# The wave equation on lattices and
Oscillatory Integrals

Cheng Bi  Cheng Bi: School of Mathematical Sciences, Fudan University,
Shanghai 200433, China<EMAIL_ADDRESS>,  Jiawei Cheng  Jiawei Cheng:
School of Mathematical Sciences, Fudan University, Shanghai 200433, China
<EMAIL_ADDRESS>and  Bobo Hua  Bobo Hua: School of Mathematical
Sciences, LMNS, Fudan University, Shanghai 200433, China; Shanghai Center for
Mathematical Sciences, Fudan University, Shanghai 200433, China
<EMAIL_ADDRESS>
###### Abstract.

In this paper, we establish sharp dispersive estimates for the linear wave
equation on the lattice ${\mathbb{Z}}^{d}$ with dimension $d=4$. Combining the
singularity theory with results in uniform estimates of oscillatory integrals,
we prove that the optimal time decay rate of the fundamental solution is of
order $|t|^{-\frac{3}{2}}\log|t|$, which is the first extension of P.
Schultz’s results [38] in $d=2,3$ to the higher dimension. We also observe
that the decay rate for $d=2,3,4$ can be well interpreted by the Newton
polyhedra. Moreover, we prove $l^{p}\rightarrow l^{q}$ estimates as well as
Strichartz estimates and give applications to nonlinear wave equations.

##  1\. Introduction

Consider the following nonlinear wave equation on the $d$-dimensional lattice
$\mathbb{Z}^{d}$

(1) $\left\\{\begin{aligned} &\partial_{t}^{2}u(x,t)-\Delta
u(x,t)=F(u(x,t)),\\\
&u(x,0)=g(x)~{},~{}\partial_{t}u(x,0)=f(x),\end{aligned}\right.$

where $x=(x_{1},x_{2},\cdots,x_{d})\in\mathbb{Z}^{d}$ and $t\in{\mathbb{R}}$.
The discrete Laplacian $\Delta$ is defined by

$\Delta
u(x,t):=\sum_{y\in{\mathbb{Z}}^{d},\,d(x,y)=1}\left(u(y,t)-u(x,t)\right),$

with $d(x,y)=\sum_{k=1}^{d}|x_{k}-y_{k}|$. There are a number of physical
applications where this equation appears in a natural way, mainly to describe
the behaviour of wave propagation. One of the best known of them is the
description of the vibrations of atoms inside crystals. A fundamental model is
the monotonic chains, in which each atom vibrates as a simple harmonic
oscillator and only feels the force of its nearest neighbours, see [11, 13,
45].

We first consider dispersive inequalities for linear models, i.e. $F\equiv 0$
in (1), which play important roles in the study of nonlinear problems. For
instance, such results yield space-time estimates which are utilized for the
existence of a global solution to (1). Generally, obtaining dispersive
estimates amounts to establishing a decay estimate for the $l^{\infty}$ norm
of the solution in terms of time and the $l^{1}$ norm of the initial data. The
main step of the proof usually starts from the representation of the solution,
relies on the method of stationary phase and ends up with pointwise estimates
to the fundamental solution, also called the Green’s function.

We recall some results in the continuous setting. In Euclidean space
${\mathbb{R}}^{d}$, it is well-known that the decay rate of the solution to
wave equation is of order $|t|^{-\frac{d-1}{2}}$, while for Schrödinger
equation, it decays like $|t|^{-\frac{d}{2}}$, see [16, 29, 41]. These results
have been extended to more general framework such as Heisenberg groups and
H-type groups, see for instance H. Bahouri et al. [2, 3] and M. Del Hierro
[10]. For discrete analogs of wave equations on general graphs, see [14, 30,
31].

From now on we focus on lattices. For simplicity, let $F,g\equiv 0$, then we
get the corresponding Green’s function of (1) by the discrete Fourier
transform (see Section 2):

(2)
$G(x,t)=\frac{1}{(2\pi)^{d}}\int_{\mathbb{T}^{d}}e^{ix\cdot\xi}\,\frac{\sin(t\,\omega(\xi))}{\omega(\xi)}\,d\xi,$

where $\mathbb{T}^{d}:=[-\pi,\pi]^{d}$ is the torus and
$\omega(\xi):=\sqrt{\sum_{j=1}^{d}(2-2\cos\xi_{j})}$. To establish the optimal
dispersive estimate, we look for the value of $\beta<0$ such that

