# Decay estimates for a class of Dunkl wave equations

Cheng Luo  ,  Shyam Swarup Mondal  and  Manli Song  School of Mathematics and
Statistics, Northwestern Polytechnical University, Xi’an, Shaanxi 710129,
China<EMAIL_ADDRESS>Stat-Math unit, Indian Statistical Institute Kolkata,
BT Road, Baranagar, Kolkata 700108, India<EMAIL_ADDRESS>School of
Mathematics and Statistics, Northwestern Polytechnical University, Xi’an,
Shaanxi 710129, China<EMAIL_ADDRESS>
###### Abstract.

Let $\Delta_{\kappa}$ be the Dunkl Laplacian on $\mathbb{R}^{n}$ and
$\phi:\mathbb{R}^{+}\to\mathbb{R}$ is a smooth function. The aim of this
manuscript is twofold. First, we study the decay estimate for a class of
dispersive semigroup of the form $e^{it\phi(\sqrt{-\Delta_{\kappa}})}$. We
overcome the difficulty arising from the non-homogeneousity of $\phi$ by
frequency localization. As applications, in the next part of the paper, we
establish Strichartz estimates for some concrete wave equations associated
with the Dunkl Laplacian $\Delta_{k},$ which corresponds to
$\phi(r)=r,r^{2},r^{2}+r^{4},\sqrt{1+r^{2}},\sqrt{1+r^{4}}$, and
$r^{\mu},0<\mu\leq 2,\mu\neq 1$. More precisely, we unify and simplify all the
known dispersive estimates and extend to more general cases. Finally, using
the decay estimates, we prove the global-in-time existence of small data
Sobolev solutions for the nonlinear Klein-Gordon equation and beam equation
with the power type nonlinearities.

###### Key words and phrases:

Dunkl Laplacian, Dunkl Bosev space, Wave equation, Decay estimate, Strichartz
estimate.

###### 2010 Mathematics Subject Classification:

Primary 22E25, 33C45, 35H20, 35B40.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2.1 Dunkl operators
    2. 2.2 The Dunkl transform
    3. 2.3 Dunkl convolution operator
    4. 2.4 Dunkl Besov spaces
    5. 2.5 Technical Lemmas
  3. 3 Proof of main results
  4. 4 Applications
    1. 4.1 Wave equation
    2. 4.2 Fractional Schrödinger equation
    3. 4.3 Fourth-order Schrödinger equation
    4. 4.4 Klein-Gordon equation
    5. 4.5 Beam equation
  5. 5 Well-posedness results for nonlinear Klein-Gordon and beam equations

##  1\. Introduction

The main aim of this paper is to investigate decay estimates for a class of
dispersive equations of the form

$\begin{cases}i\partial_{t}u+\phi(\sqrt{-\Delta_{\kappa}})u=f,\\\
u|_{t=0}=u_{0},\end{cases}$ (1.1)

where $\Delta_{\kappa}$ is the Dunkl Laplacian on $\mathbb{R}^{n}$,
$f:\mathbb{R}^{n}\times\mathbb{R}\to\mathbb{C}$, and
$\phi:\mathbb{R}^{+}\to\mathbb{R}$ is a smooth function that satisfies some
growth condition.

Dispersive inequalities for evolution equations, such as the Schrödinger and
wave equations play a significant role in the study of semilinear and
quasilinear problems, which naturally appear in several physical applications.
Proving dispersion means estimating the decay in time on the evolution group
associated with the free equation. In recent times, dispersive properties of
evolution equations have become a crucial tool in the study of a variety of
questions, including local and global existence for nonlinear equations, well-
posedness of Cauchy problems for nonlinear equations in Sobolev spaces of low
order, scattering theory, and many others. In many circumstances, the
essential step in proving this time decay is based on the application of a
stationary phase theorem on an (approximate) representation of the solution.
Dispersion phenomena, when combined with an abstract functional analysis
argument, famously known as the $TT^{*}$-argument (see [35]), offer a range of
estimates involving space-time Lebesgue norms. These inequalities also known
as Strichartz estimates, have evolved into a fundamental and amazing tool in
the study of nonlinear partial differential equations over the last few
decades, see [8] and references therein.

Strichartz estimates also be interpreted as Fourier restriction estimates,
which have a vital role in classical harmonic analysis and are closely linked
to arithmetic combinatorics. Considerable attention has been devoted by
several researchers to deriving Strichartz estimates in different frameworks.
For example, in the Euclidean setting, these estimates have been proved for
several dispersive equations, such as the wave equation and Schrödinger
equation, for instance, see the pioneering works [25, 35, 51]. The theory of
Strichartz estimates has also been extended in the non-Euclidean frameworks.
For example, see [6, 29, 23, 24, 36, 48, 46, 7, 5, 47] for Strichartz
estimates on H-type groups and more generality of step $2$ stratified Lie
groups, [13, 17, 22, 42, 43, 45, 37, 39, 40, 9, 11] for metric measure spaces,
[2, 3, 4] for hyperbolic spaces, [12, 14] for compact Riemannian manifolds,
and [31] for bounded domains.

One of the essential ingredients in the derivation of Strichartz estimates is
the dispersive estimates. Our aim of this paper is to establish a decay
estimate for a class of dispersive equations as follows:

$\begin{cases}i\partial_{t}u+\phi(\sqrt{-\Delta_{\kappa}})u=f,\\\
u|_{t=0}=u_{0},\end{cases}$ (1.2)

where $\Delta_{\kappa}$ is the Dunkl Laplacian on $\mathbb{R}^{n}$,
$f:\mathbb{R}^{n}\times\mathbb{R}\to\mathbb{C}$, and
$\phi:\mathbb{R}^{+}\to\mathbb{R}$ is a smooth function satisfying

(C1) There exists $m_{1}>0$ such that for any $\alpha\geqslant 2$ and
$\alpha\in\mathbb{N}$,

$|\phi^{\prime}(r)|\sim r^{m_{1}-1},\quad|\phi^{(\alpha)}(r)|\lesssim
r^{m_{1}-\alpha},\quad r\geqslant 1.$

(C2) There exists $m_{2}>0$ such that for any $\alpha\geqslant 2$ and
$\alpha\in\mathbb{N}$,

$|\phi^{\prime}(r)|\sim r^{m_{2}-1},\quad|\phi^{(\alpha)}(r)|\lesssim
r^{m_{2}-\alpha},\quad 0<r<1.$

(C3) There exists $\alpha_{1}>0$ such that

$|\phi^{\prime\prime}(r)|\sim r^{\alpha_{1}-2},\quad r\geqslant 1.$

(C4) There exists $\alpha_{2}>0$ such that

$|\phi^{\prime\prime}(r)|\sim r^{\alpha_{2}-2},\quad 0<r<1.$

Here we use a Fourier multiplier notation to define
$\phi(\sqrt{-\Delta_{\kappa}})$ as in the following form

$\phi(\sqrt{-\Delta_{\kappa}})f=\mathcal{F}_{\kappa}^{-1}\phi(|\xi|)\mathcal{F}_{\kappa}f,$

where $\mathcal{F}_{\kappa}$ denotes Dunkl transform, defined in (2.3),
Section 2. Notice that the conditions (C1) and (C3) represent the homogeneous
order of $\phi$ in high frequency, and the conditions (C2) and (C4) reflect
the homogeneous order of $\phi$ in low frequency. If $\phi$ satisfies
conditions (C1) and (C3), then $\alpha_{1}\leq m_{1}$. Similarly, if $\phi$
satisfies (C2) and (C4), then $\alpha_{2}\geq m_{2}$.

Here, we want to emphasize that several dispersive wave equations associated
with Dunkl Laplacian reduced to type (1.2). For instance, the Dunkl
Schrödinger equation corresponds to $\phi(r)=r^{2}$, the Dunkl wave equation
corresponds to $\phi(r)=r$, the fractional Dunkl Schrödinger equation
corresponds to $\phi(r)=r^{\mu}$, $0<\mu<2,\mu\neq 1$, the fourth-order Dunkl
Schrödinger equation corresponds to $\phi(r)=r^{2}+r^{4}$, the beam equation
$\phi(r)=\sqrt{1+r^{4}}$, and the Klein-Gordon equation corresponds to
$\phi(r)=\sqrt{1+r^{2}}$, among others.

Dunkl operators are differential-difference operators that generalize the
standard partial derivatives. These operators were introduced by Charles Dunkl
[19] using a finite reflection group and a parameter function on its root
system. Applications of Dunkl operators in mathematics and mathematical
physics are numerous. For example, they play an important role in the
integration of quantum many-body problems of the Calogero-Moser-Sutherland
type and are employed in the study of probabilistic processes. Additionally,
these operators have had a lasting impression on the study of special
functions in one and multiple variables as well as orthogonal polynomials, see
[21].

For $\kappa=0,$ as the the Dunkl Laplacian $\Delta_{\kappa}$ reduces to usual
Laplacian $\Delta$ on $\mathbb{R}^{n}$, the system (1.2) reduces to

$\begin{cases}i\partial_{t}u+\phi(\sqrt{-\Delta})u=f,\\\
u|_{t=0}=u_{0}.\end{cases}$ (1.3)

In 1977, Strichartz [51] derived the priori estimates for the solution to
(1.3) in the space-time norm using the classical Fourier restriction theorem
of Stein and Tomas [49, 54, 55]. However, later, it was improved by Ginibre-
Velo [25] using a standard duality argument along with the dispersive estimate
of the form

$\displaystyle\left\|e^{it\phi(\sqrt{-\Delta})}u_{0}\right\|_{X}\lesssim|t|^{-\theta}\left\|u_{0}\right\|_{X^{*}},$
(1.4)

where $X^{*}$ is the dual space of $X$. In 2008, Guo-Peng-Wang [28] used a
unified way to study the decay for a class of dispersive semigroup
$e^{it\phi(\sqrt{-\Delta})}$ on $\mathbb{R}^{n}$ for the system (1.3). They
assumed that $\phi:\mathbb{R}^{+}\to\mathbb{R}$ is smooth satisfying (C1)-(C4)
and obtained several decay estimates in time for the dispersive semigroup
$e^{it\phi(\sqrt{-\Delta})}$ by introducing Littlewood–Paley projector
operator, see [28, Theorem 1].

Motivated by the recent works, in this paper, we aim to establish the decay
estimate for a class of dispersive equations associated with Dunkl Laplacian
(1.2). Note that, if $\phi$ is a homogeneous function of order $m$, namely,
$\phi(\lambda r)=$ $\lambda^{m}\phi(r)$ for all $\lambda>0$, the dispersive
estimate can be easily obtained by a theorem of Littman and dyadic
decomposition. However, things become complicated when the function $\phi$ is
not homogeneous because the scaling constants can not be effectively separated
from the time. Since the phases we consider are not necessarily homogeneous or
weakly dispersive, to overcome the difficulty, we use a Littlewood-Paley
decomposition (see [28]) to more accurately capture the difference between the
low and the high frequencies in different scales and catch the possible need
for extra regularity.

Let $S_{0}$ and $\tilde{\Delta}_{j}$ be the Littlewood-Paley decomposition
related to the Dunkl Laplacian $\Delta_{\kappa}$ and $N=2\gamma+n$ (see
Section 2 for detailed definition). Then the main results of this paper are as
follows:

######  Theorem 1.1.

Assume $\phi:\mathbb{R}^{+}\to\mathbb{R}$ is smooth and
$U_{t}=e^{it\phi(\sqrt{-\Delta_{\kappa}})}$. Then the subsequent outcomes are
valid.

  1. (A)

For $j\geq 0$, assume that $\phi$ satisfies (C1), then

$\|U_{t}\tilde{\Delta}_{j}u_{0}\|_{L_{\kappa}^{\infty}(\mathbb{R}^{n})}\lesssim|t|^{-\theta}2^{j\left(N-m_{1}\theta\right)}\|u_{0}\|_{L_{\kappa}^{1}(\mathbb{R}^{n})},\,0\leq\theta\leq\frac{N-1}{2}.$
(1.5)

In addition, if $\phi$ satisfies (C3), then

$\|U_{t}\tilde{\Delta}_{j}u_{0}\|_{L_{\kappa}^{\infty}(\mathbb{R}^{n})}\lesssim|t|^{-\frac{N-1+\theta}{2}}2^{j\left(N-\frac{m_{1}(N-1+\theta)}{2}-\frac{\theta(\alpha_{1}-m_{1})}{2}\right)}\|u_{0}\|_{L_{\kappa}^{1}(\mathbb{R}^{n})},\,0\leq\theta\leq
1.$ (1.6)

  2. (B)

For $j<0$, suppose $\phi$ satisfies (C2), then

$\|U_{t}\tilde{\Delta}_{j}u_{0}\|_{L_{\kappa}^{\infty}(\mathbb{R}^{n})}\lesssim|t|^{-\theta}2^{j\left(N-m_{2}\theta\right)}\|u_{0}\|_{L_{\kappa}^{1}(\mathbb{R}^{n})},\,0\leq\theta\leq\frac{N-1}{2}.$

In addition, if $\phi$ satisfies (C4), then

$\|U_{t}\tilde{\Delta}_{j}u_{0}\|_{L_{\kappa}^{\infty}(\mathbb{R}^{n})}\lesssim|t|^{-\frac{N-1+\theta}{2}}2^{j\left(N-\frac{m_{2}(N-1+\theta)}{2}-\frac{\theta(\alpha_{2}-m_{2})}{2}\right)}\|u_{0}\|_{L_{\kappa}^{1}(\mathbb{R}^{n})},\,0\leq\theta\leq
1.$

  3. (C)

If $\phi$ satisfies (C2), then

$\|U_{t}S_{0}u_{0}\|_{L_{\kappa}^{\infty}(\mathbb{R}^{n})}\lesssim(1+|t|)^{-\theta}\|u_{0}\|_{L_{\kappa}^{1}(\mathbb{R}^{n})},\,\theta=\min\left(\frac{N}{m_{2}},\frac{N-1}{2}\right).$
(1.7)

In addition, if $\phi$ satisfies (C4) and $\alpha_{2}=m_{2}$, then

$\|U_{t}S_{0}u_{0}\|_{L_{\kappa}^{\infty}(\mathbb{R}^{n})}\lesssim(1+|t|)^{-\theta}\|u_{0}\|_{L_{\kappa}^{1}(\mathbb{R}^{n})},\,\theta=\min\left(\frac{N}{m_{2}},\frac{N}{2}\right).$
(1.8)

We prove the above result in Section 3. The main difficulty in the setup of
Dunkl is that we don’t have an explicit expression for the Dunkl transform
compared to the classical Fourier transform. In this paper, we used
Littlewood-Paley decomposition to better understand the regularity is needed
at low and high frequencies in order to get a time decay. We also use the fact
that Dunkl transform for radial functions can be expressed in terms of
integrals of Bessel functions. In order to estimate these integrals, we rely
on Van der Corput’s Lemma and the decay properties of the Bessel functions.

Using the decay estimates in Theorem 1.1, we establish Strichartz estimates
for the Dunkl Schrödinger equation (corresponds to $\phi(r)=r^{2}$), the Dunkl
wave equation (corresponds to $\phi(r)=r$), the fractional Dunkl Schrödinger
equation (corresponds to $\phi(r)=r^{\mu}$), $0<\mu<2,\mu\neq 1$, the fourth-
order Dunkl Schrödinger equation (corresponds to $\phi(r)=r^{2}+r^{4}$), the
beam equation (corresponds to $\phi(r)=\sqrt{1+r^{4}}$), and the Klein-Gordon
equation (corresponds to $\phi(r)=\sqrt{1+r^{2}}$). We also prove the global
well-posedness for the nonlinear Klein-Gordon equation and beam equation for
small initial data.

There are many situations where dispersion phenomena fail, for instance in
compact manifolds, on some domains, on the Heisenberg group. However, in some
cases, Strichartz estimates (in weak forms) or smoothing properties can be
established using other approaches that are not based on dispersive inequality
techniques. The Fourier restriction theorem and Kato smoothing effect
techniques stand out among them. Recently, the authors in [33] proved
Strichartz estimates for the Dunkl Schrödinger equation (which corresponds to
$\phi(r)=r^{2}$) using the Fourier-Dunkl transform restriction theorem for
paraboloid surfaces. See also [37, 39].

######  Remark 1.2.

Since the Dunkl Laplacian $\Delta_{\kappa}$ reduces to the Euclidean Laplacian
$\Delta$ for $\kappa=0$, we recover the results proved in Guo-Peng-Wang [28].

Apart from the introduction, this paper is organized as follows: In Section 2,
we recall harmonic analysis associated with the Dunkl operators, Dunkl
transforms, and their certain properties. We also recall some useful technical
lemmas for proving the decay estimates and Strichartz estimates. In Section 3,
we prove our main result of this paper, i.e., Theorem 1.1. We provide
Strichartz estimates for some concrete wave equations associated with the
Dunkl Laplacian $\Delta_{\kappa}$ in Section 4. In Section 5, we prove global
well-posedness for the nonlinear Klein-Gordon equation and beam equation for
small initial data.

Notations:

  * •

$A\lesssim B$ means that $A\leqslant CB$, and $A\sim B$ stands for $C_{1}B\leq
A\leq C_{2}B$, where $C$, $C_{1}$, $C_{2}$ denote positive universal
constants.

  * •

$\Delta_{\kappa}$ denotes Dunkl-Laplacian on $\mathbb{R}^{n}$.

  * •

$S_{0}$ and $\tilde{\Delta}_{j}$ be the Littlewood-Paley decomposition related
to $\Delta_{\kappa}$.

  * •

$J_{\nu}$ denotes the Bessel function of order $\nu.$

  * •

$B^{s,\kappa}_{p,q}$ denotes the Dunkl Besov space.

##  2\. Preliminaries

In this section, we recall some basic definitions and important properties of
the Dunkl operators and briefly overview the related harmonic analysis to make
the paper self-contained. A complete account of harmonic analysis related to
Dunkl operators can be found in [44, 10, 11, 39, 41, 53].

###  2.1. Dunkl operators

The basic ingredient in the theory of Dunkl operators are root systems and
finite reflection groups. We start this section by the definition of root
system.

Let $\langle\cdot,\cdot\rangle$ denotes the standard Euclidean scalar product
in $\mathbb{R}^{n}$. For $x\in\mathbb{R}^{n}$, we denote $|x|$ as $|x|=\langle
x,x\rangle^{1/2}$. For $\alpha\in\mathbb{R}^{n}\backslash\\{0\\},$ we denote
$r_{\alpha}$ as the reflection with respect to the hyperplane
$\langle\alpha\rangle^{\perp}$, orthogonal to $\alpha$ and is defined by

$r_{\alpha}(x):=x-2\frac{\langle\alpha,x\rangle}{|\alpha|^{2}}\alpha,\quad
x\in\mathbb{R}^{n}.$

A finite set $\mathcal{R}$ in $\mathbb{R}^{n}\backslash\\{0\\}$ is said to be
a root system if the following holds:

  1. (1)

$r_{\alpha}(\mathcal{R})=\mathcal{R}$ for all $\alpha\in\mathcal{R}$,

  2. (2)

$\mathcal{R}\cap\mathbb{R}\alpha=\\{\pm\alpha\\}$ for all
$\alpha\in\mathcal{R}$.

