\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2005(2005), No. 74, pp. 1--17.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2005 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2005/74\hfil Damped wave equations with odd initial data]
{$L^p$-$L^q$ estimates for damped wave equations with odd initial data}
\author[T. Narazaki\hfil EJDE-2005/74\hfilneg]
{Takashi Narazaki} 

\address{Takashi Narazaki\hfill\break
 Department of Mathematical Sciences, Tokai University,
 Hiratsuka 259-1292, Japan}
\email{narazaki@ss.u-tokai.ac.jp}

\date{}
\thanks{Submitted July 5, 2003. Published July 5, 2005.}
\thanks{Partially supported by Grand-in-Aid 16540205 from Science Research JSPS}
\subjclass[2000]{35B40, 35L05}
\keywords{Damped wave equation; $L^p$-$L^q$ estimate; odd initial data}

\begin{abstract}
 We study the Cauchy problem for the damped wave equation.
 In a previous paper \cite{Narazaki} the author has shown the
 $L^{p}$-$L^{q}$ estimates between the solutions of the damped
 wave equation and the solutions of the corresponding heat equation.
 In this paper, we show new $L^{p}$-$L^{q}$ estimates for the damped
 wave equation with odd initial data.
\end{abstract}
\dedicatory{Dedicated to the memory of Professor Tsutomu Arai}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}{Proposition}[section]
\newtheorem{lemma}{Lemma}[section]
\newtheorem{corollary}{Corollary}[section]

\section{Introduction}

Consider the Cauchy problem for the damped wave equation
\begin{equation}
\partial _{t} ^{2} u - \Delta u + 2 a \partial _{t} u = 0, \quad
 (t, x) \in (0, \infty) \times R ^{n}
\label{linear_equation}
\end{equation}
with initial data
\begin{equation}
u(0, x ) = \varphi _{0} ( x ), \quad \partial _{t} u(0, x ) =
\varphi _{1} (x), \quad x \in R ^{n}, \label{initial_data}
\end{equation}
where $ a $ is a positive constant, $\partial _{t} =  \partial /
\partial t$, $\partial _{j} = \partial / \partial x _{j} $ for
$j=1, 2, \dots, n$ and $\Delta = \partial _{1} ^{2} + \dots +
\partial _{n} ^{2} $ is the Laplace operator in $ R ^{n} $.
Here and after we denote $\partial _{x} ^{\alpha} =
\partial _{1} ^{\alpha _{1} } \dots
\partial _{n} ^{\alpha _{n} }$ and
$|\alpha| = \alpha _{1} + \dots + \alpha _{n} $ for a multi-index
of non-negative integers $ \alpha = (\alpha _{1} , \dots, \alpha
_{n} )$, and $\nabla h = (\partial _{1} h, \dots,
\partial _{n} h)$.

Several authors have indicated the diffusive structure of problem
(\ref{linear_equation})--(\ref{initial_data}) as $ t \to \infty$;
see for example \cite{Bellout,Karch,Li,Marcati,Narazaki,Nishihara}.
 Recently the author has
shown the $ L ^{p} $-$L ^{q} $ estimates of the difference between
the solution of problem
(\ref{linear_equation})--(\ref{initial_data}) and the solution of
the corresponding heat equation
\begin{equation}
2 a \partial _{t} \phi - \Delta \phi = 0, \quad (t, x) \in (0,
\infty) \times R ^{n} \label{heat_equation}
\end{equation}
with initial data
\begin{equation}
\phi(0, x ) = \varphi _{0} ( x ) + \varphi _{1}(x)/2a, \quad x \in
R ^{n}. \label{heat_initial}
\end{equation}

We use the standard function spaces $L ^{p} = L ^{p} ( R^{n} )$,
$L ^{p} = H ^{0} _{p}$ and $H ^{s} _{p} = H ^{s} _{p} ( R ^{n} ) =
( 1 - \Delta ) ^{-s/2} L ^{p}$ equipped with the norms
\[
\| f \| _{ H ^{s} _{p}} \equiv \| f \| _{s, p} \equiv \| \mathcal{
F} ^{-1} ( (1 + | \xi | ^{2} ) ^{s/2} \hat{f} ) | |
_{p},
\]
where $\| f \| _{p} $ denotes the usual $L ^{p}$-norm. $\mathcal{
F}$ denotes the Fourier transformation:
\[ ( \mathcal{F} f ) (\xi) \equiv \widehat{f}(\xi)
\equiv \big( \frac{1}{2\pi} \big) ^{n/2} \int _{ R ^{n}}
 e ^{-i x \cdot \xi} f(x) \, d x,
\]
$\mathcal{F} ^{-1}$ denotes an inverse of $\mathcal{F} $, and $
* $ denotes the convolution with respect to $ x $;
\[
( f * g )( x ) = \int _{ R ^{n}} f(x-y) g(y) \, dy .
\]
Let $X_1 \cap \dots \cap X_m$ be the normed space equipped with
norm $\| \cdot \|_{X_1 \cap \dots \cap X_m} \equiv \| \cdot
\|_{X_1} +\dots+ \| \cdot \|_{X_m} $ for normed spaces $X_1, \dots
, X_m$, and let $[ \mu ]$ denote the greatest integer that does
not exceed $\mu$.

To illustrate the decay profiles of problem
(\ref{linear_equation})--(\ref{initial_data}) we set
$\varphi_0(x)=\varphi_1(x)=x_1 \dots x_d \exp( -a|x|^2/2)$, where
$d \in [0,n]$ is an integer. Let $u$ and $\phi$ be the solutions
of problem (\ref{linear_equation})--(\ref{initial_data}) and
problem (\ref{heat_equation})--(\ref{heat_initial}), respectively.
Since
\[
\phi (t, x)=(1+1/2a)(t+1)^{-n/2-d} x _{1} \dots x _{d} \exp \big(
-\frac{a |x| ^{2}}{ 2(t+1)} \big),
\]
it follows that
\[
\| \phi(t, \cdot) \|_{p} = C(1+t)^{-n/2(1-1/p)-d/2}, \quad 1 \le p
\le \infty, t > 0.
\]
Hence, Theorems \ref{thm1.1}--\ref{thm1.2} below show that
\begin{equation}
\widetilde{C}_1 (1+t)^{-n(1-1/p)-d/2} \le \| u(t, \cdot) \|_{p}
\le \widetilde{C}_2(1+t)^{-n/2(1-1/p)-d/2} \label{add1}
\end{equation}
for any $p \in [1,\infty]$ and sufficiently large $t >0$, where
$\widetilde{C}_1$ and $\widetilde{C}_2$ are positive constants
that depend only on $n$, $d$, $p$ and $a$. When $d=0$,
(\ref{add1}) indicates that the optimal decay rate of $L ^{p}$
norm of the solution to (\ref{linear_equation}) is
$(1+t)^{-n/2(1-1/p)}$ as $t \to \infty$. When $d \ge 1$,
(\ref{add1}) also shows that the solution decays faster than
solutions with general initial data. This faster decay seems to be
caused by the fact $(\partial/\partial \xi) ^{\alpha} \widehat{
u}(t, 0)=0$ for $|\alpha|<d$. When the initial data are odd in the
sense of (\ref{odd}) below, the solution $u$ of
(\ref{linear_equation}) satisfies $(\partial/\partial \xi)
^{\alpha} \widehat{ u}(t, 0)=0$ for $|\alpha|<d$. Hence, we may
expect a new $L^p$-$L^q$ estimates of the solutions of problem
(\ref{linear_equation})--(\ref{initial_data}), when initial data
are odd.

The aim in this paper is to show the new $ L ^{p} $-$ L ^{q} $
estimates to the solutions of problem
(\ref{linear_equation})--(\ref{initial_data}), when the initial
data $ ( \varphi _{0}, \varphi _{1})$ are odd in the sense of
(\ref{odd}). These new $ L ^{p} $-$ L ^{q} $ estimates imply the
new decay estimates to the solution of problem
(\ref{linear_equation})--(\ref{initial_data}).

Let $d \in [1, n]$ be an integer, and
$x \equiv (x', x'') \equiv
 ( x _{1}, \dots, x _{d}, x _{d+1}, \dots , x _{n} )$.
A function $f(x)$ defined on $R ^{n}$ is said to be odd with
respect to $x'$ when it satisfies
\begin{equation}
f ( x _{1}, \dots, - x _{j}, \dots, x _{n} ) = - f(x _{1}, \dots,
x _{j}, \dots, x _{n} ), \quad (j=1, \dots, d). \label{odd}
\end{equation}
Define the weight function $P(\cdot)$ by
\begin{equation}
P(x) = (1 + x _{1} ^{2} ) ^{1/2} \cdot \dots \cdot (1 + x _{d}
^{2} ) ^{1/2}. \label{P_teigi}
\end{equation}

Our first result is as follows.


\begin{theorem}[Estimate of the low frequency part] \label{thm1.1}
Let $1 \le q \le p \le \infty $, $\epsilon>0$, and let $b>0$ be
constants. Let $v$ be the solution of \eqref{linear_equation} with
initial data
\[
v ( 0, x) = v _{0} ( x ), \quad \partial _{t} v( 0, x ) = v _{1} (
x ), \quad  x \in  R ^{n}.
\]
Let $V$ be the solution of (\ref{heat_equation}) with initial data
\[
V(0, x ) = v _{0} ( x ) + v _{1} (x) /2a, \quad x \in R ^{n}.
\]
Assume that the function $ v _{i} $ is odd with respect to $x'$
and it satisfies
\[
P(\cdot) v _{i} \in L ^{q}, \quad
 \mathop{\rm supp}\widehat{v} _{i} \subset \{ \xi; |\xi| \le b \}
\quad (i=0, 1).
\]
Then, for any $\theta \in [0, 1]$, for a multi-index $\alpha = (
\alpha _{1}, \dots, \alpha _{n})$ and for a non-negative integer
$k$, the following estimates holds:
\begin{align*}
& \| P(\cdot) ^{\theta} \partial _{t} ^{k}
\partial _{x} ^{ \alpha}
(  v(t, \cdot) - V(t, \cdot) ) | | _{p}\\
& \le C ( 1 + t ) ^{-n \delta(p, q)
 - |\alpha|/2 - k - (1 - \theta)d/2
 - 1 + \epsilon}
(\| P(\cdot) v _{0} \| _{q} + \| P(\cdot) v _{1} \| _{q} )
\end{align*}
for some constant $C=C(p, q, \epsilon)>0$, where $ \delta(p, q) = 1/2q - 1/2p$.
When $ 1 < q < p < \infty$, $p=\infty$ and  $ q = 1$ or $p=q=2$,
we may take $\epsilon =0 $ in the above estimates.
\end{theorem}

The decay property of the solution to (\ref{heat_equation}) with
odd initial data (Proposition \ref{prop3.1} below, see also \cite{Meier})
shows the following estimates.