(3) $|G(x,t)|\leq C(1+|t|)^{\beta},$

where $C>0$ only depends on $d$. The symbols $C,c$ will be used throughout to
denote positive constants, which may vary from one line to the next. Analogous
results as (3) have been studied for other dispersive equations with Cauchy
data such as the discrete Schrödinger equation (DS, in short)

$i\partial_{t}u(x,t)+\Delta u(x,t)=0$

and the discrete Klein-Gordon equation (DKG)

(4) $\partial_{t}^{2}u(x,t)-\Delta u(x,t)+m^{2}u(x,t)=0,$

where $m>0$ is the mass parameter . These models are closely related to the
discrete wave equation (DW). For the DS, due to the special form of its
Green’s function, one can separate variables in order to reduce the problem to
the case $d=1$, and obtain the sharp decay rate of $|t|^{-\frac{d}{3}}$ on
$\mathbb{Z}^{d}$, see [39, Theorem 3].

Unfortunately, the DKG and the DW in dimensions $d\geq 2$ fail to have the
separation-of-variables property, which leads to more complicated analysis as
dimension increases. For the DKG, A. Stefanov and P. G. Keverekidis [39,
Theorem 5] established a sharp decay estimate of $|t|^{-\frac{1}{3}}$ on
$\mathbb{Z}$. Later, V. Borovyk and M. Goldberg [5, Corollary 2.5] computed a
decay of $|t|^{-\frac{3}{4}}$ on ${\mathbb{Z}}^{2}$. Recently, using a
detailed analysis of singularities, J.-C. Cuenin and I. A. Ikromov [9, Theorem
1] extended the results to dimensions 2, 3 and 4.

However, we shall see that the situation in the DW is harder than that of the
DKG. Progress in the DW mainly comes from P. Schultz [38], who settled the
case $d=2,3$ completely by analyzing (2) as well as a related oscillatory
integral,

(5)
$I(v,t):=\frac{1}{(2\pi)^{d}}\int_{\mathbb{T}^{d}}e^{it\phi(v,\xi)}\frac{1}{\omega(\xi)}\,d\xi,\;(v,t)\in{\mathbb{R}}^{d}\times{\mathbb{R}},$

with phase function $\phi(v,\xi):=v\cdot\xi-\omega(\xi)$ and amplitude
function ${1}/{\omega}$, here $v\cdot\xi=\sum_{j=1}^{d}v_{j}\xi_{j}$ is the
usual inner product. Notice that the phase and amplitude are not smooth at the
origin, which brings difficulties in the analysis. In light of (3) and the
relation $G(x,t)=-\mbox{Im}\,I(x/t,t)$, $\forall\,t\neq 0$ (see (15) below),
in many occasions it is more convenient to work on $I(v,t)$ and find optimal
value $\beta$ such that

(6) $|I(v,t)|\leqslant C(1+|t|)^{\beta}$

holds uniformly in $v\in{\mathbb{R}}^{d}$.

We first look for critical points of $\phi(v,\cdot)$, which suggests a
partition of $v$ into three regions (see Section 2.3):

(7) $\displaystyle\mbox{the exterior}\;(|v|>1),\;\;\mbox{the
vicinity}\;(|v|\approx 1)\;\;\mbox{and the interior}\;(|v|<1)$

of the light cone. The first two regions are handled by [38] in all dimensions
$d\geq 2$. However, the situation becomes complicated inside the light cone.
The reason is that there are “bad” $v$ such that some critical points of
$\phi(v,\cdot)$ are degenerate and various singularities appear. By degeneracy
we mean the rank of Hess${}_{\xi}\phi(v,\xi)$, i.e. the Hessian matrix of
$\phi(v,\cdot)$ at $\xi$, is less than $d$. In this case, the stationary phase
method breaks down and we have to consider each dimension one-by-one.

In this article, we mainly consider the degenerate cases in the third region
of (7), and establish uniform estimates of $I(v,t)$ for those “bad” $v$. More
precisely, for any $v_{0}\in{\mathbb{R}}^{d}$, since

(8) $\phi(v,\xi)=(v-v_{0})\cdot\xi+\phi(v_{0},\xi),$

we first get the decay rate of (5) with $v=v_{0}$, then we prove that the
estimate holds uniformly for $I(v,t)$ with $v$ in a small neighborhood of
$v_{0}$. Finally, we get the global uniform estimate by a finite covering on
the velocity space. Therefore, this problem can be viewed as the stability of
oscillatory integral under phase (linear) permutations.

