\documentclass[reqno]{amsart}
\usepackage{hyperref}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 118, pp. 1--13.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2013 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2013/118\hfil Strichartz estimates]
{Strichartz estimates on $\alpha$-modulation spaces}

\author[W. Guo, J. Chen \hfil EJDE-2013/118\hfilneg]
{Weichao Guo, Jiecheng Chen}  

\address{Weichao Guo \newline
Department of Mathematics, Zhejiang University,
Hangzhou 310027, China}
\email{maodunguo@163.com}

\address{Jiecheng Chen \newline
Department of Mathematics,  Zhejiang Normal University,
Jinhua 321004, China}
\email{jcchen@zjnu.edu.cn}

\thanks{Submitted  March 12, 2013. Published May 10, 2013.}
\subjclass[2000]{42B37, 46E35, 35L05}
\keywords{Strichartz estimate; Schr\"odinger equation; wave equation;
\hfill\break\indent $\alpha$-modulation space}

\begin{abstract}
 In this article, we consider some dispersive equations,
 including  Schr\"odinger equations, nonelliptic Schr\"odinger
 equations, and wave equations.  We develop some Strichartz estimates 
 in the frame of $\alpha$-modulation spaces.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks

\section{Introduction}

We study the Cauchy problem for the Schr\"odinger type equation
\begin{equation}\label{1.1}
\begin{gathered}
iu_t+(-\Delta)^{\beta/2}u=F \\
u(0,x)=u_{0},
\end{gathered}
\end{equation}
the Cauchy problem for nonelliptic Schr\"odinger equation
\begin{equation}\label{1.2}
\begin{gathered}
iv_t+\psi(D)v=F \\
v(0,x)=v_{0}
\end{gathered},
\end{equation}
where $\psi(\xi)=\Sigma_{l=1}^{n}\pm|\xi_l|^{\beta}$,
 and the Cauchy problem for wave equation
\begin{equation}\label{1.3}
\begin{gathered}
w_{tt}-\Delta w=F \\
w(0,x)=w_{0}, w_{t}(0,x)=w_{1}.
\end{gathered}
\end{equation}
The initial data belongs to the $\alpha$-modulation space $M^{0,\alpha}_{2,1}$,
and we use $F$ to denote some nonlinear terms.

We recall Duhamel's formula for above three dispersive equations.
The solution to \eqref{1.1} is
\begin{equation}
u(t,x)=e^{it(-\Delta)^{\beta/2}}u_0
-i\int_0^t e^{i(t-s)(-\Delta)^{\beta/2}}F(s)ds.
\end{equation}
The solution to \eqref{1.2} is
\begin{equation}
v(t,x)=e^{it\psi(D)}v_0-i\int_0^t e^{i(t-s)\psi(D)}F(s)ds.
\end{equation}
The solution to \eqref{1.3} is
\begin{equation}
w(t,x)=\cos(t\sqrt{-\Delta})w_0+\frac{\sin(t\sqrt{-\Delta})}{\sqrt{-\Delta}}w_1
+\int_0^t\frac{\sin((t-s)\sqrt{-\Delta})}{\sqrt{-\Delta}}F(s)ds.
\end{equation}
There are many publications about the theoretical and applied aspects 
of the Schr\"odinger equation and the wave equation; see for example 
Tao \cite{Tao_book} and Sogge \cite{Sogge_book} for a nice introduction.
We refer the reader to 
\cite{AKKL_JFA_2007,AK_LMS_2009,Chen_Fan,Chen_Fan_Sun,RSW_2012,
WH_JDE_2007,Wang_book,WZG_JFA_2006} for the study of modulation 
space and dispersive equation.

In this paper, we are concerned mainly with the Strichartz estimates
 for the solutions of the above three equations.
The original  estimates are due to Strichartz \cite{Strichartz}, 
and they became fundamental and important tools in the study of 
dispersive equations.
The theory of Strichartz estimates has also been studied by many authors. 
One is referred to \cite{GV1}, \cite{GV2} and \cite{KT_1998} for classical 
Strichartz estimates.
We also refer the readers to \cite{Cordero_Zucco} and \cite{Z_NLA_2012} 
for the Strichartz estimates in the frame of Wiener amalgam spaces 
and modulation spaces.
The following lemma is a basic Strchartz estimate, proved
 by Keel-Tao \cite{KT_1998}, that we will use it frequently in our proofs.

\begin{definition}\label{def-adi}\rm
An exponent pair $(r,p)$ is called  $\sigma$-admissible if
 $r,p\geq 2$, $(r,p,n)\neq (2,\infty,2)$
and
\begin{equation}\label{admissible pair}
\frac{1}{r}+\frac{\sigma}{p}\leq \frac{\sigma}{2}\,.
\end{equation}
If the equality holds, we say that $(r,p)$ is sharp $\sigma$-admissible,
otherwise we say that $(r,p)$ is nonsharp $\sigma$-admissible.
If $\sigma>1$ we say the sharp $\sigma$-admissible pair
\begin{equation}
(2,\frac{2\sigma}{\sigma-1})
\end{equation}
is an endpoint.
\end{definition}
Next we have the Strichartz estimates.

\begin{lemma}[\cite{KT_1998}] \label{strichartz-estimates}
Let $\{U(t)\}_{t\in \mathbb{R}}$ be a semigroup of operators that 
obey energy estimate
\begin{equation}
\|U(t)f\|_{L_x^2}\lesssim \|f\|_{L_x^2}
\end{equation}
and dispersive estimate
\begin{equation}
\|U(t)(U(s))^{*}g\|_{L_x^{\infty}}\lesssim |t-s|^{-\sigma}\|g\|_{L_x^1}.
\end{equation}
Then the estimates
\begin{gather}
\|U(t)f\|_{L_{t}^{r}L_x^p}\lesssim \|f\|_{L_x^2} ,
\\
\|\int_{\mathbb{R}}(U(s))^{*}F(s)ds\|_{L_x^2}\lesssim \|F\|_{L_{t}^{r'}L_x^{p'}} ,
\\
\|\int_{s<t}U(t)(U(s))^{*}F(s)ds\|_{L_t^r L_x^p}\lesssim\|F\|_{L_{t}^{\tilde{r}'}L_x^{\tilde{p}'}}
\end{gather}
hold for all sharp $\sigma$-admissible pairs $(r,p)$ and $(\tilde{r},\tilde{p})$. 
If $U(t)$ satisfies stronger condition
\begin{equation}
\|U(t)(U(s))^{*}g\|_{L_x^{\infty}}\lesssim (1+|t-s|)^{-\sigma}\|g\|_{L_x^1},
\end{equation}
then the above estimates hold for all $\sigma$-admissible pairs.
\end{lemma}

In 2012, Zhang\cite{Z_NLA_2012} established some Strichartz estimates 
in the frame of modulation spaces, here we will study the estimates 
in the frame of $\alpha$-modulation spaces, our theorems will cover 
the estimates in \cite{Z_NLA_2012}.
First, we recall the definition of $\alpha$-modulation space.