\begin{corollary} \label{coro1.1}
Under the assumptions of Theorem \ref{thm1.1},
\[
\| P(\cdot) ^{\theta} \partial _{t} ^{k}
\partial _{x} ^{\alpha} v(t, \cdot) \| _{p}
\le C( 1 + t ) ^{-n \delta(p, q) - |\alpha|/2 - k - (1 -
\theta)d/2} (\|P(\cdot) v _{0} \| _{q} + \| P(\cdot) v _{1} \|
_{q} ).
\]
\end{corollary}

Similar arguments to ones in \cite{Narazaki} give
the following estimates.

\begin{theorem}[Estimate of high frequency part] \label{thm1.2}
Let  $1 < q \le p < \infty$ and $\theta=0, 1$.
Assume that
 $P(\cdot) ^{\theta} w _{i} \in L ^{q}$,
$\mathop{\rm supp}\widehat{w} _{i} \subset \{ \xi; |\xi| \ge 2a \}$
for $ i=0, 1$.
Then the solution $w$ of (\ref{linear_equation}) with initial data
\[
w(0,x) = w _{0}( x ), \quad \partial _{t} w ( 0, x) = w _{1} ( x
), \quad x \in R \sp {n}
\]
satisfies
\begin{align*}
&  \| P(\cdot) ^{\theta} ( w(t, \cdot) - e ^{-at} \mathcal{
F} ^{-1} ( M _{0}( t , \cdot ) \widehat{w} _{0} + M _{1}( t ,
\cdot ) \widehat{w} _{1} ) ) \| _{p}\\
& \le C(p, q) e ^{-a t}(1 + t) ^{N} ( \| P(\cdot) ^{\theta} w
_{0} \| _{q} + \| P(\cdot) ^{\theta} w _{1} \| _{q} )
\end{align*}
for some constant $N=N(n) >0$  and $C(p, q)>0$,
where
\begin{align*}
 M _{1}( t , \xi ) &= \frac{1}{\sqrt{|\xi| ^{2} - a ^{2}}}
\Big( \sin t |\xi| \sum _{0 \le k<(n-1)/4} \frac{(-1) ^{k}}{(2k)!} t
^{2k} \Theta(\xi) ^{2k} \\
&\quad  -\cos t|\xi| \sum _{0 \le k< (n-3)/4}
\frac{(-1) ^{k}}{(2k+1)!} t ^{2k+1} \Theta(\xi) ^{2k+1} \Big),
\\
M _{0}( t, \xi) &= \cos t|\xi| \sum _{0 \le k < (n + 1
)/4} \frac{( - 1) ^{k} }{(2k)!} t ^{2k} \Theta(\xi) ^{2k}\\
&\quad  + \sin t |\xi| \sum _{0 \le k < ( n - 1)/4} \frac{(-1)
^{k}}{(2k+1)!} t ^{2k+1} \Theta(\xi) ^{2 k + 1} + a M _{1}( t ,
\xi ),
\end{align*}
and $\Theta(\xi) \equiv \Theta(|\xi|) \equiv
 |\xi| - \sqrt{ |\xi| ^{2} - a ^{2} }$.
\end{theorem}

\begin{corollary} \label{coro1.2}
Let $m=[n/2]$ and $\max (0, 1/2-1/2m) < 1/p < \min (1, 1/2 + 1/2m)$.
Under the assumptions in Theorem \ref{thm1.2}, the following estimate holds;
\[
\| P(\cdot) ^{\theta} w(t, \cdot) \| _{p} \le C e
^{-at/2} ( \| P(\cdot) ^{\theta} w _{0} \| _{1,p} + \| P(\cdot)
^{\theta} w _{1} \| _{p} ).
\]
\end{corollary}

\section{Preliminaries}

In this section we state the preliminary results necessary for the
proofs. $J_{\mu}(s) $ is the Bessel function of order $\mu$. We
shall denote $\widetilde{J} _{\mu} (s) = J_{\mu}(s) /s^{\mu}$
according to Levandosky \cite{Levandosky}. Here and after we
denote $g(s)=O(|s|^{\sigma})$ when $|g(s)| \le C |s|^{\sigma}$ for
a constant $\sigma$.

\begin{lemma} [\cite{Levandosky, Narazaki}] \label{lem2.1}
Assume that $\mu$ is not a negative integer.
Then it follows that:
\begin{enumerate}
\item $s \widetilde{J}_{\mu}'(s)=\widetilde{J}_{\mu - 1}(s)
- 2 \mu \widetilde{J}_{\mu}(s) $.
\item $\widetilde{J}_{\mu}'(s) = - s\widetilde{J}_{\mu + 1}(s) $.
\item $\widetilde{J}_{-1/2}(s)=\sqrt{\frac{\pi}{2}} \cos s$ .
\item If $\mathop{\rm Re} \mu $ is fixed, then
\begin{gather*}
|\widetilde{J}_{\mu}(s)| \le C e^{\pi | {\rm Im \ }  \mu|} ,
\quad (|s| \le 1) , \\
 J_{\mu}(s) = C s^{-1/2} \cos ( s - \frac{\mu}{2}\pi
- \frac{\pi}{4} ) + O( e^{2\pi| {\rm Im \ } \mu|} |s|^{-3/2} )
,\quad (|s| \ge 1) .
\end{gather*}
\item $r^2 \rho \widetilde{J}_{\mu + 1}(r \rho )=
-\frac{\partial}{\partial \rho}\widetilde{J}_{\mu}
(r \rho)$ .
\end{enumerate}
\end{lemma}

The following lemmas are well-known.

\begin{lemma}[\cite{Stein}]  \label{lem2.2}
Assume that $\widehat{f} \in L^p$ $(1 \le p \le 2)$ is a radial function.
Then
\[
f(x) = c \int_0 ^{\infty} g(\rho) \rho ^{n-1}
\widetilde{J}_{n/2-1}(|x| \rho) \, d\rho, \quad g(|\xi|) \equiv
\widehat{f}(\xi).
\]
\end{lemma}

\begin{lemma}[Young] \label{lem2.3}
Let $1 \le q \le p \le \infty$ satisfy $1-1/r=1/q-1/p$,
then the following estimate holds for any $f \in L^q $ and $g \in L^{r} $:
\[
\| f*g \|_p \le C  \| f \|_q \| g \|_r . \]
\end{lemma}

\begin{lemma}[Hardy-Littlewood-Sobolev] \label{lem2.4}
Let $ 1<q<p<\infty$ satisfy $ 1-1/r=1/q-1/p$.
Assume that $ |g(x)| \le A |x|^{-n/r}$,
where $A$ is a constant.
Then the following estimate holds for any $f \in L^p $:
\[
\| f*g \|_p \le C(p, q) A \| f \|_q .
\]
\end{lemma}

\section{Proof of Theorem \ref{thm1.1}}

Let $V$ be the solution of the heat equation
\begin{equation}
2 a \partial _{t} V (t, x) - \Delta V (t, x) = 0, \quad t>0, x \in
R ^{n} \label{heat_equation2}
\end{equation}
with initial data
\begin{equation}
V(0, x) = V_0 (x), \quad x \in R ^{n}. \label{heat_initial2}
\end{equation}
Assume that the function $V _{0}$ is odd with respect to $x'$ and
$V_0 \in L ^{q} $ for some $1 \le q \le \infty$. Then, $V (t,
\cdot)$ is also odd with respect to $x'$.
Arguments similar  to those in \cite{Meier} and \cite{Narazaki} give
the following result.

\begin{proposition}[Meier \cite{Meier}] \label{prop3.1}
Let $1 \le q \le p \le \infty$, $ 0 \le \theta_1, \dots, \theta_d
\le 1$ and $b>0$ be constants. Assume that $V_0$ is odd with
respect to $x'$, $ P (\cdot) V_0 \in L ^{q} $ and ${\hat V} _{0}
(\xi)=0 $ for $ |\xi| \ge b$. Let $V$ be the solution of the
Cauchy problem \eqref{heat_equation2}--\eqref{heat_initial2}.
Then, for $t>0$, $V(t, \cdot)$ is odd with respect to $x'$ and
${\hat V}(t, \xi) = 0$ for $|\xi| \ge b $. Moreover, for any
multi-index of non-negative integers $\alpha=(\alpha_1, \dots,
\alpha_n)$ and for any integer $k \ge 0$, the following estimates
hold;
\begin{align*}
&\|(1+x_1^{2}) ^{\theta_1/2} \dots (1+x_d ^{2})
^{\theta_d/2} \partial_ t ^{k} \partial_ x ^{\alpha} V(t, \cdot)
| | _{p} \\
&\le C (1 + t ) ^{-n \delta(p, q) - k - |\alpha|/2 - (1 -
\theta_1)/2 - \dots - (1-\theta_d)/2} \| P(\cdot) V_0 \|_{q},
\end{align*}
where $ \delta(p, q) = 1/2q - 1/2p$.
\end{proposition}


Choose a function $\chi_1$ of class $C ^{\infty}$ satisfying $
\chi_1(\rho) = 1 $ for $\rho \le a/2$ and $ \chi_1(\rho) = 0 $ for
$\rho \ge 2a/3$. Define the functions $\Theta_1$ and $g$ by
\begin{gather}
\Theta_{1}(\rho)= \frac{\rho^{4}}{ 2a( a
+\sqrt{a^{2}-\rho^{2}})^{2}}, \label{Theta_1_teigi}
\\
g(t, \rho) =  ( \exp ( -t \Theta_{1}(\rho)) -1) \exp \big( -\frac{t
\rho ^{2}}{4a} \big). \label{g_teigi}
\end{gather}

Here and after we denote $\chi _{1}(\xi)=\chi _{1} (|\xi|)$ and
$g(t,\xi)=g(t,|\xi|)$. For the proof of Theorem \ref{thm1.1}, we need the
following lemmas.
 Let $\mathcal{I}$ be the set of all multi-indices
$\alpha=(\alpha _{1}, \dots, \alpha _{n})$ satisfying $\alpha
_{j}=0$, $1$ for $j=1, \dots, d$ and $\alpha _{j}=0$ for $j=d+1,
\dots, n$.

