dimension, and
Burq [5] and Metcalfe [25] for even dimension. Blair-Ford-Marzuola [3]
established global Strichartz estimates for the wave equation on flat cones
$C(\mathbb{S}_{\rho}^{1})$ by using the explicit representation of the
fundamental solution.

In this paper, we consider the establishment of global-in-time Strichartz
estimates on asymptotically conic manifolds satisfying a nontrapping
condition. Here, ‘asymptotically conic’ is meant in the sense that $M^{\circ}$
can be compactified to a manifold with boundary $M$ such that $g$ becomes a
scattering metric on $M$. On the nontrapping asymptotically conic manifolds,
Hassell, Tao, and Wunsch first established an $L^{4}_{t,z}$-Strichartz
estimate for Schrödinger equation in [14] and then they [15] extended the
estimate to full admissible Strichartz exponents except endpoint $q=2$. More
precisely, they obtained the local-in-time Strichartz inequalities for non-
endpoint Schrödinger admissible pairs $(q,r)$

$\|e^{it\Delta_{g}}u_{0}\|_{L^{q}_{t}L^{r}_{z}([0,1]\times M^{\circ})}\leq
C\|u_{0}\|_{L^{2}(M^{\circ})}.$

Recently, Hassell and the author [20] improved the Strichartz inequalities by
replacing the interval $[0,1]$ by $\R$. The purpose of this article is to
extend the above investigations carried out for Schrödinger to wave equations.

Let us recall the asymptotically conic geometric setting (i.e. scattering
manifold), which is the same as in [12, 13, 17, 15, 20]. Let $(M^{\circ},g)$
be a complete noncompact Riemannian manifold of dimension $n\geq 2$ with one
end, diffeomorphic to $(0,\infty)\times Y$ where $Y$ is a smooth compact
connected manifold without boundary. Moreover, we assume $(M^{\circ},g)$ is
asymptotically conic which means that $M^{\circ}$ allows a compactification
$M$ with boundary, with $\partial M=Y$, such that the metric $g$ becomes an
asymptotically conic metric on $M$. In details, the metric $g$ in a collar
neighborhood $[0,\epsilon)_{x}\times\partial M$ near $Y$ takes the form of

(1.1)
$g=\frac{\mathrm{d}x^{2}}{x^{4}}+\frac{h(x)}{x^{2}}=\frac{\mathrm{d}x^{2}}{x^{4}}+\frac{\sum
h_{jk}(x,y)dy^{j}dy^{k}}{x^{2}},$

where $x\in C^{\infty}(M)$ is a boundary defining function for $\partial M$
and $h$ is a smooth family of metrics on $Y$. Here we use
$y=(y_{1},\cdots,y_{n-1})$ for local coordinates on $Y=\partial M$, and the
local coordinates $(x,y)$ on $M$ near $\partial M$. Away from $\partial M$, we
use $z=(z_{1},\cdots,z_{n})$ to denote the local coordinates. If
$h_{jk}(x,y)=h_{jk}(y)$ is independent of $x$, we say $M$ is perfectly conic
near infinity. Moreover if every geodesic $z(s)$ in $M$ reaches $Y$ as
$s\rightarrow\pm\infty$, we say $M$ is nontrapping. The function $r:=1/x$ near
$x=0$ can be thought of as a “radial” variable near infinity and $y$ can be
regarded as the $n-1$ “angular” variables; the metric is asymptotic to the
exact conic metric $((0,\infty)_{r}\times Y,dr^{2}+r^{2}h(0))$ as
$r\rightarrow\infty$. The Euclidean space $M^{\circ}=\mathbb{R}^{n}$ is an
example of an asymptotically conic manifold with $Y=\mathbb{S}^{n-1}$ and the
standard metric. However a metric cone itself is not an asymptotically conic
manifold because of its cone point. We remark that the Euclidean space is a
perfectly metric nontrapping cone, where the cone point is a removable
singularity.

Let $\dot{H}^{s}(M^{\circ})={(-\Delta_{g})}^{-\frac{s}{2}}L^{2}(M^{\circ})$ be
the homogeneous Sobolev space over $M^{\circ}$. Throughout this paper, pairs
of conjugate indices are written as $r,r^{\prime}$, where
$\frac{1}{r}+\frac{1}{r^{\prime}}=1$ with $1\leq r\leq\infty$. Our main result
concerning Strichartz estimates is the following.

######  Theorem 1.1 (Global-in-time Strichartz estimate).

Let $(M^{\circ},g)$ be an asymptotically conic non-trapping manifold of
dimension $n\geq 3$. Let $\mathrm{H}=-\Delta_{g}$ and suppose that $u$ is the
solution to the Cauchy problem

(1.2) $\begin{cases}\partial_{t}^{2}u+\mathrm{H}u=F(t,z),\quad(t,z)\in I\times
M^{\circ};\\\ u(0)=u_{0}(z),~{}\partial_{t}u(0)=u_{1}(z),\end{cases}$

for some initial data $u_{0}\in\dot{H}^{s},u_{1}\in\dot{H}^{s-1}$, and the
time interval $I\subseteq\R$, then

(1.3)
$\begin{split}&\|u(t,z)\|_{L^{q}_{t}(I;L^{r}_{z}(M^{\circ}))}+\|u(t,z)\|_{C(I;\dot{H}^{s}(M^{\circ}))}\\\
&\qquad\lesssim\|u_{0}\|_{\dot{H}^{s}(M^{\circ})}+\|u_{1}\|_{\dot{H}^{s-1}(M^{\circ})}+\|F\|_{L^{\tilde{q}^{\prime}}_{t}(I;L^{\tilde{r}^{\prime}}_{z}(M^{\circ}))},\end{split}$

where the pairs $(q,r),(\tilde{q},\tilde{r})\in[2,\infty]^{2}$ satisfy the
wave-admissible condition

(1.4)
$\frac{2}{q}+\frac{n-1}{r}\leq\frac{n-1}{2},\quad(q,r,n)\neq(2,\infty,3).$

and the gap condition

(1.5)
$\frac{1}{q}+\frac{n}{r}=\frac{n}{2}-s=\frac{1}{\tilde{q}^{\prime}}+\frac{n}{\tilde{r}^{\prime}}-2.$

######  Remark 1.2.

We remark that the estimates are sharp from the sharpness in [22] for the
Euclidean space. There is no loss of derivatives. We can take the interval
$I=\R$ which means the estimates are global in time.

We sketch the proof as follows. Our strategy is to use the abstract Strichartz
estimate proved in Keel-Tao [22] and our previous argument [20] for
Schrödinger. Thus, with $U(t)$ denoting the (abstract) propagator, we need to
show uniform $L^{2}\rightarrow L^{2}$ estimate for $U(t)$, and
$L^{1}\rightarrow L^{\infty}$ type dispersive estimate on the $U(t)U(s)^{*}$
with a bound of the form $O(|t-s|^{-(n-1)/2})$. In the flat Euclidean setting,
the estimates are considerably simpler because of the explicit formula of the
spectral measure. But in our general setting, the estimates turn out to be
more complicated. It follows from [17] that the Schrödinger propagator
$e^{it\Delta_{g}}$ fails to satisfy such a dispersive estimate at any pair of
conjugate points $(z,z^{\prime})\in M^{\circ}\times M^{\circ}$ (i.e. pairs
$(z,z^{\prime})$ where a geodesic emanating from $z$ has a conjugate point at
$z^{\prime}$), so we need localize the propagator such that the conjugating
points are separated. One may avoid the conjugated points in a sufficiently
short time by using the finite speed of propagation $U(t)(z,z^{\prime})$. If
we do this, we would only obtain the local-in-time Strichartz estimates. We
instead overcome the difficulties caused by conjugate points by
microlocalizing the spectral measure [20], which is in the same spirit of the
proof in [13] of a _restriction estimate_ for the spectral measure, that is,
an estimate of the form

$\big{\|}dE_{\sqrt{\mathbf{H}}}(\lambda)\big{\|}_{L^{p}(M^{\circ})\to
L^{p^{\prime}}(M^{\circ})}\leq
C\lambda^{n(\frac{1}{p}-\frac{1}{p^{\prime}})-1},\quad 1\leq
p\leq\frac{2(n+1)}{n+3}.$