For a given root system $\mathcal{R}$, the subgroup $G\subset O(n,\mathbb{R})$
generated by the reflections
$\left\\{r_{\alpha}\mid\alpha\in\mathcal{R}\right\\}$ is called the finite
Coxeter group associated with $\mathcal{R}$. The dimension of
$span\mathcal{R}$ is called the rank of $\mathcal{R}$. For a detailed study on
the theory of finite reflection groups, we refer to [30]. Let
$\mathcal{R}^{+}:=\\{\alpha\in\mathcal{R}:\langle\alpha,\beta\rangle>0\\}$ for
some
$\beta\in\mathbb{R}^{n}\backslash\bigcup_{\alpha\in\mathcal{R}}\langle\alpha\rangle^{\perp}$,
be a fix positive root system.

Some typical examples of such a system are the Weyl groups such as the
symmetric group $S_{n}$ for the type $A_{n-1}$ root system and the
hyperoctahedral group for the type $B_{n}$ root system. In addition,
$H_{3},H_{4}$ (icosahedral groups) and, $I_{2}(n)$ (symmetry group of the
regular $n$-gon) are also the Coxeter groups.

A multiplicity function for $G$ is a function
$\kappa:\mathcal{R}\rightarrow\mathbb{C}$ which is constant on $G$-orbits.
Setting $\kappa_{\alpha}:=\kappa(\alpha)$ for $\alpha\in\mathcal{R},$ from the
definition of $G$-invariant, we have $\kappa_{g\alpha}=\kappa_{\alpha}$ for
all $g\in G$. We say $\kappa$ is non-negative if $\kappa_{\alpha}\geq 0$ for
all $\alpha\in\mathcal{R}$. The $\mathbb{C}$-vector space of non-negative
multiplicity functions on $\mathcal{R}$ is denoted by $\mathcal{K}^{+}$. Let
us denote $\gamma$ as
$\gamma=\gamma(\kappa):=\sum\limits_{\alpha\in\mathcal{R}^{+}}\kappa(\alpha)$.
For convenience, we denote $N=2\gamma+n$ throughout the paper.

For $\xi\in\mathbb{C}^{n}$ and $\kappa\in\mathcal{K}^{+},$ Dunkl in 1989
introduced a family of first-order differential-difference operators
$T_{\xi}:=T_{\xi}(\kappa)$ by

$\displaystyle
T_{\xi}(\kappa)f(x):=\partial_{\xi}f(x)+\sum_{\alpha\in\mathcal{R}^{+}}\kappa_{\alpha}\langle\alpha,\xi\rangle\frac{f(x)-f\left(r_{\alpha}x\right)}{\langle\alpha,x\rangle},\quad
f\in C^{1}\left(\mathbb{R}^{n}\right),$ (2.1)

where $\partial_{\xi}$ denotes the directional derivative corresponding to
$\xi$. The operator $T_{\xi}$ defined in (2.1), formally known as Dunkl
operator and is one of the most important developments in the theory of
special functions associated with root systems [18]. They commute pairwise and
are skew-symmetric with respect to the $G$-invariant measure
$h_{\kappa}(x)dx$, where the weight function

$h_{\kappa}(x):=\prod\limits_{\alpha\in\mathcal{R}^{+}}|\langle\alpha,x\rangle|^{2\kappa_{\alpha}}$

is of homogeneous degree $2\gamma$. Thanks to the $G$-invariance of the
multiplicity function, the definition of the Dunkl operator is independent of
the choice of the positive subsystem $\mathcal{R}^{+}$. In [19], it is shown
that for any $\kappa\in\mathcal{K}^{+}$, there is a unique linear isomorphism
$V_{\kappa}$ (Dunkl’s intertwining operator) on the space
$\mathcal{P}(\mathbb{R}^{n})$ of polynomials on $\mathbb{R}^{n}$ such that

  1. (1)

$V_{\kappa}\left(\mathcal{P}_{m}(\mathbb{R}^{n})\right)=\mathcal{P}_{m}(\mathbb{R}^{n})$
for all $m\in\mathbb{N}$,

  2. (2)

$V_{\kappa}|_{\mathcal{P}_{0}(\mathbb{R}^{n})}=id$,

  3. (3)

$T_{\xi}(\kappa)V_{\kappa}=V_{\kappa}\partial_{\xi}$,

where $\mathcal{P}_{m}(\mathbb{R}^{n})$ denotes the space of homogeneous
polynomials of degree $m$.

For any finite reflection group $G$ and any $k\in\mathcal{K}^{+}$, Rösler in
[44] proved that there exists a unique positive Radon probability measure
$\rho_{x}^{\kappa}$ on $\mathbb{R}^{n}$ such that

$V_{\kappa}f(x)=\int_{\mathbb{R}^{n}}f(\xi)d\rho_{x}^{\kappa}(\xi).$ (2.2)

The measure $\rho_{x}^{k}$ depends on $x\in\mathbb{R}^{n}$ and its support is
contained in the ball
$B\left(\|x\|\right):=\\{\xi\in\mathbb{R}^{n}:\|\xi\|\leq\|x\|\\}$. In view of
the Laplace type representation (2.2), Dunkl’s intertwining operator
$V_{\kappa}$ can be extended to a larger class of function spaces.

Let $\\{\xi_{1},\xi_{2},\cdots,\xi_{n}\\}$ be an orthonormal basis of
$(\mathbb{R}^{n},\langle\cdot,\cdot\rangle)$. Then the Dunkl Laplacian
operator $\Delta_{\kappa}$ is defined as

$\Delta_{\kappa}=\sum_{j=1}^{n}T^{2}_{\xi_{j}}(\kappa).$

The definition of $\Delta_{\kappa}$ is independent of the choice of the
orthonormal basis of $\mathbb{R}^{n}.$ In fact, one can see that the operator
$\Delta_{\kappa}$ also can be expressed as

$\Delta_{\kappa}f(x)=\Delta
f(x)+\sum_{\alpha\in\mathcal{R}^{+}}\kappa_{\alpha}\left\\{\frac{2\langle\nabla
f(x),\alpha\rangle}{\langle\alpha,x\rangle}-|\alpha|^{2}\frac{f(x)-f(r_{\alpha}x)}{\langle\alpha,x\rangle^{2}}\right\\},\quad
f\in C^{1}(\mathbb{R}^{n}),$

where $\nabla$ and $\Delta$ are the usual gradiant and Laplacian operator on
$\mathbb{R}^{n}$, respectively. Observe that, for $\kappa\equiv 0$, the Dunkl
Laplacian operator $\Delta_{\kappa}$ reduces to the classical Euclidean
Laplacian $\Delta$.

For every $y\in\mathbb{R}^{n}$, the system

$\begin{cases}T_{\xi}u(x,y)=\langle
y,\xi\rangle\,u(x,y),\quad\xi\in\mathbb{R}^{n},\\\ u(0,y)=1,\end{cases}$

admits a unique analytic solution on $\mathbb{R}^{n}$, which we denote as
$K(x,y)$ and is known as the Dunkl kernel in general. The kernel has a unique
holomorphic extension to $\mathbb{C}^{n}\times\mathbb{C}^{n}$ and is given by

$K(x,y)=V_{\kappa}(e^{\langle\cdot,y\rangle})(x)=\int_{\mathbb{R}^{n}}e^{\langle\xi,y\rangle}d\rho_{x}^{\kappa}(\xi),\quad\forall
x,y\in\mathbb{R}^{n},$

where the positive Radon probability measure $\rho_{x}^{k}$ depends on
$x\in\mathbb{R}^{n}$ and its support is contained in the ball
$B\left(\|x\|\right):=\\{\xi\in\mathbb{R}^{n}:\|\xi\|\leq\|x\|\\}$. When
$\kappa\equiv 0$, the Dunkl kernel $K(x,y)$ reduces to the exponential
$e^{\langle x,y\rangle}$. For $x,y\in\mathbb{R}^{n}$ and
$z,w\in\mathbb{C}^{n}$, the Dunkl kernel satisfies the following properties:

  1. (i)

$K(z,w)=K(w,z)$ and $K(\lambda z,w)=K(z,\lambda w),\quad\lambda\in\mathbb{C}$.

  2. (ii)

For any $\nu\in\mathbb{N}^{n}$, we have

$|D_{z}^{\nu}K(x,z)|\leq|x|^{|\nu|}e^{|\operatorname{Re}(z)||x|},$

where
$D_{z}^{\nu}=\frac{\partial^{|\nu|}}{\partial{z_{1}}^{\nu_{1}}\cdots{z_{n}}^{\nu_{n}}}$
and $|\nu|=\nu_{1}+\cdots+\nu_{n}$. In particular, for any
$x,y\in\mathbb{R}^{n}$, we have

$|K(x,iy)|\leq 1.$

  3. (iii)

$K(ix,y)=\overline{K(-ix,y)}$ and $K(gx,gy)=K(x,y),\quad g\in G$.

###  2.2. The Dunkl transform

For $1\leq p<\infty$, let $L_{\kappa}^{p}\left(\mathbb{R}^{n}\right)$ be the
space of $L^{p}$-functions on $\mathbb{R}^{n}$ with respect to the weight
$h_{\kappa}(x)$ with the $L_{\kappa}^{p}\left(\mathbb{R}^{n}\right)$-norm

$\|f\|_{L_{\kappa}^{p}\left(\mathbb{R}^{n}\right)}=\left(\int_{\mathbb{R}^{n}}|f(x)|^{p}h_{\kappa}(x)dx\right)^{\frac{1}{p}},$

and
$\|f\|_{L_{\kappa}^{\infty}\left(\mathbb{R}^{n}\right)}=ess\sup\limits_{x\in\mathbb{R}^{n}}|f(x)|$.
The Dunkl transform of $f\in L_{\kappa}^{1}\left(\mathbb{R}^{n}\right)$ is
defined by

$\mathcal{F}_{\kappa}f(y)=\frac{1}{c_{\kappa}}\int_{\mathbb{R}^{n}}f(x)K(x,-iy)h_{\kappa}(x)dx,$
(2.3)

where $c_{\kappa}$ is the Mehta-type constant defined by

$c_{\kappa}=\int_{\mathbb{R}^{n}}e^{-\frac{|x|^{2}}{2}}h_{\kappa}(x)dx.$

When $\kappa\equiv 0$, the Dunkl transform coincides with the classical
Fourier transform. The Dunkl transform plays the same role as the Fourier
transform in classical Fourier analysis and enjoys properties similar to those
of the classical Fourier transform. With $N=2\gamma+n$, here we list some
basic properties of the Dunkl transform (see [20, 32, 16]).

  1. (i)

For all $f\in L_{\kappa}^{1}\left(\mathbb{R}^{n}\right)$, we have

$\|\mathcal{F}_{\kappa}f\|_{L_{\kappa}^{\infty}\left(\mathbb{R}^{n}\right)}\leq\frac{1}{c_{\kappa}}\|f\|_{L_{\kappa}^{1}\left(\mathbb{R}^{n}\right)}.$

  2. (ii)

For any function $f$ in the Schwartz space
$\mathcal{S}\left(\mathbb{R}^{n}\right)$, we have

$\mathcal{F}_{\kappa}(T_{\xi}f)(y)=i\langle\xi,y\rangle\mathcal{F}_{\kappa}f(y),\quad
y,\xi\in\mathbb{R}^{n}.$

In particular, it follows that

$\mathcal{F}_{\kappa}(\Delta_{\kappa}f)(y)=-|y|^{2}\mathcal{F}_{\kappa}f(y),\quad
y\in\mathbb{R}^{n}.$

  3. (iii)

For all $f\in L_{\kappa}^{1}\left(\mathbb{R}^{n}\right)$, we have

$\mathcal{F}_{\kappa}(f(\cdot/\lambda))(y)=\lambda^{N}\mathcal{F}_{\kappa}f(\lambda
y),\quad y\in\mathbb{R}^{n},$

for all $\lambda>0$.

  4. (iv)

The Dunkl transform $\mathcal{F}_{\kappa}$ is a homeomorphism of the Schwartz
space $\mathcal{S}\left(\mathbb{R}^{n}\right)$ and its inverse is given by
$\mathcal{F}_{\kappa}^{-1}g(x)=\mathcal{F}_{\kappa}g(-x),$ for all
$g\in\mathcal{S}\left(\mathbb{R}^{n}\right)$. In addition, for all
$f\in\mathcal{S}\left(\mathbb{R}^{n}\right)$, it satisfies

$\int_{\mathbb{R}^{n}}|f(x)|^{2}h_{\kappa}(x)dx=\int_{\mathbb{R}^{n}}|\mathcal{F}_{\kappa}f(y)|^{2}h_{\kappa}(y)dy.$

In particular, the Dunkl transform extends to an isometric isomorphism on
$L_{\kappa}^{2}\left(\mathbb{R}^{n}\right)$. The Dunkl transform can be
extended to the space of tempered distributions
$\mathcal{S}^{\prime}\left(\mathbb{R}^{n}\right)$ and is also a homeomorphism
of $\mathcal{S}^{\prime}\left(\mathbb{R}^{n}\right)$.

  5. (v)

For every $f\in L_{\kappa}^{1}\left(\mathbb{R}^{n}\right)$ such that
$\mathcal{F}_{\kappa}f\in L_{\kappa}^{1}\left(\mathbb{R}^{n}\right)$, we have
the inversion formula

$f(x)=\frac{1}{c_{\kappa}}\int_{\mathbb{R}^{n}}\mathcal{F}_{\kappa}f(y)K(ix,y)h_{\kappa}(y)dy,\quad
a.e.\;x\in\mathbb{R}^{n}.$

  6. (vi)

If $f$ is a radial function in $L_{\kappa}^{1}\left(\mathbb{R}^{n}\right)$
such that $f(x)=\tilde{f}(|x|)$, then

$\mathcal{F}_{\kappa}f(y)=\frac{1}{\Gamma(N/2)}\int_{0}^{\infty}\tilde{f}(r)\frac{J_{\frac{N-2}{2}}(r|y|)}{\left(r|y|\right)^{\frac{N-2}{2}}}r^{N-1}dr,$
(2.4)

where $J_{\nu}$ denotes the Bessel function of order $\nu>-\frac{1}{2}$.

###  2.3. Dunkl convolution operator

For given $x\in\mathbb{R}^{n}$, the Dunkl translation operator
$f\mapsto\tau_{x}f$ is defined on $\mathcal{S}\left(\mathbb{R}^{n}\right)$ by

$\mathcal{F}_{\kappa}(\tau_{x}f)(y)=K(x,iy)\mathcal{F}_{\kappa}f(y),\quad
y\in\mathbb{R}^{n}.$

At the moment, the explicit formula of the Dunkl translation operator is known
only in two cases. One is when $f(x)=\tilde{f}(|x|)$ is a continuous radial
function in $L_{k}^{2}(\mathbb{R}^{n})$, the Dunkl translation operator is
represented by (see [44, 15])

$\tau_{x}f(y)=V_{\kappa}[\tilde{f}\left(\sqrt{|y|^{2}+|x|^{2}-2\langle
y,\cdot\rangle}\right)](x)=\int_{\mathbb{R}^{n}}\tilde{f}\left(\sqrt{|y|^{2}+|x|^{2}-2\langle
y,\xi\rangle}\right)d\rho^{\kappa}_{x}(\xi).$

The Dunkl translation operator can also be extended to the space of tempered
distributions $\mathcal{S}^{\prime}\left(\mathbb{R}^{n}\right)$. It turns out
to be rather difficult to extend important results in classical Fourier
analysis to the setting of the Dunkl transform in general. One of the
difficulties comes from the fact that the Dunkl translation operator is not
positive in general. In fact, even
$L_{\kappa}^{p}(\mathbb{R}^{n})$-boundedness of $\tau_{x}$ is not established
in general. On the other hand, when $f$ is a radial function in
$L_{\kappa}^{p}(\mathbb{R}^{n})$, $1\leq p\leq\infty$, the following holds
(see [26, 52])

$\|\tau_{x}f\|_{L_{\kappa}^{p}(\mathbb{R}^{n})}\leq\|f\|_{L_{\kappa}^{p}(\mathbb{R}^{n})}.$

Using the Dunkl translation operator, we define the Dunkl convolution of
functions $f,g\in\mathcal{S}\left(\mathbb{R}^{n}\right)$ by

$f*_{\kappa}g(x)=\int_{\mathbb{R}^{n}}\tau_{x}f(-y)g(y)h_{\kappa}(y)dy,\quad
x\in\mathbb{R}^{n}.$

The Dunkl convolution satisfies the following properties (see [52]).

  1. (1)

$\mathcal{F}_{\kappa}(f*_{\kappa}g)=\mathcal{F}_{\kappa}(f)\mathcal{F}_{\kappa}(g)$
and $f*_{\kappa}g=g*_{\kappa}f$.

  2. (2)

Young’s inequality: Let $1\leq p,q,r\leq\infty$ such that
$1+\frac{1}{r}=\frac{1}{p}+\frac{1}{q}$. If $f\in
L^{p}_{\kappa}(\mathbb{R}^{n})$ and $g$ is a radial function of
$L_{\kappa}^{q}(\mathbb{R}^{n})$, then $f*_{\kappa}g\in
L_{\kappa}^{r}(\mathbb{R}^{n})$ and we have

$\|f*_{\kappa}g\|_{L_{\kappa}^{r}(\mathbb{R}^{n})}\leq\|f\|_{L_{\kappa}^{p}(\mathbb{R}^{n})}\|g\|_{L_{\kappa}^{q}(\mathbb{R}^{n})}.$
(2.5)

###  2.4. Dunkl Besov spaces

In this subsection, we recall the definition of the Dunkl Besov spaces using
Littlewood-Paley decomposition related to the Dunkl Laplacian
$\Delta_{\kappa}$ (see [34, 38]). Dunkl-Besov spaces will be a convenient tool
to prove many of our results.