\begin{lemma} \label{lem3.1}
Let $1 \le q \le p \le \infty$ and $b>0$ be constants, and let
$\chi_{11}$ be a function of class $C ^{\infty}$ satisfying $\chi
_{11}(\xi) = 0 $ for $|\xi| \ge b$. Then, the estimates
\[
\| P(\cdot) \mathcal{F} ^{-1} ( \chi_{11}\widehat{h} ) \| _{p}
\le C _{b} \sup_{\xi} \sum_{|\alpha| \le n+d+1}
|\partial_{\xi} ^{\alpha} \chi_{11}(\xi) | \|P(\cdot) h \| _{q}
\]
hold for any $h$ satisfying $P(\cdot) h \in L^q$.
\end{lemma}

\begin{proof} Since $P(x) \le C_{1} \sum _{ \alpha \in
\mathcal{I} } |x ^{\alpha}| \le C _{2} P(x)$ and $\mathcal{F} (
x ^{\alpha} f )(\xi)= c _{\alpha}
\partial _{\xi} ^{\alpha} {\hat f}(\xi)$, it follows that
\begin{equation}
 \| P(\cdot)\mathcal{F}^{-1} ( \chi_{11} \widehat{h} )\| _{p}
\le  C \sum _{\alpha \in \mathcal{I}} \|
  x ^{\alpha} \mathcal{F} ^{-1} ( \chi _{11} {\hat h} )\|_{p}
\le C \sum_{\alpha \in \mathcal{I}} \sum _{\beta+ \gamma=
\alpha} \| \mathcal{F}^{-1} (
\partial_{\xi}^{\beta} \chi_{11} \partial_{\xi} ^{\gamma}
\widehat{h} ) \| _{p}. \label{lemma3_1_1}
\end{equation}
Since $(1+|x|)^{-(n+1)} \in L ^{1}$ and $\mathop{\rm supp}\chi _{11} \subset
\{ \xi: |\xi| \le b\}$, it follows that
\begin{align*}
\|\mathcal{F} ^{-1} (
\partial_{\xi} ^{\beta} \chi_{11} )\|_{L ^{1} \cap L ^{\infty}}
& \le  C \|
(1+|x|)^{n+1} \mathcal{F} ^{-1} (
\partial_{\xi} ^{\beta} \chi_{11} )\|_{\infty} \\
& \le  C \sum_{ |\alpha| \le n+1} \| \partial _{\xi} ^{\alpha}
\partial _{\xi} ^{\beta} \chi _{11} \| _{L ^{1}}
\\
& \le  C \sup _{\xi} \sum _{|\alpha| \le n+1}  \big|
\partial _{\xi} ^{\alpha}\partial _{\xi} ^{\beta} \chi _{11} (\xi)
\big|.
\end{align*}
Hence, for any $\beta$  satisfying $|\beta| \le d$,
\begin{equation}
\|\mathcal{F} ^{-1} (
\partial_{\xi} ^{\beta} \chi_{11} )
| |_{L^1 \cap L^{\infty}}
\le C \sup_{\xi} \sum_{|\alpha| \le n+d+1} |
\partial_{\xi} ^{\alpha} \chi_{11}(\xi) |.
\label{lemma3_1_2}
\end{equation}
Since
\begin{gather*}
\mathcal{F}^{-1} ( \partial_{\xi} ^{\beta} \chi_{11}
\partial_{\xi} ^{\gamma} \widehat{h} ) = c \mathcal{F}
^{-1} ( \partial_{\xi} ^{\beta} \chi_{11} )* \mathcal{F} ^{-1} (
\partial_{\xi} ^{\gamma} {\hat h} ),
\\
\| \mathcal{F} ^{-1} ( \partial_{\xi} ^{\gamma} {\hat h} ) \|
_{q} \le C \| P(\cdot) h \| _{q} , \quad \gamma \in \mathcal{I},
\end{gather*}
Lemma \ref{lem2.3} and estimates (\ref{lemma3_1_1})--(\ref{lemma3_1_2})
give the desired estimate.
\end{proof}


Note that the function
\[
I(t, x)=\mathcal{F} ^{-1} ( \chi_1 g(t, \cdot) )(x)
=(\frac{1}{2\pi} )^{n/2}\int_{R^n} e^{ix \cdot \xi} \chi
_{1}(\xi)g(t, \xi) \, d\xi
\]
is a radial function and  belongs to $\mathcal{S}(R ^{n})$ for any
$t \ge 0$.

\begin{lemma} \label{lem3.2}
For any $t >0$, the following two estimates hold
\begin{gather}
\sup _{x} | I(t, x) | \le C(1+t) ^{-n/2 -1},
\label{lemma3_2_1} \\
\sup _{x} |(1+ |x|) ^{n+1/2} I(t, x) | \le C(1+t) ^{ -
3/4}. \label{lemma3_2_2}
\end{gather}
\end{lemma}

\begin{proof} We prove only the case where $n=1$.
For the proof when $ n \ge 2$, see  \cite[Proposition 3.1]{Narazaki}.
Since
\begin{equation}
I(t, x)= \sqrt{\frac{2}{\pi}} \int _{0} ^{\infty} \chi _{1}(\rho)
g (t, \rho) \cos \rho |x| \, d\rho \label{lemma3_2_1pr}
\end{equation}
and
\[
| g (t, \rho) | \le C t \rho ^{4} \exp \big( - \frac{t \rho
^{2}}{4a} \big), \quad ( 0 \le \rho \le 2a/3),
\]
easy calculations show that
\[
| I(t, x)| \le C \int _{0} ^{2a/3} t \rho ^{4} \exp \big(
- \frac{t \rho ^{2}}{4a} \big) \, d\rho \le C ( 1+ t) ^{-3/2}.
\]
Thus we have proved estimate (\ref{lemma3_2_1}). Since
\[
\cos \rho |x| = -\frac{1}{ |x| ^{2} } ( \frac{\partial}{\partial
\rho} ) ^{2} \cos \rho |x|,
\]
Using integration by parts in (\ref{lemma3_2_1pr}),
\begin{equation}
| I(t, x) | \le \frac{C}{ x ^{2}} \int _{0} ^{\infty}
\big|( \frac{\partial}{\partial \rho} ) ^{2} ( \chi _{1} (\rho)
g(t, \rho) ) \big| \, d\rho \le \frac{C}{ x ^{2}} (1+t) ^{-1/2},
\label{lemma3_2_2pr}
\end{equation}
where we have used
\[
| \frac{\partial g}{\partial \rho} (t, \rho) |+
|\frac{\partial^{2} g}{\partial \rho ^{2}} (t, \rho) |
 \le C \exp \big( - \frac{t \rho ^{2}}{8a} \big), \quad (0 \le \rho \le a).
\]
Estimates (\ref{lemma3_2_1}) and (\ref{lemma3_2_2pr}) show that
\[
| I(t, x) | \le  \frac{C}{1 + x ^{2} } (1+t) ^{-1/2}.
\]
Therefore,
\[
| I(t, x) | \le  C ( (1+t) ^{ - 3/2} ) ^{1/4} (\frac{1}{ 1 + x ^{2} }
(1+t) ^{-1/2}) ^{3/4}
\le  \frac{C}{ (1+|x|) ^{3/2}} (1+t) ^{-3/4}.
\]
Thus we have proved estimate (\ref{lemma3_2_2}).
\end{proof}

\begin{lemma} \label{lem3.3}
Let $ 1 \le q \le p \le \infty$, and let $k$ be a non-negative integer.
Then
\[
\| \partial _{t} ^{k} I(t, \cdot) * f \| _{p} \le C(1+t)^{-n
\delta(p, q) - k  -1+ \epsilon}
 \|f\| _{q}, \quad t \ge 0
\]
for any $ \epsilon > 0 $, where
$C=C(p, q, \epsilon, k)>0$ and $\delta(p, q) =1/2q - 1/2p$.
We may take $\epsilon=0$ when $1<q<p<\infty$,
$p=\infty$ and $q=1$ or $p=q=2$.
\end{lemma}

\begin{proof}
Consider the case where $k=0$. Lemma \ref{lem3.2} shows
\begin{equation}
\| I(t, \cdot) * f \| _{\infty} \le C(1+t) ^{ -n/2 - 1} \|
f\|_{1}. \label{lemma3_3_1pr}
\end{equation}
Since
\[
| \chi_1(\xi) g( t, \xi)| \le C|\chi_1(\xi)| |\xi| ^{4} t \exp \big(
-\frac{|\xi|^{2} t}{4a} \big) \le C \min ( \frac{1}{t }, \ t ) \le
\frac{C}{1+t },
\]
it follows that
\begin{equation}
 \| I(t, \cdot) * f\| _{2} \le C
\| \chi_1  g_1(t, \cdot) \widehat{f}\|_{2} \le C (1+t) ^{ -1}
\|f\|_{2}. \label{lemma3_3_2pr}
\end{equation}
Set $r \in [1, \infty]$ by $1 - 1/r = 2\delta(p, q)$, and set
$\theta=2n/((2n+1)r) \in [0, 1)$, then Lemma \ref{lem3.2}
shows
\[ | I(t, x)|= | I(t, x)|^{\theta}| I(t, x)|^{1-\theta}
\le C (1+t) ^{- n \delta(p, q) - 1} |x| ^{-n/r}.
\]
Hence Lemma \ref{lem2.4} show
\begin{equation}
\| I(t, \cdot) * f\| _{p} \le C(1+t) ^{-n \delta(p, q) -1}
\|f\|_{q}, \quad (1 < q < p < \infty). \label{lemma3_3_3pr}
\end{equation}
Since Lemma \ref{lem3.2} also shows $|I(t, x)| \le C(1 + t) ^{- 1 +
\epsilon} (1 + |x|)^{-n-2\epsilon}$ for $0 < \epsilon \le 1/4$, it
follows that
\[
\|I(t,\cdot)\|_1 \le C(\epsilon )(1 + t) ^{- 1 + \epsilon}, \quad
(0 < \epsilon \le 1/4).
\]
Therefore, Lemma \ref{lem2.3} gives
\begin{equation}
\| I(t, \cdot) * f\| _{p} \le C(\epsilon ) (1+t) ^{- 1 + \epsilon}
\|f\|_{p}, \quad (1 \le p \le \infty). \label{lemma3_3_4pr}
\end{equation}
Estimates (\ref{lemma3_3_1pr})--(\ref{lemma3_3_4pr}) give the desired estimate
when $k=0$.