We notice that each decay rate in $d=2,3,4$ can be well interpreted by the
Newton polyhedron of $\phi$, see Section 4.2. Moreover, for $d=3,4$ the decay
is governed by a single velocity
$v=(\frac{\pi}{2},\cdots,\frac{\pi}{2})\in{\mathbb{R}}^{d}$, corresponding to
the most degenerate case of Hess${}_{\xi}\phi(v,\cdot)$, see Remark 2.13. This
fact may be useful in predicting decay rates in higher dimensions, and we
obtain the decay rates of (5) in this most degenerate case for all odd
dimensions $d\geq 3$ using the method of Newton polyhedron, see Conjecture 1.5
and Theorem 4.5 below.

The uniform estimate of oscillatory integral is noteworthy in itself. Roughly
speaking, we consider the following integral

(9) $J(t,S+P,\,\psi):=\int_{{\mathbb{R}}^{d}}e^{it(S(x)+P(x))}\psi(x)\,dx,$

with proper phase $S$, amplitude $\psi$ and perturbation $P$. We consider the
natural question about the semicontinuity of the decay rates, that is, whether
the decay of $J(t,S,\,\psi)$ in $t$ could extend to $J(t,S+P,\,\psi)$ when $P$
is “small” enough. Unidimensional uniform estimate goes back to I. M.
Vinogradov [43] and J. G. van der Corput [8]. In the sequel we consider the
two-dimensional case. The concept of adapted coordinate system is important.
A. N. Varchenko [42] proved the existence of such coordinate system for
analytic functions (without multiple components) in his pioneering work
studying the connection between oscillatory integral and Newton polyhedra.
Based on this, V. N. Karpushkin [27] gave an affirmative answer to the
question above for the real-analytic phase. Later, Ikromov and D. Müller [23]
extended the results to smooth phase of finite type when $P$ is linear.
Besides, it is also effective to analyze the singularities of the phase
function, see for instance Arnold et al. [1] and J. J. Duistermaat [12].

In higher dimensions, however, examples in [42] violate the full analogue of
the conjecture about semicontinuity and show that the adapted coordinates may
not exist. Moreover, even for $P=0$, it is still extremely difficult to
determine the precise asymptotic behaviour of (9) in many cases. Some results
of this subject are motivated by Newton polyhedra, see for instance [42], D.
H. Phong and E. M. Stein [35], J. Kamimoto and T. Nose [25], T. C. Collins et
al. [7] and M.Greenblatt [17].

Since the seminal paper of Schultz [38] on dispersive estimates for the DW on
${\mathbb{Z}}^{d}$ with $d=2,3$, whether these results can be extended to
higher dimensional lattices has been left as a major open problem. In this
article, we extend the results of [38] to $d=4$, serving as the first step
towards the above open problem. Our main result is the following.

######  Theorem 1.1.

There exists $C>0$ independent of $x$ and $t$, such that

$|G(x,t)|\leqslant C(1+|t|)^{-\frac{3}{2}}\log(2+|t|)$

holds for all $(x,t)\in{\mathbb{Z}}^{4}\times{\mathbb{R}}$.

######  Remark 1.2.

The estimate is sharp in the sense that there exist some vector
$v_{0}\in{\mathbb{R}}^{4}$ and a constant $A>0$ such that (cf. [28])

$|I(v_{0},t)|\geqslant A\,t^{-\frac{3}{2}}\log
t,\;\mbox{as}\;\;t\rightarrow\infty.$

Our proof strategy is to choose proper change of coordinates to express $\phi$
in a simple way. Then we apply results in uniform estimates of the oscillatory
integrals, which appeared in Karpushkin [26] first, to reduce the phase
function to a quasi-homogeneous polynomial (see Definition 2.4). Finally we
investigate the singularity type and obtain the decay rate. Similar argument
may also be used in the study of dispersive equations on general graphs. In
the terminology of Arnold [1], the stable singularities which will appear in
the proof are $A_{k}$ ($k\geq 1$) and $D_{4}^{-}$, whose normal forms are
$x^{k+1}$ and $x_{1}^{2}x_{2}-x_{2}^{3}$, respectively. We will also handle
the stability of oscillatory integral with phase $x_{1}x_{2}x_{3}$, which was
first considered in [26, Section 7].