\begin{definition}\rm
Let $\rho(\xi)$ be a smooth radial bump supported in the ball 
$|\xi|<2$, satisfying $\rho(\xi)=1$ as $|\xi|\leq 1$.
For any $k\in \mathbb{Z}^n$, we set
\begin{equation}
\rho_k^\alpha(\xi)
=\rho\Big(\frac{\xi-\langle k\rangle^{\alpha/(1-\alpha)}k}
{C\langle k\rangle^{\alpha/(1-\alpha)}} \Big),
\end{equation}
and denote
\begin{equation}
\eta_k^\alpha(\xi)=\rho_k^\alpha(\xi)
\Big(\sum_{l\in \mathbb{Z}^n}\rho_l^\alpha(\xi)\Big)^{-1}.
\end{equation}
For any $k\in \mathbb{Z}^n$, we define
\begin{equation}
\Box_k^{\alpha}=\mathscr{F}^{-1}\eta_k^\alpha\mathscr{F}.
\end{equation}
When $\alpha \in [0,1)$, the $\alpha$-modulation space associated 
with the above decomposition is defined by
\[
M_{p,q}^{s,\alpha}(\mathbb{R}^n)
=\big\{f\in \mathcal {S}'(\mathbb{R}^{n}): 
\|f\|_{M_{p,q}^{s,\alpha}(\mathbb{R}^n)}
=\Big( \sum_{k\in \mathbb{Z}^{n}}\langle k\rangle ^{sq/(1-\alpha)}
\|\Box_k^{\alpha} f\|_{p}^{q}\Big)^{1/q}<\infty \big\}
\]
with the usual modifications when $q=\infty$.
\end{definition}

The $\alpha$-modulation space was introduced by
 Gr\"obner \cite{P_introduction}, and it is an intermediate space
between modulation space
and Besov space, when $\alpha=0$ it is the usual modulation space,
and the Besov space can be regarded as the case $\alpha\to 1$.
A comprehensive study of $\alpha$-modulation space has been 
done in \cite{Wang_Han}.
We also need some frequency decomposition spaces which are similar
 with the spaces defined in \cite{WH_JDE_2007}.

\begin{definition} \rm
If $X=L_{t}^{r}L_x^p(\mathbb{R}\times \mathbb{R}^n)$ $(1\leq r,p,q < \infty)$, 
we denote
\[
l_{\Box^{\alpha}}^{s,q}(X)=\big\{u\in \mathcal {S}'(\mathbb{R}^{n+1})
:\|u\|_{l_{\Box^{\alpha}}^{s,q}(X)}
=\Big(\sum_{k\in \mathbb{Z}^n} \langle k\rangle ^{sq/(1-\alpha)}\|
\Box_k^{\alpha} u\|_{X}^q\Big)^{1/q}<\infty \big\}
\]
with the usual modifications when $q=\infty$.
\end{definition}


Using Minkowski's inequality, we can verify that for $r\geq q$,
\begin{equation}
\|f\|_{L_t^r(\mathbb{R},M^{\alpha,s}_{p,q})}
\leq
\|f\|_{l^{s,q}_{\Box^{\alpha}}(L_{t}^{r}L_x^p)}\,.
\end{equation}

In Section 2, we will give some basic definitions and properties associated 
with  $\alpha$-modulation spaces, and recall some basic estimates of 
oscillatory integral which are useful in our proof. We will give the proof 
of main theorems in Section 3.
Standard techniques involving $TT^{*}$ method and duality argument 
will be used to establish some Strichartz estimates.
In the case when $(r,p)$ and $(\tilde{r},\tilde{p})$ are sharp,
we will use a dilation argument, based on some basic Strichartz estimates.
Now, we present our main results.
First the Strichartz estimates for Schr\"odinger equation:

\begin{theorem}\label{S-estimate1}
Suppose $s\in \mathbb{R}$, $q\geq 1$, $\alpha \in [0,1)$, 
$\beta \in (0,2]$ and $\beta \neq 1$, $(p,r)$ and $(\tilde{p},\tilde{r})$ 
are both $\frac{n}{2}$-admissible pairs,
then the solution of \eqref{1.1} satisfies
\begin{equation}
\|u(t,x)\|_{l_{\Box^{\alpha}}^{s,q}(L_t^rL_x^p)}\lesssim\|u_0\|_{M_{2,q}^{s+\delta(r,p),\alpha}}
+\|F\|_{l_{\Box^{\alpha}}^{s+\delta(r,p)+\delta(\tilde{r},\tilde{p}),q}(L_t^{\tilde{r}'}L_x^{\tilde{p}'})}
\end{equation}
where $\delta(r,p)=\alpha(\frac{n}{2}-\frac{2}{r}-\frac{n}{p})
+(2-\beta)\frac{1}{r}$.
More precisely, we have
\begin{gather}\label{S-estimate1.1}
\|e^{it(-\Delta)^{\beta/2}}u_0\|_{l_{\Box^{\alpha}}^{s,q}(L_t^rL_x^p)}
\lesssim \|u_0\|_{M_{2,q}^{s+\delta(r,p),\alpha}},\\
\label{S-estimate1.2}
\|\int_{s<t} e^{i(t-s)(-\Delta)^{\beta/2}}F(s)ds
\|_{l_{\Box^{\alpha}}^{s,q}(L_t^rL_x^p)} 
\lesssim \|F\|_{l_{\Box^{\alpha}}^{s+\delta(r,p)
+\delta(\tilde{r},\tilde{p}),q}(L_t^{\tilde{r}'}L_x^{\tilde{p}'})}.
\end{gather}
\end{theorem}

Next we have the Strichartz estimates for nonelliptic Schr\"odinger equation:

\begin{theorem} \label{S-estimate2}
Suppose $s\in \mathbb{R}$, $q\geq 1$, $\alpha \in [0,1)$,
 $\beta \in (0,2]$ and $\beta \neq 1$, $(p,r)$ and $(\tilde{p},\tilde{r})$ 
are both $\frac{n}{2}$-admissible pairs,
then the solution of \eqref{1.2} satisfies
\begin{equation}
\|v(t,x)\|_{l_{\Box^{\alpha}}^{s,q}(L_t^rL_x^p)}
\lesssim\|v_0\|_{M_{2,q}^{s+\delta(r,p),\alpha}}
+\|F\|_{l_{\Box^{\alpha}}^{s+\delta(r,p)+\delta(\tilde{r},
\tilde{p}),q}(L_t^{\tilde{r}'}L_x^{\tilde{p}'})}
\end{equation}
where $\delta(r,p)=\alpha(\frac{n}{2}-\frac{2}{r}-\frac{n}{p})
+(2-\beta)\frac{1}{r}$.
More precisely, we have
\begin{gather}\label{S-estimate2.1}
\|e^{it\psi(D)}v_0\|_{l_{\Box^{\alpha}}^{s,q}(L_t^rL_x^p)} 
\lesssim \|v_0\|_{M_{2,q}^{s+\delta(r,p),\alpha}},\\
\label{S-estimate2.2}
\|\int_{s<t} e^{i(t-s)\psi(D)}F(s)ds\|_{l_{\Box^{\alpha}}^{s,q}(L_t^rL_x^p)} 
\lesssim \|F\|_{l_{\Box^{\alpha}}^{s+\delta(r,p)+\delta(\tilde{r},
\tilde{p}),q}(L_t^{\tilde{r}'}L_x^{\tilde{p}'})}.
\end{gather}
\end{theorem}