Now consider the case where $k \ge 1$.
Easy calculations show that
\begin{equation}
\partial _{t} ^{k} \widehat{I}(t, \xi)
= |\xi|^{2k} \Big(B_{k,1}(\xi) \widehat{I}(t, \xi) + B_{k,2}(\xi)
|\xi|^{2} \chi_{1} (\xi) \exp \big(-\frac{|\xi|^{2} t}{4a} \big)\Big),
\label{lemma3_3_5pr}
\end{equation}
where $B_{k,1}, B_{k,2} \in C^{\infty}$ satisfying $B _{k,1}(\xi)
= B _{k,2}(\xi)=0$ when $|\xi| \ge 2a/3$. Since $\mathcal{F}
^{-1} B _{k, i} \in \mathcal{S}( R ^{n} )$ for $i=1, 2$, the
well-known estimate
\[
\| \mathcal{F} ^{-1} \Big( |\xi| ^{2k + 2} \chi _{1} \exp
\big(-\frac{|\xi|^{2} t}{4a} \big) \widehat{f} \big) \| _{p} \le C(1+t) ^{-n
\delta(p, q) - k - 1} \|f\| _{q}
\]
hold for $1 \le q \le p \le \infty$.
Hence, the estimates when $k=0$ and (\ref{lemma3_3_5pr}) give
the desired estimate in the case where $k \ge 1$.
\end{proof}

 From  Proposition \ref{prop3.1} and Lemma \ref{lem3.3}, we obtain the next lemma.

\begin{lemma} \label{lem3.4}
Let $ 1 \le q \le p \le \infty$, $0 \le \theta \le 1$ and
$\epsilon > 0$. Assume that $f$ is odd with respect to $x'$,
$P(\cdot) f \in L^{q}$ and $\widehat{f}(\xi)=0$ for $|\xi| \ge
a/2$. We set
\[
\widehat{h}(t, \xi) = \exp \big( - \frac{|\xi|^{2} t}{4a}
\big)\widehat{f}(\xi), \quad t \ge 0.
\]
Then, for any integer $k \ge 0$ and a multi-index $\alpha$, estimates
\[
\| P(\cdot) ^{\theta} \partial _{t} ^{k}
\partial_x ^{\alpha} ( I(t,\cdot) * h(t, \cdot) )
\| _{p} \le C(1+t) ^{- n \delta(p, q) - k - |\alpha|/2 -(1-\theta)
d/2-1+\epsilon} \| P(\cdot) f \| _{q}
\]
hold, where $C = C(p, q, \epsilon, k, \alpha)$ and $\delta(p, q)=1/2q - 1/2p$.
In the above estimates we may take $\epsilon=0$ when $1<q<p<\infty$,
$p=\infty$ and $q=1$ or $p=q=2$.
\end{lemma}

\begin{proof}
Consider the case where $\theta=0$. Since
\[
\partial _{t} ^{k} \partial_x ^{\alpha} ( I(t, x) * h(t, x) )
= \sum_{ k_1 + k_2 =k} c(k_1, k_2) \partial _{t} ^{k_1} I(t, x) *
\partial _{t} ^{k_2} \partial_x ^{\alpha} h(t, x),
\]
Proposition \ref{prop3.1} and Lemma \ref{lem3.3} show that
\begin{equation}
\begin{aligned}
\| \partial _{t} ^{k} \partial_x ^{\alpha}
( I(t,\cdot) * h(t, \cdot) ) \| _{p}
&  \le C \sum_{ k_1 + k_2 =k} (1+t) ^{-n\delta(p,
q)-k_1-1+\epsilon} \| \partial _{t} ^{k_2}
\partial_x ^{\alpha} h(t, \cdot) \| _{q} \\
& \le C(1 + t) ^{-n \delta(p, q) - k - |\alpha|/2 - d/2 - 1
+ \epsilon} \| P (\cdot) f \| _{q}.
\end{aligned}\label{Lemma3_4_1pr}
\end{equation}
Thus we have obtained the desired estimate when $\theta=0$.

Now we show the estimate of $\|P(\cdot) I(t, \cdot)*h(t, \cdot)\|_p$.
Easy calculations show
\begin{equation}
\chi _{1} (\xi)
\partial_{\xi} ^{\beta}
( \exp \big(-t\Theta_1)-1 \big) =\xi ^{\beta} \sum_{j=1} ^{|\beta|} ct
^{j}|\xi| ^{2\sigma(j, \beta)} \Psi_j(|\xi|^{2}) \exp
(-t\Theta_1), \label{Lemma3_4_2pr}
\end{equation}
for $\beta \in \mathcal{I}$ with $|\beta| \ge 1$, where
$\sigma(j,\beta)=\max ( 2j - |\beta|, 0 )$ and $\Psi_j$ is a
function of class $C ^{\infty}$ satisfying $ \Psi_j(|\xi|^{2})=0$
for $|\xi|^{2} \ge 2a/3$ for $j= 1, \dots, |\beta|$,
\begin{equation}
\partial_{\xi} ^{\gamma}
\exp \big( -\frac{t|\xi|^{2}}{4a} \big)=c t ^{|\gamma|}\xi ^{\gamma} \exp
\big( -\frac{t|\xi|^{2}}{4a} \big), \label{Lemma3_4_3pr}
\end{equation}
for $\gamma \in \mathcal{I} $, and
\begin{equation}
\exp (-t\Theta_1)\exp \big(- \frac{t|\xi|^{2}}{4a} \big) =g(t, \xi)+\exp
\big( - \frac{t|\xi|^{2}}{4a} \big). \label{Lemma3_4_4pr}
\end{equation}
Let $\alpha \in \mathcal{I}$ be fixed.
Since $\chi_1(\xi)=1$ on $\mathop{\rm supp}\widehat{h}(t, \cdot)$ for any $t \ge 0$,
(\ref{Lemma3_4_2pr})--(\ref{Lemma3_4_4pr}) imply
\begin{equation}
\begin{aligned}
& \mathcal{F} ( x ^{\alpha} I(t, \cdot) * h(t, \cdot) ) \\
& = \sum_{\beta + \gamma + \mu = \alpha} c _{\beta,
\gamma, \mu} \chi _{1}(|\xi|)
\partial_{\xi} ^{\beta} ( \exp (-t\Theta_1)-1 )
\partial_{\xi} ^{\gamma} \exp \big( -\frac{t|\xi|^{2}}{4a} \big)
\partial_{\xi} ^{\mu}\widehat{h}(t, \xi)\\
& = \sum _{\gamma+ \mu = \alpha} c _{\gamma, \mu}
\xi ^{\gamma} t ^{|\gamma|} g(t, \xi) \partial_{\xi} ^{\mu}
\widehat{h}(t, \xi) + \sum _{\beta + \gamma+ \mu = \alpha, |\beta|
\ge 1} \sum _{j=1} ^{|\beta|} c _{\beta, \gamma, \mu, j} \xi
^{\beta + \gamma }  \\
&\quad \times |\xi| ^{2\sigma(j, \beta)} t ^{j+|\gamma|}
\Big( g(t, \xi) + \exp \big(-\frac{t|\xi|^{2}}{4a} \big) \Big)
\partial_{\xi} ^{\mu} \widehat{h}(t, \xi).
\end{aligned} \label{Lemma3_4_5pr}
\end{equation}
Hence, Proposition \ref{prop3.1}, Lemma \ref{lem3.1}--\ref{lem3.2}
 and (\ref{Lemma3_4_5pr}) imply
\begin{align*}
& \| x ^{\alpha} I(t, \cdot)* h(t, \cdot)\| _{p} \le C
\sum _{ \beta + \gamma + \mu =\alpha} (1+ t) ^{|\beta|/2 +
|\gamma|/2 - n\delta(p, q) - 1 + \epsilon} \| x ^{\mu} h(t, \cdot)
\| _{q}\\
& \le C \sum _{ \beta + \gamma + \mu =\alpha} (1+ t) ^{|\beta|/2 +
|\gamma|/2 - n\delta(p, q) - 1 + \epsilon -(d-|\mu|)/2} \|
P(\cdot) f \| _{q}\\
 & \le C (1+ t) ^{- n \delta(p, q) - 1 + \epsilon}
\| P(\cdot) f \| _{q}.
\end{align*}
Therefore, we obtain the following estimate for $1 \le q \le p \le \infty$:
\begin{equation}
\| P(\cdot) I(t,\cdot)* h(t, \cdot) \| _{p} \le C (1+ t) ^{-
n\delta(p, q) - 1 + \epsilon} \| P(\cdot) f \| _{q}.
\label{Lemma3_4_6pr}
\end{equation}
Since
\begin{align*}
&\mathcal{F} \big\{\partial _{t} ^{j} \partial_x ^{\alpha} ( I(t,\cdot)*
h(t, \cdot))\big\}(\xi)\\
&=\{ g_1(t,\xi)m_1(\xi) +
m_2(\xi)|\xi| ^{2}\} |\xi| ^{2j} \xi ^{\alpha} \exp \big(
-\frac{t|\xi|^{2}}{4a}\big) \widehat{h}(t,\xi)
\end{align*}
for some functions $m_1$ and $m_2$ defined on $R ^{n}$ of class $C
^{\infty}$ with compact support, estimate (\ref{Lemma3_4_6pr})
shows that
\begin{equation}
\| P(\cdot) \partial_ t ^{j} \partial_x ^{\alpha} ( I(t,
\cdot)*h(t, \cdot) ) \| _{p} \le C (1+t) ^{- n \delta(p, q) - j -
|\alpha|/2 - 1 + \epsilon} \|P(\cdot) f \| _{q} .
\label{Lemma3_4_7pr}
\end{equation}
Since
\begin{align*}
&\| P(\cdot) ^{\theta} \partial _{t} ^{j}
\partial_x ^{\alpha} ( I(t, \cdot)*h(t, \cdot) )
\| _{p} \\
& \le \| P(\cdot) \partial _{t} ^{j}
\partial_x ^{\alpha} ( I(t, \cdot)* h(t, \cdot) )
\| _{p} ^{\theta} \| \partial _{t} ^{j}
\partial_x ^{\alpha} ( I(t, \cdot)*h(t, \cdot) )
\| _{p} ^{1 - \theta}
\end{align*}
for $0 \le \theta \le 1$, estimates (\ref{Lemma3_4_1pr}) and
(\ref{Lemma3_4_7pr}) give the desired estimate.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm1.1}]
The Fourier transformation of (\ref{linear_equation}) yields
\begin{equation}
\widehat{v}(t, \xi)
 = e^{-at} \Big( \cos t\sqrt{|\xi|^2-a^2}
\widehat{v}_0(\xi)+ \frac{\sin t
\sqrt{|\xi|^2-a^2}}{\sqrt{|\xi|^2-a^2}} (a\widehat{v}_{0}(\xi)+
\widehat{v}_{1}(\xi)) \big) \label{Theorem1_1_1pr}
\end{equation}
for $t \ge 0$. Since
\[
\cos \sqrt{z}= \sum _{k=0} ^{\infty} \frac{(-z) ^{k}}{(2k)!},
\quad \frac{\sin \sqrt{z}}{\sqrt{z}}= \sum_{k=0} ^{\infty}
\frac{(-z) ^{k}}{(2k+1)!}, \quad z \in \mathbb{C},
\]
$\cos t\sqrt{|\xi|^2-a^2}$ and $\sin
t\sqrt{|\xi|^2-a^2}/\sqrt{|\xi|^2-a^2}$ are smooth functions of
$(t, \xi)$ in $R \times R^{n}$, and they satisfy
\begin{gather*}
\cos t\sqrt{|\xi|^2-a^2}= \sum _{k=0} ^{\infty} \frac{(-1) ^{k}
t^{2k}}{(2k)!}(|\xi|^2-a^2)^k,\\
\frac{\sin t\sqrt{|\xi|^2-a^2}}{\sqrt{|\xi|^2-a^2}}= \sum_{k=0}
^{\infty} \frac{(-1) ^{k}t^{2k+1}}{(2k+1)!}(|\xi|^2-a^2)^k.
\end{gather*}
First we consider the case where  $b \ge a/2$ and
$\mathop{\rm supp}\widehat{v}_0 \cup
\mathop{\rm supp}\widehat{v}_1 \subset \{ \xi;a/3 \le |\xi| \le b \}$.
Since
\[
| \partial_{\xi}^{\alpha} ( |\xi|^2-a^2 )^k | \le
C_{\alpha} \frac{k!}{(k-|\alpha|)!}(1+|\xi|^{|\alpha|}) |
|\xi|^2-a^2 |^{k-|\alpha|}, \quad |\alpha| \le k,
\]
it follows that
\begin{equation}
\begin{aligned}
& \big| \partial_{\xi} ^{\alpha} \big( \frac{ \sin
t\sqrt{|\xi|^2-a^2}}{ \sqrt{|\xi|^2-a^2}} \big) \big| +|
\partial_{\xi} ^{\alpha} \cos t\sqrt{|\xi|^2-a^2} |  \\
 & =\Big| \partial_{ \xi} ^{\alpha} \sum_{k=0} ^{\infty}
\frac{(-1)^k }{(2k+1)!} t^{2k+1} ( |\xi|^2-a^2 )^{k} \Big| +
\Big|\partial_{\xi} ^{\alpha} \sum_{k=0} ^{\infty} \frac{(-1)^k }{(2k)!}
t^{2k} ( |\xi|^2-a^2 )^{k} \Big| \\
& \le C_{\alpha} (1+ t^{2|\alpha|+1})(1+ |\xi|^{\alpha}) \exp (
t| |\xi|^2-a^2 |^{1/2}) \\
 & \le C_{\alpha} (1+ t^{2|\alpha|+1}) \exp ( at \sqrt{15}/4 ), \quad
a/4 \le |\xi| \le 5a/4, \quad t \ge 0.
\end{aligned} \label{Theorem1_1_2pr}
\end{equation}
 From the estimates
\[
|\partial_{\xi} ^{\alpha}\sqrt{|\xi|^2-a^2 }|+
\big|\partial_{\xi} ^{\alpha}\big(
\frac{1}{\sqrt{|\xi|^2-a^2 }} \big) \big| \le C_{\alpha}, \quad
|\xi| \ge 5a/4, \quad|\alpha| \ge 1,
\]
we obtain
\begin{equation}
\big| \partial_{\xi} ^{\alpha} \big(\frac{ \sin t\sqrt{|\xi|^2-a^2}}{
\sqrt{|\xi|^2-a^2}} \big) \big|
+|\partial_{\xi} ^{\alpha} \cos t\sqrt{|\xi|^2-a^2}|
\le C_{\alpha}(1+t^{|\alpha|})
\label{Theorem1_1_3pr}
\end{equation}
for $t \ge 0$ and $|\xi| \ge 5a/4$.