In [9], Cuenin and Ikromov considered the DKG in dimensions $d=2,3,4$. Their
proof relied also on the analysis of singularities. The corresponding
oscillatory integral is

(10)
$\tilde{I}(v,t)=\int_{\mathbb{T}^{d}}e^{it(v\cdot\xi-\tilde{\omega}(\xi))}\frac{1}{\tilde{\omega}(\xi)}\,d\xi$

where $\tilde{\omega}(\xi)=\sqrt{1+\sum(2-2\cos\xi_{j})}$. The situation of
the DKG is simpler than that of the DW. On one hand, compared to (5), the
phase and amplitude in (10) are more regular since “1” appears in
$\tilde{\omega}$. In fact, “1” corresponds to the mass of the system, hence
the DW is the vanishing mass limit of the DKG, i.e. $m\rightarrow 0$ in (4).
On the other hand, in order to establish Strichartz estimates in Section 5.2,
we need to overcome some difficulties which are mainly caused by the operator
$\frac{1}{\sqrt{-\Delta}}$ (see (51) below). In the case of the DKG, the
$l^{p}$ boundedness of $\frac{1}{\sqrt{\boldsymbol{1}-\Delta}}$ is
straightforward, while for the DW, we need to investigate more properties of
the Fourier transform of $\frac{1}{\omega}$.

In contrast to the continuous setting, there exists a nontrivial Tychnoff-type
solution to (1) with $F,g,f\equiv 0$. In particular, this solution does not
belong to $l^{2}$, hence it is not a “physical” solution, see [18]. In this
paper, by a solution to (1) we mean the semigroup solution in $l^{2}$ as in
(14).

A direct corollary of Theorem 1.1 leads to the $l^{1}\rightarrow l^{\infty}$
estimate for the solution. Then by interpolation we can prove the
$l^{p}\rightarrow l^{q}$ estimates. In the sequel, by “$a^{-}$” we mean that
one may choose any $\epsilon>0$ to replace $a$ with $a-\epsilon$ for some
$a>0$.

######  Theorem 1.3.

Suppose that $u$ is the solution to (1) on ${\mathbb{Z}}^{4}$ with $F,g\equiv
0$. If $1\leqslant p<q\leqslant+\infty$ with $1/p-1/q\geqslant 1/2$, then

$\|u(t,\cdot)\|_{l^{q}}\leqslant C_{p,q}(1+|t|)^{-\beta_{p,q}}\|f\|_{l^{p}}$

with index
$\beta_{p,q}:=3^{-}\left(\frac{1}{p}-\frac{1}{q}-\frac{1}{2}\right).$
Furthermore, if $(\frac{1}{p},\frac{1}{q})$ lies on the segment with vertices
$(\frac{3}{4},\frac{1}{2})$ and $(1,0)$, then

$\|u(t,\cdot)\|_{l^{q}}\leqslant C_{p,q}(1+|t|)^{-\zeta_{q}}\|f\|_{l^{p}}$

with index
$\zeta_{q}:=\left(\frac{3}{2}\right)^{-}\left(1-\frac{2}{q}\right)$.

The following Strichartz estimate on the lattice graph is proved for the first
time by a combination of Theorem 1.1, the $l^{p}\rightarrow l^{q}$ boundedness
of $\frac{1}{\sqrt{-\Delta}}$ and the result of M. Keel and T. Tao [29]. The
analog in the case $d=3$ can be derived similarly and will be given in Theorem
5.8.

######  Theorem 1.4.

Suppose $u$ is the solution to (1) on ${\mathbb{Z}}^{4}$. Then

$\|u\|_{L^{q}_{t}l^{r}_{x}}\leqslant
C_{q,r,\tilde{q},\tilde{r}}\left(\|g\|_{l^{2}}+\|f\|_{l^{\frac{4}{3}}}+\|F\|_{L_{t}^{\tilde{q}^{\prime}}l^{\tilde{r}}_{x}}\right)$

with pairs $(q,r)$ and $(\tilde{q},\tilde{r})$ satisfying

(11) $q,r\geqslant
2,~{}\frac{1}{q}<\frac{3}{2}\left(\frac{1}{2}-\frac{1}{r}\right)$

and

(12) $\tilde{q}\geqslant
2,~{}1<\tilde{r}\leqslant\frac{4}{3},~{}\frac{1}{\tilde{q}}<\frac{3}{2}\left(\frac{1}{\tilde{r}}-\frac{3}{4}\right).$

We remark that there is another kind of Strichartz estimates, Theorem 5.2,
which follows from Theorem 1.3. A detailed comparison will be shown in Remark
5.7. As an application to the nonlinear DW, these estimates can be used to
prove the global existence of the solution under a sufficient condition on the
nonlinearity of $F(u)$, see Theorem 5.10 and 5.11.