Fix a non-increasing smooth function $R$ on $[0,+\infty)$ such that $R=1$ in
$[0,1/2]$ and $R=0$ in $[1,+\infty)$. Let $\psi(r)=R(r/2)-R(r)$ and
$0\leq\psi\leq 1$ such that $\hbox{supp\,}\psi\subseteq C_{0}=[1/2,2]$.
Moreover, we observe that

$\displaystyle R(r)+\sum_{j=0}^{\infty}\psi(2^{-j}r)$ $\displaystyle=1,\quad
r\geq 0,$ $\displaystyle\underset{j\in\mathbb{Z}}{\sum}\psi(2^{-j}r)$
$\displaystyle=1,\quad r>0.$

For $u\in\mathcal{S}^{\prime}\left(\mathbb{R}^{n}\right)$, we define the
Littlewood-Paley decomposition related to the Dunkl Laplacian by
$S_{0}u=R(\sqrt{-\Delta_{\kappa}})u=\mathcal{F}_{\kappa}^{-1}\left(R(|\cdot|)\mathcal{F}_{\kappa}u\right)$
and
$\tilde{\Delta}_{j}u=\psi(2^{-j}\sqrt{-\Delta_{\kappa}})u=\mathcal{F}_{\kappa}^{-1}\left(\psi(2^{-j}|\cdot|)\mathcal{F}_{\kappa}u\right),$
for all $j\in\mathbb{Z}$. Moreover, we have

$u=S_{0}u+\sum_{j=0}^{\infty}\tilde{\Delta}_{j}u\;\in\;\mathcal{S}^{\prime}\left(\mathbb{R}^{n}\right).$

Moreover, we also have

$S_{0}u=S_{0}(\sum_{j\in\mathbb{Z}}\tilde{\Delta}_{j})u=\sum_{j\leq
0}S_{0}\tilde{\Delta}_{j}u.$

In addition, from Bernstein’s inequalities [34], for any
$\sigma\in\mathbb{R}$, $j\in\mathbb{Z}$, $1\leq p,q\leq\infty$ and
$u\in\mathcal{S}^{\prime}\left(\mathbb{R}^{n}\right)$, we have

$\displaystyle\left\|(-\Delta_{\kappa})^{\frac{\sigma}{2}}\tilde{\Delta}_{j}u\right\|_{L_{\kappa}^{p}(\mathbb{R}^{n})}$
$\displaystyle\lesssim
2^{j\sigma}\|\tilde{\Delta}_{j}u\|_{L_{\kappa}^{p}(\mathbb{R}^{n})},$ (2.6)
$\displaystyle\left\|\tilde{\Delta}_{j}u\right\|_{L_{\kappa}^{q}(\mathbb{R}^{n})}$
$\displaystyle\lesssim
2^{jN\left(\frac{1}{p}-\frac{1}{q}\right)}\|\tilde{\Delta}_{j}u\|_{L_{\kappa}^{p}(\mathbb{R}^{n})}.$
(2.7)

For $1\leq p,q\leq\infty$ and $s\in\mathbb{R}$, the Dunkl Besov space
$B^{s,\kappa}_{p,q}$ is defined as the set of tempered distributions
$u\in\mathcal{S}^{\prime}\left(\mathbb{R}^{n}\right)$ such that
$\|u\|_{B^{s,\kappa}_{p,q}}<\infty$ with

$\|u\|_{B^{s,\kappa}_{p,q}}=\|S_{0}u\|_{L_{\kappa}^{p}(\mathbb{R}^{n})}+\begin{cases}\left(\sum\limits_{j=0}^{\infty}2^{jsq}\|\tilde{\Delta}_{j}u\|_{L_{\kappa}^{p}(\mathbb{R}^{n})}^{q}\right)^{\frac{1}{q}},\quad
1\leq q<\infty\\\ \sup_{j\geq
0}2^{js}\|\tilde{\Delta}_{j}u\|_{L_{\kappa}^{p}(\mathbb{R}^{n})}\quad,\quad
q=\infty.\end{cases}$

Also, for $1\leq p\leq\infty$ and $s\in\mathbb{R}$, the Dunkl Sobolev space
$H_{p}^{s,\kappa}(\mathbb{R}^{n})$ is defined as the set of tempered
distributions $u\in\mathcal{S}^{\prime}\left(\mathbb{R}^{n}\right)$ such that
$\mathcal{F}^{-1}_{\kappa}\left((1+|\cdot|^{2})^{\frac{s}{2}}\mathcal{F}_{\kappa}u\right)\in
L_{\kappa}^{p}(\mathbb{R}^{n})$ with the norm

$\|u\|_{H_{p}^{s,\kappa}(\mathbb{R}^{n})}=\|\mathcal{F}^{-1}_{\kappa}\left((1+|\cdot|^{2})^{\frac{s}{2}}\mathcal{F}_{\kappa}u\right)\|_{L_{\kappa}^{p}(\mathbb{R}^{n})}.$

Throughout the paper, we denote $H^{s,\kappa}(\mathbb{R}^{n})$ for
$H_{2}^{s,\kappa}(\mathbb{R}^{n})$. Moreover, we have the equivalent property
that

$u\in H^{s,\kappa}(\mathbb{R}^{n})\Leftrightarrow u\in
B^{s,\kappa}_{2,2}(\mathbb{R}^{n}).$

Now we list some important properties related to the Besov space
$B^{s,\kappa}_{p,q}(\mathbb{R}^{n})$ in the following proposition.

######  Proposition 2.1.

[34] Let $p,q\in[1,\infty]$ and $s\in\mathbb{R}$. Then

  1. (1)

The space $B^{s,\kappa}_{p,q}(\mathbb{R}^{n})$ is a Banach space with the norm
$||\cdot||_{B^{s,\kappa}_{p,q}(\mathbb{R}^{n})}$.

  2. (2)

The definition of $B^{s,k}_{p,q}(\mathbb{R}^{n})$ does not depend on the
choice of the function $R$ in the Littlewood-Paley decomposition.

  3. (3)

The dual space of $B^{s,\kappa}_{p,q}(\mathbb{R}^{n})$ is
$B^{-s,\kappa}_{p^{\prime},q^{\prime}}(\mathbb{R}^{n})$;

  4. (4)

For any $\sigma>0$, we have the continuous inclusion
$B^{s+\sigma,\kappa}_{p,q}(\mathbb{R}^{n})\subseteq
B^{s,\kappa}_{p,q}(\mathbb{R}^{n})$.

  5. (5)

Let $u\in\mathcal{S}^{\prime}\left(\mathbb{R}^{n}\right)$ and
$\sigma\in\mathbb{R}$, then $u\in B^{s,\kappa}_{p,q}(\mathbb{R}^{n})$ if and
only if $(-\Delta_{\kappa})^{\sigma/2}u\in
B^{s-\sigma,\kappa}_{p,q}(\mathbb{R}^{n})$.

  6. (6)

For any $1\leq p_{1}\leq p_{2}\leq\infty$, $1\leq q_{1}\leq q_{2}\leq\infty,$
and $s_{1}\leq s_{2}\in\mathbb{R}$ such that
$s_{1}-\frac{N}{p_{1}}=s_{2}-\frac{N}{p_{2}}$, we also have the continuous
inclusion $B^{s_{1},\kappa}_{p_{1},q_{1}}(\mathbb{R}^{n})\subseteq
B^{s_{2},\kappa}_{p_{2},q_{2}}(\mathbb{R}^{n}).$

  7. (7)

For all $p\in[2,\infty]$, we have the continuous inclusion
$B^{0,\kappa}_{p,2}(\mathbb{R}^{n})\subseteq L_{\kappa}^{p}(\mathbb{R}^{n})$.

  8. (8)

$B^{0,\kappa}_{2,2}(\mathbb{R}^{n})=L_{\kappa}^{2}(\mathbb{R}^{n})$.

###  2.5. Technical Lemmas

In this subsection, we recall several important properties to prove the main
result of this paper. From the Dunkl transform and Young’s inequality, the
$L_{\kappa}^{1}-L_{\kappa}^{\infty}$ decay estimates are reduced to a list of
oscillatory integrals related to Bessel functions. In order to estimate the
oscillatory integrals, we recall the stationary phase lemmas and the
properties of Bessel functions. We first recall the van der Corput lemma.

######  Lemma 2.2 (Van der Corput lemma [50]).

Let $g\in C^{\infty}([a,b])$ be real-valued such that
$|g^{\prime\prime}(x)|\geq\delta$ for any $x\in[a,b]$ with $\delta>0$. Then
for any function $\psi\in C^{\infty}([a,b])$, there exists a constant $C$
(does not depend on $\delta,a,b,g$ or $\psi$) such that

$\left|\int_{a}^{b}e^{{ig(x)}}\psi(x)\,dx\right|\leq
C{\delta^{-1/2}}\left(\|\psi\|_{\infty}+\|\psi^{\prime}\|_{1}\right).$

Let $J_{\nu}$ be the Bessel function of order $\nu>-\frac{1}{2}$, defined as

$J_{\nu}(r)=\frac{(\frac{r}{2})^{\nu}}{\Gamma(\nu+\frac{1}{2})\pi^{\frac{1}{2}}}\int_{-1}^{1}e^{ir\tau}(1-\tau^{2})^{\nu-\frac{1}{2}}d\tau.$

We list some properties of $J_{\nu}$ in the following lemma.

######  Lemma 2.3 (see [27]).

For $r>0$ and $\nu>-\frac{1}{2}$, we have

$\displaystyle(1)J_{\nu}(r)\leq C_{\nu}r^{\nu},$ (2.8)
$\displaystyle(2)\frac{d}{dr}\left(r^{-\nu}J_{\nu}(r)\right)=-r^{-\nu}J_{\nu+1}(r),$
(2.9) $\displaystyle(3)J_{\nu}(r)\leq C_{\nu}r^{-\frac{1}{2}},$ (2.10)
$\displaystyle(4)J_{\nu}(r)=\frac{(\frac{r}{2})^{\nu}}{\Gamma(\nu+\frac{1}{2})\Gamma(\frac{1}{2})}\left[ie^{-ir}\int_{0}^{\infty}e^{-rt}(t^{2}+2it)^{\nu-\frac{1}{2}}dt-
ie^{ir}\int_{0}^{\infty}e^{-rt}(t^{2}-2it)^{\nu-\frac{1}{2}}dt\right].$ (2.11)

######  Remark 2.4.

From (2.8) and (2.9) of Lemma 2.3, we can easily obtain that for any $0\leq
s\leq 2$ and $\beta\in\mathbb{N}_{0}$,

$\left|\frac{d^{\beta}}{dr^{\beta}}\left(\psi(r)\frac{J_{\frac{N-2}{2}}(rs)}{(rs)^{\frac{N-2}{2}}}r^{N-1}\right)\right|\leq
C_{\beta},$ (2.12)

where $\psi$ is the function in Section 2.4. Again from (2.11), we have the
identity

$\frac{J_{\frac{N-2}{2}}(r)}{r^{\frac{N-2}{2}}}=C\left(e^{ir}h(r)+e^{-ir}\overline{h(r)}\right),$
(2.13)

where

$h(r)=-i\int_{0}^{\infty}e^{-rt}(t^{2}-2it)^{\frac{N-3}{2}}dt.$

Moreover, integrating by parts (if necessary) and using Van der Corput’s
Lemma, for any $\beta\in\mathbb{N}_{0}$, one can get

$\left|\frac{d^{\beta}}{dr^{\beta}}h(r)\right|\leq
C_{\beta}(1+r)^{-\frac{N-1}{2}-\beta},$ (2.14)

for all $\beta\in\mathbb{N}_{0}$. Thus, from (2.14), for any $s>2$ and
$\beta\in\mathbb{N}_{0}$, we have

$\left|\frac{d^{\beta}}{dr^{\beta}}\left(\psi(r)h(rs)r^{N-1}\right)\right|\leq
C_{\beta}s^{-\frac{N-1}{2}}.$ (2.15)

We also exploit the following estimates, which can be easily proved by
comparing the sums with the corresponding integrals.

######  Lemma 2.5.

Fix $\beta>0$. There exists $C_{\beta}>0$ such that for any $A>0$, we have

$\displaystyle\sum_{j\in\mathbb{Z},2^{j}\leq A}2^{j\beta}$ $\displaystyle\leq
C_{\beta}A^{\beta},$ $\displaystyle\sum_{j\in\mathbb{Z},2^{j}>A}2^{-j\beta}$
$\displaystyle\leq C_{\beta}A^{-\beta}.$

Finally, we apply the following duality arguments (see [25]) to prove the
Strichartz estimates for some concrete equations related to the Dunkl
Laplacian.

######  Lemma 2.6.

Let $H$ be a Hilbert space, $X$ be a Banach space, $X^{*}$ is the dual of $X$,
and $D$ be a vector space densely contained in $X$. Let
$A\in\mathcal{L}_{a}(D,H)$ and $A^{*}\in\mathcal{L}_{a}(H,D_{a}^{*})$ be its
adjoint operator, defined by

$\langle A^{*}v,f\rangle_{D}=\langle v,Af\rangle_{H},\quad\forall f\in
D,\quad\forall v\in H,$

where $\mathcal{L}_{a}(Y,Z)$ is the space of linear maps from a vector space
$Y$ to a vector space $Z$, $D^{*}_{a}$ is the algebraic dual of $D$,
$\langle\varphi,f\rangle_{D}$ is the pairing between $D^{*}_{a}$ and $D$, and
$\langle\cdot,\cdot\rangle_{H}$ is the scalar product in $H$. Then the
following three conditions are equivalent:

  1. (1)

There exists $0\leqslant a\leqslant\infty$ such that for all $f\in D$,

$\|Af\|\leqslant a\|f\|_{X}.$

  2. (2)

$\mathfrak{R}(A^{*})\subset X^{*}$ and there exists $0\leqslant
a\leqslant\infty$, such that for all $v\in H$,

$\|A^{*}v\|_{X^{*}}\leqslant a\|v\|.$

  3. (3)

$\mathfrak{R}(A^{*}A)\subset X^{*}$,and there exists $a$, $0\leqslant
a\leqslant\infty$, such that for all $f\in D$,

$\|A^{*}Af\|_{X^{*}}\leqslant a^{2}\|f\|_{X},$

where $\|\cdot\|$ denote the norm in $H$.

The constant $a$ is the same in all three parts. If one of those conditions is
satisfied, the operators $A$ and $A^{*}A$ extend by continuity to bounded
operators from $X$ to $H$ and from $X$ to $X^{*}$, respectively.

######  Lemma 2.7.

Let $H$, $D$ and two triplets $(X_{i},A_{i},a_{i}),i=1,2$, satisfy any of the
conditions of Lemma 2.6. Then for all choices of $i,j=1,2$,
$\mathfrak{R}(A_{i}^{*}A_{j})\subset X_{i}^{*}$, and for all $f\in D$, we have

$\|A_{i}^{*}A_{j}f\|_{X_{i}^{*}}\leqslant a_{i}a_{j}\|f\|_{X_{j}}.$

######  Lemma 2.8.

Let $H$ be a Hilbert space, $I$ be an interval of $\mathbb{R}$,
$X\subset\mathcal{S}^{\prime}(I\times\mathbb{R}^{n})$ be a Banach space which
is stable under time restriction, and let $X$ and $A$ satisfy (any of) the
conditions of Lemma 2.6. Then the operator $A^{*}A$ is a bounded operator from
$L_{t}^{1}(I,H)$ to $X^{*}$ and from $X$ to $L_{t}^{\infty}(I,H)$.

##  3\. Proof of main results

This section is devoted to proving Theorem 1.1, i.e., decay estimate for a
class of dispersive semigroup $e^{it\phi(\sqrt{-\Delta_{\kappa}})}$ on
$\mathbb{R}^{n}$.

###### Proof of Theorem 1.1.

In order to prove part (A) first, we note that

$U_{t}\tilde{\Delta}_{j}u_{0}=\mathcal{F}_{\kappa}^{-1}\left(e^{it\phi(|\cdot|)}\psi(2^{-j}|\cdot|)\mathcal{F}_{\kappa}u_{0}\right)=\mathcal{F}_{\kappa}^{-1}\left(e^{it\phi(|\cdot|)}\psi(2^{-j}|\cdot|)\right)*_{\kappa}u_{0}.$
(3.1)

It follows from Young’s inequality (2.5) that

$\|U_{t}\tilde{\Delta}_{j}u_{0}\|_{L_{\kappa}^{\infty}(\mathbb{R}^{n})}\leq\|\mathcal{F}_{\kappa}^{-1}\left(e^{it\phi(|\cdot|)}\psi(2^{-j}|\cdot|)\right)\|_{L_{\kappa}^{\infty}(\mathbb{R}^{n})}\|u_{0}\|_{L_{\kappa}^{1}(\mathbb{R}^{n})}.$

Now our aim is to estimate
$I_{j}(x)=\mathcal{F}_{\kappa}^{-1}\left(e^{it\phi(|\cdot|)}\psi(2^{-j}|\cdot|)\right)(x)$.
From the Dunkl transform formula of a radial function (2.4), we have

$\displaystyle I_{j}(x)$
$\displaystyle=\mathcal{F}_{\kappa}^{-1}\left(e^{it\phi(|\cdot|)}\psi(2^{-j}|\cdot|)\right)(x)$
(3.2)
$\displaystyle=\frac{1}{\Gamma(N/2)}\int_{0}^{\infty}e^{it\phi(r)}\psi(2^{-j}r)\frac{J_{\frac{N-2}{2}}(r|x|)}{\left(r|x|\right)^{\frac{N-2}{2}}}r^{N-1}dr$
$\displaystyle=\frac{{2^{jN}}}{\Gamma(N/2)}\int_{0}^{\infty}e^{it\phi(2^{j}r)}\psi(r)\frac{J_{\frac{N-2}{2}}(2^{j}r|x|)}{\left(2^{j}r|x|\right)^{\frac{N-2}{2}}}r^{N-1}dr$
$\displaystyle=II_{j}(2^{j}s),$

where $s=|x|$ and

$II_{j}(s)=\frac{2^{jN}}{\Gamma(N/2)}\int_{0}^{\infty}e^{it\phi(2^{j}r)}\psi(r)\frac{J_{\frac{N-2}{2}}(rs)}{\left(rs\right)^{\frac{N-2}{2}}}r^{N-1}dr.$

We will continue the discussion in two cases. First, we use the vanishing
property at the origin (2.8) and the recurring property for the Bessel
functions (2.9). On the other hand, we apply the decay property of the Bessel
function (2.13).

Case 1. $s\leq 2$: By denoting
$D_{r}=\frac{1}{i2^{j}t\phi^{\prime}(2^{j}r)}\frac{d}{dr}$, we notice that

$D_{r}\left(e^{it\phi(2^{j}r)}\right)=e^{it\phi(2^{j}r)},\;D_{r}^{*}f=\frac{i}{2^{j}t}\frac{d}{dr}\left(\frac{1}{\phi^{\prime}(2^{j}r)}f\right).$

For $\beta\in\mathbb{N}$ and $r\in[1/2,2]$, it follows from the hypothesis
(C1) that

$\frac{d^{\beta}}{dr^{\beta}}\left(\frac{1}{\phi^{\prime}(2^{j}r)}\right)\leq
C_{\beta}2^{-j(m_{1}-1)}.$ (3.3)

Using integration by part, we have

$\displaystyle II_{j}(s)$
$\displaystyle=\frac{{2^{jN}}}{\Gamma(N/2)}\int_{0}^{\infty}e^{it\phi(2^{j}r)}\psi(r)\frac{J_{\frac{N-2}{2}}(rs)}{\left(rs\right)^{\frac{N-2}{2}}}r^{N-1}dr$
$\displaystyle=\frac{{2^{jN}}}{\Gamma(N/2)}\int_{0}^{\infty}D_{r}\left(e^{it\phi(2^{2j}r)}\right)\psi(r)\frac{J_{\frac{N-2}{2}}(rs)}{\left(rs\right)^{\frac{N-2}{2}}}r^{N-1}dr$
$\displaystyle=\frac{{2^{jN}}}{\Gamma(N/2)}\cdot\frac{i}{2^{j}t}\int_{0}^{\infty}e^{it\phi(2^{j}r)}\frac{d}{dr}\left(\frac{1}{\phi^{\prime}(2^{j}r)}\psi(r)\frac{J_{\frac{N-2}{2}}(rs)}{\left(rs\right)^{\frac{N-2}{2}}}r^{N-1}\right)dr$
$\displaystyle=\frac{{2^{jN}}}{\Gamma(N/2)}\cdot\left(\frac{i}{2^{j}t}\right)^{Q}\sum_{\ell=0}^{Q}\sum_{(\beta_{1},\beta_{2},\cdots,\beta_{Q})\in\Lambda_{Q}^{\ell}}C_{Q,\ell}\int_{0}^{\infty}e^{it\phi(2^{j}r)}$
$\displaystyle\qquad\qquad\times\prod_{i=1}^{Q}\frac{d^{\beta_{i}}}{dr^{\beta_{i}}}\left(\frac{1}{\phi^{\prime}(2^{j}r)}\right)\frac{d^{Q-\ell}}{dr^{Q-\ell}}\left(\psi(r)\frac{J_{\frac{N-2}{2}}(rs)}{\left(rs\right)^{\frac{N-2}{2}}}r^{N-1}\right)\,dr,$

for any $Q\in\mathbb{N}$, where
$\Lambda_{Q}^{\ell}=\\{(\beta_{1},\beta_{2},\cdots,\beta_{Q})\in\\{0,1,2,\cdots,\ell\\}^{Q}:\sum\limits_{i=1}^{Q}\beta_{i}=\ell\\}$.
Now using the recurring property for the Bessel functions (2.9), the vanishing
property at the origin (2.8), and the estimate (3.3), we have

$|II_{j}(s)|\leq C_{Q,N,k}|t|^{-Q}2^{j(N-m_{1}Q)},$ (3.4)

for any $Q\in\mathbb{N}$. Using Riesz-Thorin interpolation between different
$Q\in\mathbb{N}$ in (3.5), for any $\theta\geq 0$, we have

$|II_{j}(s)|\lesssim|t|^{-\theta}2^{j(N-m_{1}\theta)}.$ (3.5)

This completes the proof of (A) in this case.