Choose a smooth function $\chi_{12}$ satisfying
$ \chi_{12}(\xi){\hat v}_i(\xi)={\hat v}_i(\xi)$ $(i=0, 1)$ and \\
$\mathop{\rm supp}\chi_{12} \subset \{ \xi : a/4< |\xi| <b+1 \}$.
(\ref{Theorem1_1_2pr})--(\ref{Theorem1_1_3pr}) and Lemma \ref{lem3.1} with
\[
\chi _{11}(\xi)= \chi_{12}(\xi) \cos t\sqrt{|\xi|^2-a^2},\quad
\chi_{11}(\xi)= \chi_{12}(\xi) \frac{\sin
t\sqrt{|\xi|^2-a^2}}{\sqrt{|\xi|^2-a^2}}
\]
give the following estimate
\begin{equation}
\begin{aligned}
&\big\| P(\cdot) \partial_x ^{\beta} \mathcal{F}^{-1}
\chi_{12} \Big( \cos t \sqrt{|\xi|^2-a^2}{\hat v}_0 +\frac{\sin
t\sqrt{|\xi|^2-a^2}}{\sqrt{|\xi|^2-a^2}} (a{\hat v}_0 + {\hat
v}_1) \Big) \|_p  \\
&\le C_{\beta}(1+t)^{2n+2d+3} (\|P(\cdot)v_0 \|_q
+\|P(\cdot)v_1 \|_q )
\end{aligned}\label{Theorem1_1_4pr}
\end{equation}
for $t \ge 0$ and $\beta$.
Since ${\hat v}(t, \xi)=\chi_{12} {\hat v}(t, \xi)$,
(\ref{Theorem1_1_1pr}) and (\ref{Theorem1_1_4pr}) shows
\begin{equation}
\| P(\cdot) \partial_x ^{\beta} v(t, \cdot) \|_p \le C_{\beta} e^{
- \lambda t} ( \| P(\cdot) v _{0} \| _{q} + \| P(\cdot) v _{1}\|
_{q} ), \quad t \ge 0 \label{Theorem1_1_5pr}
\end{equation}
for $1 \le q \le p \le \infty$, where
$\lambda \equiv (4-\sqrt{15})a/5$.

Set $v _{k}(t, x)= \partial _{t} ^{k} v(t, x)$ for $k=0, 1, 2,
\dots$. Then $v _{k}(t, x)$  satisfies
\[
\partial_t v _{k}(t, x) + 2a v _{k}(t, x)=
\Delta v _{k-1}(t, x), \quad v _{k}(0, x)=\partial _{t} ^{k} v
(0,x), \quad t>0, x \in R^n,
\]
hence
\begin{equation}
v _{k}(t, x)=e^{-2at}v _{k}(0, x)+\int_0 ^t e^{-2a(t-\tau)}\Delta
v _{k-1}(\tau, x) \, d \tau, \quad t \ge 0, x \in R^n
\label{Theorem1_1_6pr}
\end{equation}
for $k=1,2, \dots$. Moreover, it satisfies ${\hat v} _{k} (0,
\xi)=0$ for $|\xi| \ge b$, and
\begin{equation}
\| P(\cdot) \partial _{x} ^{\beta} v _{k} (0, \cdot) |
| _{q} \le C _{k,\beta} \big( \| P(\cdot) v _ 0 \|_q + \|
P(\cdot) v _ 1 \|_q\big) \quad 1 \le q \le \infty,
\label{Theorem1_1_7pr}
\end{equation}
for $k=1,2, \dots$.  Therefore, (\ref{Theorem1_1_5pr})--
(\ref{Theorem1_1_7pr}) show
\begin{equation}
\| P(\cdot) \partial_t ^k \partial_x ^{\beta} v(t, \cdot) \|_p \le
C_{k,\beta} e^{ - \lambda t} \big( \| P(\cdot) v _{0}\| _{q} + \|
P(\cdot) v _{1}\| _{q} \big), \quad t \ge 0 \label{Theorem1_1_8pr}
\end{equation}
for $1 \le q \le p \le \infty$, $k=0,1,2, \dots$ and $\beta$.
Since ${\hat v}_i(\xi)=\chi_{12}(\xi) {\hat v}_i(\xi)$ $(i=0,1)$,
the solution formula
\[
{\hat V}(t, \xi)= \exp \big( -\frac{t|\xi|^2}{2a} \big)
\big( {\hat v}_0(\xi)
+ \frac{1}{2a} {\hat v}_1(\xi) \big), \quad t \ge 0,
\]
shows that ${\hat V}(t, \xi)=\chi_{12} {\hat V}(t, \xi)$.
Hence, the similar arguments to the above estimates and Lemma \ref{lem3.1} shows
\begin{equation}
\| P(\cdot) \partial_t ^k \partial_x ^{\beta} V(t, \cdot) \|_p \le
C_{k, \beta} e^{-\lambda t} ( \| P(\cdot) v_0 \|_q+ \| P(\cdot)v_1
\|_q ), \quad t \ge 0 \label{Theorem1_1_9pr}
\end{equation}
for $1 \le q \le p \le \infty$, $k=0,1,2,\dots$ and $\beta$. Since
$P(x) \ge 1$, (\ref{Theorem1_1_8pr})--(\ref{Theorem1_1_9pr}) give
the desired result in Theorem \ref{thm1.1}, in the case where $b \ge a/2$
and $\mathop{\rm supp}\widehat{v}_0\cup\mathop{\rm supp}\widehat{v}_1 \subset \{
\xi : a/3 \le |\xi| \le b \}$.