Most of our results focus on $d=4$. However, we also investigate the most
degenerate case in higher dimensions (see Section 3.2) and conjecture the
following estimate is true.

######  Conjecture 1.5.

For all odd $d\in{\mathbb{N}}$, $d\geq 5$, there exists $C=C(d)>0$ such that

(13) $|G(x,t)|\leqslant C\,(1+|t|)^{-\frac{2d+1}{6}}$

holds for all $(x,t)\in{\mathbb{Z}}^{d}\times{\mathbb{R}}$.

In Theorem 4.5 below, we partially prove Conjecture 1.5 using the Newton
polyhedra and a suitable change of coordinates. More precisely, let $x=vt$, we
shall prove that (13) is true for $G(tv,t)$ with
$v=(\frac{\pi}{2},\cdots,\frac{\pi}{2})$. This is better than [9, Remark
1(b)], in which the power of the logarithm term is $(d-4)$ for $d\geq 5$.
Notice also that there is no logarithm term appearing in (13), which is
different from the case $d=4$.

This paper is organized as follows. In Section 2 we give basic concepts and
results. Theorem 1.1 is proven in Section 3. In Section 4 we recall some facts
of Newton polyhedra and discuss results for the DW in dimensions $d=2,3,4$ and
all odd $d\geqslant 5$. In Section 5, we prove $l^{p}\rightarrow l^{q}$
estimates, Strichartz estimates and the existence of solution for nonlinear
equations.

##  2\. Preliminaries

###  2.1. Basics on $l^{p}({\mathbb{Z}}^{d})$ and the DW

We denote by $\mathbb{Z}^{d}$ the standard integer lattice graph in
$\mathbb{R}^{d}$. Recall that for $p\in[1,\infty]$, $l^{p}({\mathbb{Z}}^{d})$
is the $l^{p}$-space of functions on ${\mathbb{Z}}^{d}$ with respect to the
counting measure, which is a Banach space with the norm

$||h||_{l^{p}}:=\left\\{\begin{aligned}
&\left(\sum_{x\in{\mathbb{Z}}^{d}}|h(x)|^{p}\right)^{\frac{1}{p}},\
p\in[1,\infty),\\\ &\sup_{x\in{\mathbb{Z}}^{d}}|h(x)|,\
p=\infty\end{aligned}\right.$

We shall also use $|h|_{p}$ to denote the $l^{p}$ norm of $h$ for notational
convenience. The mixed space-time Lebesgue spaces $L^{q}_{t}l^{r}$ are Banach
spaces endowed with the norms

$||F||_{L^{q}_{t}l^{r}}:=\left(\int_{{\mathbb{R}}}\left(\sum_{x\in{\mathbb{Z}}^{d}}|F(x,t)|^{r}\right)^{\frac{q}{r}}\right)^{\frac{1}{q}}$

for $1\leqslant q,r<\infty$ with natural modifications for the case $q=\infty$
or $r=\infty$. Moreover, for appropriate functions $h_{1},h_{2}$ on
$\mathbb{Z}^{d}$ we define the convolution product of them as

$h_{1}*h_{2}(x):=\sum_{y\in\mathbb{Z}^{d}}h_{1}(x-y)h_{2}(y).$

The $l^{p}$ spaces are analogous to the $L^{p}$ spaces of functions defined on
${\mathbb{R}}^{d}$. Many results of the $L^{p}$ spaces extend to the lattice
such as the Hölder inequality, Young’s inequality for convolution and Riesz-
Thorin interpolation theorem. One major difference is that the $l^{p}$ spaces
are nested:

$l^{p}\subset
l^{q}~{}~{}\mathrm{and}~{}~{}|h|_{q}\leqslant|h|_{p}~{}~{}\mathrm{if}~{}~{}1\leqslant
p\leqslant q\leqslant\infty.$

To obtain the solution for linear equation, we adopt the discrete analogs to
the methods for wave equations in $\mathbb{R}^{d}$. An important tool is the
discrete Fourier transform of a proper function, given by

$\mathcal{F}(h)(\xi)=\hat{h}(\xi):=\sum_{x\in\mathbb{Z}^{d}}e^{-i\xi\cdot
x}h(x),~{}~{}\xi\in\mathbb{T}^{d}.$