Next we have the Strichartz estimates for wave equation:

\begin{theorem}\label{S-estimate3}
Suppose $s\in \mathbb{R}$, $q\geq 1$, $\alpha \in [0,1)$, $(p,r)$ and 
$(\tilde{p},\tilde{r})$ are both $\frac{n-1}{2}-admissible$ pairs,
if $\frac{n}{2}-\frac{n}{p}-\frac{1}{r}-1>0$ and 
$n-1-\frac{n}{p}-\frac{1}{r}-\frac{n}{\tilde{p}}-\frac{1}{\tilde{r}}>0$,
then the solution of \eqref{1.3} satisfies
\[
\|w\|_{l_{\Box^{\alpha}}^{s,q}(L_t^rL_x^p)}
\lesssim\|w_0\|_{M_{2,q}^{s+\theta(r,p),\alpha}}
+\|w_1\|_{M_{2,q}^{s+\theta(r,p)-1,\alpha}}
+\|F\|_{l_{\Box^{\alpha}}^{s+\theta(r,p)
+\theta(\tilde{r},\tilde{p})-1,q}(L_t^{\tilde{r}'}L_x^{\tilde{p}'})}
\]
where $\theta(r,p) =\alpha\frac{n}{n-1}
(\frac{n-1}{2}-\frac{2}{r}-\frac{n-1}{p})
+\frac{n+1}{n-1}\frac{1}{r}$.
More precisely, we have
\begin{gather}\label{S-estimate3.1}
\|\cos(t\sqrt{-\Delta})w_0\|_{l_{\Box^{\alpha}}^{s,q}(L_t^rL_x^p)}
 \lesssim \|w_0\|_{M_{2,q}^{s+\theta(r,p),\alpha}}, \\
\label{S-estimate3.2}
\|\frac{\sin(t\sqrt{-\Delta})}{\sqrt{-\Delta}}w_1\|_{l_{\Box^{\alpha}}^{s,q}
(L_t^rL_x^p)} \lesssim \|w_1\|_{M_{2,q}^{s+\theta(r,p)-1,\alpha}},\\
\label{S-estimate3.3}
\|\int_{s<t}\frac{\sin((t-s)\sqrt{-\Delta})}{\sqrt{-\Delta}}F(s)ds
\|_{l_{\Box^{\alpha}}^{s,q}(L_t^rL_x^p)}
\lesssim
\|F\|_{l_{\Box^{\alpha}}^{s+\theta(r,p)+\theta(\tilde{r},
\tilde{p})-1,q}(L_t^{\tilde{r}'}L_x^{\tilde{p}'})}.
\end{gather}
\end{theorem}
We must point out that our results can't be deduced directly by a simple 
interpolation between the modulation space($\alpha=0$) and the Besov space, 
since in \cite{Wang_Han}, the authors have pointed
out that $\alpha-$modulation space can't be reformulated by interpolations 
between modulation and Besov spaces at least for some special cases.

\section{Preliminaries}

We will often use the notation $X\lesssim  Y$ whenever there exists some 
constant $C$ so that $X\leq CY$, $C$ can depend on 
$n,p,r,\tilde{p},\tilde{r},\alpha,\beta$.
For $k=(k_1,k_2,\dots, k_n)\in \mathbb{Z}^{n}, 
\langle k\rangle: =(1+|k|^{2})^{\frac{1}{2}}$.
We denote by $\mathcal {S}:= \mathcal {S}(\mathbb{R}^{n})$ the Schwartz space
and $\mathcal {S}':=\mathcal {S}'(\mathbb{R}^{n})$ the tempered distribution 
space.
We use $L_{t}^{r}(\mathbb{R})$ to denote the Banach space of functions 
$f:\mathbb{R} \to \mathbb{C}$ whose norm
\begin{equation}
\|f\|_{L_{t}^{r}(\mathbb{R})}
:= \Big(\int_{\mathbb{R}}|f(t)|^{r}dt\Big)^{1/r}
\end{equation}
is finite, with the usual modifications when $r=\infty$.
We use $L_{t}^{r}L_x^p(\mathbb{R}\times \mathbb{R}^n)$ to denote 
the spacetime norm
\begin{equation}
\|F\|_{L_{t}^{r}L_x^p(\mathbb{R}\times \mathbb{R}^n)}
=\Big(\int_{\mathbb{R}} \|F\|_{L_x^p}^r dt\Big)^{1/r}
\end{equation}
with the usual modifications when $p,q$ or $r$ is infinity.

In our proofs, the almost orthogonality property will be used frequently, this
property is independent of $\alpha$.
Here we give a proof which is different from \cite{Wang_Han} and seems
 more regular. Firstly, we establish following position lemma.

\begin{lemma} \label{position-lemma}
For every $x,y\in \mathbb{R}^n$, we have the following
\begin{align*}
\min(\langle x\rangle^{\alpha/(1-\alpha)},
 \langle y\rangle^{\alpha/(1-\alpha)})|x-y|
&\lesssim|\langle x\rangle^{\alpha/(1-\alpha)} x
 -\langle y\rangle^{\alpha/(1-\alpha)} y| \\
&\lesssim \max(\langle x\rangle^{\alpha/(1-\alpha)},\langle
 y\rangle^{\alpha/(1-\alpha)})|x-y|.
\end{align*}
Particularly, if we choose $y=k$, $|x-y|=r$, then
\begin{align*}
\min(\langle x\rangle^{\alpha/(1-\alpha)},\langle k\rangle^{\alpha/(1-\alpha)})r
&\lesssim|\langle x\rangle^{\alpha/(1-\alpha)} x
 -\langle k\rangle^{\alpha/(1-\alpha)} k|\\
&\lesssim \max(\langle x\rangle^{\alpha/(1-\alpha)},
\langle k\rangle^{\alpha/(1-\alpha)})r,
\end{align*}
so we have
\begin{equation}
|\langle x\rangle^{\alpha/(1-\alpha)} x-\langle k\rangle^{\alpha/(1-\alpha)} k|
\thicksim
\langle k\rangle^{\alpha/(1-\alpha)}
\quad\text{if $|k|\longrightarrow \infty$   and $|x-k|=r$}.
\end{equation}
\end{lemma}

\begin{proof}
By the symmetry of $x$ and $y$, we only prove the case $|x|\leq |y|$.
Let
\begin{equation}
h(t)=|x-ty|^2,
\end{equation}
and derivative it, we have
\begin{equation}
h'(t)=\sum_{i=1}^n 2ty_i^2-\sum_{i=1}^n 2x_iy_i.
\end{equation}
Using the fact $|x|\leq |y|$ and Cauchy-Schwartz inequality,
we verify that
\begin{equation}
h'(t)\geq 0
\end{equation}
when $t\geq 1$.
Using this inequality, we conclude
\begin{equation}
\begin{split}
|\langle x\rangle^{\alpha/(1-\alpha)} x-\langle y\rangle^{\alpha/(1-\alpha)} y|
&=\langle x\rangle^{\alpha/(1-\alpha)}| x-\frac{\langle y\rangle^{\alpha/(1-\alpha)}}{\langle x\rangle^{\alpha/(1-\alpha)}} y|
\\
&\geq \langle x\rangle^{\alpha/(1-\alpha)}|x-y|\\
&=\min(\langle x\rangle^{\alpha/(1-\alpha)},
 \langle y\rangle^{\alpha/(1-\alpha)})|x-y|.
\end{split}
\end{equation}
On the other hand,
\[
|\langle x\rangle^{\alpha/(1-\alpha)} x-\langle y\rangle^{\alpha/(1-\alpha)} y|
\leq
|\langle x\rangle^{\alpha/(1-\alpha)}-\langle y\rangle^{\alpha/(1-\alpha)}||x|
+|\langle y\rangle^{\alpha/(1-\alpha)}||x-y|.
\]
By mean-valued theorem and the fact $|x|\leq |y|$, one can verify that
\begin{equation}
|\langle x\rangle^{\alpha/(1-\alpha)}-\langle y\rangle^{\alpha/(1-\alpha)}|
\lesssim
|x-y|\langle y\rangle^{\frac{\alpha}{1-\alpha}-1}.
\end{equation}
Since $|x|\leq |y| \leq \langle y\rangle$, so we have
\begin{equation}
|\langle x\rangle^{\alpha/(1-\alpha)}-\langle y\rangle^{\alpha/(1-\alpha)}||x|
\lesssim
|x-y|\langle y\rangle^{\alpha/(1-\alpha)},
\end{equation}
so
\begin{align*}
|\langle x\rangle^{\alpha/(1-\alpha)} x-\langle y\rangle^{\alpha/(1-\alpha)} y|
&\lesssim
|\langle x\rangle^{\alpha/(1-\alpha)}-\langle y\rangle^{\alpha/(1-\alpha)}||x|
+|\langle y\rangle^{\alpha/(1-\alpha)}||x-y|
\\
&\lesssim
|\langle y\rangle^{\alpha/(1-\alpha)}||x-y|\\
&= \max(\langle x\rangle^{\alpha/(1-\alpha)},
\langle y\rangle^{\alpha/(1-\alpha)})|x-y|.
\end{align*}
\end{proof}