However, the microlocalized spectral measure
$Q_{i}(\lambda)dE_{\sqrt{\mathbf{H}}}(\lambda)Q_{i}(\lambda)^{*}$ only has a
size estimate in [13], where $Q_{i}(\lambda)$ is a member of a partition of
the identity operator in $L^{2}(M^{\circ})$. To obtain the dispersive
estimate, the authors [20] refined the microlocalized spectral measure by
capturing its oscillatory behavior. Thus we efficiently exploit the
oscillation of the ‘spectral multiplier’ $e^{it\lambda^{2}}$ and
microlocalized spectral measure to prove the dispersive estimate for
Schrödinger. However, the multiplier $e^{it\lambda}$ corresponding to the wave
equation has much less oscillation than the Schrödinger multiplier
$e^{it\lambda^{2}}$ at high frequency, so we need to modify the argument.
Because of this, we have to resort to a Littlewood-Paley squarefunction
estimate on this setting. We remark that the authors [20] avoid using the
Littlewood-Paley squarefunction estimate in the Schrödinger case. We prove the
Littlewood-Paley squarefunction estimate on this setting by using a spectral
multiplier estimate in Alexopoulos [1] and Stein’s [28] classical argument
involving Rademacher functions. The crucial ingredient is to obtain the
Gaussian upper bounds on the heat kernel on this setting. We show the Gaussian
upper bounds on the heat kernel by using the local-in-time heat kernel bounds
in Cheng-Li-Yau [7], and Guillarmou-Hassell-Sikora’s [13] restriction estimate
for low frequency which implies the long-time bounds. Having the
squarefunction estimate, we reduce Theorem 1.1 to prove a frequency-localized
estimate. To do this, we define a microlocalized half-wave propagator and
prove that it satisfies $L^{2}\rightarrow L^{2}$-bounded and dispersive
estimate. We prove the homogeneous Strichartz estimates for the microlocalized
half-wave propagator by using a semiclassical version of Keel-Tao’s argument.
The Strichartz estimate for $e^{it\sqrt{\mathrm{H}}}$ then follows by summing
each microlocalizing piece. The inhomogeneous Strichartz estimates follow from
the homogeneous estimates and the Christ-Kiselev lemma. Compared with the
establishment of Schrödinger inhomogeneous Strichartz estimate in [20], we do
not require additional argument since one must have $q>\tilde{q}^{\prime}$ if
both $(q,r)$ and $(\tilde{q},\tilde{r})$ satisfy (1.4) and (1.5).

As an application of the Strichartz estimates, we note that these inequalities
can be utilized to generalize a theorem of Lindblad-Sogge [24] on the
asymptotically conic non-trapping manifolds. More precisely, we prove the
well-posedness and scattering of the following semi-linear wave equation,

(1.6)
$\begin{cases}\partial_{t}^{2}u+\mathrm{H}u=\gamma|u|^{p-1}u,\qquad(t,z)\in\R\times
M^{\circ},\gamma\in\\{1,-1\\},\\\
u(t,z)|_{t=0}=u_{0}(z),\quad\partial_{t}u(t,z)|_{t=0}=u_{1}(z).\end{cases}$

In the case of flat Euclidean space, there are many results on the
understanding of the global existence and scattering. We refer the readers to
[24, 29] and references therein. Blair-Ford-Marzuola [3] also considered
similar results for the wave equation on flat cones
$C(\mathbb{S}_{\rho}^{1})$. Due to better understanding the spectral measure,
we can extend the result to high dimension. We here are mostly interested in
the range of exponents $p\in[p_{\text{conf}},1+\frac{4}{n-2}]$ and the initial
data is in $\dot{H}^{s_{c}}(M^{\circ})\times\dot{H}^{s_{c}-1}(M^{\circ})$,
where $p_{\text{conf}}=1+\frac{4}{n-1}$ and $s_{c}=\frac{n}{2}-\frac{2}{p-1}$.

Our main result concerning well-posedness and scattering is the following.

######  Theorem 1.3.

Let $(M^{\circ},g)$ be a non-trapping asymptotically conic manifold of
dimension $n\geq 3$. Suppose $p\in[p_{\mathrm{conf}},1+\frac{4}{n-2}]$ and
$(u_{0},u_{1})\in\dot{H}^{s_{c}}(M^{\circ})\times\dot{H}^{s_{c}-1}(M^{\circ})$,
then there exist $T>0$ and a unique solution $u$ to (1.6) satisfying

(1.7) $u\in C_{t}([0,T];\dot{H}^{s_{c}}(M^{\circ}))\cap
L^{q_{0}}([0,T];L^{q_{0}}(M^{\circ})),$

where $q_{0}=(p-1)(n+1)/2$. In addition, if there is a small constant
$\epsilon(p)$ such that

(1.8) $\|u_{0}\|_{\dot{H}^{s_{c}}}+\|u_{1}\|_{\dot{H}^{s_{c}-1}}<\epsilon(p),$

then there is a unique global and scattering solution $u$ to (1.6) satisfying

(1.9) $u\in C_{t}(\R;H^{s_{c}}(M^{\circ}))\cap
L^{q_{0}}(\R;L^{q_{0}}(M^{\circ})).$

This paper is organized as follows. In Section 2 we review the results of the
microlocalized spectral measure and prove the square function inequalities on
this setting. Section 3 is devoted to the proofs of the microlocalized
dispersive estimates and $L^{2}$-estimates. In Section 4, we prove the
homogeneous and inhomogeneous Strichartz estimates. Finally, we apply the
Strichartz estimates to show Theorem 1.3.

Acknowledgments: The author would like to thank Jean-Marc Bouclet, Andrew
Hassell and Changxing Miao for their helpful discussions and encouragement. He
also would like to thank the anonymous referee for careful reading the
manuscript and for giving useful comments. This research was supported by
PFMEC(20121101120044), Beijing Natural Science Foundation(1144014), National
Natural Science Foundation of China (11401024) and Discovery Grant DP120102019
from the Australian Research Council.

##  2\. The microlocalized spectral measure and Littlewood-Paley
squarefunction estimate

In this section, we briefly recall the key elements of the microlocalized
spectral measure, which was constructed by Hassell and the author [20] to
capture both its size and the oscillatory behavior. We also prove the
Littlewood-Paley squarefunction estimates on this setting that we require in
subsequence section.

###  2.1. The microlocalized spectral measure

In the free Euclidean space, the half wave propagator has an explicit formula
by using the Fourier transform, but in the asymptotically conical manifold it
turns out to be quite complicated. From the results of [12, 16], we have known
that the Schwartz kernel of the spectral measure can be described as a
Legendrian distribution on the compactification of the space $M\times M$
uniformly with respect to the spectral parameter $\lambda$. As pointed out in
introduction, we really need to choose an operator partition of unity to
microlocalize the spectral measure such that the spectral measure can be
expressed in a formula capturing not only the size also the oscillatory
behavior. This was constructed and proved in [20]. For convenience, we recall
and slightly modify the statement to adapt our following application.

######  Proposition 2.1.

Let $(M^{\circ},g)$ and $\mathrm{H}$ be in Theorem 1.1. For fixed
$\lambda_{0}>0$, then there exists an operator partition of unity on
$L^{2}(M)$

(2.1)
$\begin{split}\mathrm{Id}=\sum_{i=0}^{N_{l}}Q^{\mathrm{low}}_{i}(\lambda)\quad\text{for}~{}0<\lambda\leq
2\lambda_{0};\\\
\mathrm{Id}=\big{(}\sum_{i=1}^{N^{\prime}}+\sum_{i=N^{\prime}+1}^{N_{h}}\big{)}Q^{\mathrm{high}}_{i}(\lambda)\quad\text{for}~{}\lambda\geq\lambda_{0}/2,\end{split}$

where the $Q^{\mathrm{low}}_{i}$ and $Q^{\mathrm{high}}_{i}$ are uniformly
bounded as operators on $L^{2}$ and $N_{l}$ and $N_{h}$ are bounded
independent of $\lambda$, such that

$\bullet$ when $Q(\lambda)$ is equal to either $Q^{\mathrm{low}}_{0}(\lambda)$
or $Q^{\mathrm{low}}_{1}(\lambda)$; or $Q(\lambda)$ is equal to
$Q^{\mathrm{high}}_{1}(\lambda)$, we have

(2.2)
$\begin{split}\Big{|}\big{(}\frac{d}{d\lambda}\big{)}^{\alpha}\big{(}Q(\lambda)dE_{\sqrt{\mathrm{H}}}(\lambda){Q}^{*}(\lambda)\big{)}\Big{|}\leq
C_{\alpha}\lambda^{n-1-\alpha}\quad\forall\alpha\in\mathbb{N}.\end{split}$

$\bullet$ when $Q(\lambda)$ is equal to $Q^{\mathrm{low}}_{i}(\lambda)$ or
$Q^{\mathrm{high}}_{i}(\lambda)$ for $i\geq 2$, we have

(2.3)
$(Q(\lambda)dE_{\sqrt{\mathrm{H}}}(\lambda)Q^{*}(\lambda))(z,z^{\prime})=\lambda^{n-1}e^{\pm
i\lambda d(z,z^{\prime})}a(\lambda,z,z^{\prime}).$

Here $d(\cdot,\cdot)$ is the Riemannian distance on $M^{\circ}$, and $a$
satisfies

(2.4) $|\partial_{\lambda}^{\alpha}a(\lambda,z,z^{\prime})|\leq
C_{\alpha}\lambda^{-\alpha}(1+\lambda d(z,z^{\prime}))^{-\frac{n-1}{2}}.$

Having this result, we can exploit the oscillations both in the multiplier
$e^{i(t-s)\lambda}$ and in $e^{\pm i\lambda d(z,z^{\prime})}$ to obtain the
required dispersive estimate for the $TT^{*}$ version of the microlocalized
propagator.