The inverse transform is defined as, for proper $h$,

$\mathcal{F}^{-1}(h)(x)=\check{h}(x):=\frac{1}{(2\pi)^{d}}\int_{\mathbb{T}^{d}}e^{i\xi\cdot
x}h(\xi)d\xi,~{}~{}x\in\mathbb{Z}^{d}.$

See e.g. [44] for more facts about discrete Fourier analysis. Applying the
Fourier transform to both sides of (1), we get

$\left\\{\begin{aligned}
&\partial_{t}^{2}\hat{u}(\xi,t)=-\left(\sum_{j=1}^{d}\left(2-2\cos\xi_{j}\right)\right)\hat{u}(\xi,t)=-\omega(\xi)^{2}\hat{u}(\xi,t),\\\
&\hat{u}(\xi,0)=\hat{g}(\xi),~{}~{}\partial_{t}\hat{u}(\xi,0)=\hat{f}(\xi).\end{aligned}\right.$

For any fixed $\xi$, the solution to this ordinary differential equation is

$\hat{u}(\xi,t)=\cos(t\omega)\hat{g}(\xi)+\frac{\sin(t\omega)}{\omega}\hat{f}(\xi).$

Therefore we have

$u(x,t)=\frac{1}{(2\pi)^{d}}\int_{\mathbb{T}^{d}}e^{i\xi\cdot
x}\left(\cos(t\omega)\hat{g}(\xi)+\frac{\sin(t\omega)}{\omega}\hat{f}(\xi)\right)d\xi.$

In the notion of operator theory, for any $f,g\in l^{2}$,

(14)
$u(t)=\cos(t\sqrt{-\Delta})g+\frac{\sin(t\sqrt{-\Delta})}{\sqrt{-\Delta}}f.$

From now on we only consider the solution for zero initial position and a
given initial velocity, i.e. $g\equiv 0$, unless otherwise stated. Indeed, the
other case for $f\equiv 0$ can be treated similarly and the same result holds
as the former case, see Section 5.2. Under this assumption, the solution $u$
is given by $u=f*G$ with the Green’s function $G(x,t)$ defined in (2). We
point out that $G$ is related to the oscillatory integral (5). In fact, noting
that $\omega(\xi)=\omega(-\xi)$, we have $I(x/t,t)=I(-x/t,t)$, $t\neq 0$.
Hence for $x=vt$,

(15) $\displaystyle G(x,t)=-\mbox{Im}\,I(x/t,t)=-\mbox{Im}\,I(v,t).$

###  2.2. Results on uniform estimates.

Before analyzing $I(v,t)$, we recall some useful notions and results which
were initiated from [26]. For any $r>0$, we denote by
$B_{{\mathbb{R}}^{d}}(0,r):=\\{x=(x_{1},\cdots,x_{d})\in{\mathbb{R}}^{d}:|x|=(\sum_{j=1}^{d}x_{j}^{2})^{\frac{1}{2}}<r\\}$
the usual Euclidean ball, and
$B_{{\mathbb{C}}^{d}}(0,r):=\\{\boldsymbol{w}=(w_{1},\cdots,w_{d})\in{\mathbb{C}}^{d}:|\boldsymbol{w}|=(\sum_{j=1}^{d}|w_{j}|^{2})^{\frac{1}{2}}<r\\}$.

######  Definition 2.1.

For $r,s>0$, we define the space $\mathcal{H}_{r}(s)$. A function
$P\in\mathcal{H}_{r}(s)$ means that $P$ is real-analytic in
$B_{{\mathbb{R}}^{d}}(0,r)$, which has a unique holomorphic extension
$\widetilde{P}$ on $B_{{\mathbb{C}}^{d}}(0,r)$ such that $\widetilde{P}$ is
continuous on $\overline{B}_{{\mathbb{C}}^{d}}(0,r)$ and
$|\widetilde{P}(\boldsymbol{w})|<s$,
$\forall\,\boldsymbol{w}\in\overline{B}_{{\mathbb{C}}^{d}}(0,r)$.

For any given $r,s>0$, real polynomials with coefficients sufficiently small
are in $\mathcal{H}_{r}(s)$.