By the position lemma, we can conclude that there exists a positive 
constant $c_1$ independent of $x$ and $k$, such that
\begin{equation}\label{regular}
c_1\langle k\rangle^{\alpha/(1-\alpha)}|x-k|
\leq
|\langle x\rangle^{\alpha/(1-\alpha)} x-\langle k\rangle^{\alpha/(1-\alpha)} k|.
\end{equation}

In fact, if $|k|>2|x|$, we have
\begin{equation}
|x-k|\leq |x|+|k|\leq \frac{3}{2}|k|,
\end{equation}
then
\begin{align*}
|\langle x\rangle^{\alpha/(1-\alpha)} x-\langle k\rangle^{\alpha/(1-\alpha)} k|
&\geq
|\langle k\rangle^{\alpha/(1-\alpha)} k|-|\langle x\rangle^{\alpha/(1-\alpha)} x|
\\
&\geq
|\langle k\rangle^{\alpha/(1-\alpha)} k|-\frac{1}{2}|\langle k\rangle^{\alpha/(1-\alpha)} k|
=\frac{1}{2}|\langle k\rangle^{\alpha/(1-\alpha)} k|
\\
&\gtrsim
\frac{1}{2}\cdot \frac{2}{3}|\langle k\rangle^{\alpha/(1-\alpha)}||x-k|=\frac{1}{3}|\langle k\rangle^{\alpha/(1-\alpha)}||x-k|.
\end{align*}
If $|k|\leq 2|x|$, then
\begin{equation}
\begin{split}
|\langle x\rangle^{\alpha/(1-\alpha)} x-\langle k\rangle^{\alpha/(1-\alpha)} k|
&\geq
\min(\langle x\rangle^{\alpha/(1-\alpha)},\langle k\rangle^{\alpha/(1-\alpha)})|x-k|
\\
&\geq
\min(c_\alpha\langle k\rangle^{\alpha/(1-\alpha)},\langle k\rangle^{\alpha/(1-\alpha)})|x-k|
\\
&\gtrsim
\langle k\rangle^{\alpha/(1-\alpha)}|x-k|.
\end{split}
\end{equation}
Similarly, for a fixed constant $G>0$, we can find
a constant $c_2$ only depend on $G$, such that if $|x-k|<G$,
\begin{equation}
|\langle x\rangle^{\alpha/(1-\alpha)} x-\langle k\rangle^{\alpha/(1-\alpha)} k|
\leq
c_2\langle k\rangle^{\alpha/(1-\alpha)}|x-k|.
\end{equation}
This can be easily concluded by position lemma.

We set a map $J_\alpha$ from $\mathbb{R}^n$ to $\mathbb{R}^n$
\begin{equation}
J_\alpha(x)=\langle x\rangle^{\alpha/(1-\alpha)}x.
\end{equation}
Using inequality \eqref{regular}, we can take $R$ sufficiently large such that
$Rc_1>2C$, thus
\begin{equation}
\operatorname{supp}\eta_k^\alpha
\subset
B(\langle k\rangle^{\alpha/(1-\alpha)}k,c_1\langle k\rangle^{\alpha/(1-\alpha)}R)
\subset J_\alpha(B(k,R)).
\end{equation}
Similarly, we can choose $r$ small enough such that
$rc_2<C$, thus
\begin{equation}
J_\alpha(B(k,r))
\subset
B(\langle k\rangle^{\alpha/(1-\alpha)}k,c_2\langle k\rangle^{\alpha/(1-\alpha)}r)
\subset
\operatorname{supp}\eta_k^\alpha.
\end{equation}
So we have
\begin{equation}
J_\alpha(B(k,r)) \subset
\operatorname{supp}\eta_k^\alpha
\subset J_\alpha(B(k,R)),
\end{equation}
and
\begin{equation}
\begin{split}
\{(i,j): \operatorname{supp}\eta_i^\alpha\cap\operatorname{supp}\eta_j^\alpha
 \neq \emptyset\}
&\subset\{(i,j):J_{\alpha}(B(i,R))\cap J_{\alpha}(B(j,R))\ \neq \emptyset\}
\\
&=\{(i,j): (B(i,R)\cap B(j,R)) \neq \emptyset \}.
\end{split}
\end{equation}
So, in some sense, the $\alpha$-modulation space is as regular as modulation 
space up to a transform $J_{\alpha}$.
We recall some estimates of oscillatory integrals, which can be deduced
 by principle of stationary phase. One can find
the methods in \cite{Stein} and \cite{Sogge_book}.
We use $\varphi(\xi)$ to denote the symbol of the Littlewood-Paley 
operator $\Delta_0$.

\begin{lemma}\label{estimate1}
If $0<\beta \neq 1$, then
\begin{equation}
\Big|\int_{\mathbb{R}^n}e^{it|\xi|^{\beta}}\varphi(\xi)e^{ix\cdot \xi}d\xi\big|
\lesssim |t|^{-n/2}.
\end{equation}
If $\beta=1$, then
\begin{equation}
\Big|\int_{\mathbb{R}^n}e^{it|\xi|}\varphi(\xi)e^{ix\cdot \xi}d\xi\Big|
\lesssim |t|^{(n-1)/2}.
\end{equation}
\end{lemma}

We omit the proof of the above lemma, and refer the reader to \cite{Z_NLA_2012}. 
This lemma can be also concluded by a lemma by Littman \cite{Wang_book}.
Then we will show another inequality which will be used in the proof of 
Theorem \ref{1.2}, one can find the proof
in \cite{Z_NLA_2012} for the case that $\alpha=0$.

\begin{lemma}\label{estimate2}
If $0<\beta\leq 2$ and $\beta \neq 1$, then
\begin{equation}
\Big|\int_{\mathbb{R}^n}e^{it\sum_{l=1}^n\pm |\xi_l|^{\beta}}
\eta_k^{\alpha}(\xi)e^{ix\cdot \xi}d\xi\Big|
\lesssim
\langle k\rangle^{\frac{2-\beta}{1-\alpha}\frac{n}{2}}|t|^{-n/2}.
\end{equation}
\end{lemma}