###  2.2. The Littlewood-Paley squarefunction estimate

In this subsection, we prove the Littlewood-Paley squarefunction estimate for
the asymptotically conic manifold, which allows us to reduce Theorem 1.1 to a
frequency-localized estimate (see Proposition 4.2).

Let $\varphi\in C_{0}^{\infty}(\mathbb{R}\setminus\\{0\\})$ take values in
$[0,1]$ and be supported in $[1/2,2]$ such that

(2.5) $1=\sum_{j\in\Z}\varphi(2^{-j}\lambda),\quad\lambda>0.$

Define $\varphi_{0}(\lambda)=\sum_{j\leq 0}\varphi(2^{-j}\lambda)$. Then the
result about the Littlewood-Paley squarefunction estimate reads as follows:

######  Proposition 2.2.

Let $(M^{\circ},g)$ be an asymptotically conic manifold, trapping or not, and
$\mathrm{H}=-\Delta_{g}$ is the Laplace-Beltrami operator on $(M^{\circ},g)$.
Then for $1<p<\infty$, there exist constants $c_{p}$ and $C_{p}$ depending on
$p$ such that

(2.6)
$c_{p}\|f\|_{L^{p}(M^{\circ})}\leq\big{\|}\big{(}\sum_{j\in\Z}|\varphi(2^{-j}\sqrt{\mathrm{H}})f|^{2}\big{)}^{\frac{1}{2}}\big{\|}_{L^{p}(M^{\circ})}\leq
C_{p}\|f\|_{L^{p}(M^{\circ})}.$

######  Remark 2.3.

To our knowledge, such squarefunction estimates are new in the case of
asymptotically conic manifolds, though the proof is considerably simpler due
to the heat kernel bounds in Cheng-Li-Yau [7], Guillarmou-Hassell-Sikora’s
[13] restriction estimate for low frequency and the spectral multiplier
estimates in Alexopoulos [1]. In the general noncompact manifolds with ends,
Bouclet [4] proved a weak version square function inequality which was given
by for $1<p<\infty$

(2.7) $\|f\|_{L^{p}}\lesssim\big{\|}\big{(}\sum_{j\geq
0}|\varphi(2^{-2j}\mathrm{H})f|^{2}\big{)}^{\frac{1}{2}}\big{\|}_{L^{p}}+\|f\|_{L^{2}}.$

Bouclet also pointed out that the usual square function inequalities may fail
on asymptotically hyperbolic manifolds and improved (2.7) for asymptotically
conic manifolds by showing

(2.8) $\|\varphi_{0}(\mathrm{H})f\|_{L^{p}}+\big{\|}\big{(}\sum_{j\geq
0}|\varphi(2^{-2j}\mathrm{H})f|^{2}\big{)}^{\frac{1}{2}}\big{\|}_{L^{p}}\sim\|f\|_{L^{p}}.$

One can see that the squarefunction estimate in (2.6) involves the low
frequency in contrast to (2.8).

###### Proof.

This proof follows from the Stein’s [28] classical argument (in $\R^{n}$)
involving Rademacher functions and an appropriate Mikhlin-Hörmander multiplier
theorem. Now we provide details as follows. We notice that the asymptotically
conic manifolds are a relatively well-behaved class of manifolds. In
particular, all section curvatures of $(M^{\circ},g)$ approach zero as $x$
goes to zero, and thus $(M^{\circ},g)$ has bounded sectional curvature and has
low bounds for the injectivity radius. Now we need a theorem in Cheng-Li-Yau
[7] and recall it for convenience. For complete Riemannian manifolds
$M^{\circ}$ of bounded sectional curvature and injectivity radius bounded
below, Cheng-Li-Yau’s theorem gives the following local-in-time Gaussian upper
bound for the heat kernel

######  Lemma 2.4.

There exist nonzero constants $c$ and $C$ such that the heat kernel on
$M^{\circ}$, denoted $H(t,z,z^{\prime})$, satisfies the Gaussian upper bound
of the form for $t\in[0,T]$

(2.9) $H(t,z,z^{\prime})\leq
Ct^{-\frac{n}{2}}\exp\Big{(}-\frac{d(z,z^{\prime})^{2}}{ct}\Big{)},$

where $d(z,z^{\prime})$ is the distance between $z$ and $z^{\prime}$ on
$M^{\circ}$.

We claim that the global-in-time Gaussian upper bound for the heat kernel also
holds, that is

(2.10)
$H(t,z,z^{\prime})\lesssim\frac{1}{|B(z,\sqrt{t})|}\exp\Big{(}-\frac{d(z,z^{\prime})^{2}}{ct}\Big{)}$

holds for all $t>0$, where $|B(z,\sqrt{t})|$ is the volume of the ball of
radius $\sqrt{t}$ at $z$. By (2.9), we only consider the case $t\geq 1$. To
prove this, we write

$H(t,z,z^{\prime})=e^{-t\mathrm{H}}(z,z^{\prime})=\int_{0}^{\infty}e^{-t\lambda^{2}}dE_{\sqrt{\mathrm{H}}}(\lambda).$

Choose $\chi\in C_{c}^{\infty}(\R)$, such that $\chi(\lambda)=1$ for
$\lambda\leq 1$, we decompose

$\begin{split}&H(t,z,z^{\prime})\\\
&=\int_{0}^{\infty}e^{-t\lambda^{2}}\chi(\lambda)dE_{\sqrt{\mathrm{H}}}(\lambda)+\int_{0}^{\infty}e^{-t\lambda^{2}}(1-\chi)(\lambda)dE_{\sqrt{\mathrm{H}}}(\lambda)\\\
&=:I+II.\end{split}$

By using [13, Theorem 1.3], we see for $\lambda\leq 1$

$\begin{split}|dE_{\sqrt{\mathrm{H}}}(\lambda)(z,z^{\prime})|\leq
C\lambda^{n-1}.\end{split}$

Hence $I\leq Ct^{-\frac{n}{2}}$. To treat $II$, we need the following lemma

######  Lemma 2.5.

If the local-in-time heat kernel bound $\|e^{-t\mathrm{H}}\|_{L^{1}\rightarrow
L^{2}}\leq Ct^{-\frac{n}{4}}$ holds for $t\leq 1$, then the following spectral
projection estimate holds for $\mu\geq 1$,

$\|E_{\sqrt{\mathrm{H}}}([0,\mu])\|_{L^{1}\rightarrow L^{2}}\leq C\mu^{n/2}.$

###### Proof.

Let $t=\mu^{-2}$. Notice $1_{[0,\mu]}(s)\leq e\exp(-\frac{s^{2}}{\mu^{2}})$,
then spectral projection estimate is proved by writing
$E_{\sqrt{\mathrm{H}}}([0,\mu])=E_{\sqrt{\mathrm{H}}}([0,\mu])e^{\mathrm{H}/\mu^{2}}e^{-\mathrm{H}/\mu^{2}}$.
Indeed, we have

$\begin{split}\|E_{\sqrt{\mathrm{H}}}([0,\mu])\|_{L^{1}\rightarrow
L^{2}}\leq\|E_{\sqrt{\mathrm{H}}}([0,\mu])e^{\mathrm{H}/\mu^{2}}\|_{L^{2}\rightarrow
L^{2}}\|e^{-\mathrm{H}/\mu^{2}}\|_{L^{1}\rightarrow L^{2}}\leq
C\mu^{n/2}.\end{split}$