Case 2. $s>2$. In this case, we shall apply the decay property of Bessel
functions. From the identity (2.13), we can write

$\displaystyle II_{j}(s)$
$\displaystyle=C2^{jN}\int_{0}^{\infty}e^{it\phi(2^{j}r)}\psi(r)\left(e^{irs}h(rs)+e^{-irs}\overline{h(rs)}\right)r^{N-1}\,dr$
$\displaystyle=C2^{jN}\int_{0}^{\infty}e^{i\left(t\phi(2^{j}r)+rs\right)}\psi(r)h(rs)r^{N-1}\,dr$
$\displaystyle\qquad+C2^{jN}\int_{0}^{\infty}e^{i\left(t\phi(2^{j}r)-rs\right)}\psi(r)\overline{h(rs)}r^{N-1}\,dr$
$\displaystyle=II^{+}_{j}(s)+II^{-}_{j}(s),$

where

$\displaystyle II^{+}_{j}(s)$
$\displaystyle=C2^{jN}\int_{0}^{\infty}e^{i\left(t\phi(2^{j}r)+rs\right)}\psi(r)h(rs)r^{N-1}\,dr=C2^{jN}\int_{0}^{\infty}e^{it\phi_{+}(2^{j}r)}\psi(r)h(rs)r^{N-1}\,dr,$
$\displaystyle II^{-}_{j}(s)$
$\displaystyle=C2^{jN}\int_{0}^{\infty}e^{i\left(t\phi(2^{j}r)-rs\right)}\psi(r)\overline{h(r|s|)}r^{N-1}\,dr=C2^{jN}\int_{0}^{\infty}e^{it\phi_{-}(2^{j}r)}\psi(r)\overline{h(rs)}r^{N-1}\,dr,$

with $\phi_{\pm}(r)=\phi(r)\pm\frac{rs}{2^{j}t}$. Without loss of generality,
we can assume that $t>0$ and $\phi^{\prime}(r)>0$. Now we will estimate
$II^{+}_{j}(s)$ and $II^{-}_{j}(s)$ separately.

For $II^{+}_{j}(s)$, note that $\phi^{\prime}_{+}(r)\geq\phi^{\prime}(r)$ and
(3.3) also holds true if we replace $\phi$ by $\phi_{+}$. Then, proceeding
similarly as Case 1, combined with (2.15), for any $\theta\geq 0,$ we obtain

$|II^{+}_{j}(s)|\lesssim|t|^{-\theta}2^{j(N-m_{1}\theta)}.$ (3.6)

For $II^{-}_{j}(s)$,
$\phi^{\prime}_{-}(2^{j}r)=\phi^{\prime}(2^{j}r)-\frac{s}{2^{j}t}$. Noticing
that if $s=2^{j}t\phi^{\prime}(2^{j}r)$, then $\phi^{\prime}_{-}(2^{j}r)=0$.
We continue our discussion by considering the following three sub-cases:

  * •

When $s\geq 2\sup\limits_{r\in[1/2,2]}2^{j}t\phi^{\prime}(2^{j}r)$:
In this case, we have
$|\phi^{\prime}_{-}(2^{j}r)|=\frac{1}{2^{j}t}\left(s-2^{j}t\phi^{\prime}(2^{j}r)\right)\geq\phi^{\prime}(2^{j}r)$
and also the estimate (3.3) hold true if we replace $\phi$ by $\phi_{-}$.
Then, analogous to $II^{+}_{j}(s)$, for any $\theta\geq 0$, we obtain

$|II^{-}_{j}(s)|\lesssim|t|^{-\theta}2^{j(N-m_{1}\theta)}.$ (3.7)

  * •

When $s\leq\frac{1}{2}\inf\limits_{r\in[1/2,2]}2^{j}t\phi^{\prime}(2^{j}r)$:
In this case, we have
$\phi^{\prime}_{-}(2^{j}r)=\frac{1}{2^{j}t}\left(2^{j}t\phi^{\prime}(2^{j}r)-s\right)\geq\frac{1}{2}\phi^{\prime}(2^{j}r)$
and (3.3) also hold true if we replace $\phi$ by $\phi_{-}$. Analogous to
$II^{+}_{j}(s)$, for any $\theta\geq 0$, we obtain

$|II^{-}_{j}(s)|\lesssim|t|^{-\theta}2^{j(N-m_{1}\theta)}.$ (3.8)

  * •

When $\frac{1}{2}\inf\limits_{r\in[1/2,2]}2^{j}t\phi^{\prime}(2^{j}r)\leq
s\leq 2\sup\limits_{r\in[1/2,2]}2^{j}t\phi^{\prime}(2^{j}r)$:
In this case, we see that $s\sim t2^{jm_{1}}$. Moreover, from (2.15), it
follows that

$\displaystyle|II^{-}_{j}(s)|$ $\displaystyle\lesssim 2^{jN}s^{-(N-1)/2}$
(3.9) $\displaystyle\lesssim|t|^{-(N-1)/2}2^{j\left(N-m_{1}(N-1)/2\right)}.$

Now using the estimate (2.14) in $II^{-}_{j}(s)$, a simple calculation yields
the following easy estimate

$|II^{-}_{j}(s)|\lesssim 2^{jN}.$ (3.10)

Applying interpolation between (3.9) and (3.10), we obtain

$|II^{-}_{j}(s)|\lesssim|t|^{-\theta}2^{j\left(N-m_{1}\theta\right)}$ (3.11)

for all $0\leq\theta\leq(N-1)/2$. Combining (3.11) and (3.5), we complete the
proof of (1.5) in $(A)$.

If (C3) holds in addition, we have
$\left|\frac{d^{2}}{dr^{2}}\left(\phi_{-}(2^{j}r)\right)\right|=2^{2j}|\phi^{\prime\prime}(2^{j}r)|\sim
2^{j\alpha_{1}}$ in the last sub-case discussed above. It follows from Lemma
2.2 and (2.15) that

$\displaystyle|II^{-}_{j}(s)|\lesssim
2^{jN}|t2^{j\alpha_{1}}|^{-1/2}s^{-(N-1)/2}\lesssim|t|^{-N/2}2^{j\left(N-m_{1}N/2-(\alpha_{1}-m_{1})/2\right)}.$
(3.12)

For any $0\leq\theta\leq 1$, since $(N-1+\theta)/2=(1-\theta)(N-1)/2+\theta
N/2$, by an interpolation between (3.11) (with $\theta=(N-1)/2$) and (3.12),
we obtain

$|II^{-}_{j}(s)|\lesssim|t|^{-(N-1+\theta)/2}2^{j\left(N-m_{1}(N-1+\theta)/2-\theta(\alpha_{1}-m_{1})/2\right)}.$
(3.13)

Noting that $\alpha_{1}\leq m_{1}$, from (3.5), (3.6), (3.7), (3.8) and
(3.13), we have

$|II_{j}(s)|\lesssim|t|^{-(N-1+\theta)/2}2^{j\left(N-m_{1}(N-1+\theta)/2-\theta(\alpha_{1}-m_{1})/2\right)},$
(3.14)

for all $0\leq\theta\leq 1$. This completes the proof of (1.6) in part (A).

Now we will prove part (C), as the proof of part (B) is similar to part (A).
To prove part (C), take any fixed $\theta$ satisfying
$0\leq\theta\leq\min\left(\frac{N}{m_{2}},\frac{N-1}{2}\right)$. First, we
consider the case when $\theta=\frac{N-1}{2}<\frac{N}{m_{2}}$, i.e.,
$N-m_{2}\theta>0$. Now applying part (B), we obtain

$\displaystyle\|U_{t}S_{0}u_{0}\|_{L_{\kappa}^{\infty}(\mathbb{R}^{n})}$
$\displaystyle\lesssim\left\|\sum_{j\leq
0}U_{t}\tilde{\Delta}_{j}S_{0}u_{0}\right\|_{L_{\kappa}^{\infty}(\mathbb{R}^{n})}$
$\displaystyle\leq\sum_{j\leq
0}\left\|U_{t}\tilde{\Delta}_{j}S_{0}u_{0}\right\|_{L_{\kappa}^{\infty}(\mathbb{R}^{n})}$
$\displaystyle\lesssim\sum_{j\leq
0}|t|^{-\theta}2^{j(N-m_{2}\theta)}\left\|S_{0}u_{0}\right\|_{L_{\kappa}^{1}(\mathbb{R}^{n})}$
$\displaystyle\lesssim|t|^{-\theta}\left\|S_{0}u_{0}\right\|_{L_{\kappa}^{1}(\mathbb{R}^{n})}\lesssim|t|^{-\frac{(N-1)}{2}}\|u_{0}\|_{L_{\kappa}^{1}(\mathbb{R}^{n})}.$

Therefore, it suffices to consider the case
$\theta=\frac{N}{m_{2}}\leq\frac{N-1}{2}$. By (3.1) and (3.2), we have

$U_{t}S_{0}u_{0}=\sum_{j\leq
0}U_{t}\tilde{\Delta}_{j}(S_{0}u_{0})=\left(\sum_{j\leq
0}II_{j}\left(2^{j}|\cdot|\right)\right)*_{\kappa}(S_{0}u_{0}).$ (3.15)

Argued similarly as (a) but for the proof of (b), we know that if $j_{0}<0$
and $s\sim|t|2^{j_{0}m_{2}}>2$, then

$\displaystyle\left|II_{j_{0}}(2^{j_{0}}s)\right|$
$\displaystyle\lesssim|t|^{-(N-1)/2}2^{j_{0}\left(N-m_{2}(N-1)/2\right)}$
(3.16)
$\displaystyle\lesssim|t|^{-(N-1)/2}\left|\frac{s}{t}\right|^{\left(N-m_{2}(N-1)/2\right)\frac{1}{m_{2}}}$
$\displaystyle\lesssim|t|^{-\frac{N}{m_{2}}}.$

When $|j-j_{0}|>C\gg 1$, it holds true that

$\left|II_{j}(2^{j}s)\right|\lesssim|t|^{-\alpha}2^{j(N-m_{2}\alpha)},\;\forall\alpha\geq
0.$ (3.17)

Hence, by (3.16) and (3.17), we get

$\displaystyle\quad\sum_{j\leq 0}\left|II_{j}\left(2^{j}s\right)\right|$
$\displaystyle\leq\left(\sum_{|j-j_{0}|\leq
C}+\sum_{|j-j_{0}|>C,\;2^{j}\leq|t|^{-\frac{1}{m_{2}}}}+\sum_{|j-j_{0}|>C,\;2^{j}>|t|^{-\frac{1}{m_{2}}}}\right)\left|II_{j}\left(2^{j}s\right)\right|$
$\displaystyle\lesssim|t|^{-\frac{N}{m_{2}}}+\sum_{2^{j}\leq|t|^{-\frac{1}{m_{2}}}}2^{jN}+\sum_{2^{j}>|t|^{-\frac{1}{m_{2}}}}|t|^{-\alpha}2^{j(N-m_{2}\alpha)}$
$\displaystyle\lesssim|t|^{-\frac{N}{m_{2}}},$ (3.18)

where the last inequality is obtained by applying Lemma 2.5 and choosing
$\alpha>\frac{N}{m_{2}}$. By Young’s inequality (2.5), combined with (3.15)
and (3.18), we obtain

$\displaystyle\|U_{t}S_{0}u_{0}\|_{L_{\kappa}^{\infty}(\mathbb{R}^{n})}$
$\displaystyle\leq\left\|\sum_{j\leq
0}II_{j}\left(2^{j}|\cdot|\right)\right\|_{L_{\kappa}^{\infty}(\mathbb{R}^{n})}\|S_{0}u_{0}\|_{L_{\kappa}^{1}(\mathbb{R}^{n})}$
$\displaystyle\lesssim|t|^{-\frac{N}{m_{2}}}\|S_{0}u_{0}\|_{L_{\kappa}^{1}(\mathbb{R}^{n})}\lesssim|t|^{-\frac{N}{m_{2}}}\|u_{0}\|_{L_{\kappa}^{1}(\mathbb{R}^{n})},$

which proves (1.7). In addition, if (C4) holds, we can argue similarly as
previously. ∎

##  4\. Applications

In this section, we provide several applications of Theorem 1.1 by considering
some particular smooth function $\phi$. In particular, we prove Strichartz
estimates for some concrete wave equations related to the Dunkl Laplacian by
using Theorem 1.1. Above all, we introduce the general Strichartz estimates
for $U_{t}=e^{it\phi(\sqrt{-\Delta_{k}})}$. We start this section by recalling
the following versions of the Hardy-Littlewood-Sobolev inequality:

######  Lemma 4.1 ([28]).

Assume $\theta_{1},\theta_{2}\in\mathbb{R}$. Let

$k(y)=\begin{cases}|y|^{-\theta_{1}},\,&|y|\leq 1,\\\
|y|^{-\theta_{2}},\,&|y|>1.\end{cases}$

Assume that one of the following condition holds:

  1. (1)

$0<\theta_{1}=\theta_{2}<n$, $1<p<q<\infty$ and
$1-\frac{1}{p}+\frac{1}{q}=\frac{\theta_{1}}{n}$,

  2. (2)

$\theta_{1}<\theta_{2}$, $0<\theta_{1}<n$, $1<p<q<\infty$ and
$1-\frac{1}{p}+\frac{1}{q}=\frac{\theta_{1}}{n}$,

  3. (3)

$\theta_{1}<\theta_{2}$, $0<\theta_{2}<n$, $1<p<q<\infty$ and
$1-\frac{1}{p}+\frac{1}{q}=\frac{\theta_{2}}{n}$,

  4. (4)

$\theta_{1}<\theta_{2}$, $1\leq p\leq q\leq\infty$ and
$\frac{\theta_{1}}{n}<1-\frac{1}{p}+\frac{1}{q}<\frac{\theta_{2}}{n}$.

We have

$\left\|\int_{\mathbb{R}^{n}}f(\cdot-y)k(y)dy\right\|_{L^{q}(\mathbb{R}^{n})}\lesssim\|f\|_{L^{p}(\mathbb{R}^{n})}.$

######  Definition 4.2.

For $\theta_{1},\theta_{2}\in\mathbb{R}$ with $\theta_{1}<\theta_{2}$, we say
$r\in E(\theta_{1},\theta_{2})$ if one of the following conditions holds:

  1. (1)

$0<\theta_{1}=\theta_{2}<1$ and $r=\frac{2}{\theta_{1}}$,

  2. (2)

$\theta_{1}<\theta_{2}$, $0<\theta_{1}<1$ and $r=\frac{2}{\theta_{1}}$,

  3. (3)

$\theta_{1}<\theta_{2}$, $0<\theta_{2}<1$ and $r=\frac{2}{\theta_{2}}$,

  4. (4)

$\theta_{1}<\theta_{2}$, $2\leq r\leq\infty$ and
$\theta_{1}<\frac{2}{r}<\theta_{2}$.

For $2\leq p\leq\infty$, $s=s(p)\in\mathbb{R}$ and $\theta_{1}\leq\theta_{2}$,
we assume that

$\|U_{t}f\|_{B_{p,2}^{s,\kappa}(\mathbb{R}^{n})}\lesssim
k(t)\|f\|_{B_{p^{\prime},2}^{-s,\kappa}(\mathbb{R}^{n})},\quad\text{where
}k(t)=\begin{cases}|t|^{-\theta_{1}},\,&|t|\leq 1,\\\
|t|^{-\theta_{2}},\,&|t|>1.\end{cases}$ (4.1)

Once we obtain this dispersive estimate with different decay rates between
$|t|>1$ and $|t|\leq 1$, we can prove the following proposition by using Lemma
4.1 and standard duality arguments in Lemma 2.6-2.8.

######  Proposition 4.3 (see [28, 35]).

Assume $U_{t}$ satisfies (4.1). Then for $r\in E(\theta_{1},\theta_{2})$ and
$T>0$, we have

$\displaystyle\|U_{t}f\|_{L^{r}\left(\mathbb{R},B^{s,\kappa}_{p,2}(\mathbb{R}^{n})\right)}$
$\displaystyle\lesssim\|f\|_{L_{\kappa}^{2}(\mathbb{R}^{n})},$
$\displaystyle\|\int_{0}^{t}U_{t-\tau}g(\tau,\cdot)d\tau\|_{L^{r}\left([-T,T],B^{s,\kappa}_{p,2}(\mathbb{R}^{n})\right)}$
$\displaystyle\lesssim\|g\|_{L^{r^{\prime}}\left([-T,T],B^{-s,\kappa}_{p^{\prime},2}(\mathbb{R}^{n})\right)},$
$\displaystyle\|\int_{0}^{t}U_{t-\tau}g(\tau,\cdot)d\tau\|_{L^{\infty}\left([-T,T],H^{s,\kappa}(\mathbb{R}^{n})\right)}$
$\displaystyle\lesssim\|g\|_{L^{r^{\prime}}\left([-T,T],B_{p^{\prime},2}^{-s,\kappa}(\mathbb{R}^{n})\right)},$
$\displaystyle\|\int_{0}^{t}U_{t-\tau}g(\tau,\cdot)d\tau\|_{L^{r}\left([-T,T],B_{p,2}^{s,\kappa}(\mathbb{R}^{n})\right)}$
$\displaystyle\lesssim\|g\|_{L^{1}\left([-T,T],L_{\kappa}^{2}(\mathbb{R}^{n})\right)}.$

Note that the endpoint case $\theta_{i}=1$ also holds true and can be seen
from Keel-Tao [35].

###  4.1. Wave equation

First, we consider the Dunkl-wave equation as

$\begin{cases}\partial_{t}^{2}u-\Delta_{\kappa}u=g,\\\ u|_{t=0}=u_{0},\\\
\partial_{t}u|_{t=0}=u_{1}.\end{cases}$ (4.2)

By Duhamel’s principle, the solution is formally given by

$u(t)=\frac{d{\mathcal{W}}_{t}}{dt}u_{0}+{\mathcal{W}}_{t}u_{1}-\int_{0}^{t}{\mathcal{W}}_{t-\tau}g(\tau,\cdot)\,d\tau,$

where

$\mathcal{W}_{t}=\frac{\sin(t\sqrt{-\Delta_{\kappa}})}{\sqrt{-\Delta_{\kappa}}},\quad\frac{d\mathcal{W}_{t}}{dt}=\cos(t\sqrt{-\Delta_{\kappa}}).$

This reduces to the semigroup $W_{t}=e^{it\sqrt{-\Delta_{\kappa}}}$, which
corresponds to the case $\phi(r)=r$. Thus $\phi$ satisfies (C1) and (C2) with
$m_{1}=m_{2}=1$. Using Theorem 1.1, we obtain the following decay estimate for
the wave operator.