\begin{proof}
We only prove the case $\beta < 2$.
One can easily find a one dimension smooth bump function $\phi(\xi)$
such that $\prod_{l=1}^n\phi(\frac{\xi_l-\langle 
k\rangle^{\alpha/(1-\alpha)}k_l }{\langle k\rangle^{\alpha/(1-\alpha)}})
\eta_k^{\alpha}=\eta_k^{\alpha}$
for every $k\in \mathbb{Z}^n$. Then we have
\begin{equation}
\begin{split}
&\Big|\int_{\mathbb{R}^n}e^{it\sum_{l=1}^n\pm |\xi_l|^{\beta}}
 \prod_{l=1}^n\phi(\frac{\xi_l-\langle 
k\rangle^{\alpha/(1-\alpha)}k_l }{\langle k\rangle^{\alpha/(1-\alpha)}})
e^{ix\cdot \xi}d\xi_l\Big|
\\
&=\prod_{l=1}^n\Big|\int_{\mathbb{R}}e^{i(x_l\cdot \xi_l\pm t 
|\xi_l|^{\beta})} \phi(\frac{\xi_l-\langle k
\rangle^{\alpha/(1-\alpha)}k_l }{\langle k\rangle^{\alpha/(1-\alpha)}})d\xi_l
\Big|
\end{split}
\end{equation}
and
\begin{align*}
&\Big|\int_{\mathbb{R}}e^{i(x_l\cdot \xi_l\pm t |\xi_l|^{\beta})} 
\phi(\frac{\xi_l-\langle k\rangle^{\alpha/(1-\alpha)}k_l }{\langle 
k\rangle^{\alpha/(1-\alpha)}})d\xi_l\Big|
\\
\lesssim
&\Big|\int_{\mathbb{R}^{+}}e^{iP_{+}(\xi_l)}\phi(\frac{\xi_l-\langle 
 k\rangle^{\alpha/(1-\alpha)}k_l }{\langle k\rangle^{\alpha/(1-\alpha)}})d\xi_l
  \Big|
+\Big|\int_{\mathbb{R}^{+}}e^{iP_{-}(\xi_l)}\phi
(\frac{-\xi_l-\langle k\rangle^{\alpha/(1-\alpha)}k_l }{\langle 
k\rangle^{\alpha/(1-\alpha)}})d\xi_l \Big|.
\end{align*}
When $k_l\neq 0$ ($k_l$ is large), we have 
$|P''_{\pm}(\xi_l)|\gtrsim  
t\big( \langle k\rangle^{\alpha/(1-\alpha)}k_l \big)^{\beta-2}$,
 we use Van de Corputs lemma to deduce
\begin{equation}
\begin{split}
&\Big|\int_{\mathbb{R}}e^{i(x_l\cdot \xi_l\pm t |\xi_l|^{\beta})} 
\phi(\frac{\xi_l-\langle k\rangle^{\alpha/(1-\alpha)}k_l }{\langle 
k\rangle^{\alpha/(1-\alpha)}})d\xi_l\Big|
\\
&\lesssim
\Big\|\Big(\phi\big(\frac{\xi_l-\langle k\rangle^{\alpha/(1-\alpha)}k_l }{\langle 
 k\rangle^{\alpha/(1-\alpha)}}\big)\Big)'\|_{L^1}
\left( \langle k\rangle^{\alpha/(1-\alpha)}k_l \right)^{(2-\beta)/2}
|t|^{-1/2}
\\
&\lesssim \left( \langle k\rangle^{\alpha/(1-\alpha)}k_l \right)^{(2-\beta)/2}
|t|^{-1/2}
\lesssim
\left( \langle k\rangle^{\frac{1}{1-\alpha}} \right)^{(2-\beta)/2}
|t|^{-1/2}.
\end{split}
\end{equation}
When $k_l= 0$ ($k_l$  small), we have
\begin{align*}
&\Big|\int_{\mathbb{R}}e^{i(x_l\cdot \xi_l\pm t |\xi_l|^{\beta})} 
\phi(\frac{\xi_l }{\langle k\rangle^{\alpha/(1-\alpha)}})d\xi_l\Big|\\
&=
\langle k\rangle^{\alpha/(1-\alpha)}
\Big|\int_{\mathbb{R}}e^{i(\langle k\rangle^{\alpha/(1-\alpha)}
x_l\cdot \xi_l\pm \langle k\rangle^{\frac{\alpha\beta}{1-\alpha}}
t |\xi_l|^{\beta})} \phi(\xi_l )d\xi_l\Big|.
\end{align*}
Let $\varphi_j(\xi)=\varphi(\xi/2^j)$ be the symbol of one dimension 
Littlewood-Paley operator $\Delta_j$,
we use Lemma \ref{estimate1}(one dimension version) and dilation to 
deduce that
\[
\langle k\rangle^{\alpha/(1-\alpha)}
\Big|\int_{\mathbb{R}}e^{i(\langle k\rangle^{\alpha/(1-\alpha)}
x_l\cdot \xi_l\pm \langle k\rangle^{\frac{\alpha\beta}{1-\alpha}}
t |\xi_l|^{\beta})} \varphi_j(\xi_l )d\xi_l\Big|
\lesssim
2^{j(1-\frac{\beta}{2})}\langle k\rangle^{\frac{\alpha}{1-\alpha}
\frac{2-\beta}{2}}|t|^{-1/2},
\]
then 
\begin{align*}
&\langle k\rangle^{\alpha/(1-\alpha)}
\Big|\int_{\mathbb{R}}e^{i(\langle k\rangle^{\alpha/(1-\alpha)}
x_l\cdot \xi_l\pm \langle k\rangle^{\frac{\alpha\beta}{1-\alpha}}t 
|\xi_l|^{\beta})} \phi(\xi_l )d\xi_l\Big|\\
&\lesssim
\sum_{j=-\infty}^{0}2^{j(1-\frac{\beta}{2})}\langle k\rangle^{\frac{\alpha}{1-\alpha}\frac{2-\beta}{2}}|t|^{-1/2}
\\
&\lesssim
\langle k\rangle^{\frac{\alpha}{1-\alpha}\frac{2-\beta}{2}}|t|^{-1/2}
\\
&\lesssim
\left( \langle k\rangle^{\frac{1}{1-\alpha}} \right)^{(2-\beta)/2}|t|^{-1/2}.
\end{align*}
Using above estimates and the fact that 
$\|\mathscr{F}^{-1}\eta_k^{\alpha}\|_{L^1}\lesssim 1$, we complete the proof.
\end{proof}