Now we consider the case where $\mathop{\rm supp}\widehat{v}_{0} \cup
\widehat{v}_{1} \subset \{ \xi: |\xi| \le a/2 \}$. The solution
formula (\ref{Theorem1_1_1pr}) shows that
\begin{equation}
\widehat{v}(t, \xi) = \widehat{V}(t,\xi)+
\frac{1}{2}\widehat{\phi}_{1}(t, \xi) +\widehat{\phi}_{2}(t,
\xi)+\widehat{\phi}_{3}(t, \xi), \label{Theorem1_1_10pr}
\end{equation}
where
\begin{gather*}
\widehat{\phi}_{1}(t, \xi)=g(t, \xi)\chi_{1}(\xi) \exp \big(
-\frac{t|\xi|^{2}}{4a} \big) \big( \widehat{v}_{0}(\xi) + \frac{a
\widehat{v}_{0}(\xi)+ \widehat{v}_{1}(\xi)}{ \sqrt{a^{2} -
|\xi|^{2}}} \big),
\\
\widehat{\phi}_{2}(t, \xi) = \exp \big( -\frac{t|\xi|^{2}}{2a} \big)
\frac{ |\xi|^{2} \chi_{1}(\xi)}{
\sqrt{a^{2}-|\xi|^{2}}(a+\sqrt{a^{2}-|\xi|^{2}})} \cdot \frac{a
\widehat{v}_{0}(\xi) + \widehat{v}_{1}(\xi)}{2a},
\\
\widehat{\phi}_{3}(t, \xi) = \frac{1}{2} \exp \big( -at
-t\sqrt{a^{2}-|\xi|^{2}} \big) \chi_{1}(\xi) \big( \widehat{v}_{0}(\xi)-
\frac{a \widehat{v}_{0}(\xi) +
\widehat{v}_{1}(\xi)}{\sqrt{a^{2}-|\xi|^{2}}} \big).
\end{gather*}
It follows that $\widehat{v}_i(\xi) = \chi_1(\xi)
\widehat{v}_i(\xi)$ for $i=0, 1$, the function $ \xi \mapsto
\chi_1(\xi)/\sqrt{a ^{2} - |\xi|^{2}}$ is a radial function that
belongs to $S( R ^{n} )$, and the function $v_i$ $(i=0, 1)$ is odd
with respect to $x'$. Hence the function
\[
\mathcal{F}^{-1}\big( \widehat{v}_0+ \frac{a \widehat{v}_0+
\widehat{v}_1}{\sqrt{a^{2}-|\xi|^{2}}} \big)
=v_0+ c\mathcal{F}^{-1}\big(
\frac{\chi_1(\xi)}{\sqrt{a^{2}-|\xi|^{2}}} \big)*( a v_0+v_1 )
\]
is also odd with respect to $x'$, and moreover, Lemma \ref{lem2.3} shows
\begin{equation}
\| P(\cdot) \mathcal{F}^{-1} ( \widehat{v}_0 + \frac{a
\widehat{v}_0 +\widehat{v}_1}{\sqrt{a^{2}-|\xi|^{2}}}) \|_q \le C
( \| P(\cdot) v_0 \|_q +\| P(\cdot) v_1 \|_q )
\label{Theorem1_1_11pr}
\end{equation}
for $t \ge 0$.
Set
\[
\widehat{h}(t, \xi)= \exp \big( -\frac{t|\xi|^{2} }{4a} \big) \big(
\widehat{v}_0 (\xi) + \frac{ a \widehat{v}_0(\xi) +
\widehat{v}_1(\xi)}{\sqrt{a^{2}-|\xi|^{2}}} \big), \quad t \ge 0.
\]
Since $\phi_1(t, \cdot)=c I(t, \cdot)*h(t, \cdot)$,
Lemma \ref{lem3.4} and estimate (\ref{Theorem1_1_11pr}) show that
\begin{equation}
\begin{aligned}
& \| P(\cdot) ^{\theta} \partial _{t} ^{k}
\partial_x ^{\alpha} \phi_1(t, \cdot)\|_p \\
 & \le C (1+t) ^{- n \delta(p,q) - k - |\alpha|/2
- (1-\theta)d/2-1+\epsilon} ( \|P(\cdot)
v_0\|_q +\|P(\cdot)  v_1\|_q )
\end{aligned} \label{Theorem1_1_12pr}
\end{equation}
for $t \ge 0$, $1 \le q \le p \le \infty$ and $\theta=0, 1$.
Proposition \ref{prop3.1} and Lemma \ref{lem3.1} show
\begin{equation}
\begin{aligned}
& \| P(\cdot) ^{\theta}
\partial _{t} ^{k} \partial_x ^{\alpha}
\phi_2(t, \cdot) \|_p + \| P(\cdot) ^{\theta} \partial _{t} ^{k}
\partial_x ^{\alpha} \phi_3(t, \cdot) \|_p \\
&\le C (1+t) ^{- n \delta(p, q) - k - |\alpha|/2 - (1-\theta)d/2 -
1 + \epsilon} ( \|P(\cdot) v_0\|_q +\|P(\cdot) v_1\|_q )
\end{aligned} \label{Theorem1_1_13pr}
\end{equation}
for $t \ge 0$, $1 \le q \le p \le \infty$ and $\theta=0, 1$.
Hence, (\ref{Theorem1_1_10pr}) and estimates
(\ref{Theorem1_1_12pr})--(\ref{Theorem1_1_13pr})
give the desired estimate.
\end{proof}

\section{Proof of Theorem \ref{thm1.2}}

Let $N$ be a positive integer. Then the function
\begin{equation}
h _{N} ( y ) = e^{iy} - \sum _{k=0} ^{N} \frac{(iy)^{k}}{k!}
\label{h_N_teigi}
\end{equation}
satisfies $| \partial_{y} ^{k} h _{N} (y)| \le C |y| ^{N - k}$, for
$k \in [0, N]$. Let $\chi _{2}$
be a radial function of class $C^{\infty}$ that satisfies $ \chi
_{2}( \xi)= 0$ for $|\xi| \le 3a/2$, and $ \chi _{2}( \xi)= 1$ for
$|\xi| \ge 2a$. Here and after we denote $\chi _{2}(\rho)=\chi
_{2}(\rho \omega)$ for $\rho \ge 0$ and $\omega \in R^n$,
$|\omega|=1$.

Define the function
\[
II _{N} (t, x ) = \mathcal{F}^{-1} ( \chi _{2}(\cdot) h _{N} (t
\Theta(\cdot)) e^{it|\xi|}  )(x)\,.
\]
Then Lemma \ref{lem2.2} shows that
\begin{equation}
II _{N} (t, x) = c \int _{0} ^{\infty} \chi _{2}(\rho) h _{N} (t
\Theta (\rho) ) \rho^{n - 1} \widetilde{J} _{-1 + n/2}( \rho |x| )
\, d \rho. \label{II_N_seishitsu}
\end{equation}

\begin{lemma}[{cf. in \cite[Lemma 4.1]{Narazaki}}]  \label{lem4.1}
Let $N \ge n+1$ and $ m = [ n/2 ]$, then
\begin{enumerate}
\item $\| II _{N} (t, \cdot) \| _{\infty} \le C | t | ^{N}$,
\item $\| II _{N} (t, \cdot) \| _{1} \le C ( | t | ^{N}
+ |t | ^{ N + m + 2})$.
\end{enumerate}
\end{lemma}

\begin{proof}
 (1) Since $ | h _{N} (t \Theta(\rho) ) |
\le C | t | ^{N} \Theta(\rho) ^{N} \le C |t| ^{N}/ \rho^{N}$, for
$\rho \ge 3a/2$, Lemma \ref{lem2.1} (4) and (\ref{II_N_seishitsu}) show the
desired estimate
\[
| II _{N} (t, x) | \le C \int _{3a/2} ^{\infty}
\frac{|t|^{N} }{\rho^{N - n + 1}} \, d \rho \le C | t |^{N}.
\]
(2) Since
\[
| ( \frac{d}{dy} ) ^{k} h _{N} (y) | \le C |y|^{N - k},
\quad | ( \frac{d}{d \rho} ) ^{k} \Theta (\rho) | \le C
\rho^{- k - 1}
\]
for $ \rho \ge 3a/2$ and $0 \le k \le N$, easy calculations show
\begin{equation}
| ( \frac{\partial}{\partial \rho} ) ^{k} h _{N} ( t
\Theta(\rho) ) | \le C |t|^{N} \rho^{- k - N}, \quad (0 \le
k \le N, \rho \ge 3a/2). \label{Lemma4_1_1pr}
\end{equation}
The differential operator $X$ defined by
\[
X v(t, \rho) = \frac{\partial}{\partial \rho} ( \frac{1}{\rho}
v(t, \rho) )
\]
satisfies
\begin{equation}
X ^{k} ( v(t, \rho) \rho^{l}) = \sum _{j=0} ^{k} c _{j k l}
\partial _{\rho}^{j} v(t, \rho) \rho^{l - 2k + j}.
\label{Lemma4_1_2pr}
\end{equation}
Then (\ref{Lemma4_1_1pr})--(\ref{Lemma4_1_2pr}) read
\begin{equation}
( \frac{\partial}{\partial \rho} )^{l} ( \rho ^{i} X ^{k} (
\chi _{2} (\rho) h _{N} (t\Theta (\rho)) e ^{it\rho} \rho ^{n - 1}
) ) \Big| _{0} ^{\infty} = 0 \label{Lemma4_1_3pr}
\end{equation}
for $i=0, 1$, $0 \le k \le m$ and $ 0 \le l \le 2$.
Hence, Lemma \ref{lem2.1} (5),
(\ref{II_N_seishitsu}), (\ref{Lemma4_1_3pr}) and
integration by parts give
\begin{align*}
II _{N}(t, x)&= \frac{c}{|x|^{2}} \int _{0} ^{\infty}
\chi _{2}(\rho) h _{N}  (t \Theta(\rho)) e^{it\rho} \rho ^{n - 1}
\frac{1}{\rho} ( \frac{\partial}{\partial \rho} ) \widetilde{J}
_{n/2 - 2}(\rho |x|) \, d\rho \\
&= \frac{c}{|x|^{2} } \int _{0} ^{\infty} X
( \chi _{2}(\rho) h _{N} (t \Theta(\rho)) e^{it\rho} \rho ^{n - 1}
) \widetilde{J} _{n/2 - 2}(\rho |x|) \, d\rho,
\end{align*}
when $(n/2 - 2)$ is not a negative integer.
Repeating the above integration by parts, we obtain
\begin{equation}
II _{N}(t, x)= \frac{c}{|x| ^{2\mu}} \int _{0} ^{\infty} X ^{\mu}
( \chi _{2}(\rho) h _{N} (t \Theta(\rho)) e ^{it\rho} \rho ^{ n -
1} ) \widetilde{J} _{n/2 - 1 - \mu}(\rho |x|) \, d\rho,
\label{Lemma4_1_4pr}
\end{equation}
where $\mu=[(n - 1)/2]$.
In the case where $ n = 2m $, equation (\ref{Lemma4_1_4pr}) reads
\begin{equation}
II _{N}(t, x)= \frac{c}{|x| ^{n - 2}} \int _{0} ^{\infty} X ^{m -
1} ( \chi _{2}(\rho) h _{N} (t \Theta(\rho)) e ^{it\rho} \rho ^{n
- 1} ) J_{0}(\rho |x|) \, d\rho . \label{Lemma4_1_5pr}
\end{equation}
Lemma \ref{lem2.1} (1) shows that
\[
J _{0}(\rho |x|) = 2 \widetilde{J} _{1}(\rho |x|)+ \rho (
\frac{\partial}{\partial \rho} ) \widetilde{J} _{1}(\rho |x|),
\]
hence  (\ref{Lemma4_1_3pr}), (\ref{Lemma4_1_5pr}) and integration by parts give
\begin{equation} \label{Lemma4_1_6pr}
\begin{aligned}
&| II _{N} (t, x) | \\
& \le \frac{c}{|x| ^{n-2}} \sum _{k=0} ^{1} \Big|
\int _{0} ^{\infty} \rho ^{k} ( \frac{\partial}{\partial \rho} )
^{k} X ^{m-1}(\chi _{2}(\rho) h _{N} (t\Theta(\rho)) e ^{it\rho}
\rho ^{n-1}) \widetilde{J} _{1} (\rho | x | ) \, d \rho \Big| .
\end{aligned}
\end{equation}
Since Lemma \ref{lem2.1} (4) shows
\[
| \widetilde{J} _{1}(\rho |x|) - c \rho ^{-3/2}|x| ^{-3/2}
\cos ( \rho |x|- \frac{3\pi}{4} ) | \le C \rho ^{-5/2}|x|
^{-5/2},
\]
estimate (\ref{Lemma4_1_6pr}) and integration by parts show
\begin{equation}  \label{Lemma4_1_7pr}
\begin{aligned}
& |II _{N}(t, x)|\\
& \le \frac{C}{|x| ^{n+1/2}} \sum _{ k = 0} ^{2} \int _{0} ^{\infty}
\rho^{k - 5/2} \big| ( \frac{\partial}{\partial \rho} ) ^{k} X
^{m - 1}(\chi _{2}(\rho) h _{N} (t\Theta(\rho)) e ^{it\rho} \rho
^{n - 1}) \big| \, d\rho,
\end{aligned}
\end{equation}
where we have used
\[
\cos ( \rho |x|- \frac{3\pi}{4} )= \frac{1}{|x|}
\frac{\partial}{\partial \rho} \sin ( \rho |x|- \frac{3\pi}{4} ).
\]
Hence, estimate (\ref{Lemma4_1_2pr}), (\ref{Lemma4_1_7pr}) and
(\ref{Lemma4_1_1pr}) show
\begin{equation}
| II _{N} (t, x) | \le  \frac{C}{|x| ^{n+1/2}} |t| ^{N}
(1 + |t| ^{m+2}). \label{Lemma4_1_8pr}
\end{equation}
Since
\[
\| II _{N} (t, \cdot) \| _{1} =\int _{|x| \le 1} | II _{N}
(t, x) | \, dx +\int _{|x| \ge 1} | II _{N} (t, x)
| \, dx,
\]
estimate (\ref{Lemma4_1_8pr}) and Lemma \ref{lem4.1} (1) give
the desired estimate in Lemma \ref{lem4.1} (2) when $ n = 2m$.