######  Proposition 4.4.

Assume $2\leq p\leq\infty$, $1\leq q\leq\infty$,
$\delta=\frac{1}{2}-\frac{1}{p}$ and $2s\leq-(N+1)\delta$, then we have

$\|W_{t}f\|_{B_{p,q}^{s,\kappa}(\mathbb{R}^{n})}\lesssim
k(t)\|f\|_{B_{p^{\prime},q}^{-s,\kappa}(\mathbb{R}^{n})},\quad
k(t)=\begin{cases}|t|^{-2\max(s+N\delta,0)},\,&|t|\leq 1,\\\
|t|^{-(N-1)\delta},\,&|t|\geq 1.\end{cases}$

###### Proof.

First, we prove the case $|t|\geq 1$. It follows from (1.7) of Theorem 1.1 by
taking $\theta=\frac{N-1}{2}$ and Plancherel’s identity that

$\displaystyle\|W_{t}S_{0}f\|_{L_{\kappa}^{\infty}(\mathbb{R}^{n})}$
$\displaystyle\lesssim|t|^{-\frac{N-1}{2}}\|S_{0}f\|_{L_{\kappa}^{1}(\mathbb{R}^{n})},$
$\displaystyle\|W_{t}S_{0}f\|_{L_{\kappa}^{2}(\mathbb{R}^{n})}$
$\displaystyle\leq\|S_{0}f\|_{L_{\kappa}^{2}(\mathbb{R}^{n})}.$

From (1.5) of Theorem 1.1 by taking $\theta=\frac{N-1}{2}$ and Plancherel’s
identity, for $j\geq 0$, we have

$\displaystyle\left\|W_{t}\tilde{\Delta}_{j}f\right\|_{L_{\kappa}^{\infty}(\mathbb{R}^{n})}$
$\displaystyle\lesssim|t|^{-\frac{N-1}{2}}2^{j\frac{N+1}{2}}\|\tilde{\Delta}_{j}f\|_{L_{\kappa}^{1}(\mathbb{R}^{n})},$
$\displaystyle\left\|W_{t}\tilde{\Delta}_{j}f\right\|_{L_{\kappa}^{2}(\mathbb{R}^{n})}$
$\displaystyle\leq\|\tilde{\Delta}_{j}f\|_{L_{\kappa}^{2}(\mathbb{R}^{n})}.$

Applying the Riesz-Thorin interpolation theorem, for any $j\geq 0$, we have

$\displaystyle\|W_{t}S_{0}f\|_{L_{\kappa}^{p}(\mathbb{R}^{n})}$
$\displaystyle\lesssim|t|^{-(N-1)\delta}\|S_{0}f\|_{L_{\kappa}^{p^{\prime}}(\mathbb{R}^{n})},$
$\displaystyle\|W_{t}\tilde{\Delta}_{j}f\|_{L_{\kappa}^{p}(\mathbb{R}^{n})}$
$\displaystyle\lesssim|t|^{-(N-1)\delta}2^{j(N+1)\delta}\|\tilde{\Delta}_{j}f\|_{L_{\kappa}^{p^{\prime}}(\mathbb{R}^{n})}\leq|t|^{-(N-1)\delta}2^{-2js}\|\tilde{\Delta}_{j}f\|_{L_{\kappa}^{p^{\prime}}(\mathbb{R}^{n})},$

where in the last inequality we used the fact that $2s\leq-(N+1)\delta$ and
$\delta=\frac{1}{2}-\frac{1}{p}$. Therefore

$\|W_{t}f\|_{B_{p,q}^{s,\kappa}(\mathbb{R}^{n})}\lesssim|t|^{-(N-1)\delta}\|f\|_{B_{p^{\prime},q}^{-s,\kappa}(\mathbb{R}^{n})},\,|t|\geq
1.$ (4.3)

Next, we consider the case $|t|\leq 1$. Putting $\theta\geq 0$ in (1.7) of
Theorem 1.1 and applying Plancherel’s identity and Riesz-Thorin interpolation
theorem, we get

$\|W_{t}S_{0}f\|_{L_{\kappa}^{p}(\mathbb{R}^{n})}\lesssim\|S_{0}f\|_{L_{\kappa}^{p^{\prime}}(\mathbb{R}^{n})}.$

Again for $j\geq 0$, from (1.5) of Theorem 1.1 with Plancherel’s identity and
Riesz-Thorin interpolation theorem, we obtain

$\|W_{t}\tilde{\Delta}_{j}f\|_{L_{\kappa}^{p}(\mathbb{R}^{n})}\lesssim|t|^{-2\theta\delta}2^{2j(N-\theta)\delta}\|\tilde{\Delta}_{j}f\|_{L_{\kappa}^{p^{\prime}}(\mathbb{R}^{n})},$

for $0\leq\theta\leq\frac{N-1}{2}$. Therefore

$2^{js}\|W_{t}\tilde{\Delta}_{j}f\|_{L_{\kappa}^{p}(\mathbb{R}^{n})}\lesssim|t|^{-2\theta\delta}2^{2j(s+(N-\theta)\delta)}2^{-js}\|\tilde{\Delta}_{j}f\|_{L_{\kappa}^{p^{\prime}}(\mathbb{R}^{n})}.$

If $-N\delta\leq s\leq-\frac{(N+1)\delta}{2}$, then we can choose
$0\leq\theta\leq\frac{N-1}{2}$ such that $s+N\delta=\theta\delta$.
Furthermore, for $s<-N\delta$, we will choose $\theta=0$. Combing all the
scenarios, we have

$2^{js}\|W_{t}\tilde{\Delta}_{j}f\|_{L_{\kappa}^{p}(\mathbb{R}^{n})}\lesssim|t|^{-2\max(s+N\delta,0)}2^{-js}\|\tilde{\Delta}_{j}f\|_{L_{\kappa}^{p^{\prime}}(\mathbb{R}^{n})}$

and hence

$\|W_{t}f\|_{B_{p,q}^{s,\kappa}(\mathbb{R}^{n})}\lesssim|t|^{-2\max(s+N\delta,0)}\|f\|_{B_{p^{\prime},q}^{-s,\kappa}(\mathbb{R}^{n})},$
(4.4)

for $|t|\leq 1$. Thus from (4.3) and (4.4), we finally have

$\|W_{t}f\|_{B_{p,q}^{s,\kappa}(\mathbb{R}^{n})}\lesssim
k(t)\|f\|_{B_{p^{\prime},q}^{-s,\kappa}(\mathbb{R}^{n})},\quad
k(t)=\begin{cases}|t|^{-2\max(s+N\delta,0)},\,&|t|\leq 1,\\\
|t|^{-(N-1)\delta},\,&|t|\geq 1.\end{cases}$

∎

Now using Proposition 4.3, we obtain the following estimates for the Dunkl-
wave operator.

######  Theorem 4.5.

Let $N>1$. For $i=1,2$, let $p_{i},r_{i}\in[2,\infty]$ and
$s_{i}\in\mathbb{R}$ such that

$\displaystyle(i)\,\frac{2}{r_{i}}+\frac{N-1}{p_{i}}\leq\frac{N-1}{2},$
$\displaystyle(ii)\,s_{i}\leq-\frac{N+1}{2}\left(\frac{1}{2}-\frac{1}{p_{i}}\right),$

except for $(N,p_{i},r_{i})=(3,\infty,2)$. Then for any $T>0$, the Dunkl-wave
operator $W_{t}$ satisfies

$\|W_{t}u_{0}\|_{L^{r_{1}}(\mathbb{R},B_{p_{1},2}^{s_{1},\kappa}(\mathbb{R}^{n}))}\lesssim\|u_{0}\|_{L_{\kappa}^{2}(\mathbb{R}^{n})},$
$\left\|\int_{0}^{t}W_{t-\tau}g(\tau,\cdot)\,d\tau\right\|_{L^{r_{1}}\left([-T,T],B_{p_{1},2}^{s_{1},\kappa}(\mathbb{R}^{n})\right)}\lesssim\|g\|_{L^{r^{\prime}_{2}}\left([-T,T],B_{p^{\prime}_{2},2}^{-s_{2},\kappa}(\mathbb{R}^{n})\right)}.$

It immediately reduces the Strichartz inequality for the solution of the
Dunkl-wave equation (4.2).

######  Corollary 4.6.

Under the same hypotheses as in Theorem 4.5, the solution $u$ of the Dunkl-
wave equation (4.2) satisfies the following estimate

$\|u\|_{L^{r_{1}}\left([-T,T],B_{p_{1},2}^{s_{1}+1,\kappa}(\mathbb{R}^{n})\right)}\lesssim\|u_{0}\|_{H^{1,\kappa}(\mathbb{R}^{n})}+\|u_{1}\|_{L_{\kappa}^{2}(\mathbb{R}^{n})}+\|g\|_{L^{r^{\prime}_{2}}\left([-T,T],B_{p^{\prime}_{2},2}^{-s_{2},\kappa}(\mathbb{R}^{n})\right)}.$

Finally, by the embedding relation between the Besov space and Lebesgue space
in Proposition 2.1, we have the following Strichartz estimates on Lebesgue
spaces.

######  Corollary 4.7.

Let $N>1$. Suppose $r_{i}\geq 2$ and $2\leq p_{i}<\infty$ such that

$\frac{1}{r_{1}}+\frac{N}{p_{1}}=\frac{N}{2}-1,\quad\frac{2}{r_{1}}+\frac{N-1}{p_{1}}\leq\frac{N-1}{2}\quad
and\quad\frac{1}{r_{2}}+\frac{N}{p_{2}}\leq\frac{N}{2},$

except for $(N,p_{1},r_{1},p_{2},r_{2})=(3,\infty,2,3,2)$. If $u$ is the
solution of the Dunkl-wave equation (4.2), for any $T>0$, we have the
following estimate

$\|u\|_{L^{r_{1}}\left([-T,T],L_{\kappa}^{p_{1}}(\mathbb{R}^{n})\right)}\lesssim\|u_{0}\|_{H^{1,\kappa}(\mathbb{R}^{n})}+\|u_{1}\|_{L_{\kappa}^{2}(\mathbb{R}^{n})}+\|g\|_{L^{r^{\prime}_{2}}\left([-T,T],L_{\kappa}^{p_{2}^{\prime}}(\mathbb{R}^{n})\right)}.\\\
$

###### Proof.

From Corollary 4.6, it follows that

$\|u\|_{L^{r_{1}}\left([-T,T],B_{\tilde{p}_{1},2}^{s_{1}+1,\kappa}(\mathbb{R}^{n})\right)}\lesssim\|u_{0}\|_{H^{1,\kappa}(\mathbb{R}^{n})}+\|u_{1}\|_{L_{\kappa}^{2}(\mathbb{R}^{n})}+\|g\|_{L^{r^{\prime}_{2}}\left([-T,T],B_{\tilde{p}^{\prime}_{2},2}^{-s_{2},\kappa},(\mathbb{R}^{n})\right)}$

where $\tilde{p}_{i},r_{i}\in[2,\infty]$ and

$\frac{2}{r_{1}}+\frac{N-1}{\tilde{p}_{1}}=\frac{N-1}{2},\;\;\frac{2}{r_{2}}+\frac{N-1}{\tilde{p}_{2}}\leq\frac{N-1}{2},~{}~{}\text{
and
}~{}~{}s_{i}=-\frac{N+1}{2}\left(\frac{1}{2}-\frac{1}{\tilde{p}_{i}}\right),\;i=1,2,$

except for $(N,\tilde{p}_{i},{r}_{i})=(3,\infty,2)$. If
$p_{i},r_{i}\in[2,\infty],i=1,2$ with the sole exception of the pair
$(N,p_{1},r_{1},p_{2},r_{2})=(3,\infty,2,3,2)$ satisfies the condition

$\frac{1}{r_{1}}+\frac{N}{p_{1}}=\frac{N}{2}-1,\;\;\frac{2}{r_{1}}+\frac{N-1}{p_{1}}\leq\frac{N-1}{2}~{}~{}\text{
and }~{}~{}\frac{1}{r_{2}}+\frac{N}{p_{2}}\leq\frac{N}{2},$

then we the following relations

$s_{1}+1\leq 0,\;\;s_{1}+1-\frac{N}{\tilde{p}_{1}}=0-\frac{N}{p_{1}},$

and

$-s_{2}\geq 0,\;\;-s_{2}-\frac{N}{\tilde{p}_{2}}=0-\frac{N}{p_{2}}.$

Therefore, using the embedding relation between the Besov space and Lebesgue
space in Proposition 2.1, we obtain
$B_{\tilde{p}_{1},2}^{s_{1}+1,\kappa}(\mathbb{R}^{n})\subseteq
L^{p_{1}}_{\kappa}(\mathbb{R}^{n})$ and
$L_{\kappa}^{p_{2}^{\prime}}(\mathbb{R}^{n})\subseteq
B_{\tilde{p}^{\prime}_{2},2}^{-s_{2},\kappa}(\mathbb{R}^{n})$ and this
completes our desired result. ∎

###  4.2. Fractional Schrödinger equation

Consider the fractional Dunkl-Schrödinger equation ($0<\mu\leq 2$ and $\mu\neq
1$)

$\begin{cases}i\partial_{t}u+(-\Delta_{\kappa})^{\frac{\mu}{2}}u=g,\\\
u|_{t=0}=u_{0}.\end{cases}$ (4.5)

By Duhamel’s principle, the solution is formally given by

$u(t)=\mathcal{S}^{\mu}_{t}u_{0}-i\int_{0}^{t}\mathcal{S}^{\mu}_{t-\tau}g(\tau,\cdot)\,d\tau,$

where $\mathcal{S}^{\mu}_{t}=e^{it(-\Delta_{\kappa})^{\frac{\mu}{2}}}$ and it
corresponds to the case when $\phi(r)=r^{\mu}$. By a simple calculation, we
see that

$\phi^{\prime}(r)=\mu
r^{\mu-1},\quad\phi^{\prime\prime}(r)=\mu(\mu-1)r^{\mu-2}.$

This shows that $\phi$ satisfies (C1)-(C4) with
$m_{1}=\alpha_{1}=m_{2}=\alpha_{2}=\mu$. Using Theorem 1.1, we obtain the
following decay estimate for $\mathcal{S}^{\mu}_{t}$.

######  Proposition 4.8.

Let $0<\mu\leq 2$ and $\mu\neq 1$. Assume $2\leq p\leq\infty$, $1\leq
q\leq\infty$, $\delta=\frac{1}{2}-\frac{1}{p}$ and
$s\leq-N\left(1-\frac{\mu}{2}\right)\delta$, then we have

$\|\mathcal{S}^{\mu}_{t}f\|_{B_{p,q}^{s,\kappa}(\mathbb{R}^{n})}\lesssim
k(t)\|f\|_{B_{p^{\prime},q}^{-s,\kappa}(\mathbb{R}^{n})},\quad
k(t)=\begin{cases}|t|^{-\frac{2}{\mu}\max(s+N\delta,0)},\,&|t|\leq 1,\\\
|t|^{-N\delta},\,&|t|\geq 1.\end{cases}$

###### Proof.

First, we prove the case $|t|\geq 1$. From (1.8) of Theorem 1.1 with
$\theta=\frac{N}{2}$, Plancherel’s identity yields

$\displaystyle\|\mathcal{S}^{\mu}_{t}S_{0}f\|_{L_{\kappa}^{\infty}(\mathbb{R}^{n})}$
$\displaystyle\lesssim|t|^{-\frac{N}{2}}\|S_{0}f\|_{L_{\kappa}^{1}(\mathbb{R}^{n})},$
$\displaystyle\|\mathcal{S}^{\mu}_{t}S_{0}f\|_{L_{\kappa}^{2}(\mathbb{R}^{n})}$
$\displaystyle\leq\|S_{0}f\|_{L_{\kappa}^{2}(\mathbb{R}^{n})}.$

Again from (1.6) of Theorem 1.1 by taking $\theta=1$, for $j\geq 0$,
Plancherel’s identity gives

$\displaystyle\left\|\mathcal{S}^{\mu}_{t}\tilde{\Delta}_{j}f\right\|_{L_{\kappa}^{\infty}(\mathbb{R}^{n})}$
$\displaystyle\lesssim|t|^{-\frac{N}{2}}2^{jN(1-\frac{\mu}{2})}\|\tilde{\Delta}_{j}f\|_{L_{\kappa}^{1}(\mathbb{R}^{n})},$
$\displaystyle\left\|\mathcal{S}^{\mu}_{t}\tilde{\Delta}_{j}f\right\|_{L_{\kappa}^{2}(\mathbb{R}^{n})}$
$\displaystyle\leq\|\tilde{\Delta}_{j}f\|_{L_{\kappa}^{2}(\mathbb{R}^{n})}.$

Now applying the Riesz-Thorin interpolation theorem, for any $j\geq 0$, we
obtain

$\displaystyle\|\mathcal{S}^{\mu}_{t}S_{0}f\|_{L_{\kappa}^{p}(\mathbb{R}^{n})}$
$\displaystyle\lesssim|t|^{-N\delta}\|S_{0}f\|_{L_{\kappa}^{p^{\prime}}(\mathbb{R}^{n})},$
$\displaystyle\|\mathcal{S}^{\mu}_{t}\tilde{\Delta}_{j}f\|_{L_{\kappa}^{p}(\mathbb{R}^{n})}$
$\displaystyle\lesssim|t|^{-N\delta}2^{2jN\left(1-\frac{\mu}{2}\right)\delta}\|\tilde{\Delta}_{j}f\|_{L_{\kappa}^{p^{\prime}}(\mathbb{R}^{n})}\leq|t|^{-N\delta}2^{-2js}\|\tilde{\Delta}_{j}f\|_{L_{\kappa}^{p^{\prime}}(\mathbb{R}^{n})}.$

This implies that

$\|\mathcal{S}^{\mu}_{t}f\|_{B_{p,q}^{s,\kappa}(\mathbb{R}^{n})}\lesssim|t|^{-N\delta}\|f\|_{B_{p^{\prime},q}^{-s,\kappa}(\mathbb{R}^{n})},\,|t|\geq
1.$

Next, we discuss the case $|t|\leq 1$. For $\theta=0$ in (1.8) of Theorem 1.1,
applying $\theta=0$, Plancherel’s identity and Riesz-Thorin interpolation
theorem, we get

$\|\mathcal{S}^{\mu}_{t}S_{0}f\|_{L_{\kappa}^{p}(\mathbb{R}^{n})}\lesssim\|S_{0}f\|_{L_{\kappa}^{p^{\prime}}(\mathbb{R}^{n})}.$

For $j\geq 0$, it follows from (1.5) and (1.6) of Theorem 1.1, Plancherel’s
identity and Riesz-Thorin interpolation theorem that for
$0\leq\theta\leq\frac{N}{2}$, we have

$\|\mathcal{S}^{\mu}_{t}\tilde{\Delta}_{j}f\|_{L_{\kappa}^{p}(\mathbb{R}^{n})}\lesssim|t|^{-2\theta\delta}2^{2j(N-\theta\mu)\delta}\|\tilde{\Delta}_{j}f\|_{L_{\kappa}^{p^{\prime}}(\mathbb{R}^{n})}.$

This implies that

$2^{js}\|\mathcal{S}^{\mu}_{t}\tilde{\Delta}_{j}f\|_{L_{\kappa}^{p}(\mathbb{R}^{n})}\lesssim|t|^{-2\theta\delta}2^{2j(s+(N-\theta\mu)\delta)}2^{-js}\|\tilde{\Delta}_{j}f\|_{L_{\kappa}^{p^{\prime}}(\mathbb{R}^{n})}.$

If $-N\delta\leq s\leq-N\left(1-\frac{\mu}{2}\right)\delta$, then we can
choose $0\leq\theta\leq\frac{N}{2}$ such that $s+N\delta=\mu\theta\delta$. If
$s<-N\delta$, we take $\theta=0$. Therefore

$2^{js}\|\mathcal{S}^{\mu}_{t}\tilde{\Delta}_{j}f\|_{L_{\kappa}^{p}(\mathbb{R}^{n})}\lesssim|t|^{-\frac{2}{\mu}\max(s+N\delta,0)}2^{-js}\|\tilde{\Delta}_{j}f\|_{L_{\kappa}^{p^{\prime}}(\mathbb{R}^{n})}$

and

$\|\mathcal{S}^{\mu}_{t}f\|_{B_{p,q}^{s,\kappa}(\mathbb{R}^{n})}\lesssim|t|^{-\frac{2}{\mu}\max(s+N\delta,0)}\|f\|_{B_{p^{\prime},q}^{-s,\kappa}(\mathbb{R}^{n})},\,|t|\leq
1,$

which completes the proof of the proposition. ∎

By Proposition 4.3, we obtain the following estimates for the fractional
Schrödinger operator.