\section{Proof of main results}

\subsection*{Proof of Theorem \ref{S-estimate1}}
We  prove only the case that $\beta<2$.
Let $\varphi_j=\varphi(\frac{\xi}{2^j})$ be the symbol of the 
littlewood-paley operator $\Delta_j$.
Using dilation and Lemma \ref{estimate1}, we can deduce that
\begin{equation}
\Big|\int_{\mathbb{R}^n}e^{it|\xi|^{\beta}}\varphi(\frac{\xi}{2^j})
e^{ix\cdot \xi}d\xi\Big|\lesssim 2^{jn(1-\frac{\beta}{2})}|t|^{-n/2},
\end{equation}
so we have
\begin{equation}
\begin{split}
\|\Box_0^{\alpha}e^{it(-\Delta)^{\beta/2}}f\|_{L_x^{\infty}}
\lesssim
&\sum_{j\leq c}\|\Delta_j\Box_0^{\alpha}e^{it(-\Delta)^{\beta/2}}f\|_{L_x^{\infty}}
\\
\lesssim
&\sum_{j\leq c}2^{jn(1-\frac{\beta}{2})}|t|^{-n/2}\|\Box_0^{\alpha}f\|_{L_x^{1}}\\
&\lesssim |t|^{-n/2}|\Box_0^{\alpha}f\|_{L_x^{1}} \lesssim |t|^{-n/2}\|f\|_{L_x^{1}}.
\end{split}
\end{equation}
For $k\neq 0$, we have
\begin{equation}
\begin{split}
\|\Box_k^{\alpha}e^{it(-\Delta)^{\beta/2}}f\|_{L_x^{\infty}}
\lesssim
&\sum_{\operatorname{supp}\varphi_j\cap \operatorname{supp} \eta_k^{\alpha}\neq \emptyset}\|\Delta_j\Box_k^{\alpha}e^{it(-\Delta)^{\beta/2}}f\|_{L_x^{\infty}}
\\
\lesssim
&\sum_{\operatorname{supp}\varphi_j\cap \operatorname{supp} \eta_k^{\alpha}\neq \emptyset}2^{jn(1-\frac{\beta}{2})}|t|^{-n/2}\|\Box_k^{\alpha}f\|_{L_x^{1}}.
\end{split}
\end{equation}
Using the almost orthogonality property and 
$2^j\sim \langle k\rangle^{\frac{1}{1-\alpha}}$, we can deduce that
\begin{equation}
\|\Box_k^{\alpha}e^{it(-\Delta)^{\beta/2}}f\|_{L_x^{\infty}}
\lesssim
\left(\langle k\rangle^{\frac{\beta-2}{1-\alpha}}|t|\right)^{-n/2}\|f\|_{L_x^{1}}.
\end{equation}
On the other hand, we have
\begin{equation}
\Big|\int_{\mathbb{R}^n}e^{it|\xi|^{\beta}}\eta_k^{\alpha}(\xi)
e^{ix\cdot \xi}d\xi\Big|
\lesssim \|\eta_k^{\alpha}\|_{L^1}
\lesssim \langle k\rangle^{\frac{\alpha}{1-\alpha}n}
=\left(\langle k\rangle^{\frac{\alpha}{1-\alpha}(-2)}\right)^{-n/2}.
\end{equation}
The above two estimates imply that
\begin{equation}\label{dispersive estimate 1}
\|\Box_k^{\alpha}e^{it(-\Delta)^{\beta/2}}f\|_{L_x^{\infty}}
\lesssim
\left(\langle k\rangle^{\frac{\alpha}{1-\alpha}(-2)}
+\langle k\rangle^{\frac{\beta-2}{1-\alpha}}|t|\right)^{-n/2}\|f\|_{L_x^1}.
\end{equation}
When $k=0$, we have
\begin{equation}
\|\Box_0^{\alpha}e^{it(-\Delta)^{\beta/2}}f\|_{L_x^{\infty}}
\lesssim \left(1+|t|\right)^{-n/2}\|f\|_{L_x^1}
\end{equation}
and the energy estimate
\begin{equation}
\|\Box_0^{\alpha}e^{it(-\Delta)^{\beta/2}}f\|_{L_x^{2}}
\lesssim \|f\|_{L_x^{2}},
\end{equation}
we can use Lemma \ref{strichartz-estimates} to deduce that
\begin{gather*}
\|\Box_0^{\alpha}e^{it(-\Delta)^{\beta/2}}f\|_{L_{t}^{r}L_x^p}
\lesssim \|f\|_{L_x^2} ,
\\
\|\Box_0^{\alpha}\int_{s<t}e^{i(t-s)(-\Delta)^{\beta/2}}F(s)ds
\|_{L_t^r L_x^p}\lesssim\|F\|_{L_{t}^{\tilde{r}'}L_x^{\tilde{p}'}} .
\end{gather*}
Using Lemma \ref{estimate1}, one can easily verify that
\begin{equation}
\|\Delta_0 e^{it(-\Delta)^{\beta/2}}f\|_{L_x^{\infty}}
\lesssim (1+|t|)^{-n/2}
\end{equation}
and deduce that
\begin{gather*}
\|\Delta_0 e^{it(-\Delta)^{\beta/2}}f\|_{L_{t}^{r}L_x^p}\lesssim \|f\|_{L_x^2} ,
\\
\|\Delta_0 \int_{s<t}e^{i(t-s)(-\Delta)^{\beta/2}}F(s)ds\|_{L_t^r L_x^p}
\lesssim\|F\|_{L_{t}^{\tilde{r}'}L_x^{\tilde{p}'}} .
\end{gather*}
Using dilation, we obtain
\begin{gather}\label{ls1}
\|\Delta_j e^{it(-\Delta)^{\beta/2}}f\|_{L_{t}^{r}L_x^p}\lesssim 2^{j
\left((\frac{n}{2}-\frac{n}{p}-\frac{2}{r})+\frac{2-\beta}{r}\right)}
\|f\|_{L_x^2} , \\
\label{ls2}
\begin{split}
&\|\Delta_j \int_{s<t}e^{i(t-s)(-\Delta)^{\beta/2}}F(s)ds\|_{L_t^r L_x^p}
\\
&\lesssim 2^{j\left((\frac{n}{2}-\frac{2}{r}-\frac{n}{p})
+(\frac{n}{2}-\frac{2}{\tilde{r}}-\frac{n}{\tilde{p}})+(2-\beta)
(\frac{1}{r}+\frac{1}{\tilde{r}})\right)}
\|F\|_{L_{t}^{\tilde{r}'}L_x^{\tilde{p}'}} .
\end{split}
\end{gather}
When $(r,p)$ is sharp $\frac{n}{2}$-admissible, we have
\begin{equation}
\begin{split}
\|\Box_k^{\alpha}e^{it(-\Delta)^{\beta/2}}f\|_{L_{t}^{r}L_x^p}
\lesssim
&\sum_{\operatorname{supp}\varphi_j\cap \operatorname{supp} 
\eta_k^{\alpha}\neq \emptyset}\|\Delta_j e^{it(-\Delta)^{\beta/2}}
f\|_{L_{t}^{r}L_x^p}
\\
&\lesssim \sum_{\operatorname{supp}\varphi_j\cap \operatorname{supp} 
\eta_k^{\alpha}\neq \emptyset} 2^{j\frac{2-\beta}{r}}\|f\|_{L_x^2}.
\end{split}
\end{equation}
Using the almost orthogonality property and 
$2^j\sim \langle k\rangle^{\frac{1}{1-\alpha}}$, we can deduce that
\begin{equation}
\|\Box_k^{\alpha}e^{it(-\Delta)^{\beta/2}}f\|_{L_{t}^{r}L_x^p}
\lesssim \langle k\rangle^{\frac{1}{1-\alpha}(\frac{2-\beta}{r})}\|f\|_{L_x^2}.
\end{equation}
Similarly, we can deduce that
\begin{equation}\label{inhomogeous sharp 1}
\|\Box_k^{\alpha} \int_{s<t}e^{i(t-s)(-\Delta)^{\beta/2}}F(s)ds\|_{L_t^r L_x^p}
\lesssim
\langle k\rangle^{\frac{1}{1-\alpha}(2-\beta)(\frac{1}{r}
+\frac{1}{\tilde{r}})}\|F\|_{L_{t}^{\tilde{r}'}L_x^{\tilde{p}'}}
\end{equation}
for all sharp $\frac{n}{2}$-admissible pairs $(r,p)$.
When $(r,p)$ is nonsharp $\frac{n}{2}$-admissible; that is,
$\frac{2}{r}+\frac{n}{p}<\frac{n}{2}$.
Combining with \ref{dispersive estimate 1} and energy estimate, we have
\begin{equation}
\|\Box_k^{\alpha}e^{it(-\Delta)^{\beta/2}}f\|_{L_x^{p}}
\lesssim
\left(\langle k\rangle^{\frac{\alpha}{1-\alpha}(-2)}+\langle k\rangle^{\frac{\beta-2}{1-\alpha}}|t|\right)^{-\frac{n}{2}(1-\frac{2}{p})}\|f\|_{L_x^{p'}}.
\end{equation}
So
\begin{equation}
\begin{split}
&\|\Box_k^{\alpha}\int_{\mathbb{R}}e^{i(t-s)(-\Delta)^{\beta/2}}
 F(s)ds\|_{L_t^r L_x^p}
\\
&\lesssim \big\|\int_{\mathbb{R}}\left(\langle k
\rangle^{\frac{\alpha}{1-\alpha}(-2)}
+\langle k\rangle^{\frac{\beta-2}{1-\alpha}}|t-s|
\right)^{-\frac{n}{2}(1-\frac{2}{p})}\|F(s)\|_{L_x^{p'}}ds\big\|_{L_t^r}
\\
&\lesssim \|\left(\langle k\rangle^{\frac{\alpha}{1-\alpha}(-2)}
+\langle k\rangle^{\frac{\beta-2}{1-\alpha}}|t|
\right)^{-\frac{n}{2}(1-\frac{2}{p})}\|_{L_t^{r/2}}
\|F(s)\|_{L_t^{r'}L_x^{p'}}.
\end{split}
\end{equation}
One can check that
\begin{equation}
\big\|\left(\langle k\rangle^{\frac{\alpha}{1-\alpha}(-2)}
+\langle k\rangle^{\frac{\beta-2}{1-\alpha}}|t|
\right)^{-\frac{n}{2}(1-\frac{2}{p})}\big\|_{L_t^{r/2}}
\lesssim
\langle k\rangle^{\frac{2\delta(r,p)}{1-\alpha}},
\end{equation}
then deduce that
\begin{equation}
\|\Box_k^{\alpha}\int_{\mathbb{R}}e^{i(t-s)(-\Delta)^{\beta/2}}F(s)ds
\|_{L_t^r L_x^p}
\lesssim
\langle k\rangle^{\frac{2\delta(r,p)}{1-\alpha}}\|F(s)\|_{L_t^{r'}L_x^{p'}}.
\end{equation}
Then homogeneous estimate\eqref{S-estimate1.1} follows by using $TT^{*}$ method, 
standard duality argument and the almost orthogonality property of 
$\alpha$-modulation space.
For the inhomogeneous part, for every $\frac{n}{2}$-admissible pairs $(r,p)$ 
and $(\tilde{r},\tilde{p})$, there exist two constant $p_1$ and $\tilde{p}_1$
such that $(r,p_1)$ and $(\tilde{r},\tilde{p_1})$ are sharp. 
Combining the following inequalities
\begin{gather*}
\|\Box_k^{\alpha}f\|_{L_x^p}
\lesssim
\langle k\rangle^{\frac{\alpha}{1-\alpha}n(\frac{1}{p_1}-\frac{1}{p})}\|
\Box_k^{\alpha}f\|_{L_x^{p_1}}, 
\\
\|\Box_k^{\alpha}f\|_{L_x^{\tilde{p_1}'}}
\lesssim
\langle k\rangle^{\frac{\alpha}{1-\alpha}n(\frac{1}{\tilde{p}_1}
-\frac{1}{\tilde{p}})}\|\Box_k^{\alpha}f\|_{L_x^{\tilde{p}'}}
\end{gather*}
with \eqref{inhomogeous sharp 1},
we have
\begin{equation}
\begin{split}
&\|\Box_k^{\alpha}\int_{s<t} e^{i(t-s)(-\Delta)^{\beta/2}}F(s)ds\|_{L_t^rL_x^p}
\\
&\lesssim \langle k\rangle^{\frac{\alpha}{1-\alpha}n(\frac{1}{p_1}-\frac{1}{p})}\|\Box_k^{\alpha}\int_0^t e^{i(t-s)(-\Delta)^{\beta/2}}F(s)ds\|_{L_t^rL_x^{p_1}}
\\
&\lesssim
 \langle k\rangle^{\frac{\alpha}{1-\alpha}n(\frac{1}{p_1}-\frac{1}{p})}
\langle k\rangle^{\frac{1}{1-\alpha}(2-\beta)(\frac{1}{r}+\frac{1}{\tilde{r}})}
\|F\|_{L_{t}^{\tilde{r}'}L_x^{\tilde{p_1}'}}
\\
&\lesssim
\langle k\rangle^{\frac{\alpha}{1-\alpha}n(\frac{1}{p_1}-\frac{1}{p})}
\langle k\rangle^{\frac{1}{1-\alpha}(2-\beta)(\frac{1}{r}+\frac{1}{\tilde{r}})}
\langle k\rangle^{\frac{\alpha}{1-\alpha}n(\frac{1}{\tilde{p}_1}-\frac{1}{\tilde{p}})}
\|F\|_{L_t^{\tilde{r}'}L_x^{\tilde{p}'}}.
\end{split}
\end{equation}
Recall that
$\frac{n}{p_1}=\frac{n}{2}-\frac{2}{r}$ and 
$\frac{n}{\tilde{p_1}}=\frac{n}{2}-\frac{2}{\tilde{r}}$, so we have
\begin{equation}
\|\Box_k^{\alpha}\int_{s<t} e^{i(t-s)(-\Delta)^{\beta/2}}F(s)ds\|_{L_t^rL_x^p}
\lesssim
\langle k\rangle^{\frac{\delta(r,p)+\delta(\tilde{r},\tilde{p})}{1-\alpha}}\|F\|_{L_t^{\tilde{r}'}L_x^{\tilde{p}'}}.
\end{equation}
Then the inhomogeneous estimate \eqref{S-estimate1.2} follows by 
the definition and almost orthogonality property of $\alpha$-modulation space.