Now let us consider the case where $n=2m+1$.
Since Lemma \ref{lem2.1} (3) shows
\[
\widetilde{J} _{-1/2}(\rho |x|)=\sqrt{ \frac{\pi}{2} } \cos \rho
|x| = -\sqrt{ \frac{\pi}{2} } \frac{1}{|x| ^{2}} (
\frac{\partial}{\partial \rho} ) ^{2} \cos \rho |x|,
\]
(\ref{Lemma4_1_2pr})--(\ref{Lemma4_1_4pr}) and integration by parts give
\begin{equation}
\begin{aligned}
|II _{N} (t, x)|
& =\frac{c}{|x|^{n+1}} \big| \int _{0} ^{\infty} X ^{m-1} ( \chi _{2}(\rho) h
_{N} (t\Theta (\rho))e ^{it\rho} \rho ^{n-1} ) (
\frac{\partial}{\partial \rho} )^2 \cos \rho | x | \, d\rho
\big|  \\
& = \frac{c}{|x| ^{n+1}} \int _{0}
^{\infty} \big| ( \frac{\partial}{\partial \rho} ) ^{2} X ^{m} (
\chi _{2} (\rho) h _{N}( t \Theta(\rho) ) e ^{it\rho} \rho ^{n-1}
) \cos \rho |x| \big| \, d \rho
 \\
&  \le  \frac{C}{|x| ^{n+1}}|t| ^{N}( 1 + |t| ^{m+2}).
\end{aligned}\label{Lemma4_1_9pr}
\end{equation}
Estimates (\ref{Lemma4_1_9pr}) and Lemma \ref{lem4.1} (1) give the desired estimate
when $ n = 2m + 1$.
\end{proof}

\begin{corollary} \label{coro4.1}
Let $1 \le q \le p \le \infty$.
Under the assumptions in Lemma \ref{lem4.1}, the following estimates hold;
\[
\| II _{N} (t, \cdot)* g \| _{p} \le C |t| ^{N} ( 1 + |t| ^{m+2})
\| g \| _{q}, \quad g \in L ^{q}.
\]
\end{corollary}

\begin{proof} Set $r \in [0, \infty]$ by $ 1 - 1/r = 1/q - 1/p$.
Lemma \ref{lem4.1} shows
\[
\| II _{N} (t, \cdot) \| _{r} \le \| II _{N} (t, \cdot) \| _{1}
^{1/r} \| II _{N} (t, \cdot) \| _{\infty} ^{1 - 1/r} \le C |t|
^{N} ( 1 + |t| ^{m+2}),
\]
hence Lemma \ref{lem2.3} gives the desired estimate.
\end{proof}

Note that Corollary \ref{coro4.1} shows the following estimates:

\begin{lemma} \label{lem4.2}
Let $ N \ge n + d + 1$, $1 < q \le p < \infty$ and $ m = [n/2] $.
Assume that $f \in L^{q} $ and $\mathop{\rm supp}\widehat{f} \subset \{ \xi;
|\xi| \ge 2a \}$, then
\[
\| P(\cdot) \mathcal{F} ^{-1} ( \chi _{2} h _{N} (t\Theta )
\widehat{f} ) | | _{p} \le C |t|^{N} ( 1 + |t|
^{m+d+2} ) \| P(\cdot) f | | _{q} .
\]
\end{lemma}

\begin{proof} Let $\alpha \in \mathcal{I}$ be fixed.
Since $\chi _{2}(\xi)=1$ on $\mathop{\rm supp}\widehat{f}$,
\begin{equation}
\begin{aligned}
 &\| x ^{\alpha} \mathcal{F} ^{-1} \Big( \chi _{2} h _{N} (
t\Theta ) e^{it|\xi|} \widehat{f} \Big) \| _{p}
= c \| \mathcal{F}^{-1} \Big( \chi _{2}\partial_{\xi} ^{\alpha}
\big( h _{N} ( t\Theta) e ^{it|\xi|} \widehat{f} \big) \Big) \| _{p} \\
&\le C  \sum _{\beta +\gamma + \mu = \alpha} \| \mathcal{F}
^{-1}\Big( \chi _{2}
\partial _{\xi} ^{\beta} h _{N} ( t\Theta) \chi _{2}
\partial _{\xi} ^{\gamma} e ^{it|\xi|}
\partial _{\xi} ^{\mu} \widehat{f} \Big) \| _{p}.
\end{aligned} \label{Lemma4_2_1pr}
\end{equation}
Easy calculations show
\begin{equation}
\chi _{2}(\xi)\partial _{\xi} ^{\beta} h _{N}(t \Theta(\xi))= \sum
_{0 \le k \le |\beta|} c t ^{k} H _{\beta, k, 1} (\xi) h _{N-k}(t
\Theta(\xi)) \label{Lemma4_2_2pr}
\end{equation}
when $|\beta| \ge 1$, and
\begin{equation}
\chi _{2}(\xi)\partial _{\xi} ^{\gamma} e ^{it|\xi|} = \sum _{0
\le k \le |\gamma|} c t ^{k} H _{\gamma, k, 2}(\xi) e ^{it|\xi|}
\label{Lemma4_2_3pr}
\end{equation}
when  $|\gamma| \ge 1$, where
\[
H _{\beta, k, 1}(\xi) = \chi _{2}(\xi) \sum _{\widetilde{\beta}
_{1}+ \dots + {\tilde \beta}_k = \beta, | \widetilde{\beta} _{1}|
\ge 1, \dots, |\widetilde{\beta}_k| \ge 1} c \partial _{\xi} ^{
\widetilde{\beta} _{1}} \Theta(\xi) \dots
\partial _{\xi} ^{ \widetilde{\beta}_k} \Theta (\xi),
\]
and
\[
H _{\gamma, k, 2}(\xi) = \chi _{2}(\xi) \sum _{ \widetilde{\gamma}
_{1}+\dots+ \widetilde{\gamma}_k=\gamma, | \widetilde{\gamma}
_{1}| \ge 1, \dots, | \widetilde{\gamma}_k| \ge 1} c \partial
_{\xi} ^{ \widetilde{\gamma} _{1}}|\xi| \dots
\partial _{\xi} ^{ \widetilde{\gamma}_k} |\xi|.
\]
Since $H _{\beta, k, 1}, H _{\gamma, k,2} \in C ^{\infty}( R
^{n})$ satisfying
\[
| \partial _{\xi} ^{\nu} H _{\beta, k, 1} (\xi)  | +
| \partial _{\xi} ^{\nu} H _{\gamma, k, 2} (\xi) | \le
C _{\nu, \beta, \gamma, k}
\]
for any $k$, $\beta, \gamma \in \mathcal{I}$ with $|\beta| \ge 1$
and $|\gamma| \ge 1$ and any multi-index $\nu$,
H\"{o}rmander's multiplier theorem(see \cite{Duoandikoetxea}
for example) shows that
\begin{equation}
\|\mathcal{F} ^{-1}( H _{\beta, k, 1}\widehat{g} ) \| _{p}
+\|\mathcal{F} ^{-1}(H _{\gamma, k, 2} \widehat{g} ) \| _{p} \le
C _{\beta, \gamma, p, k}\| g \| _{p} \label{Lemma4_2_4pr}
\end{equation}
for $1 < p < \infty$ and $k \ge 0$ when $\beta, \gamma \in \mathcal{I}$
satisfy $| \beta | \ge 1 $ and $| \gamma | \ge 1$.

Since $N - k \ge n + 1$ when $0 \le k \le d$,
(\ref{Lemma4_2_1pr})--(\ref{Lemma4_2_4pr}) and Corollary \ref{coro4.1} show that
\begin{equation}
\begin{aligned}
 \| x ^{\alpha} \mathcal{F} ^{-1}( \chi _{2} h _{N} (t
\Theta) e ^{it|\xi|}\widehat{f} ) \| _{p}
& \le C \sum _{0 \le k + l \le d, \mu \in \mathcal{I}} |t| ^{ k
+ l} \| \mathcal{F} ^{-1}( \chi _{2} h _{N-k}(t \Theta) e
^{it|\xi|} \partial _{\xi} ^{\mu} \widehat{f} ) \| _{p}
\\
&\le C \sum _{ k + l \le d, \mu \in \mathcal{I}} |t| ^{ k + l} \|
II _{N-k}(t, \cdot) * ( x ^{\mu} f ) \| _{p} \\
 & \le C \sum _{ l = 0} ^{d} | t | ^{ N + l}( 1 + |t| ^{m+2} ) \sum _{\mu
\in \mathcal{I}} \| x ^{\mu} f\| _{q} \nonumber \\
& \le C | t| ^{N}( 1 + | t | ^{m + d + 2} ) \| P(\cdot) f \| _{q}
\end{aligned} \label{Lemma4_2_5pr}
\end{equation}
for $ 1 <q \le p < \infty$.
Since
\[
\| P(\cdot) \psi \| _{p} \le C \sum _{\alpha \in \mathcal{I}} \|
x ^{\alpha}  \psi \| _{p},
\]
estimate (\ref{Lemma4_2_5pr}) gives the desired estimate.
\end{proof}

For $t \in R$ and a constant $k$, define the operator $T _{k} (t)$
by
\begin{equation}
T _{k} ( t ) h = \mathcal{F} ^{-1} \big( \chi _{2} |\xi| ^{-k} e
^{it|\xi|} \widehat{h} \big) . \label{T_teigi}
\end{equation}
Set  $m = [n/2 ] $ and $[ 2/q - 1] _{+} =\max( 2/q-1, 0)$.