######  Definition 2.2.

Suppose that $h:\mathbb{R}^{d}\rightarrow\mathbb{R}$ is real-analytic at 0 and
$(\beta,p)\in{\mathbb{R}}\times{\mathbb{N}}$. We write

$M(h)\curlyeqprec(\beta,p)$

if for all $r>0$ sufficiently small we can find $\epsilon>0$, $C>0$ and a
neighbourhood $A\subset B_{{\mathbb{R}}^{d}}(0,r)$ of the origin such that

(16)
$\bigg{|}\int_{\mathbb{R}^{d}}e^{it(h(x)+P(x))}\varphi(x)dx\bigg{|}\leqslant
C(1+|t|)^{\beta}\log^{p}(|t|+2)\|\varphi\|_{C^{N}(A)}$

holds for all $t\in{\mathbb{R}}$, $\varphi\in C^{\infty}_{0}(A)$ and
$P\in\mathcal{H}_{r}(\epsilon)$. Here $N\in{\mathbb{N}}$ only depends on $d$,
$\|\varphi\|_{C^{N}(A)}=\sup\\{|\partial^{\gamma}\varphi(x)|:x\in
A,\gamma\in{\mathbb{N}}^{d},|\gamma|\leq N\\}.$

From this definition, integrating by parts directly gives that:

######  Lemma 2.3.

If $h:{\mathbb{R}}^{d}\rightarrow{\mathbb{R}}$ is real analytic at $0$ and
$\nabla h(0)\neq 0$, then

$M(h)\curlyeqprec(-n,0),\quad\forall\,n\in{\mathbb{N}}.$

By $M(h,a)\curlyeqprec(\beta,p)$ we mean that $h_{a}(x):=h(x+a)$ has a uniform
estimate at 0 with exponent $(\beta,p)$, i.e. $M(h_{a})\curlyeqprec(\beta,p)$.
For integers $n_{1},n_{2}\geq 1$, two functions
$h_{1}:\mathbb{R}^{n_{1}}\rightarrow\mathbb{R}$,
$h_{2}:\mathbb{R}^{n_{2}}\rightarrow\mathbb{R}$ and a given pair
$(\beta_{1},p_{1})$, we write $M(h_{1})\curlyeqprec
M(h_{2})+(\beta_{1},p_{1})$ if $M(h_{2})\curlyeqprec(\beta,p)$ implies that
$M(h_{1})\curlyeqprec(\beta+\beta_{1},p+p_{1})$. If $(\beta_{1},p_{1})=(0,0)$,
we write $M(h_{1})\curlyeqprec M(h_{2})$. Meanwhile, we say
$\alpha\in{\mathbb{R}}^{d}$ is a weight if
$\alpha=(\alpha_{1},\cdots,\alpha_{d})$ and $\alpha_{j}>0$ for all $j$. For
$x\in{\mathbb{R}}^{d}$ and $k>0$, we also write
$k^{\alpha}x:=(k^{\alpha_{1}}x_{1},\cdots,k^{\alpha_{d}}x_{d})$.

######  Definition 2.4.

A polynomial $h:\mathbb{R}^{d}\rightarrow\mathbb{R}$ is called quasi-
homogeneous of degree $\varrho\geq 0$ with respect to the weight $\alpha$, if
$h(k^{\alpha}x)=k^{\varrho}h(x)$ holds for all $k>0$ and
$x\in{\mathbb{R}}^{d}$. Let $\mathcal{E}_{\alpha,d}$ be the set of all quasi-
homogeneous polynomials of degree 1 with respect to $\alpha$, and
$H_{\alpha,d}$ denotes the set containing functions real-analytic at $0$,
whose Taylor series consist of monomials which are quasi-homogeneous of degree
greater than $1$ with respect to $\alpha$.

The following useful lemmas, first appeared in [26], enable us to reduce a
general phase to a much easier one, i.e. the “principle part” of it. We give a
proof of Lemma 2.5 and an alternative proof of Lemma 2.6, which rely on a
simple scaling argument and the method of stationary phase, respectively.

######  Lemma 2.5.

Let $\alpha\in{\mathbb{R}}^{d}$ be a weight. If $h\in\mathcal{E}_{\alpha,d}$
and $P\in H_{\alpha,d}$, then $M(h+P)\curlyeqprec M(h).$

###### Proof.

The idea is the same as in [26, Lemma 1]. If $M(h,0)\curlyeqprec(\beta,p)$, in
view of Definition 2.2, we need to find $N\in{\mathbb{N}}$, and for all $r>0$,
to find $\epsilon=\epsilon(r)>0,\ C=C({r})>0$ and
$A=A(r)\subset{\mathbb{R}}^{n}\cap B_{{\mathbb{C}}^{d}}(0,r)$ such that