\begin{remark} \rm
In the case $(r,p)$ is nonsharp $\frac{n}{2}$-admissible, one can also deduce
the estimates of $\|\Box_k^{\alpha}e^{it(-\Delta)^{\beta/2}}f\|_{L_{t}^{r}L_x^p}$ 
and $\|\Box_k^{\alpha} \int_{s<t}e^{i(t-s)(-\Delta)^{\beta/2}}F(s)ds
\|_{L_t^r L_x^p}$
by using \eqref{ls1} and \eqref{ls2} respectively,
but it will lose more regularity.
\end{remark}


\subsection*{Proof of Theorem \ref{S-estimate2}} 
We prove only the case $\beta<2$.
Using Lemma \ref{estimate2} and the fact that
\begin{equation}
\Big|\int_{\mathbb{R}^n}e^{it\sum_{l=1}^n\pm |\xi_l|^{\beta}}
\eta_k^{\alpha}(\xi)e^{ix\cdot \xi}d\xi\Big|
\lesssim \|\eta_k^{\alpha}\|_{L^1}
\lesssim \langle k\rangle^{\frac{\alpha}{1-\alpha}n}
=\left(\langle k\rangle^{\frac{\alpha}{1-\alpha}(-2)}\right)^{-n/2},
\end{equation}
we have
\begin{equation}\label{dispersive estimate 2}
\|\Box_k^{\alpha}e^{it\psi(D)}f\|_{L_x^{\infty}}
\lesssim
\left(\langle k\rangle^{\frac{\alpha}{1-\alpha}(-2)}
+\langle k\rangle^{\frac{\beta-2}{1-\alpha}}|t|\right)^{-n/2}\|f\|_{L_x^1}.
\end{equation}
When $k=0$ we can deduce the estimates as the proof of Theorem \ref{S-estimate1}.
When $k\neq 0$, we can also obtain the homogeneous estimates \eqref{S-estimate2.1}
 for nonsharp pair $(r,p)$ by $TT^{*}$ method and standard duality argument.