######  Theorem 4.9.

Let $0<\mu\leq 2$ and $\mu\neq 1$. For $i=1,2$, let $p_{i},r_{i}\in[2,\infty]$
and $s_{i}\in\mathbb{R}$ such that

$\displaystyle(i)\,\frac{2}{r_{i}}+\frac{N}{p_{i}}\leq\frac{N}{2},$
$\displaystyle(ii)\,s_{i}\leq-N\left(1-\frac{\mu}{2}\right)\left(\frac{1}{2}-\frac{1}{p_{i}}\right),$

except for $(N,p_{i},r_{i})=(2,\infty,2)$. Then for any $T>0$, the fractional
Schrödinger operator $\mathcal{S}^{\mu}_{t}$ satisfies the estimates

$\|\mathcal{S}^{\mu}_{t}u_{0}\|_{L^{r_{1}}(\mathbb{R},B_{p_{1},2}^{s_{1},\kappa}(\mathbb{R}^{n}))}\lesssim\|u_{0}\|_{L_{\kappa}^{2}(\mathbb{R}^{n})},$
(4.6)
$\left\|\int_{0}^{t}\mathcal{S}^{\mu}_{t-\tau}g(\tau,\cdot)\,d\tau\right\|_{L^{r_{1}}\left([-T,T],B_{p_{1},2}^{s_{1},\kappa}(\mathbb{R}^{n})\right)}\lesssim\|g\|_{L^{r^{\prime}_{2}}\left([-T,T],B_{p^{\prime}_{2},2}^{-s_{2},\kappa}(\mathbb{R}^{n})\right)}.$
(4.7)

The above estimate immediately reduces the following Strichartz inequality for
the solution of the fractional Schrödinger equation.

######  Corollary 4.10.

Under the same hypotheses as in Theorem 4.9, the solution $u$ of the
fractional Schrödinger equation (4.5) satisfies the following estimate

$\|u\|_{L^{r_{1}}\left([-T,T],B_{p_{1},2}^{s_{1},\kappa}(\mathbb{R}^{n})\right)}\lesssim\|u_{0}\|_{L_{\kappa}^{2}(\mathbb{R}^{n})}+\|g\|_{L^{r^{\prime}_{2}}\left([-T,T],B_{p^{\prime}_{2},2}^{-s_{2},\kappa}(\mathbb{R}^{n})\right)}.$

Finally, by the embedding relation between the Besov space and Lebesgue space
in Proposition 2.1, we have the Strichartz inequalities on Lebesgue spaces.

######  Corollary 4.11.

Let $0<\mu<2$ and $\mu\neq 1$. Suppose $p_{i},r_{i}\geq 2$ such that

$\frac{1}{p_{i}}\geq\frac{1}{2}-\frac{2}{(2-\mu)N}\quad
and\quad\frac{\mu}{r_{i}}+\frac{N}{p_{i}}\leq\frac{N}{2}-1.$

If $u$ is the solution of the fractional Schrödinger equation (4.5), for any
$T>0$, we have the following estimate

$\|u\|_{L^{r_{1}}\left([-T,T],L_{\kappa}^{p_{1}}(\mathbb{R}^{n})\right)}\lesssim\|u_{0}\|_{H^{1,\kappa}(\mathbb{R}^{n})}+\|g\|_{L^{r^{\prime}_{2}}\left([-T,T],L_{\kappa}^{p_{2}^{\prime}}(\mathbb{R}^{n})\right)}.$

In particular, when $\mu=2$, fractional Schrödinger operator reduces to the
Dunkl-Schrödinger equation

$\begin{cases}i\partial_{t}u+\Delta_{\kappa}u=g,\\\
u|_{t=0}=u_{0}.\end{cases}$ (4.8)

By the embedding relation between the Besov space and Lebesgue space in
Proposition 2.1 along with Theorem 4.9, we obtain the Strichartz inequalities
on Lebesgue space for the Dunkl-Schrödinger equation considered in [39, 40,
33].

######  Corollary 4.12.

Suppose for $j=1,2$, $r_{i},p_{i}\geq 2$, $(N,p_{i},r_{i})\neq(2,\infty,2)$
and

$\frac{2}{r_{i}}+\frac{N}{p_{i}}\leq\frac{N}{2}.$

If $u$ is the solution of the Dunkl-Schrödinger equation (4.8), then for any
$T>0$, we have the following estimate

$\|u\|_{L^{r_{1}}\left([-T,T],L_{\kappa}^{p_{1}}(\mathbb{R}^{n})\right)}\lesssim\|u_{0}\|_{L_{\kappa}^{2}(\mathbb{R}^{n})}+\|g\|_{L^{r^{\prime}_{2}}\left([-T,T],L_{\kappa}^{p_{2}^{\prime}}(\mathbb{R}^{n})\right)}.$

###  4.3. Fourth-order Schrödinger equation

Consider the fourth-order Schrödinger equation

$\begin{cases}i\partial_{t}u+\Delta_{\kappa}^{2}u-\Delta_{\kappa}u=g,\\\
u|_{t=0}=u_{0}.\end{cases}$ (4.9)

By Duhamel’s principle, the solution is formally given by

$u(t)=\mathcal{U}_{t}u_{0}-i\int_{0}^{t}\mathcal{U}_{t-\tau}g(\tau,\cdot)\,d\tau,$

where $\mathcal{U}_{t}=e^{it(\Delta_{\kappa}^{2}-\Delta_{\kappa})}$ and it
corresponds to the case when $\phi(r)=r^{4}+r^{2}$. By a simple calculation,
we see that

$\phi^{\prime}(r)=4r^{3}+2r,\quad\phi^{\prime\prime}(r)=12r^{2}+2.$

Thus $\phi$ satisfies (C1)–(C4) with $m_{1}=\alpha_{1}=4$,
$m_{2}=\alpha_{2}=2$. Using Theorem 1.1, we obtain the following decay
estimate for $\mathcal{U}_{t}$.

######  Proposition 4.13.

Assume $2\leq p\leq\infty$, $1\leq q\leq\infty$,
$\delta=\frac{1}{2}-\frac{1}{p}$ and $s\leq N\delta$.
Then we have

$\|\mathcal{U}_{t}f\|_{B_{p,q}^{s,\kappa}(\mathbb{R}^{n})}\lesssim
k(t)\|f\|_{B_{p^{\prime},q}^{-s,\kappa}(\mathbb{R}^{n})},\quad
k(t)=\begin{cases}|t|^{-\frac{1}{2}\max(s+N\delta,0)},\,&|t|\leq 1,\\\
|t|^{-N\delta},\,&|t|\geq 1.\end{cases}$

###### Proof.

First, we prove the case $|t|\geq 1$. For $\theta=\frac{N}{2}$ in (1.8) of
Theorem 1.1, Plancherel’s identity gives

$\displaystyle\|\mathcal{U}_{t}S_{0}f\|_{L_{\kappa}^{\infty}(\mathbb{R}^{n})}$
$\displaystyle\lesssim|t|^{-\frac{N}{2}}\|S_{0}f\|_{L_{\kappa}^{1}(\mathbb{R}^{n})},$
$\displaystyle\|\mathcal{U}_{t}S_{0}f\|_{L_{\kappa}^{2}(\mathbb{R}^{n})}$
$\displaystyle\leq\|S_{0}f\|_{L_{\kappa}^{2}(\mathbb{R}^{n})}.$

Applying Riesz-Thorin interpolation theorem, we have

$\|\mathcal{U}_{t}S_{0}f\|_{L_{\kappa}^{p}(\mathbb{R}^{n})}\lesssim|t|^{-N\delta}\|S_{0}f\|_{L_{\kappa}^{p^{\prime}}(\mathbb{R}^{n})}.$
(4.10)

Again for $j\geq 0$, it follows from (1.5) and (1.6) of Theorem 1.1 that

$\|\mathcal{U}_{t}\tilde{\Delta}_{j}f|_{L_{\kappa}^{\infty}(\mathbb{R}^{n})}\lesssim|t|^{-\theta}2^{j\left(N-4\theta\right)}\|\tilde{\Delta}_{j}f\|_{L_{\kappa}^{1}(\mathbb{R}^{n})},\,0\leq\theta\leq\frac{N}{2}.$
(4.11)

Taking $\theta=\frac{N}{2}$ in (4.11), combined with Plancherel’s identity and
Riesz-Thorin interpolation theorem, it indicates

$\|\mathcal{U}_{t}\tilde{\Delta}_{j}f\|_{L_{\kappa}^{p}(\mathbb{R}^{n})}\lesssim|t|^{-N\delta}2^{-2jN\delta}\|\tilde{\Delta}_{j}f\|_{L_{\kappa}^{p^{\prime}}(\mathbb{R}^{n})}\leq|t|^{-N\delta}2^{-2js}\|\tilde{\Delta}_{j}f\|_{L_{\kappa}^{p^{\prime}}},$
(4.12)

for any $j\geq 0$. From (4.10) and (4.12), we obtain our desired result of the
proposition in this case.

Next, we discuss the case $|t|\leq 1$. For $\theta=0$ in (1.7) of Theorem 1.1,
Plancherel’s identity and Riesz-Thorin interpolation theorem gives

$\|\mathcal{U}_{t}S_{0}f\|_{L^{p}_{\kappa}(\mathbb{R}^{n})}\lesssim\|S_{0}f\|_{L^{p^{\prime}}_{\kappa}(\mathbb{R}^{n})}.$

Again for $j\geq 0$, from (4.11), applying Plancherel’s identity and Riesz-
Thorin interpolation theorem, for $0\leq\theta\leq\frac{N}{2}$, we obtain

$\|\mathcal{U}_{t}\tilde{\Delta}_{j}f\|_{L^{p}_{\kappa}(\mathbb{R}^{n})}\lesssim|t|^{-2\theta\delta}2^{2j(N-4\theta)\delta}\|\tilde{\Delta}_{j}f\|_{L^{p^{\prime}}_{\kappa}(\mathbb{R}^{n})},$

which reduces to

$2^{js}\|\mathcal{U}_{t}\tilde{\Delta}_{j}f\|_{L^{p}_{\kappa}(\mathbb{R}^{n})}\lesssim|t|^{-2\theta\delta}2^{2j\left(s+(N-4\theta)\delta\right)}2^{-js}\|\tilde{\Delta}_{j}f\|_{L^{p^{\prime}}_{\kappa}(\mathbb{R}^{n})}.$

If $-N\delta\leq s\leq N\delta$, then we can choose
$0\leq\theta\leq\frac{N}{2}$ such that $s+N\delta=4\theta\delta$. If
$s<-N\delta$, we take $\theta=0$. Therefore

$2^{js}\|\mathcal{U}_{t}\tilde{\Delta}_{j}f\|_{L^{p}_{\kappa}(\mathbb{R}^{n})}\lesssim|t|^{-\frac{1}{2}\max(s+N\delta,0)}2^{-js}\|\tilde{\Delta}_{j}f\|_{L^{p^{\prime}}_{\kappa}(\mathbb{R}^{n})},$

and hence

$\|\mathcal{U}_{t}f\|_{B_{p,q}^{s,\kappa}(\mathbb{R}^{n})}\lesssim|t|^{-\frac{1}{2}\max(s+N\delta,0)}\|f\|_{B_{p,q}^{-s,\kappa}(\mathbb{R}^{n})},\,|t|\leq
1.$

This completes the proof of the proposition. ∎

###  4.4. Klein-Gordon equation

Moreover, we consider the Klein-Gordon equation

$\begin{cases}\partial_{t}^{2}u-\Delta_{\kappa}u+u=g,\\\ u|_{t=0}=u_{0},\\\
\partial_{t}u|_{t=0}=u_{1}.\end{cases}$ (4.13)

By Duhamel’s principle, the solution is formally given by

$u(t)=\frac{d\mathcal{K}_{t}}{dt}u_{0}+\mathcal{K}_{t}u_{1}-\int_{0}^{t}\mathcal{K}_{t-\tau}g(\tau,\cdot)\,d\tau,$

where

$\mathcal{K}_{t}=\frac{\sin(t\sqrt{I-\Delta_{\kappa}})}{\sqrt{I-\Delta_{\kappa}}},\quad\frac{d\mathcal{K}_{t}}{dt}=\cos(t\sqrt{I-\Delta_{\kappa}}).$

So we naturally introduce the operator $K_{t}=e^{it\sqrt{I-\Delta_{\kappa}}}$,
which corresponds to the case when $\phi(r)=\sqrt{1+r^{2}}$. By a simple
calculation have

$\phi^{\prime}(r)=r(1+r^{2})^{-\frac{1}{2}},\;\phi^{\prime\prime}(r)=(1+r^{2})^{-\frac{3}{2}}.$

This shows that $\phi$ satisfies (C1)–(C4) with $m_{1}=1,\alpha_{1}=-1,$ and
$m_{2}=\alpha_{2}=2$. Using Theorem 1.1, we obtain the following decay
estimate for $K_{t}$.

######  Proposition 4.14.

Assume $2\leq p\leq\infty$, $1\leq q\leq\infty$, and
$\delta=\frac{1}{2}-\frac{1}{p}$.

  1. (1)

Let $0\leq\theta\leq 1$ and $2s=-\left(N+1+\theta\right)\delta$, then we have

$\|K_{t}f\|_{B_{p,q}^{s,\kappa}(\mathbb{R}^{n})}\lesssim|t|^{-(N-1+\theta)\delta}\|f\|_{B_{p^{\prime},q}^{-s,\kappa}(\mathbb{R}^{n})}.$

  2. (2)

Let $0\leq\theta\leq N-1$ and $2s=-\left(N+1+\theta\right)\delta$, then we
have

$\|K_{t}f\|_{B_{p,q}^{s,\kappa}(\mathbb{R}^{n})}\lesssim|t|^{-(N-1-\theta)\delta}\|f\|_{B_{p^{\prime},q}^{-s,\kappa}(\mathbb{R}^{n})}.$

  3. (3)

In particular, for $0\leq\theta\leq 1$ and
$2s\leq-\left(N+1+\theta\right)\delta$, we have

$\|K_{t}f\|_{B_{p,q}^{s,\kappa}(\mathbb{R}^{n})}\lesssim
k(t)\|f\|_{B_{p^{\prime},q}^{-s,\kappa}(\mathbb{R}^{n})},\quad
k(t)=\begin{cases}|t|^{-2\max(s+N\delta,0)},\,&|t|\leq 1,\\\
|t|^{-(N-1+\theta)\delta},\,&|t|\geq 1.\end{cases}$

###### Proof.

First, we prove part (1). From (1.6) and (1.8) of Theorem 1.1 with
Plancherel’s identity and Riesz-Thorin interpolation theorem, for
$0\leq\theta\leq 1$, we obtain

$\displaystyle\|K_{t}\tilde{\Delta}_{j}f\|_{L_{\kappa}^{p}(\mathbb{R}^{n})}$
$\displaystyle\lesssim|t|^{-(N-1+\theta)\delta}2^{j\left(N+1+\theta\right)\delta}\|\tilde{\Delta}_{j}f\|_{L_{\kappa}^{p^{\prime}}(\mathbb{R}^{n})},\quad
j\geq 0,$ $\displaystyle\|K_{t}S_{0}f\|_{L_{\kappa}^{p}(\mathbb{R}^{n})}$
$\displaystyle\lesssim|t|^{-(N-1+\theta)\delta}\|S_{0}f\|_{L_{\kappa}^{p^{\prime}}(\mathbb{R}^{n})}.$

Hence, we complete the proof of part (1) from the fact that
$2s=-\left(N+1+\theta\right)\delta$.

Next, we prove part (2). Again from (1.5) and (1.7) of Theorem 1.1 with the
help of Plancherel’s identity and Riesz-Thorin interpolation theorem, for
$0\leq\theta\leq N-1$, we get

$\displaystyle\|K_{t}S_{0}f\|_{L_{\kappa}^{p}(\mathbb{R}^{n})}$
$\displaystyle\lesssim|t|^{-(N-1-\theta)\delta}\|S_{0}f\|_{L_{\kappa}^{p^{\prime}}(\mathbb{R}^{n})},$
$\displaystyle\|K_{t}\tilde{\Delta}_{j}f\|_{L_{\kappa}^{p}(\mathbb{R}^{n})}$
$\displaystyle\lesssim|t|^{-(N-1-\theta)\delta}2^{j(N+1+\theta)\delta}\|\tilde{\Delta}_{j}f\|_{L_{\kappa}^{p^{\prime}}(\mathbb{R}^{n})}.$

From the fact that $2s=-(N+1+\theta)\delta$, we have

$\|K_{t}f\|_{B_{p,q}^{s,\kappa}(\mathbb{R}^{n})}\lesssim|t|^{-(N-1-\theta)\delta}\|f\|_{B_{p^{\prime},q}^{-s,\kappa}(\mathbb{R}^{n})}$

and this completes the proof of part (2). To prove (3), we divide into the
following three cases.

Case 1. $|t|\geq 1$. Following the proof of part (1), we immediately get the
required estimates.