∎

Now we turn to estimate $II$. From the local-in-time heat kernel estimate
(2.9), one has $\|e^{-t\mathrm{H}}\|_{L^{1}\rightarrow L^{\infty}}\leq
Ct^{-\frac{n}{2}}$ for $t\leq 1$. By using a $TT^{*}$ argument,
$\|e^{-t\mathrm{H}}\|_{L^{1}\rightarrow L^{2}}\leq Ct^{-\frac{n}{4}}$ for
$t\leq 1$. Hence by Lemma 2.5
$\|E_{\sqrt{\mathrm{H}}}([0,\lambda])\|_{L^{1}\rightarrow L^{2}}\leq
C\lambda^{n/2}$ for $\lambda\geq 1$, which implies
$\|E_{\sqrt{\mathrm{H}}}([0,\lambda])\|_{L^{1}\rightarrow L^{\infty}}\leq
C\lambda^{n}$. Therefore we have for $t\geq 1$

$\begin{split}\|II\|_{L^{1}\rightarrow L^{\infty}}&\leq\sum_{k\geq
0}\int_{0}^{\infty}\frac{d}{d\lambda}\left(e^{-t\lambda^{2}}\phi_{k}\left(\lambda\right)(1-\chi)(\lambda)\right)\left\|E_{\sqrt{\mathrm{H}}}(\lambda)\right\|_{L^{1}\rightarrow
L^{\infty}}d\lambda\\\ &\leq Ce^{-t/2}\leq Ct^{-\frac{n}{2}}.\end{split}$

Hence we have proved for all $t>0$

$H(t,z,z^{\prime})\lesssim t^{-\frac{n}{2}}.$

We use a theorem of Grigor’yan [11, Theorem 1.1] that establishes Gaussian
upper bounds for arbitrary Riemannian manifolds. His conclusion implies that
if $H(t,z,z^{\prime})$ satisfies on-diagonal bounds

$H(t,z,z)\lesssim t^{-\frac{n}{2}},\quad H(t,z^{\prime},z^{\prime})\lesssim
t^{-\frac{n}{2}},$

then we have

$H(t,z,z^{\prime})\lesssim
t^{-\frac{n}{2}}\exp\Big{(}-\frac{d(z,z^{\prime})^{2}}{ct}\Big{)}.$

Since $|B(z,\sqrt{t})|\sim t^{\frac{n}{2}}$, this gives

(2.11)
$H(t,z,z^{\prime})\lesssim\frac{1}{|B(z,\sqrt{t})|}\exp\Big{(}-\frac{d(z,z^{\prime})^{2}}{ct}\Big{)}.$

Now we need a result of Alexopoulos [1, Theorem 6.1], which outlines how his
results on Markov chains can be extended to treat differential operators on
manifolds where the associated heat kernel satisfies Gaussian upper bounds. We
remark here that the asymptotically conic manifold satisfies the doubling
condition in contrast to the hyperbolic case. Given (2.11), Alexopoulos’
theorem implies that any spectral multiplier $m(\sqrt{\mathrm{H}})$ satisfying
the usual Hörmander condition maps $L^{p}(M)\rightarrow L^{p}(M)$ for any
$p\in(1,\infty)$. Furthermore, this boundedness holds true for function $m\in
C^{N}(\R)$ which satisfies the weaker Mihlin-type condition for
$N\geq\frac{n}{2}+1$

(2.12) $\sup_{0\leq k\leq
N}\sup_{\lambda\in\R}\Big{|}\big{(}\lambda\partial_{\lambda}\big{)}^{k}m(\lambda)\Big{|}\leq
C<\infty.$

We now want to apply this result to a family of multipliers
$m^{\pm}(s,\sqrt{\mathrm{H}}),0\leq s\leq 1$ defined using the Rademacher
functions. Let us introduce the Rademacher functions defined as follows:

(i) the function $r_{0}(s)$ is defined by $r_{0}(s)=1$ on $[0,1/2]$ and
$r_{0}(s)=-1$ on $(1/2,1)$, and then extended to the real line by periodicity,
i.e. $r_{0}(s+1)=r_{0}(s)$;

(ii) for $k\in\N\setminus\\{0\\}$, $r_{k}(s)=r_{0}(2^{k}s)$.

Given any square integrable sequence of scalars $\\{a_{k}\\}_{k\geq 0}$,
consider the function $m(s)=\sum_{k\geq 0}a_{k}r_{k}(s)$. By a lemma in [28,
Appendix D], for any $p\in(1,\infty)$ there exist constants $c_{p}$ and
$C_{p}$ such that

(2.13)
$c_{p}\|m(s)\|_{L^{p}([0,1])}\leq\|m(s)\|_{L^{2}([0,1])}=\Big{(}\sum_{k\geq
0}|a_{k}|^{2}\Big{)}^{\frac{1}{2}}\leq C_{p}\|m(s)\|_{L^{p}([0,1])}.$

Now define

$m^{\pm}(s,\lambda)=\sum_{j\geq 0}r_{j}(s)\varphi_{\pm j}(\lambda)$

where $\varphi_{\pm j}(\lambda)=\varphi(2^{\mp j}\lambda)$. Then we define the
operator $m^{\pm}(s,\sqrt{\mathrm{H}})$ through the spectral measure
$dE_{\sqrt{\mathrm{H}}}(\lambda)$:

(2.14)
$m^{\pm}(s,\sqrt{\mathrm{H}})=\int_{0}^{\infty}m^{\pm}(s,\lambda)dE_{\sqrt{\mathrm{H}}}(\lambda).$

We note that this is well-defined by the spectral theory. It can be verified
that $m^{\pm}(s,\lambda)$ satisfies the condition (2.12), and we can take the
constant $C$ independent of $s$. Therefore we have that for $1<p<\infty$ and
$f$ in $L^{p}$ by (2.13)

$\begin{split}&\Big{\|}\Big{(}\sum_{j\geq 0}\big{|}\varphi_{\pm
j}(\sqrt{\mathrm{H}})f\big{|}^{2}\Big{)}^{\frac{1}{2}}\Big{\|}^{p}_{L^{p}}\lesssim\Big{\|}\sum_{j\geq
0}\varphi_{\pm
j}(\sqrt{\mathrm{H}})f(z)r_{k}(s)\Big{\|}^{p}_{L^{p}(M;L^{p}([0,1]))}\\\
&\lesssim\int_{M^{\circ}}\int_{0}^{1}\Big{|}m^{\pm}(s,\sqrt{\mathrm{H}})f(z)\Big{|}^{p}dsdg(z)\lesssim\|f\|^{p}_{L^{p}}.\end{split}$

Therefore we prove

(2.15)
$\begin{split}&\Big{\|}\Big{(}\sum_{j\in\Z}\big{|}\varphi_{j}(\sqrt{\mathrm{H}})f\big{|}^{2}\Big{)}^{\frac{1}{2}}\Big{\|}_{L^{p}}\lesssim\|f\|_{L^{p}}.\end{split}$

To see the other inequality, we first define
$\widetilde{\varphi}_{j}(\lambda)=\sum_{i=j-1}^{j+1}\varphi_{i}(\lambda)$,
then the above also is true when $\varphi_{j}(\lambda)$ is replaced by
$\widetilde{\varphi}_{j}(\lambda)$. Let $f_{1}\in L^{p}$ and $f_{2}\in
L^{p^{\prime}}$, we see by Hölder’s inequality and (2.15)

$\begin{split}\Big{|}\int_{M^{\circ}}f_{1}(z)\overline{f_{2}(z)}dg(z)\Big{|}&=\Big{|}\int_{M^{\circ}}\sum_{j\in\Z}\big{(}\widetilde{\varphi}_{j}(\sqrt{\mathrm{H}})f_{1}\big{)}(z)\overline{\big{(}\varphi_{j}(\sqrt{\mathrm{H}})f_{2}\big{)}(z)}dg(z)\Big{|}\\\
&\lesssim\Big{\|}\big{(}\sum_{j\in\Z}\big{|}\widetilde{\varphi}_{j}(\sqrt{\mathrm{H}})f_{1}\big{|}^{2}\big{)}^{\frac{1}{2}}\Big{\|}_{L^{p}}\Big{\|}\big{(}\sum_{j\in\Z}\big{|}\varphi_{j}(\sqrt{\mathrm{H}})f_{2}\big{|}^{2}\big{)}^{\frac{1}{2}}\Big{\|}_{L^{p^{\prime}}}\\\
&\lesssim\|f_{1}\|_{L^{p}}\Big{\|}\big{(}\sum_{j\in\Z}\big{|}\varphi_{j}(\sqrt{\mathrm{H}})f_{2}\big{|}^{2}\big{)}^{\frac{1}{2}}\Big{\|}_{L^{p^{\prime}}}.\end{split}$

By duality, we hence prove (2.6).