For the case that $(r,p)$ and $(\tilde{r},\tilde{p})$ are sharp, we use 
dilation argument, but it's a little different from the proof 
Theorem \ref{S-estimate1}.
Let $S_j=\sum_{l\leq j}\Delta_l$, $\varphi_0=\sum_{j\leq 0}\varphi_j$, 
using Lemma \ref{estimate2} with $k=0$, we can deduce
\begin{equation}
\Big|\int_{\mathbb{R}^n}e^{it\sum_{l=1}^n\pm |\xi_l|^{\beta}}
\varphi_0(\xi)e^{ix\cdot \xi}d\xi\Big|
\lesssim |t|^{-n/2},
\end{equation}
so
\begin{equation}
\|S_0 e^{it\psi(D)}f\|_{L_x^{\infty}}
\lesssim
(1+|t|)^{-n/2}\|f\|_{L_x^1}.
\end{equation}
Using Lemma \ref{strichartz-estimates}, we can obtain the following estimates:
\begin{gather*}
\|S_0 e^{it\psi(D)}f\|_{L_{t}^{r}L_x^p}\lesssim \|f\|_{L_x^2} ,
\\
\|S_0 \int_{s<t}e^{i(t-s)\psi(D)}F(s)ds\|_{L_t^r L_x^p}\lesssim\|F\|_{L_{t}^{\tilde{r}'}L_x^{\tilde{p}'}} .
\end{gather*}
A dilation argument then yields
\begin{gather*}
\|S_j e^{it\psi(D)}f\|_{L_{t}^{r}L_x^p}
\lesssim 2^{j\frac{2-\beta}{r}}\|f\|_{L_x^2} ,
\\
\|S_j \int_{s<t}e^{i(t-s)\psi(D)}F(s)ds\|_{L_t^r L_x^p}
\lesssim
2^{j(2-\beta)(\frac{1}{r}+\frac{1}{\tilde{r}})}
\|F\|_{L_{t}^{\tilde{r}'}L_x^{\tilde{p}'}}.
\end{gather*}
Then we use $S_j$ to cover $\Box_k^{\alpha}$ to deduce
\begin{gather*}
\|\Box_k^{\alpha}e^{it\psi(D)}f\|_{L_{t}^{r}L_x^p}
\lesssim
\langle k\rangle^{\frac{1}{1-\alpha}(\frac{2-\beta}{r})}\|f\|_{L_x^2},
\\
\|\Box_k^{\alpha} \int_{s<t}e^{i(t-s)\psi(D)}F(s)ds\|_{L_t^r L_x^p}
\lesssim
\langle k\rangle^{\frac{1}{1-\alpha}(2-\beta)(\frac{1}{r}+\frac{1}{\tilde{r}})}\|F\|_{L_{t}^{\tilde{r}'}L_x^{\tilde{p}'}}
\end{gather*}
for any sharp $\frac{n}{2}$-admissible pairs $(r,p)$ and $(\tilde{r},\tilde{p})$.

The remain case is that inhomogeneous estimate \eqref{S-estimate2.2} 
for nonsharp admissible pairs, it
can be deduced like the proof of Lemma \ref{1.1}, we omit the details.

\subsection*{Proof of Theorem \ref{1.3}}
We  need to prove only the following estimates:
\begin{gather} \label{S-estimate3.1.0}
\|\Box_k^{\alpha}e^{it\sqrt{-\Delta}}w_0\|_{L_t^rL_x^p} 
\lesssim \langle k\rangle^{\frac{\theta(r,p)}{1-\alpha}} \|w_0\|_{L_x^2},
\\ \label{S-estimate3.2.0}
\|\Box_k^{\alpha}\frac{e^{it\sqrt{-\Delta}}}{\sqrt{-\Delta}}w_1\|_{L_t^rL_x^p} 
\lesssim \langle k\rangle^{\frac{\theta(r,p)-1}{1-\alpha}}\|w_1\|_{L_x^2},
\\ \label{S-estimate3.3.0}
\|\Box_k^{\alpha}\int_{s<t}\frac{e^{i(t-s)\sqrt{-\Delta}}}{\sqrt{-\Delta}}F(s)ds\|_{L_t^rL_x^p}
\lesssim
\langle k\rangle^{\frac{\theta(r,p)+\theta(\tilde{r},\tilde{p})-1}{1-\alpha}}
\|F\|_{L_t^{\tilde{r}'}L_x^{\tilde{p}'}}.
\end{gather}
Using the same techniques as before, we can deduce \eqref{S-estimate3.1.0} and
\begin{equation*}
\|\Box_k^{\alpha}\int_{s<t}e^{i(t-s)\sqrt{-\Delta}}F(s)ds\|_{L_t^rL_x^p}
\lesssim
\langle k\rangle^{\frac{\theta(r,p)+\theta(\tilde{r},\tilde{p})}{1-\alpha}}
\|F\|_{L_t^{\tilde{r}'}L_x^{\tilde{p}'}}.
\end{equation*}
If $k\neq 0$, then \eqref{S-estimate3.2.0} and \eqref{S-estimate3.3.0} 
will then follow by
\begin{equation*}
\|\Box_k^{\alpha}(-\Delta)^{-1/2}f\|_{L_x^2}
\lesssim \langle k\rangle^{-\frac{1}{1-\alpha}}\|f\|_{L_x^2}.
\end{equation*}
If $k=0$, we do not have energy estimate for the operator 
$\frac{e^{it\sqrt{-\Delta}}}{\sqrt{-\Delta}}$, so
we can not use the $TT^{*}$ argument. One can deduce
\begin{gather*}
\Big|\int_{\mathbb{R}^n}e^{it|\xi|}|\xi|^{-1}\varphi(\xi)
e^{ix\cdot \xi}d\xi\Big|\lesssim |t|^{(n-1)/2}, 
\\
\Big|\int_{\mathbb{R}^n}e^{it|\xi|}|\xi|^{-1/2}\varphi(\xi)e^{ix\cdot
 \xi}d\xi\Big|\lesssim |t|^{(n-1)/2}
\end{gather*}
by principle of stationary phase as in Lemma \ref{estimate1}, 
then Lemma \ref{strichartz-estimates} will yield
\begin{gather*}
\|\Delta_0\frac{e^{it\sqrt{-\Delta}}}{\sqrt{-\Delta}}f\|_{L_t^rL_x^p}
\lesssim \|f\|_{L_x^2}, \\
\|\Delta_0\int_{s<t}\frac{e^{i(t-s)\sqrt{-\Delta}}}
{\sqrt{-\Delta}}Fds\|_{L_t^rL_x^p}
\lesssim \|F\|_{L_t^{\tilde{r}'}L_x^{\tilde{p}'}}.
\end{gather*}
Using a dilation argument, we obtain
\begin{gather*}
\|\Delta_j\frac{e^{it\sqrt{-\Delta}}}{\sqrt{-\Delta}}f\|_{L_t^rL_x^p}
\lesssim
2^{j(\frac{n}{2}-\frac{n}{p}-\frac{1}{r}-1)}\|f\|_{L_x^2},
\\
\|\Delta_j\int_{s<t}\frac{e^{i(t-s)\sqrt{-\Delta}}}
 {\sqrt{-\Delta}}Fds\|_{L_t^rL_x^p}
\lesssim
2^{j(n-1-\frac{n}{p}-\frac{1}{r}-\frac{n}{\tilde{p}}
-\frac{1}{\tilde{r}})}\|F\|_{L_t^{\tilde{r}'}L_x^{\tilde{p}'}}.
\end{gather*}
If $\frac{n}{2}-\frac{n}{p}-\frac{1}{r}-1>0$ and 
$n-1-\frac{n}{p}-\frac{1}{r}-\frac{n}{\tilde{p}}-\frac{1}{\tilde{r}}>0$,
we can use $\Delta_j$ to cover $\Box_0^{\alpha}$ and get the estimates
 \eqref{S-estimate3.2.0} and \eqref{S-estimate3.3.0} for $k=0$.

\begin{remark} \rm
If we take $\alpha=0$ in Theorem $1.1-1.3$, we obtain the Strichartz 
estimates in the frame of modulation spaces.
The Strichartz estimates of Schr\"odinger equation in the frame 
of $\alpha$ modulation spaces is the case that $\beta=2$.
\end{remark}

\subsection*{Acknowledgments}
This work was supported by grant Y604563 from the 
NSFZJ of China, and grant 11271330 from the NSF of China.

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\end{document}