Case 2. $|t|\leq 1$ and $s+N\delta\geq 0$. In this case, our required estimate
will follow immediately from part (2). Indeed, if $2s\leq-(N+1+\theta)\delta$
for some $0\leq\theta\leq 1$, then $0\leq 2(s+N\delta)\leq(N-1-\theta)\delta$,
which also provides us $0\leq\theta\leq N-1$. Therefore from part (2), we
obtain

$\|K_{t}f\|_{B_{p,q}^{s,\kappa}(\mathbb{R}^{n})}\lesssim|t|^{-(N-1-\theta)\delta}\|f\|_{B_{p^{\prime},q}^{-s,\kappa}(\mathbb{R}^{n})}\leq|t|^{-2(s+N\delta)}\|f\|_{B_{p^{\prime},q}^{-s,\kappa}(\mathbb{R}^{n})}.$

Case 3. $|t|\leq 1$ and $s+N\delta<0$. In this case taking $\theta_{1}=N-1$ in
(2), for $s_{1}=-N\delta>s$, we get

$\|K_{t}f\|_{B_{p,q}^{s_{1},\kappa}(\mathbb{R}^{n})}\lesssim\|f\|_{B_{p^{\prime},q}^{-s_{1},\kappa}(\mathbb{R}^{n})}.$

Since $s_{1}>s$, using the inclusion relation (4) between Besov spaces in
Proposition 2.1, we immediately obtain

$\|K_{t}f\|_{B_{p,q}^{s,\kappa}(\mathbb{R}^{n})}\|K_{t}f\|_{B_{p,q}^{s_{1},\kappa}(\mathbb{R}^{n})}\lesssim\|f\|_{B_{p^{\prime},q}^{-s_{1},\kappa}(\mathbb{R}^{n})}\leq\|f\|_{B_{p^{\prime},q}^{-s,\kappa}(\mathbb{R}^{n})}$

and this completes the proof of part (3). ∎

###  4.5. Beam equation

Finally, we consider the beam equation

$\begin{cases}\partial_{t}^{2}u+\Delta_{\kappa}^{2}u+u=g,\\\
u|_{t=0}=u_{0},\\\ \partial_{t}u|_{t=0}=u_{1}.\end{cases}$ (4.14)

By Duhamel’s principle, the solution is formally given by

$u(t)=\frac{d\mathcal{B}_{t}}{dt}u_{0}+\mathcal{B}_{t}u_{1}-\int_{0}^{t}\mathcal{B}_{t-\tau}g(\tau,\cdot)\,d\tau,$

where

$\mathcal{B}_{t}=\frac{\sin(t\sqrt{I+\Delta_{\kappa}^{2}})}{\sqrt{I+\Delta_{\kappa}^{2}}},\quad\frac{d\mathcal{B}_{t}}{dt}=\cos(t\sqrt{I+\Delta_{\kappa}^{2}}).$

So we naturally introduce the operator $B_{t}=e^{it\sqrt{I+\Delta_{k}^{2}}}$,
which corresponds to the case when $\phi(r)=\sqrt{1+r^{4}}$. By a simple
calculation, we see that

$\phi^{\prime}(r)=2r^{3}(1+r^{4})^{-\frac{1}{2}},\quad\phi^{\prime\prime}(r)=(6r^{2}+2r^{6})(1+r^{4})^{-\frac{3}{2}}.$

This shows that $\phi$ satisfies (C1)–(C4) with
$m_{1}=\alpha_{1}=2,m_{2}=\alpha_{2}=4$. Using Theorem 1.1, we obtain the
following decay estimate for $B_{t}$.

######  Proposition 4.15.

Let $2\leq p\leq\infty$, $1\leq q\leq\infty$,
$\delta=\frac{1}{2}-\frac{1}{p},$ and $s\leq 0$. Then

$\|B_{t}f\|_{B_{p,q}^{s,\kappa}(\mathbb{R}^{n})}\lesssim
k(t)\|f\|_{B_{p^{\prime},q}^{-s,\kappa}(\mathbb{R}^{n})},\quad
k(t)=\begin{cases}|t|^{-\max(s+N\delta,0)},\,&|t|\leq 1,\\\
|t|^{-\frac{N\delta}{2}},\,&|t|\geq 1.\end{cases}$

###### Proof.

First, we prove the case $|t|\geq 1$. For $\theta=\frac{N}{4}$ in (1.8) of
Theorem 1.1 with the help of Plancherel’s identity and Riesz-Thorin
interpolation theorem, we have

$\|B_{t}S_{0}f\|_{L_{\kappa}^{p}(\mathbb{R}^{n})}\lesssim|t|^{-\frac{N\delta}{2}}\|S_{0}f\|_{L_{\kappa}^{p^{\prime}}(\mathbb{R}^{n})}.$
(4.15)

Again for $j\geq 0$, it follows from (1.5) and (1.6) of Theorem 1.1 that

$\|B_{t}\tilde{\Delta}_{j}f\|_{L_{\kappa}^{\infty}(\mathbb{R}^{n})}\lesssim|t|^{-\theta}2^{j\left(N-2\theta\right)}\|\tilde{\Delta}_{j}f\|_{L_{\kappa}^{1}(\mathbb{R}^{n})},\,0\leq\theta\leq\frac{N}{2}.$
(4.16)

Taking $\theta=\frac{N}{2}$ in (4.16), combined with Plancherel’s identity and
Riesz-Thorin interpolation theorem, we obtain

$\|B_{t}\tilde{\Delta}_{j}f\|_{L_{\kappa}^{p}(\mathbb{R}^{n})}\lesssim|t|^{-N\delta}\|\tilde{\Delta}_{j}f\|_{L_{\kappa}^{p^{\prime}}(\mathbb{R}^{n})}\leq|t|^{-\frac{N\delta}{2}}2^{-2js}\|\tilde{\Delta}_{j}f\|_{L_{\kappa}^{p^{\prime}}(\mathbb{R}^{n})},$
(4.17)

for any $j\geq 0$. From (4.15) and (4.17), we get our desired result.

Next, we consider the case $|t|<1$. By taking $\theta=0$ in (1.8) of Theorem
1.1 along with Plancherel’s identity and Riesz-Thorin interpolation theorem,
we have

$\|B_{t}S_{0}f\|_{L_{\kappa}^{p}(\mathbb{R}^{n})}\lesssim\|S_{0}f\|_{L_{\kappa}^{p^{\prime}}(\mathbb{R}^{n})}.$

For $j\geq 0$, it follows from (4.16), Plancherel’s identity, and Riesz-Thorin
interpolation theorem, for $0\leq\theta\leq\frac{N}{2}$, we have

$\|B_{t}\tilde{\Delta}_{j}f\|_{L_{\kappa}^{p}(\mathbb{R}^{n})}\lesssim|t|^{-2\theta\delta}2^{2j\left(N-2\theta\right)\delta}\|\tilde{\Delta}_{j}f\|_{L_{\kappa}^{p^{\prime}}(\mathbb{R}^{n})},$

and thus

$2^{js}\|B_{t}\tilde{\Delta}_{j}f\|_{L_{\kappa}^{p}(\mathbb{R}^{n})}\lesssim|t|^{-2\theta\delta}2^{2j\left(s+\left(N-2\theta\right)\delta\right)}2^{-js}\|\tilde{\Delta}_{j}f\|_{L_{\kappa}^{p^{\prime}}(\mathbb{R}^{n})}.$

If $-N\delta\leq s\leq 0$, we can choose $0\leq\theta\leq\frac{N}{2}$ such
that $s+N\delta=2\theta\delta$. If $s<-N\delta$, we can choose $\theta=0$.
Thus, we have

$2^{js}\|B_{t}\tilde{\Delta}_{j}f\|_{L_{\kappa}^{p}(\mathbb{R}^{n})}\lesssim|t|^{-\max(s+N\delta,0)}2^{-js}\|\tilde{\Delta}_{j}f\|_{L_{\kappa}^{p^{\prime}}(\mathbb{R}^{n})},$

which completes the proof of the proposition. ∎

##  5\. Well-posedness results for nonlinear Klein-Gordon and beam equations

Using the decay estimates for the nonlinear Klein-Gordon and beam equation
with the different decay rates between $|t|>1$ and $|t|\leq 1$ in the previous
section, we prove the global well-posedness results for nonlinear problems.
First, we consider the Cauchy problem for the nonlinear Klein-Gordon equation

$\begin{cases}\partial_{t}^{2}u-\Delta_{\kappa}u+u=F(u),\\\ u|_{t=0}=u_{0},\\\
\partial_{t}u|_{t=0}=u_{1},\end{cases}$ (5.1)

with a power-like nonlinearity $F(u)$ satisfies

$F(0)=0,\quad|F(u)-F(v)|\leq C(|u|^{\alpha}+|v|^{\alpha})|u-v|$ (5.2)

for some $\alpha>0$, where $C$ is a positive constant. By Duhamel’s principle,
the solution of (5.1) is formally given by

$u=\frac{d\mathcal{K}_{t}}{dt}u_{0}+\mathcal{K}_{t}u_{1}-\int_{0}^{t}\mathcal{K}_{t-\tau}F(u)\,d\tau,$

where
$\mathcal{K}_{t}=\frac{\sin(t\sqrt{I-\Delta_{\kappa}})}{\sqrt{I-\Delta_{\kappa}}}$.
Similarly, as in the Laplacian in Euclidean space, we find the critical index
as

$\alpha_{K}(N)=\frac{2-N+\sqrt{N^{2}+12N+4}}{2N}.$

Using the classical fixed point argument, we obtain the global well-posedness
theorem for the nonlinear Klein-Gordon equation.

######  Theorem 5.1.

Let $\alpha_{K}(N)<\alpha<\frac{4}{N}$,
$\sigma(\alpha)=\frac{\alpha(N+2)}{2(\alpha+2)}$, and $(u_{0},u_{1})\in
H^{\sigma(\alpha),\kappa}_{\frac{\alpha+2}{\alpha+1}}(\mathbb{R}^{n})\times
H^{\sigma(\alpha)-1,\kappa}_{\frac{\alpha+2}{\alpha+1}}(\mathbb{R}^{n})$ with
sufficient small norm. Then the equation (5.1) has a unique solution

$u\in C(\mathbb{R},H^{\frac{\alpha}{\alpha+2},\kappa}(\mathbb{R}^{n}))\cap
L^{\alpha+1}(\mathbb{R},L_{\kappa}^{\alpha+2}(\mathbb{R}^{n})).$

###### Proof.

By Proposition 2.1 and 4.14, we have

$\displaystyle\|\frac{d\mathcal{K}_{t}}{dt}u_{0}\|_{L^{\alpha+2}_{k}(\mathbb{R}^{n})}$
$\displaystyle\leq\|\frac{d\mathcal{K}_{t}}{dt}u_{0}\|_{B^{0,\kappa}_{\alpha+2,2}(\mathbb{R}^{n})}\lesssim
k(t)\|u_{0}\|_{B^{\sigma(\alpha),\kappa}_{\frac{\alpha+2}{\alpha+1},2}(\mathbb{R}^{n})}\lesssim
k(t)\|u_{0}\|_{H^{\sigma(\alpha),\kappa}_{\frac{\alpha+2}{\alpha+1}}(\mathbb{R}^{n})},$
$\displaystyle\|\mathcal{K}_{t}u_{1}\|_{L^{\alpha+2}_{k}(\mathbb{R}^{n})}$
$\displaystyle\leq\|\mathcal{K}_{t}u_{1}\|_{B^{0,\kappa}_{\alpha+2,2}(\mathbb{R}^{n})}\lesssim
k(t)\|u_{1}\|_{B^{\sigma(\alpha)-1,\kappa}_{\frac{\alpha+2}{\alpha+1},2}(\mathbb{R}^{n})}\lesssim
k(t)\|u_{1}\|_{H^{\sigma(\alpha)-1,\kappa}_{\frac{\alpha+2}{\alpha+1}}(\mathbb{R}^{n})},$
(5.3)

where

$k(t)=\begin{cases}|t|^{-\max(\frac{\alpha(N-2)}{2(\alpha+2)},0)},\,&|t|\leq
1,\\\ |t|^{-\frac{\alpha N}{2(\alpha+2)}},\,&|t|\geq 1.\end{cases}$

For $\alpha_{K}(N)<\alpha<\frac{4}{N}$, we notice that

$(\alpha+1)\frac{\alpha(N-2)}{2(\alpha+2)}<1,\;\;(\alpha+1)\frac{\alpha
N}{2(\alpha+2)}>1,\;\;\text{and }\;\sigma(\alpha)<1.$

This implies that $k\in L^{\alpha+1}(\mathbb{R})$ and

$\displaystyle\|\frac{d\mathcal{K}_{t}}{dt}u_{0}\|_{L^{\alpha+1}(\mathbb{R},L^{\alpha+2}_{k}(\mathbb{R}^{n}))}$
$\displaystyle\leq
C\|u_{0}\|_{H^{\sigma(\alpha),\kappa}_{\frac{\alpha+2}{\alpha+1}}(\mathbb{R}^{n})},$
$\displaystyle\|\mathcal{K}_{t}u_{1}\|_{L^{\alpha+1}(\mathbb{R},L^{\alpha+2}_{k}(\mathbb{R}^{n}))}$
$\displaystyle\leq
C\|u_{1}\|_{H^{\sigma(\alpha)-1,\kappa}_{\frac{\alpha+2}{\alpha+1}}(\mathbb{R}^{n})}.$

Therefore, from (5.3), Proposition 2.1, and Young’s inequality, it follows
that

$\displaystyle\left\|\int_{0}^{t}\mathcal{K}_{t-\tau}F(u)\,d\tau\right\|_{L^{\alpha+1}(\mathbb{R},L^{\alpha+2}_{k}(\mathbb{R}^{n}))}$
$\displaystyle\leq\left\|\int_{0}^{t}\|\mathcal{K}_{t-\tau}F(u)\|_{L^{\alpha+2}_{k}(\mathbb{R}^{n})}\,d\tau\right\|_{L^{\alpha+1}(\mathbb{R})}$
$\displaystyle\leq
C\left\|\int_{0}^{t}k(t-\tau)\|F(u)\|_{H^{\sigma(\alpha)-1,\kappa}_{\frac{\alpha+2}{\alpha+1}}(\mathbb{R}^{n})}\,d\tau\right\|_{L^{\alpha+1}(\mathbb{R})}$
$\displaystyle\leq
C\left\|\int_{0}^{t}k(t-\tau)\|F(u)\|_{L_{k}^{\frac{\alpha+2}{\alpha+1}}(\mathbb{R}^{n})}\,d\tau\right\|_{L^{\alpha+1}(\mathbb{R})}$
$\displaystyle\leq
C\|k\|_{L^{\alpha+1}(\mathbb{R})}\|F(u)\|_{L^{1}(\mathbb{R},L_{k}^{\frac{\alpha+2}{\alpha+1}}(\mathbb{R}^{n}))}$
$\displaystyle\leq
C\left\||u|^{\alpha+1}\right\|_{L^{1}(\mathbb{R},L_{k}^{\frac{\alpha+2}{\alpha+1}}(\mathbb{R}^{n}))}$
$\displaystyle=C\|u\|^{\alpha+1}_{L^{\alpha+1}(\mathbb{R},L_{k}^{\alpha+2}(\mathbb{R}^{n}))}.$

Taking
$M=2C(\|u_{0}\|_{H^{\sigma(\alpha),\kappa}_{\frac{\alpha+2}{\alpha+1}}(\mathbb{R}^{n})}+\|u_{1}\|_{H^{\sigma(\alpha)-1,\kappa}_{\frac{\alpha+2}{\alpha+1}}(\mathbb{R}^{n})})$,
we set the evolution space $X$ as

$X=\\{u\in
L^{\alpha+1}(\mathbb{R},L_{k}^{\alpha+2}(\mathbb{R}^{n})):\|u\|_{L^{\alpha+1}(\mathbb{R},L_{k}^{\alpha+2}(\mathbb{R}^{n}))}\leq
M\\}.$

We say that a function $u$ is a mild solution to (5.1) if $u$ is a fixed point
for the integral operator

$\mathcal{T}:u\in
X\mapsto\mathcal{T}u(t,x):=\frac{d\mathcal{K}_{t}}{dt}u_{0}+A_{t}u_{1}-\int_{0}^{t}\mathcal{K}_{t-\tau}F(u)\,d\tau.$

We prove the global-in-time existence and uniqueness of small data Sobolev
solutions of low regularity to the Cauchy problem (5.1) by finding a unique
fixed point to the operator $\mathcal{T}$. It means that there exists a unique
global solution $u$ to the equation $\mathcal{T}u=u\in X$, which also gives
the solution to (5.1). In order to prove that the operator $\mathcal{T}$ has a
uniquely determined fixed point, we use Banach’s fixed point argument with
respect to the norm on $X$ as defined above.

Now for any $u,v\in X$, we have

$\|\mathcal{T}u\|_{L^{\alpha+1}(\mathbb{R},L_{k}^{\alpha+2}(\mathbb{R}^{n}))}\leq\frac{M}{2}+CM^{\alpha+1},$

and

$\displaystyle\quad\|\mathcal{T}u-\mathcal{T}v\|_{L^{\alpha+1}(\mathbb{R},L_{k}^{\alpha+2}(\mathbb{R}^{n}))}$
$\displaystyle=\left\|\int_{0}^{t}\mathcal{K}_{t-\tau}\left(F(u)-F(v)\right)\,d\tau\right\|_{L^{\alpha+1}(\mathbb{R},L_{k}^{\alpha+2}(\mathbb{R}^{n}))}$
$\displaystyle\leq
C\left\|F(u)-F(v)\right\|_{L^{1}(\mathbb{R},L_{k}^{\frac{\alpha+2}{\alpha+1}}(\mathbb{R}^{n}))}$
$\displaystyle\leq
C\left\|(|u|^{\alpha}+|v|^{\alpha})|u-v|\right\|_{L^{1}(\mathbb{R},L_{k}^{\frac{\alpha+2}{\alpha+1}}(\mathbb{R}^{n}))}$
$\displaystyle=C\left\|\left\|(|u|^{\alpha}+|v|^{\alpha})|u-v|\right\|_{L_{k}^{\frac{\alpha+2}{\alpha+1}}(\mathbb{R}^{n})}\right\|_{L^{1}(\mathbb{R})}$
$\displaystyle\leq
C\left\|\left\||u|^{\alpha}+|v|^{\alpha}\right\|_{L_{k}^{\frac{\alpha+2}{\alpha}}(\mathbb{R}^{n})}\left\|u-v\right\|_{L_{k}^{\alpha+2}(\mathbb{R}^{n})}\right\|_{L^{1}(\mathbb{R})}$
$\displaystyle\leq
C\left\||u|^{\alpha}+|v|^{\alpha}\right\|_{L^{\frac{\alpha+1}{\alpha}}(\mathbb{R},L_{k}^{\frac{\alpha+2}{\alpha}}(\mathbb{R}^{n}))}\left\|u-v\right\|_{L^{\alpha+1}(\mathbb{R},L_{k}^{\alpha+2}(\mathbb{R}^{n}))}$
$\displaystyle\leq
C\left(\left\||u|^{\alpha}\right\|_{L^{\frac{\alpha+1}{\alpha}}(\mathbb{R},L_{k}^{\frac{\alpha+2}{\alpha}}(\mathbb{R}^{n}))}+\left\||v|^{\alpha}\right\|_{L^{\frac{\alpha+1}{\alpha}}(\mathbb{R},L_{k}^{\frac{\alpha+2}{\alpha}}(\mathbb{R}^{n}))}\right)\left\|u-v\right\|_{L^{\alpha+1}(\mathbb{R},L_{k}^{\alpha+2}(\mathbb{R}^{n}))}$
$\displaystyle=C\left(\|u\|_{L^{\alpha+1}(\mathbb{R},L_{k}^{\alpha+2}(\mathbb{R}^{n}))}^{\alpha}+\|v\|_{L^{\alpha+1}(\mathbb{R},L_{k}^{\alpha+2}(\mathbb{R}^{n}))}^{\alpha}\right)\left\|u-v\right\|_{L^{\alpha+1}(\mathbb{R},L_{k}^{\alpha+2}(\mathbb{R}^{n}))}$
$\displaystyle\leq
2CM^{\alpha}\left\|u-v\right\|_{L^{\alpha+1}(\mathbb{R},L_{k}^{\alpha+2}(\mathbb{R}^{n}))}.$