∎

##  3\. $L^{2}$-estimates and dispersive estimates

In this section, we prove the $L^{2}$-estimates and dispersive estimates
needed for the abstract Keel-Tao argument. We begin by defining microlocalized
propagators and then show the definition makes sense. We do this by showing
that each microlocalized propagator is a bounded operator on $L^{2}$. This
serves both to make the definition of each microlocalized propagator
allowable, and to establish the $L^{2}\to L^{2}$ estimate needed for the
abstract Keel-Tao argument. We point out here that the microlocalized
propagators are different from the ones defined in [20], which allow us to
easily show the $L^{2}\to L^{2}$ estimate by spectral theory on Hilbert space
but we need a square function inequality in the establishment of the
Strichartz estimate. Since the microlocalized propagators avoid the conjugate
points, we can prove the $TT^{*}$ version dispersive estimates.

###  3.1. Microlocalized propagator and $L^{2}$-estimates

We start by dividing the half wave propagator into a low-energy piece and a
high-energy piece. Choose $\chi\in C_{c}^{\infty}(\R)$, such that $\chi(t)=1$
for $t\leq 1$. We define

(3.1)
$U^{\mathrm{low}}(t)=\int_{0}^{\infty}e^{it\lambda}\chi(\lambda)dE_{\sqrt{\mathrm{H}}}(\lambda),\quad
U^{\mathrm{high}}(t)=\int_{0}^{\infty}e^{it\lambda}(1-\chi)(\lambda)dE_{\sqrt{\mathrm{H}}}(\lambda).$

Using the partition of unity $1=\sum_{j\in\Z}\varphi(2^{-j}\lambda)$ we define

(3.2)
$\begin{split}U^{\mathrm{low}}_{j}(t)&=\int_{0}^{\infty}e^{it\lambda}\varphi(2^{-j}\lambda)\chi(\lambda)dE_{\sqrt{\mathrm{H}}}(\lambda),\\\
U^{\mathrm{high}}_{j}(t)&=\int_{0}^{\infty}e^{it\lambda}\varphi(2^{-j}\lambda)(1-\chi)(\lambda)dE_{\sqrt{\mathrm{H}}}(\lambda).\end{split}$

Further using the low-energy and high-energy operator partition of identity
operator in Proposition 2.1, we define

(3.3)
$\begin{gathered}U_{i,j}(t)=\int_{0}^{\infty}e^{it\lambda}\varphi(2^{-j}\lambda)\chi(\lambda)Q_{i}^{\mathrm{low}}(\lambda)dE_{\sqrt{\mathrm{H}}}(\lambda),\quad
0\leq i\leq N_{l};\\\
U_{i,j}(t)=\int_{0}^{\infty}e^{it\lambda}\varphi(2^{-j}\lambda)(1-\chi)(\lambda)Q_{i-N_{l}}^{\mathrm{high}}(\lambda)dE_{\sqrt{\mathrm{H}}}(\lambda),~{}N_{l}+1\leq
i\leq N:=N_{l}+N_{h}.\end{gathered}$

Now we show this definition is unambiguous. To do so, it suffices to show the
above integrals are well defined over any compact interval in $(0,\infty)$.
Suppose that $A(\lambda)$ is a family of bounded operators on
$L^{2}(M^{\circ})$, compactly supported in $[a,b]$ and $C^{1}$ in
$\lambda\in(0,\infty)$. Integrating by parts, the integral of

$\int_{a}^{b}A(\lambda)dE_{\sqrt{\mathrm{H}}}(\lambda)$

is given by

(3.4)
$E_{\mathrm{\sqrt{H}}}(b)A(b)-E_{\mathrm{\sqrt{\mathrm{H}}}}(b)A(a)-\int_{a}^{b}\frac{d}{d\lambda}A(\lambda)E_{\sqrt{\mathrm{H}}}(\lambda)\,d\lambda.$

Now we need the following lemma which is the consequence of [20, Lemma 2.3,
Lemma 3.1].

######  Lemma 3.1.

Each $Q^{\mathrm{low}}_{i}(\lambda)$ and each operator
$\lambda\partial_{\lambda}Q^{\mathrm{low}}_{i}(\lambda)$ is bounded on
$L^{2}(M^{\circ})$ uniformly in $\lambda$. The same statements are true for
the high energy operators $Q^{\mathrm{high}}_{i}(\lambda)$.

###### Proof.

We use the notation in [12, 20, 16]. The uniform boundedness of the scattering
pseudodifferential operator
$Q^{\mathrm{low}}_{i}(\lambda)\in\Psi^{-\infty}_{k}(M,M^{2}_{k,b})$ is
straightforward to prove using the fact that the order is $-\infty$. This
implies that the kernel is smooth and uniformly bounded on iterated blowup
space $M^{2}_{k,\mathrm{sc}}$, as a multiple of the half density bundle
$\Omega_{k,b}^{\frac{1}{2}}$. This bundle has a nonzero section given, in the
region where $x\leq C\lambda$, by
$\lambda^{n}|dgdg^{\prime}|^{1/2}|d\lambda/\lambda|^{1/2}$, where the
$|d\lambda/\lambda|^{1/2}$ is a purely formal factor, included to make a half-
density on the whole space $M^{2}_{k,b}$, including in the
$\lambda$-direction. On the other hand, the kernels are chosen to have support
in a neighborhood of the diagonal, which is equivalent to the region where
$d(z,z^{\prime})\leq C\lambda^{-1}$. It follows that the kernel is bounded by
a multiple of the characteristic function of the set $\\{(z,z^{\prime})\mid
d(z,z^{\prime})\leq C\lambda^{-1}\\}$ times the Riemannian half-density.
Moreover, the same is true for $\lambda
d_{\lambda}Q_{i}^{\mathrm{low}}(\lambda)$, due to the smoothness of the kernel
on $M^{2}_{k,\mathrm{sc}}$. Since the volume of each ball of radius $r$ on
$M^{\circ}$ is between $cr^{n}$ and $Cr^{n}$, Schur’s test shows that such
kernels are bounded on $L^{2}(M^{\circ})$ uniformly in $\lambda$.

The high energy operators $Q^{\mathrm{high}}(\lambda)$ are semiclassical
pseudodifferential operators of semiclassical order 0 and differential order
$-\infty$. Therefore, they take the form

$\lambda^{n}\int
e^{i\lambda(z-z^{\prime})\cdot\zeta}a(z,\zeta,\lambda^{-1})\,d\zeta$

in the interior, or

$\lambda^{n}\int
e^{i\lambda((y-y^{\prime})\cdot\eta+(\sigma-1)\nu/x}a(x,y,\eta,\nu,\lambda^{-1})\,d\eta\,d\nu$

near the boundary. Here $a$ is smooth and compactly supported in its
arguments. Integration by parts in $\zeta$, or in $\eta,\nu$, shows that the
kernel is rapidly decreasing in $\lambda|z-z^{\prime}|$, respectively
$\lambda\sqrt{|y-y^{\prime}|^{2}/x^{2}+(\sigma-1)^{2}/x^{2}}$. Equivalently,
the kernel is rapidly decreasing in $\lambda d(z,z^{\prime})$. We see that the
kernel is point-wise bounded by $C\lambda^{n}(1+\lambda d(z,z^{\prime}))^{-N}$
for any $N$. The same is true for $\lambda
d_{\lambda}Q_{i}^{\mathrm{high}}(\lambda)$. Again Schur’s test shows that such
kernels are bounded on $L^{2}(M^{\circ})$ uniformly in $\lambda$. ∎

In view of this lemma, we can take
$A(\lambda)=e^{it\lambda}\chi(\lambda)\varphi(2^{-j})Q^{\mathrm{low}}_{i}(\lambda)$
(for $0\leq i\leq N_{l}$), or
$e^{it\lambda}\varphi(2^{-j})(1-\chi)(\lambda)Q^{\mathrm{high}}_{i-N_{l}}(\lambda)$
(for $N_{l}+1\leq i\leq N$), this means that the integrals are well-defined
over any compact interval in $(0,\infty)$, hence the operators $U_{i,j}(t)$
are well-defined. Now we see these operators are bounded on $L^{2}$. We only
consider the low frequency part since a similar argument also gives the
boundedness on $L^{2}$ for high energy part. We have for $0\leq i\leq N_{l}$,
by [20, Lemma 5.3],