\begin{lemma}[{\cite[Proposition 4.4]{Narazaki}}] \label{lem4.3}
Let $k$ be an integer, $k \ge (n+1)/2$. For $1 < q \le p < \infty$ and
$t \ne 0$, the operator $T _{k}(t)$ is bounded from $L ^{q}$ to
$L^{p}$. Moreover, the following estimates hold:
\[
\| T _{k}(t) h \| _{p} \le C(1+|t|) ^{m} \| h \| _{q}
\]
when $ k \ge n + 1$, and
\[
\| T _{k}(t) h \| _{p} \le C(1+|t|) ^{m} |t| ^{[2/q - 1] _{+}
(k-n)} \| h \| _{q}
\]
when $(n+1)/2 \le k \le n$.
\end{lemma}

Lemma \ref{lem4.3} and arguments similar to those in the proof of
Lemma \ref{lem4.2} give the following result.

\begin{corollary} \label{coro4.2}
Let $k$ be an integer, $k \ge (n+1)/2 $.
For $1<q \le p<\infty$, $t \ne 0$, the following
estimates hold:
\[
\| P(\cdot)T _{k}(t) h \| _{p} \le C(1 + |t|) ^{ m + d }
\|P(\cdot) h\| _{q}
\]
when $k \ge n+1$, and
\[
\| P(\cdot) T _{k}(t) h \| _{p} \le C(1 + |t|) ^{ m + d} | t | ^{[
2/q - 1] _{+} (k-n)} \| h \| _{q}
\]
when $(n+1)/2 \le k \le n$.
\end{corollary}

\begin{proof}[Proof of Theorem \ref{thm1.2}]
When $k$ is a non-negative integer, H\"{o}rmander's multiplier theorem
\cite{Duoandikoetxea} shows that the function $\xi \mapsto \chi
_{2}(\xi) \Theta ^{k} |\xi| ^{k} $ is a Fourier multiplier on
$L ^{p}$ for $ 1<p<\infty$.
Hence
\begin{equation}
\| P(\cdot) \mathcal{F} ^{-1} ( \chi _{2} \Theta ^{k}
|\xi|^{k}\widehat{h} ) \| _{p} \le C \| P(\cdot) h \| _{p}, \quad
(1<p <\infty). \label{Theorem1_2_1pr}
\end{equation}
Since $\chi _{2}=1$ on
$\mathop{\rm supp}\widehat{ w} _{1} \cup \widehat{w}_{2}$,
Corollary \ref{coro4.2} and estimate (\ref{Theorem1_2_1pr}) give the
following estimates for $j = 0, 1$ and $1 < q \le p < \infty$,
\begin{equation}
\begin{aligned}
\| P(\cdot) \mathcal{F} ^{-1} (\chi _{2} t ^{k} \Theta
^{k} e ^{it|\xi|} \widehat{w} _{j} ) \| _{p}
& = |t| ^{k} \|P(\cdot) \mathcal{F} ^{-1} ( \chi _{2} \Theta ^{k} |\xi| ^{k}
\chi _{2} |\xi| ^{-k} e ^{it|\xi|} \widehat{w} _{j} ) \| _{p}\\
& \le C |t| ^{k} \| P(\cdot) T _{k}(t) w _{j} \| _{p} \\
& \le C( 1 + |t|) ^{k} ( 1 + |t|) ^{ m + d}\|P(\cdot) w _{j}\| _{q} \\
& \le C (1 + |t| ) ^{n+m+2d+1} \|P(\cdot) w _{j}\| _{q}
\end{aligned}\label{Theorem1_2_2pr}
\end{equation}
for any $t$ and any integer $k$ in $[n+1, n+d+1]$,
\begin{equation}
\begin{aligned}
\| P(\cdot) \mathcal{F} ^{-1} ( \chi _{2} t ^{k} \Theta
^{k} e ^{it|\xi|} \widehat{w} _{j} ) \| _{p}
& \le C |t| ^{k} \| P(\cdot) T _{k}(t) w _{j} \| _{p}  \\
& \le C |t| ^{k} |t| ^{[2/q-1] _{+} (k-n)}( 1 + |t|) ^{m + d} \|
P(\cdot) w _{j} \| _{q} \\
& \le C (1 + |t| ) ^{n+m+2d+1} \|P(\cdot) w _{j}\| _{q} .
\end{aligned} \label{Theorem1_2_3pr}
\end{equation}
for any $t \ne 0$ and any integer $k$ in $[(n+1)/2, n + d +1]$.
Since
\[
e ^{it \Theta} = h _{n+d+1}(t \Theta) + \sum _{k=0} ^{n+d+1}
\frac{(it \Theta)^{k} }{k!},
\]
Lemma \ref{lem4.2} with $N=n+d+1$ and estimates
(\ref{Theorem1_2_2pr})--(\ref{Theorem1_2_3pr}) show
\begin{equation}
\begin{aligned}
&\| P(\cdot) \mathcal{F} ^{-1} \Big\{ \Big( e ^{ it \Theta} -\sum _{0
\le k <(n+1)/2} (it\Theta)^{k}/k! \Big) e ^{it|\xi|}\widehat{w} _{j}
\Big\}\| _{p}\\
&\le C (1 + | t | ) ^{n+m+2d+3} \|P(\cdot) w _{j}\| _{q}
\end{aligned}\label{Theorem1_2_4pr}
\end{equation}
for any $t$, where $j=0, 1$ and $1<q \le p< \infty$.
Since $ \cos \rho=(e ^{i \rho} + e ^{-i \rho})/2$ and $ \sin
\rho=(e ^{i \rho} - e ^{-i \rho})/2i$, estimate
(\ref{Theorem1_2_4pr}) gives the following estimates:
\begin{equation}
\begin{aligned}
&\| P(\cdot) \mathcal{F} ^{-1} \Big\{ \Big(\cos t \Theta - \sum _{0
\le 2k <(n+1)/2} (-1) ^{k}(t\Theta) ^{2k} /(2k)! \Big) Y(t, \cdot)
\widehat{w} _{j} \Big\} \| _{p}\\
&\le C (1 + | t | ) ^{n+m+2d+3} \|P(\cdot) w _{j}\| _{q} ,
\end{aligned}\label{Theorem1_2_5pr}
\end{equation}
and
\begin{equation}
\begin{aligned}
&\| P(\cdot) \mathcal{F} ^{-1} \Big\{ \Big( \sin t \Theta -\sum _{0
\le 2k+1 <(n+1)/2} (-1) ^{k}(t\Theta)^{2k+1}/(2k+1)! \Big) Y(t, \cdot)
\widehat{w} _{j} \Big\}\| _{p} \\
&\le C (1+|t|)^{n+m+2d+3} \|P(\cdot) w _{j}\| _{q} ,
\end{aligned} \label{Theorem1_2_6pr}
\end{equation}
where  $j=0,1$,  $Y(t, \xi)=\cos t|\xi|$ or
$Y(t, \xi)=\sin t|\xi|$.
Since
\[
\widehat{w}(t, \xi) = e ^{-at} \big( \cos t(|\xi|- \Theta )\widehat{w}
_{0} +\frac{ \sin t(|\xi|-\Theta )}{\sqrt{|\xi|^{2}-a^{2}}} ( a
\widehat{w} _{0} + \widehat{w} _{1} ) \big),
\]
estimates (\ref{Theorem1_2_5pr})--(\ref{Theorem1_2_6pr}) give the desired
 estimate in Theorem \ref{thm1.2}.
\end{proof}

\begin{proof}[Proof of Corollary \ref{coro1.2}]
 Let $T_{k}(t)$ be the operator defined in Lemma \ref{lem4.3}.
$w(t, \cdot) \equiv T _{m} (t)f$ satisfies the Cauchy problem to the
wave equation
\[
\partial_t^2 w-\Delta w=0, \quad
w(0,\cdot)=\mathcal{F}^{-1}( \chi_2 |\xi|^{-m} \widehat{f}),
\quad
\partial_t w(0, \cdot)=i\mathcal{F}^{-1}( \chi_2 |\xi|^{-m+1}
\widehat{f})
\]
for $t>0$, $x \in R^n$.
Hence, the solution formula for the Cauchy problem to the wave
equation
\[
\partial _{t} ^{2} W - \Delta W=0
\]
shows that $T _{m}(t)$ is a bounded operator on $L ^{p}$ for any $
1 < p < \infty$, and it satisfies
\begin{equation}
\| T _{m}(t) f \| _{p} \le C _{p} (1+ |t|) ^{m} \| f \| _{p}\,.
\label{Cor1_2_1pr}
\end{equation}
For any $t$, the operator $T_0(t)$ is bounded on $L ^{2}$, and
satisfies
\begin{equation}
\| T _{0}(t) f \| _{2} \le C \| f \| _{2}\,. \label{Cor1_2_2pr}
\end{equation}
Hence, the Stein interpolation theorem between estimates
(\ref{Cor1_2_1pr})--(\ref{Cor1_2_1pr}) shows that
\begin{equation}
\| T _{1}(t) f \| _{p} \le C _{p} (1+ |t|)^{m} \| f \| _{p}
\label{Cor1_2_3pr}
\end{equation}
holds for $\max (0, 1/2 - 1/2m ) < 1/p < \min( 1, 1/2 + 1/2m)$.
For any $p \in (1, \infty)$ and $\alpha$, the functions
\[
|\xi| \partial _{\xi} ^{\alpha}( \chi _{2}(\xi) \Theta),
\quad |\xi| \partial _{\xi} ^{\alpha} \big( \frac{\chi
_{2}(\xi)}{\sqrt{|\xi| ^{2} - a ^{2}}} \big)
\]
are Fourier-multipliers on $L ^{p}$ (see \cite{Duoandikoetxea}).
Therefore estimate (\ref{Cor1_2_3pr}) and similar calculations to
ones in the proof of Lemma \ref{lem4.2} show that
\begin{equation}
\| P(\cdot) \mathcal{F} ^{-1} \big( e ^{it |\xi|} \Theta ^{k}
\frac{|\xi|}{ \sqrt{|\xi|^{2}-a^{2}}} \widehat{f}(\xi) \big) \| _{p}
\le C(1+|t|)^{m + d} \|P(\cdot) f\| _{p} \label{Cor1_2_4pr}
\end{equation}
and
\begin{equation}
\| P(\cdot) \mathcal{F} ^{-1} \big( e ^{it |\xi|} \Theta ^{l} |\xi|
\widehat{f}(\xi) \big) \| _{p} \le C( 1 + | t | ) ^{m + d} \| P(\cdot)
f \| _{p} \label{Cor1_2_5pr}
\end{equation}
for any $p$ satisfying $\max (0, 1/2 -1/2m ) < 1/p < \min( 1, 1/2
+ 1/2m)$, and for positive integers $k \ge 0$, $l \ge 1$ , provided
that $\mathop{\rm supp} \widehat{f} \subset \{ \xi: |\xi| \ge 2a \}$.
Estimate (\ref{Cor1_2_4pr})--(\ref{Cor1_2_5pr}) and Theorem \ref{thm1.2} give the
desired estimate.
\end{proof}

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\end{document}