Notice that, for $CM^{\alpha}<\frac{1}{2}$, $\mathcal{T}:X\to X$ is a
constraction mapping. Applying Banach’s fixed point theorem, there exists a
global small data Sobolev solution $u$ of the equation $u=\mathcal{T}u$ in
$X$, which also gives the solution to the equation (5.1). In addition, by the
embedding
$H^{\sigma(\alpha),\kappa}_{\frac{\alpha+2}{\alpha+1}}(\mathbb{R}^{n})\subseteq
H^{\frac{\alpha}{\alpha+2},\kappa}(\mathbb{R}^{n})$ and
$H^{\sigma(\alpha)-1,\kappa}_{\frac{\alpha+2}{\alpha+1}}(\mathbb{R}^{n})\subseteq
H^{-\frac{2}{\alpha+2},\kappa}(\mathbb{R}^{n})$, we have

$\displaystyle\|u\|_{L^{\infty}(\mathbb{R},H^{\frac{\alpha}{\alpha+2},\kappa}(\mathbb{R}^{n}))}$
$\displaystyle\leq\|\frac{d\mathcal{K}_{t}}{dt}u_{0}\|_{L^{\infty}(\mathbb{R},H^{\frac{\alpha}{\alpha+2},\kappa}(\mathbb{R}^{n}))}+\|\mathcal{K}_{t}u_{1}\|_{L^{\infty}(\mathbb{R},H^{\frac{\alpha}{\alpha+2},\kappa}(\mathbb{R}^{n}))}$
$\displaystyle\qquad\qquad+\|\int_{0}^{t}\mathcal{K}_{t-\tau}F(u)\,d\tau\|_{L^{\infty}(\mathbb{R},H^{\frac{\alpha}{\alpha+2},\kappa}(\mathbb{R}^{n}))}$
$\displaystyle\leq\|u_{0}\|_{H^{\frac{\alpha}{\alpha+2},\kappa}(\mathbb{R}^{n})}+\|u_{1}\|_{H^{-\frac{2}{\alpha+2},\kappa}(\mathbb{R}^{n})}+\|F(u)\|_{L^{1}(\mathbb{R},H^{-\frac{2}{\alpha+2},\kappa}(\mathbb{R}^{n}))}$
$\displaystyle\lesssim\|u_{0}\|_{H^{\sigma(\alpha),\kappa}_{\frac{\alpha+2}{\alpha+1}}(\mathbb{R}^{n})}+\|u_{1}\|_{H^{\sigma(\alpha)-1,\kappa}_{\frac{\alpha+2}{\alpha+1}}(\mathbb{R}^{n})}+\|F(u)\|_{L^{1}(\mathbb{R},H^{\sigma(\alpha)-1,\kappa}_{\frac{\alpha+2}{\alpha+1}})}$
$\displaystyle\lesssim\|u_{0}\|_{H^{\sigma(\alpha),\kappa}_{\frac{\alpha+2}{\alpha+1}}(\mathbb{R}^{n})}+\|u_{1}\|_{H^{\sigma(\alpha)-1,\kappa}_{\frac{\alpha+2}{\alpha+1}}(\mathbb{R}^{n})}+\||u|^{\alpha+1}\|_{L^{1}(\mathbb{R},L_{k}^{\frac{\alpha+2}{\alpha+1}}(\mathbb{R}^{n}))}$
$\displaystyle=\|u_{0}\|_{H^{\sigma(\alpha),\kappa}_{\frac{\alpha+2}{\alpha+1}}(\mathbb{R}^{n})}+\|u_{1}\|_{H^{\sigma(\alpha)-1,\kappa}_{\frac{\alpha+2}{\alpha+1}}(\mathbb{R}^{n})}+\|u\|^{\alpha}_{L^{\alpha+1}(\mathbb{R},L_{k}^{\alpha+2}(\mathbb{R}^{n}))},$

which indicates that the unique solution also belongs to
$L^{\infty}(\mathbb{R},H^{\frac{\alpha}{\alpha+2},\kappa}(\mathbb{R}^{n}))$. ∎

Argued similarly as in the nonlinear Klein-Gordon equation (5.1), we also
study the Cauchy problem for the nonlinear beam equation

$\begin{cases}\partial_{t}^{2}u+\Delta_{\kappa}^{2}u+u=F(u),\\\
u|_{t=0}=u_{0},\\\ \partial_{t}u|_{t=0}=u_{1}.\end{cases}$ (5.4)

By Duhamel’s principle, the solution is formally given by

$u(t)=\frac{d\mathcal{B}_{t}}{dt}u_{0}+\mathcal{B}_{t}u_{1}-\int_{0}^{t}\mathcal{B}_{t-\tau}F(u)\,d\tau.$
(5.5)

We also find the critical index

$\alpha_{B}(N)=\frac{4-N+\sqrt{N^{2}+24N+16}}{2N}.$

The following result is about the global well-posedness for the nonlinear beam
equation (5.5).

######  Theorem 5.2.

Let $\alpha_{B}(N)<\alpha<\frac{8}{N}$, $\sigma(\alpha)=\frac{\alpha
N}{\alpha+2}-\frac{2}{\alpha+1}$, $\sigma(\alpha)<s\leq 2$,
$s_{0}=s-\frac{\alpha N}{2(\alpha+2)}$ and $(u_{0},u_{1})\in
H^{s,\kappa}_{\frac{\alpha+2}{\alpha+1}}(\mathbb{R}^{n})\times
H^{s-2,\kappa}_{\frac{\alpha+2}{\alpha+1}}(\mathbb{R}^{n})$ with sufficient
small norm. Then the equation (5.5) has a unique solution

$u\in C(\mathbb{R},H^{s_{0},\kappa}(\mathbb{R}^{n}))\cap
L^{\alpha+1}(\mathbb{R},L_{\kappa}^{\alpha+2}(\mathbb{R}^{n})).$

###### Proof.

From Proposition 2.1 and 4.15, we have

$\displaystyle\|\frac{d\mathcal{B}_{t}}{dt}u_{0}\|_{L^{\alpha+2}_{k}(\mathbb{R}^{n})}$
$\displaystyle\leq\|\frac{d\mathcal{B}_{t}}{dt}u_{0}\|_{B^{0,\kappa}_{\alpha+2,2}(\mathbb{R}^{n})}\lesssim
k(t)\|u_{0}\|_{B^{s,\kappa}_{\frac{\alpha+2}{\alpha+1},2}(\mathbb{R}^{n})}\lesssim
k(t)\|u_{0}\|_{H^{s,\kappa}_{\frac{\alpha+2}{\alpha+1}}(\mathbb{R}^{n})},$
$\displaystyle\|\mathcal{B}_{t}u_{1}\|_{L^{\alpha+2}_{k}(\mathbb{R}^{n})}$
$\displaystyle\leq\|\mathcal{B}_{t}u_{1}\|_{B^{0,\kappa}_{\alpha+2,2}(\mathbb{R}^{n})}\lesssim
k(t)\|u_{1}\|_{B^{s-2,\kappa}_{\frac{\alpha+2}{\alpha+1},2}(\mathbb{R}^{n})}\lesssim
k(t)\|u_{1}\|_{H^{s-2,\kappa}_{\frac{\alpha+2}{\alpha+1}}(\mathbb{R}^{n})},$
(5.6)

where

$k(t)=\begin{cases}|t|^{-\max\left(\frac{\alpha
N}{2(\alpha+2)}-\frac{s}{2},0\right)},\,&|t|\leq 1,\\\ |t|^{-\frac{\alpha
N}{4(\alpha+2)}},\,&|t|\geq 1.\end{cases}$

For $\alpha_{B}(N)<\alpha<\frac{8}{N}$ and $\sigma(\alpha)<s\leq 2$, we see
that

$(\alpha+1)\left(\frac{\alpha
N}{2(\alpha+2)}-\frac{s}{2}\right)<1\quad\text{and}\quad(\alpha+1)\frac{\alpha
N}{4(\alpha+2)}>1,$

which implies that $k\in L^{\alpha+1}(\mathbb{R})$ and

$\displaystyle\|\frac{d\mathcal{B}_{t}}{dt}u_{0}\|_{L^{\alpha+1}(\mathbb{R},L^{\alpha+2}_{k}(\mathbb{R}^{n}))}$
$\displaystyle\leq
C\|u_{0}\|_{H^{s,\kappa}_{\frac{\alpha+2}{\alpha+1}}(\mathbb{R}^{n})},$
$\displaystyle\|\mathcal{B}_{t}u_{1}\|_{L^{\alpha+1}(\mathbb{R},L^{\alpha+2}_{k}(\mathbb{R}^{n}))}$
$\displaystyle\leq
C\|u_{1}\|_{H^{s-2,\kappa}_{\frac{\alpha+2}{\alpha+1}}(\mathbb{R}^{n})}.$

Therefore, from (5.6), Proposition 2.1, and Young’s inequality, it follows
that

$\displaystyle\left\|\int_{0}^{t}\mathcal{B}_{t-\tau}F(u)\,d\tau\right\|_{L^{\alpha+1}(\mathbb{R},L^{\alpha+2}_{k}(\mathbb{R}^{n}))}$
$\displaystyle\leq\left\|\int_{0}^{t}\|\mathcal{B}_{t-\tau}F(u)\|_{L^{\alpha+2}_{k}(\mathbb{R}^{n})}\,d\tau\right\|_{L^{\alpha+1}(\mathbb{R})}$
$\displaystyle\leq
C\left\|\int_{0}^{t}k(t-\tau)\|F(u)\|_{H^{s-2,\kappa}_{\frac{\alpha+2}{\alpha+1}}(\mathbb{R}^{n})}\,d\tau\right\|_{L^{\alpha+1}(\mathbb{R})}$
$\displaystyle\leq
C\left\|\int_{0}^{t}k(t-\tau)\|F(u)\|_{L_{k}^{\frac{\alpha+2}{\alpha+1}}(\mathbb{R}^{n})}\,d\tau\right\|_{L^{\alpha+1}(\mathbb{R})}$
$\displaystyle\leq
C\|k\|_{L^{\alpha+1}(\mathbb{R})}\|F(u)\|_{L^{1}(\mathbb{R},L_{k}^{\frac{\alpha+2}{\alpha+1}}(\mathbb{R}^{n}))}$
$\displaystyle\leq
C\left\||u|^{\alpha+1}\right\|_{L^{1}(\mathbb{R},L_{k}^{\frac{\alpha+2}{\alpha+1}}(\mathbb{R}^{n}))}$
$\displaystyle=C\|u\|^{\alpha+1}_{L^{\alpha+1}(\mathbb{R},L_{k}^{\alpha+2}(\mathbb{R}^{n}))}.$

Taking
$M=2C(\|u_{0}\|_{H^{s,\kappa}_{\frac{\alpha+2}{\alpha+1}}(\mathbb{R}^{n})}+\|u_{1}\|_{H^{s-2,\kappa}_{\frac{\alpha+2}{\alpha+1}}(\mathbb{R}^{n})})$,
we set the evolution spaces

$X=\\{u\in
L^{\alpha+1}(\mathbb{R},L_{k}^{\alpha+2}(\mathbb{R}^{n})):\|u\|_{L^{\alpha+1}(\mathbb{R},L_{k}^{\alpha+2}(\mathbb{R}^{n}))}\leq
M\\}.$

Define the mapping

$\mathcal{T}:u\mapsto\frac{d\mathcal{B}_{t}}{dt}u_{0}+\mathcal{B}_{t}u_{1}-\int_{0}^{t}\mathcal{B}_{t-\tau}F(u)\,d\tau,\;\forall
u\in X.$

Then for any $u,v\in X$, we have

$\|\mathcal{T}u\|_{L^{\alpha+1}(\mathbb{R},L_{k}^{\alpha+2}(\mathbb{R}^{n}))}\leq\frac{M}{2}+CM^{\alpha+1}.$

Thus

$\displaystyle\quad\|\mathcal{T}u-\mathcal{T}v\|_{L^{\alpha+1}(\mathbb{R},L_{k}^{\alpha+2}(\mathbb{R}^{n}))}$
$\displaystyle=\left\|\int_{0}^{t}\mathcal{B}_{t-\tau}\left(F(u)-F(v)\right)\,d\tau\right\|_{L^{\alpha+1}(\mathbb{R},L_{k}^{\alpha+2}(\mathbb{R}^{n}))}$
$\displaystyle\leq
C\left\|F(u)-F(v)\right\|_{L^{1}(\mathbb{R},L_{k}^{\frac{\alpha+2}{\alpha+1}}(\mathbb{R}^{n}))}$
$\displaystyle\leq
C\left\|(|u|^{\alpha}+|v|^{\alpha})|u-v|\right\|_{L^{1}(\mathbb{R},L_{k}^{\frac{\alpha+2}{\alpha+1}}(\mathbb{R}^{n}))}$
$\displaystyle=C\left\|\left\|(|u|^{\alpha}+|v|^{\alpha})|u-v|\right\|_{L_{k}^{\frac{\alpha+2}{\alpha+1}}(\mathbb{R}^{n})}\right\|_{L^{1}(\mathbb{R})}$
$\displaystyle\leq
C\left\|\left\||u|^{\alpha}+|v|^{\alpha}\right\|_{L_{k}^{\frac{\alpha+2}{\alpha}}(\mathbb{R}^{n})}\left\|u-v\right\|_{L_{k}^{\alpha+2}(\mathbb{R}^{n})}\right\|_{L^{1}(\mathbb{R})}$
$\displaystyle\leq
C\left\||u|^{\alpha}+|v|^{\alpha}\right\|_{L^{\frac{\alpha+1}{\alpha}}(\mathbb{R},L_{k}^{\frac{\alpha+2}{\alpha}}(\mathbb{R}^{n}))}\left\|u-v\right\|_{L^{\alpha+1}(\mathbb{R},L_{k}^{\alpha+2}(\mathbb{R}^{n}))}$
$\displaystyle\leq
C\left(\left\||u|^{\alpha}\right\|_{L^{\frac{\alpha+1}{\alpha}}(\mathbb{R},L_{k}^{\frac{\alpha+2}{\alpha}}(\mathbb{R}^{n}))}+\left\||v|^{\alpha}\right\|_{L^{\frac{\alpha+1}{\alpha}}(\mathbb{R},L_{k}^{\frac{\alpha+2}{\alpha}}(\mathbb{R}^{n}))}\right)\left\|u-v\right\|_{L^{\alpha+1}(\mathbb{R},L_{k}^{\alpha+2}(\mathbb{R}^{n}))}$
$\displaystyle=C\left(\|u\|_{L^{\alpha+1}(\mathbb{R},L_{k}^{\alpha+2}(\mathbb{R}^{n}))}^{\alpha}+\|v\|_{L^{\alpha+1}(\mathbb{R},L_{k}^{\alpha+2}(\mathbb{R}^{n}))}^{\alpha}\right)\left\|u-v\right\|_{L^{\alpha+1}(\mathbb{R},L_{k}^{\alpha+2}(\mathbb{R}^{n}))}$
$\displaystyle\leq
2CM^{\alpha}\left\|u-v\right\|_{L^{\alpha+1}(\mathbb{R},L_{k}^{\alpha+2}(\mathbb{R}^{n}))}.$

Notice that, for $CM^{\alpha}<\frac{1}{2}$, $\mathcal{T}:X\to X$ is a
constraction mapping. Therefore, applying Banach’s fixed point theorem, the
equation (5.4) has a unique solution $u\in X$. In addition, by the embedding
$H^{s,\kappa}_{\frac{\alpha+2}{\alpha+1}}(\mathbb{R}^{n})\subseteq
H^{s_{0},\kappa}(\mathbb{R}^{n})$ and
$H^{s-2,\kappa}_{\frac{\alpha+2}{\alpha+1}}(\mathbb{R}^{n})\subseteq
H^{s_{0}-2,\kappa}(\mathbb{R}^{n})$, we have

$\displaystyle\quad\|u\|_{L^{\infty}(\mathbb{R},H^{s_{0},\kappa}(\mathbb{R}^{n}))}$
$\displaystyle\leq\|\frac{d\mathcal{B}_{t}}{dt}u_{0}\|_{L^{\infty}(\mathbb{R},H^{s_{0},\kappa}(\mathbb{R}^{n}))}+\|\mathcal{B}_{t}u_{1}\|_{L^{\infty}(\mathbb{R},H^{s_{0},\kappa}(\mathbb{R}^{n}))}+\|\int_{0}^{t}\mathcal{B}_{t-\tau}F(u)\,d\tau\|_{L^{\infty}(\mathbb{R},H^{s_{0},\kappa}(\mathbb{R}^{n}))}$
$\displaystyle\leq\|u_{0}\|_{H^{s_{0},\kappa}(\mathbb{R}^{n})}+\|u_{1}\|_{H^{s_{0}-2,\kappa}(\mathbb{R}^{n})}+\|F(u)\|_{L^{1}(\mathbb{R},H^{s_{0}-2,\kappa})}$
$\displaystyle\lesssim\|u_{0}\|_{H^{s,\kappa}_{\frac{\alpha+2}{\alpha+1}}(\mathbb{R}^{n})}+\|u_{1}\|_{H^{s-2,\kappa}_{\frac{\alpha+2}{\alpha+1}}(\mathbb{R}^{n})}+\|F(u)\|_{L^{1}(\mathbb{R},H^{s-2,\kappa}_{\frac{\alpha+2}{\alpha+1}}(\mathbb{R}^{n}))}$
$\displaystyle\lesssim\|u_{0}\|_{H^{s,\kappa}_{\frac{\alpha+2}{\alpha+1}}(\mathbb{R}^{n})}+\|u_{1}\|_{H^{s-2,\kappa}_{\frac{\alpha+2}{\alpha+1}}(\mathbb{R}^{n})}+\||u|^{\alpha+1}\|_{L^{1}(\mathbb{R},L_{k}^{\frac{\alpha+2}{\alpha+1}}(\mathbb{R}^{n}))}$
$\displaystyle=\|u_{0}\|_{H^{s,\kappa}_{\frac{\alpha+2}{\alpha+1}}(\mathbb{R}^{n})}+\|u_{1}\|_{H^{s-2,\kappa}_{\frac{\alpha+2}{\alpha+1}}(\mathbb{R}^{n})}+\|u\|^{\alpha}_{L^{\alpha+1}(\mathbb{R},L_{k}^{\alpha+2}(\mathbb{R}^{n}))},$

which indicates that the unique solution also belongs to
$L^{\infty}(\mathbb{R},H^{s_{0},\kappa}(\mathbb{R}^{n}))$. ∎

## Statements and Declarations

The authors confirm that the data supporting the findings of this study are
available within the article and its supplementary materials.

Competing Interests: No potential competing of interest was reported by the
author.

## Acknowledgments

C. Luo and M. Song are supported by the National Natural Science Foundation of
China (Grant No. 11701452) and Guangdong Basic and Applied Basic Research
Foundation (No. 2023A1515010656). S.S. Mondal is supported by the DST-INSPIRE
Faculty Fellowship (Grant No. DST/INSPIRE/04/2023/002038).

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