(3.5)
$\begin{gathered}U_{i,j}(t)U_{i,j}(t)^{*}=\int\chi(\lambda)^{2}\varphi\big{(}\frac{\lambda}{2^{j}}\big{)}\varphi\big{(}\frac{\lambda}{2^{j}}\big{)}Q^{\mathrm{low}}_{i}(\lambda)dE_{\sqrt{\mathrm{H}}}(\lambda)Q^{\mathrm{low}}_{i}(\lambda)^{*}\\\
=-\int\frac{d}{d\lambda}\Big{(}\chi(\lambda)^{2}\varphi\big{(}\frac{\lambda}{2^{j}}\big{)}\varphi\big{(}\frac{\lambda}{2^{j}}\big{)}Q^{\mathrm{low}}_{i}(\lambda)\Big{)}E_{\sqrt{\mathrm{H}}}(\lambda)Q^{\mathrm{low}}_{i}(\lambda)^{*}\\\
-\int\chi(\lambda)^{2}\varphi\big{(}\frac{\lambda}{2^{j}}\big{)}\varphi\big{(}\frac{\lambda}{2^{j}}\big{)}Q^{\mathrm{low}}_{i}(\lambda)E_{\sqrt{\mathrm{H}}}(\lambda)\frac{d}{d\lambda}Q^{\mathrm{low}}_{i}(\lambda)^{*}.\end{gathered}$

We observe that this is independent of $t$ and we also note that the integrand
is a bounded operator on $L^{2}$, with an operator bound of the form
$C/\lambda$ where $C$ is uniform, as we see from Lemma 3.1 and the support
property of $\varphi$. The integral is therefore uniformly bounded, as we are
integrating over a dyadic interval in $\lambda$. Hence we have shown that

######  Proposition 3.2 ($L^{2}$-estimates).

Let $U_{i,j}(t)$ be defined in (3.3). Then there exists a constant $C$
independent of $t,z,z^{\prime}$ such that $\|U_{i,j}(t)\|_{L^{2}\rightarrow
L^{2}}\leq C$ for all $i\geq 0,j\in\Z$.

###  3.2. Dispersive estimates

Next we aim to establish the dispersive estimates for the microlocalized
$U_{i,j}(t)U^{*}_{i,j}(s)$. We need the following proposition.

######  Proposition 3.3 (Microlocalized dispersive estimates).

Let $Q(\lambda)$ be the operator $Q_{i}^{\mathrm{low}}$ or
$Q_{i}^{\mathrm{high}}$ constructed as in Proposition 2.1 and suppose $\phi\in
C_{c}^{\infty}([1/2,2])$ and takes value in $[0,1]$. Then the kernel estimate

(3.6)
$\begin{split}\Big{|}\int_{0}^{\infty}e^{it\lambda}\phi(2^{-j}\lambda)\big{(}Q(\lambda)&dE_{\sqrt{\mathrm{H}}}(\lambda)Q^{*}(\lambda)\big{)}(z,z^{\prime})d\lambda\Big{|}\\\
&\leq C2^{j(n+1)/2}(2^{-j}+|t|)^{-(n-1)/2}\end{split}$

holds for a constant $C$ independent of $j\in\Z$ and points $z,z^{\prime}\in
M^{\circ}$.

###### Proof.

The key to the proof is to apply Proposition 2.1. For $Q=Q_{i}^{\mathrm{low}}$
for $i=0,1$, or $Q=Q_{1}^{\mathrm{high}}$, we have by Proposition 2.1

$\Big{|}\int_{0}^{\infty}e^{it\lambda}\phi(2^{-j}\lambda)\big{(}Q(\lambda)dE_{\sqrt{\mathrm{H}}}(\lambda)Q^{*}(\lambda)\big{)}(z,z^{\prime})d\lambda\Big{|}\leq
C2^{jn}.$

We use the $N$-times integration by parts to obtain by (2.2)

$\begin{split}&\Big{|}\int_{0}^{\infty}e^{it\lambda}\phi(2^{-j}\lambda)\big{(}Q(\lambda)dE_{\sqrt{\mathrm{H}}}(\lambda)Q^{*}(\lambda)\big{)}(z,z^{\prime})d\lambda\Big{|}\\\
&\leq\Big{|}\int_{0}^{\infty}\big{(}\frac{1}{it}\frac{\partial}{\partial\lambda}\big{)}^{N}\big{(}e^{it\lambda}\big{)}\phi(2^{-j}\lambda)\big{(}Q(\lambda)dE_{\sqrt{\mathrm{H}}}(\lambda)Q^{*}(\lambda)\big{)}(z,z^{\prime})d\lambda\Big{|}\\\
&\leq C_{N}|t|^{-N}\int_{2^{j-1}}^{2^{j+1}}\lambda^{n-1-N}d\lambda\leq
C_{N}|t|^{-N}2^{j(n-N)}.\end{split}$

Therefore we obtain

(3.7)
$\begin{split}&\Big{|}\int_{0}^{\infty}e^{it\lambda}\phi(2^{-j}\lambda)\big{(}Q(\lambda)dE_{\sqrt{\mathrm{H}}}(\lambda)Q^{*}(\lambda)\big{)}(z,z^{\prime})d\lambda\Big{|}\leq
C_{N}2^{jn}(1+2^{j}|t|)^{-N}.\end{split}$

By choosing $N=(n-1)/2$, we prove (3.6). When $Q$ is equal to
$Q_{i}^{\mathrm{low}}$ or $Q_{i}^{\mathrm{high}}$ for $i\geq 2$, we see by
Proposition 2.1

$\begin{split}&\Big{|}\int_{0}^{\infty}e^{it\lambda}\phi(2^{-j}\lambda)\big{(}Q(\lambda)dE_{\sqrt{\mathrm{H}}}(\lambda)Q^{*}(\lambda)\big{)}(z,z^{\prime})d\lambda\Big{|}\\\
&=\Big{|}\int_{0}^{\infty}\left(\frac{1}{i(t-d(z,z^{\prime}))}\frac{\partial}{\partial\lambda}\right)^{N}\big{(}e^{i(t-d(z,z^{\prime}))\lambda}\big{)}\phi(2^{-j}\lambda)\lambda^{n-1}a(\lambda,z,z^{\prime})d\lambda\Big{|}\\\
&\leq
C_{N}|t-d(z,z^{\prime})|^{-N}\int_{2^{j-1}}^{2^{j+1}}\lambda^{n-1-N}(1+\lambda
d(z,z^{\prime}))^{-\frac{n-1}{2}}d\lambda\\\ &\leq
C_{N}2^{j(n-N)}|t-d(z,z^{\prime})|^{-N}(1+2^{j}d(z,z^{\prime}))^{-(n-1)/2}.\end{split}$

It follows that

(3.8)
$\begin{split}&\Big{|}\int_{0}^{\infty}e^{it\lambda}\phi(2^{-j}\lambda)\big{(}Q(\lambda)dE_{\sqrt{\mathrm{H}}}(\lambda)Q^{*}(\lambda)\big{)}(z,z^{\prime})d\lambda\Big{|}\\\
&\leq
C_{N}2^{jn}\big{(}1+2^{j}|t-d(z,z^{\prime})|\big{)}^{-N}(1+2^{j}d(z,z^{\prime}))^{-(n-1)/2}.\end{split}$

If $|t|\sim d(z,z^{\prime})$, it is clear to see (3.6). Otherwise, we have
$|t-d(z,z^{\prime})|\geq c|t|$ for some small constant $c$, then choose
$N=(n-1)/2$ to prove (3.6). ∎

######  Remark 3.4.

If $N=\frac{n-1}{2}$ is not an integer, one may need geometric mean argument
to modify the proof.

As a consequence of Proposition 3.3, we immediately have

######  Proposition 3.5.

Let $U_{i,j}(t)$ be defined in (3.3). Then there exists a constant $C$
independent of $t,z,z^{\prime}$ for all $i\geq 0,j\in\Z$ such that

(3.9) $\|U_{i,j}(t)U^{*}_{i,j}(s)\|_{L^{1}\rightarrow L^{\infty}}\leq
C2^{j(n+1)/2}(2^{-j}+|t-s|)^{-(n-1)/2}.$

##  4\. Strichartz estimates

In this section, we show the Strichartz estimates in Theorem 1.1. To obtain
the Strichartz estimates, we need a variant of Keel-Tao’s abstract Strichartz
estimate for wave equation.

###  4.1. Semiclassical Strichartz estimates

We need a variety of the abstract Keel-Tao’s Strichartz estimates theorem.
This is an analogue of the semiclassical Strichartz estimates for Schrödinger
in [23, 33].

######  Proposition 4.1.

Let $(X,\mathcal{M},\mu)$ be a $\sigma$-finite measured space and
$U:\mathbb{R}\rightarrow B(L^{2}(X,\mathcal{M},\mu))$ be a weakly measurable
map satisfying, for some constants $C$, $\alpha\geq 0$, $\sigma,h>0$,

(4.1) $\begin{split}\|U(t)\|_{L^{2}\rightarrow L^{2}}&\leq C,\quad
t\in\mathbb{R},\\\ \|U(t)U(s)^{*}f\|_{L^{\infty}}&\leq
Ch^{-\alpha}(h+|t-s|)^{-\sigma}\|f\|_{L^{1}}.\end{split}$

Then for every pair $q,r\in[1,\infty]$ such that
$(q,r,\sigma)\neq(2,\infty,1)$ and

$\frac{1}{q}+\frac{\sigma}{r}\leq\frac{\sigma}{2},\quad q\geq 2,$

there exists a constant $\tilde{C}$ only depending on $C$, $\sigma$, $q$ and
$r$ such that

(4.2)
$\Big{(}\int_{\R}\|U(t)u_{0}\|_{L^{r}}^{q}dt\Big{)}^{\frac{1}{q}}\leq\tilde{C}\Lambda(h)\|u_{0}\|_{L^{2}}$

where $\Lambda(h)=h^{-(\alpha+\sigma)(\frac{1}{2}-\frac{1}{r})+\frac{1}{q}}$.

###### Proof.

If $(q,r,\sigma)\neq(2,\infty,1)$ is on the line
$\frac{1}{q}+\frac{\sigma}{r}=\frac{\sigma}{2}$, we replace
$(|t-s|+h)^{-\sigma}$ by $|t-s|^{-\sigma}$ and then we closely follow Keel-
Tao’s argument [22, Sections 3-7] to show (4.2). So we only consider
$\frac{1}{q}+\frac{\sigma}{r}<\frac{\sigma}{2}$. By the $TT^{*}$ argument, it
suffices to show

$\begin{split}\Big{|}\iint\langle U(s)^{*}f(s),U(t)^{*}g(t)\rangle
dsdt\Big{|}\lesssim\Lambda(h)^{2}\|f\|_{L^{q^{\prime}}_{t}L^{r^{\prime}}}\|g\|_{L^{q^{\prime}}_{t}L^{r^{\prime}}}.\end{split}$

By the interpolation of the bilinear form of (4.1), we have

$\begin{split}\langle U(s)^{*}f(s),U(t)^{*}g(t)\rangle&\leq
Ch^{-\alpha(1-\frac{2}{r})}(h+|t-s|)^{-\sigma(1-\frac{2}{r})}\|f\|_{L^{r^{\prime}}}\|g\|_{L^{r^{\prime}}}.\end{split}$

Therefore we see by Hölder’s and Young’s inequalities for
$\frac{1}{q}+\frac{\sigma}{r}<\frac{\sigma}{2}$

$\begin{split}\Big{|}\iint\langle U(s)^{*}f(s),&U(t)^{*}g(t)\rangle
dsdt\Big{|}\\\ &\lesssim
h^{-\alpha(1-\frac{2}{r})}\iint(h+|t-s|)^{-\sigma(1-\frac{2}{r})}\|f(t)\|_{L^{r^{\prime}}}\|g(s)\|_{L^{r^{\prime}}}dtds\\\
&\lesssim
h^{-\alpha(1-\frac{2}{r})}h^{-\sigma(1-\frac{2}{r})+\frac{2}{q}}\|f\|_{L^{q^{\prime}}_{t}L^{r^{\prime}}}\|g\|_{L^{q^{\prime}}_{t}L^{r^{\prime}}}.\end{split}$

This proves (4.2). ∎

###  4.2. Homogeneous Strichartz estimates

To prove the homogeneous Strichartz estimates, we first reduce the estimates
to frequency localized estimates. Using the Littlewood-Paley frequency cutoff
$\varphi_{k}(\sqrt{\mathrm{H}})$, we define

(4.3) $u_{k}(t,\cdot)=\varphi_{k}(\sqrt{\mathrm{H}})u(t,\cdot).$

Notice the frequency cutoffs commute with the operator
$\mathrm{H}=-\Delta_{g}$, the frequency localized solutions
$\\{u_{k}\\}_{k\in\Z}$ satisfy the family of Cauchy problems

(4.4) $\partial_{t}^{2}u_{k}+\mathrm{H}u_{k}=0,\quad
u_{k}(0)=f_{k}(z),~{}\partial_{t}u_{k}(0)=g_{k}(z),$

where $f_{k}=\varphi_{k}(\sqrt{\mathrm{H}})u_{0}$ and
$g_{k}=\varphi_{k}(\sqrt{\mathrm{H}})u_{1}$. By the squarefunction estimates
(2.6) and Minkowski’s inequality, we obtain for $q,r\geq 2$

(4.5)
$\|u\|_{L^{q}(\R;L^{r}(M^{\circ}))}\lesssim\Big{(}\sum_{k\in\Z}\|u_{k}\|^{2}_{L^{q}(\R;L^{r}(M^{\circ}))}\Big{)}^{\frac{1}{2}}.$

Let $U(t)=e^{it\sqrt{\mathrm{H}}}$ be the half wave operator, then we write

(4.6)
$\begin{split}u_{k}(t,z)=\frac{U(t)+U(-t)}{2}f_{k}+\frac{U(t)-U(-t)}{2i\sqrt{\mathrm{H}}}g_{k}.\end{split}$

To prove the homogeneous estimates in Theorem 1.1, that is $F=0$, it suffices
to show by (4.5) and (4.6)

######  Proposition 4.2.

Let $f=\varphi_{k}(\sqrt{\mathrm{H}})f$ for $k\in\Z$, we have

(4.7) $\|U(t)f\|_{L^{q}_{t}L^{r}_{z}(\mathbb{R}\times M^{\circ})}\lesssim
2^{ks}\|f\|_{L^{2}(M^{\circ})},$

where the admissible pair $(q,r)\in[2,\infty]^{2}$ and $s$ satisfy (1.4) and
(1.5).

Now we prove this proposition. By using Proposition 3.2 and Proposition 3.5,
we have the estimates (4.1) for $U_{i,j}(t)$, where $\alpha=(n+1)/2$,
$\sigma=(n-1)/2$ and $h=2^{-j}$. Then it follows from Proposition 4.1 that

$\|U_{i,j}(t)f\|_{L^{q}_{t}(\R:L^{r}(M^{\circ}))}\lesssim
2^{j[n(\frac{1}{2}-\frac{1}{r})-\frac{1}{q}]}\|f\|_{L^{2}(M^{\circ})}.$

Notice that

$U(t)=\sum_{i=0}^{N}\sum_{j\in\Z}U_{i,j}(t),$

we can write

$U(t)f=\sum_{i}\sum_{j\in\mathbb{Z}}\int_{0}^{\infty}e^{it\lambda}\varphi(2^{-j}\lambda)Q_{i}(\lambda)dE_{\sqrt{\mathrm{H}}}(\lambda)\widetilde{\varphi}(2^{-j}\sqrt{\mathrm{H}})f$

where $\widetilde{\varphi}\in C_{0}^{\infty}(\R\setminus\\{0\\})$ takes values
in $[0,1]$ such that $\widetilde{\varphi}\varphi=\varphi$. In view of the
condition $f=\varphi(2^{-k}\sqrt{\mathrm{H}})f$, then
$\widetilde{\varphi}(2^{-j}\sqrt{\mathrm{H}})f$ vanishes if $|j-k|\gg 1$.
Hence we obtain

$\|U(t)f\|_{L^{q}_{t}(\R:L^{r}(M^{\circ}))}\lesssim
2^{k[n(\frac{1}{2}-\frac{1}{r})-\frac{1}{q}]}\|f\|_{L^{2}(M^{\circ})},$

which implies (4.7).

###  4.3. Inhomogeneous Strichartz estimates

In this subsection, we prove the inhomogeneous Strichartz estimates including
the endpoint $q=2$ for $n\geq 4$. Let
$U(t)=e^{it\sqrt{\mathrm{H}}}:L^{2}\rightarrow L^{2}$. We have already proved
that

(4.8) $\|U(t)u_{0}\|_{L^{q}_{t}L^{r}_{z}}\lesssim\|u_{0}\|_{\dot{H}^{s}}$

holds for all $(q,r,s)$ satisfying (1.4) and (1.5). For $s\in\R$ and $(q,r)$
satisfying (1.4) and (1.5), we define the operator $T_{s}$ by

(4.9) $\begin{split}T_{s}:L^{2}_{z}&\rightarrow L^{q}_{t}L^{r}_{z},\quad
f\mapsto\mathrm{H}^{-\frac{s}{2}}e^{it\sqrt{\mathrm{H}}}f.\end{split}$

Then we have by duality

(4.10)
$\begin{split}T^{*}_{1-s}:L^{\tilde{q}^{\prime}}_{t}L^{\tilde{r}^{\prime}}_{z}\rightarrow
L^{2},\quad
F(\tau,z)&\mapsto\int_{\R}\mathrm{H}^{\frac{s-1}{2}}e^{-i\tau\sqrt{\mathrm{H}}}F(\tau)d\tau,\end{split}$

where $1-s=n(\frac{1}{2}-\frac{1}{\tilde{r}})-\frac{1}{\tilde{q}}$. Therefore
we obtain

$\Big{\|}\int_{\R}U(t)U^{*}(\tau)\mathrm{H}^{-\frac{1}{2}}F(\tau)d\tau\Big{\|}_{L^{q}_{t}L^{r}_{z}}=\big{\|}T_{s}T^{*}_{1-s}F\big{\|}_{L^{q}_{t}L^{r}_{z}}\lesssim\|F\|_{L^{\tilde{q}^{\prime}}_{t}L^{\tilde{r}^{\prime}}_{z}}.$

Since $s=n(\frac{1}{2}-\frac{1}{r})-\frac{1}{q}$ and
$1-s=n(\frac{1}{2}-\frac{1}{\tilde{r}})-\frac{1}{\tilde{q}}$, thus
$(q,r),(\tilde{q},\tilde{r})$ satisfy (1.5). By the Christ-Kiselev lemma [8],
we thus obtain for $q>\tilde{q}^{\prime}$,

(4.11)
$\begin{split}\Big{\|}\int_{\tau<t}\frac{\sin{(t-\tau)\sqrt{\mathrm{H}}}}{\sqrt{\mathrm{H}}}F(\tau)d\tau\Big{\|}_{L^{q}_{t}L^{r}_{z}}\lesssim\|F\|_{L^{\tilde{q}^{\prime}}_{t}{L}^{\tilde{r}^{\prime}}_{z}}.\end{split}$

Notice that for all $(q,r),(\tilde{q},\tilde{r})$ satisfy (1.4) and (1.5), we
must have $q>\tilde{q}^{\prime}$. Therefore we have proved all inhomogeneous
Strichartz estimates including the endpoint $q=2$.

##  5\. Wellposedness and scattering

In this section, we prove Theorem 1.3. We prove the result by a contraction
mapping argument. The key point is the application of Strichartz estimates.
Let $q_{0}=(n+1)(p-1)/2$, $q_{1}=2(n+1)/(n-1)$ and $\alpha=s_{c}-\frac{1}{2}$.
For any small constant $\epsilon>0$ such that $2\epsilon<\epsilon(p)$ given by
(1.8), there exists $T>0$ such that

(5.1) $\begin{split}X:=\Big{\\{}u:~{}&u\in C_{t}(\dot{H}^{s_{c}})\cap
L^{q_{0}}([0,T];L^{q_{0}}(M^{\circ}))\cap
L^{q_{1}}([0,T];\dot{H}^{\alpha}_{q_{1}}(M^{\circ})),\\\
&\|u\|_{L^{q_{0}}([0,T];L^{q_{0}}(M^{\circ}))}+\|u\|_{L^{q_{1}}([0,T];\dot{H}^{\alpha}_{q_{1}}(M^{\circ}))}\leq
C\epsilon\Big{\\}}.\end{split}$

Consider the solution map $\Phi$ defined by

$\begin{split}\Phi(u)&=\cos(t\sqrt{\mathrm{H}})u_{0}(z)+\frac{\sin(t\sqrt{\mathrm{H}})}{\sqrt{\mathrm{H}}}u_{1}(z)+\int_{0}^{t}\frac{\sin\big{(}(t-s)\sqrt{\mathrm{H}}\big{)}}{\sqrt{\mathrm{H}}}F(u(s,z))\mathrm{d}s\\\
&=:u_{\text{hom}}+u_{\text{inh}},\end{split}$

where $F(u)=\gamma|u|^{p-1}u$. We claim the map $\Phi:X\rightarrow X$ is
contracting. Indeed, by Theorem 1.1, we obtain

(5.2) $\begin{split}\|u_{\text{hom}}\|_{C_{t}(\dot{H}^{s_{c}})\cap
L^{q_{0}}(\R;L^{q_{0}}(M^{\circ}))\cap
L^{q_{1}}(\R;\dot{H}^{\alpha}_{q_{1}}(M^{\circ}))}\leq
C\big{(}\|u_{0}\|_{\dot{H}^{s_{c}}}+\|u_{1}\|_{\dot{H}^{s_{c}-1}}\big{)}.\end{split}$

Hence we must have

(5.3)
$\begin{split}\|u_{\text{hom}}\|_{L^{q_{0}}([0,T];L^{q_{0}}(M^{\circ}))\cap
L^{q_{1}}([0,T];\dot{H}^{\alpha}_{q_{1}}(M^{\circ}))}\leq\frac{1}{2}C\epsilon\end{split}$

for $T=\infty$ if the initial data has small norm $\epsilon(p)$, or, if not,
this inequality will be satisfied for some $T>0$ by the dominated convergence
theorem. We first note that the Sobolev embedding
$L^{q_{0}}_{t}\dot{H}^{\alpha}_{r_{0}}\hookrightarrow L_{t,z}^{q_{0}}$ where
$r_{0}=2n(n+1)(p-1)/[(n^{2}-1)(p-1)-4]$. Under the condition
$p\in[p_{\mathrm{conf}},1+\frac{4}{n-2}]$, it is easy to check that the pairs
$(q_{0},r_{0}),(q_{1},q_{1})$ satisfy (1.4) and (1.5) with $s=1/2$. Applying
Theorem 1.1 with $\tilde{q}^{\prime}=\tilde{r}^{\prime}=\frac{2(n+1)}{n+3}$,
one has

(5.4) $\begin{split}\|u_{\text{inh}}\|_{C_{t}(\dot{H}^{s_{c}})\cap
L^{q_{0}}([0,T];L^{q_{0}}(M^{\circ}))\cap
L^{q_{1}}([0,T];\dot{H}^{\alpha}_{q_{1}}(M^{\circ}))}\leq
C\|F(u)\|_{L^{\tilde{q}^{\prime}}_{t}\dot{H}^{\alpha}_{\tilde{r}^{\prime}}}.\end{split}$

By the assumption on $p$, we have $0\leq\alpha\leq 1$. By using the fraction
Liebniz rule for Sobolev spaces on the asymptotically conic manifold [9,
Theorem 27], we have

(5.5)
$\begin{split}\|F(u)\|_{L^{\tilde{q}^{\prime}}_{t}\dot{H}^{\alpha}_{\tilde{r}^{\prime}}}\leq
C\|u\|^{p-1}_{L^{q_{0}}_{t,z}}\|u\|_{L^{q_{1}}_{t}\dot{H}^{\alpha}_{q_{1}}}\leq
C^{2}(C\epsilon)^{p-1}\epsilon\leq\frac{C\epsilon}{2}.\end{split}$

A similar argument as above leads to

(5.6)
$\begin{split}&\|\Phi(u_{1})-\Phi(u_{2})\|_{L^{q_{1}}([0,T];\dot{H}^{\alpha}_{q_{1}}(M^{\circ}))\cap
L^{q_{0}}([0,T];L^{q_{0}}(M^{\circ}))}\\\ &\leq
C\|F(u_{1})-F(u_{2})\|_{L^{\tilde{q}^{\prime}}_{t}\dot{H}^{\alpha}_{\tilde{r}^{\prime}}}\\\
&\leq
C^{2}(C\epsilon)^{p-1}\|u_{1}-u_{2}\|_{L^{q_{1}}([0,T];\dot{H}^{\alpha}_{q_{1}}(M^{\circ}))\cap
L^{q_{0}}([0,T];L^{q_{0}}(M^{\circ}))}\\\
&\leq\frac{1}{2}\|u_{1}-u_{2}\|_{L^{q_{1}}([0,T];\dot{H}^{\alpha}_{q_{1}}(M^{\circ}))\cap
L^{q_{0}}([0,T];L^{q_{0}}(M^{\circ}))}.\end{split}$

Therefore the solution map $\Phi$ is a contraction map on $X$ under the metric
$d(u_{1},u_{2})=\|u_{1}-u_{2}\|_{{L^{q_{1}}([0,T];\dot{H}^{\alpha}_{q_{1}}(M^{\circ}))}\cap
L^{q_{0}}([0,T];L^{q_{0}}(M^{\circ}))}$. The standard contraction argument
completes the proof of Theorem 1.3.

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