(17)
$\left|\int_{{\mathbb{R}}^{d}}e^{it(h(x)+P(x)+\Upsilon(x))}\psi(x)\,dx\right|\leq
C(1+|t|)^{\beta}\log^{p}(2+|t|)||\psi||_{C^{N}},$

holding for all $\Upsilon\in\mathcal{H}_{r}(\epsilon)$ and all $\psi\in
C_{0}^{\infty}(A)$. For any $c>0$, we define the polydisc
$\mathbf{P}(c,\alpha):=\left\\{\boldsymbol{w}\in{\mathbb{C}}^{d}:|w_{j}|<\frac{c^{\alpha_{j}}}{2}\right\\}.$
Since $M(h,0)\curlyeqprec(\beta,p)$, we can find corresponding
$\epsilon_{h,r}>0$, $C_{h,r}>0$ and region $A_{h,r}$ as in Definition 2.2 such
that $\mathbf{P}(c_{*},\alpha)\cap{\mathbb{R}}^{d}\subset A_{h,r}$ for some
$c_{*}=c_{*}({h,r})>0$. We denote by $I_{*}$ the integral in (17), using a
scaling $x=(\frac{s}{c_{*}})^{\alpha}y$, with $s=s(r)$ to be determined later,
we have

(18)
$I_{*}=\left(\frac{s}{c_{*}}\right)^{|\alpha|}\int_{{\mathbb{R}}^{d}}e^{itsc_{*}^{-1}\left(h(y)+s^{-1}c_{*}(P_{1}(y)+\Upsilon_{1}(y))\right)}\psi_{1}(y)\,dy,$

where $P_{1}(y):=P((\tfrac{s}{c_{*}})^{\alpha}y)$, while $\Upsilon_{1}$ and
$\psi_{1}$ are similar. Therefore, supp
$\psi_{1}\subset\mathbf{P}(c_{*},\alpha)\cap{\mathbb{R}}^{d}$ if
$\mbox{supp}\,\psi\subset\mathbf{P}(s,\alpha)\cap{\mathbb{R}}^{d}$. Since
$P\in H_{\alpha,d}$, by the Cauchy inequality [19, Theorem 2.2.7], we can
check that
$c_{*}s^{-1}\left(P_{1}+\Upsilon_{1}\right)\in\mathcal{H}_{r}(\epsilon_{h,r})$
after a suitable choice of $s$ and $\epsilon$. Indeed, we can write

$P(z)=\sum_{\alpha\cdot\beta>1}a_{\beta}z^{\beta},\;|a_{\beta}|\leq\frac{P_{0}}{r^{|\beta|}},\quad\mbox{for
a constant}\ P_{0}>0.$

Since $\alpha_{j}>0$ for all $j$, there exists $\sigma>0$ such that
$\alpha\cdot\beta-1\geq\sigma|\beta|$ for all $\beta$ satisfying
$\alpha\cdot\beta>1$ and further

$\displaystyle\left|\frac{c_{*}}{s}P_{1}(y)\right|$ $\displaystyle\leq
P_{0}\sum_{\alpha\cdot\beta>1}\left(\frac{s}{c_{*}}\right)^{\alpha\cdot\beta-1}\leq
P_{0}\sum_{|\beta|\geq 2}\left(\frac{s}{c_{*}}\right)^{\sigma|\beta|}\leq
M_{0}\left(\frac{s}{c_{*}}\right)^{\sigma}\leq\frac{\epsilon_{h,r}}{2},$

where $M_{0}>1$ is a constant and $s$ satisfies
$\left(s/c_{*}\right)^{\sigma}M_{0}=\min\left\\{\frac{\epsilon_{h,r}}{2},\frac{1}{10}\right\\}.$
Furthermore, we can choose $\epsilon$ such that
$c_{*}s^{-1}\Upsilon_{1}\in\mathcal{H}_{r}(\epsilon_{h,r}/2)$ for all
$\Upsilon\in\mathcal{H}_{r}(\epsilon)$. Therefore, by (18) we know that

$\displaystyle|I_{*}|$ $\displaystyle\leq
c_{r}\left(\frac{s}{c_{*}}\right)^{|\alpha|}\left(1+|tsc_{*}^{-1}|\right)^{\beta}\log^{p}(2+|tsc_{*}^{-1}|)\,\|\psi_{1}\|_{C^{N}}$
$\displaystyle\leq\tilde{c}_{r}\left(\frac{s}{d}\right)^{|\alpha|+\beta}(1+|t|)^{\beta}\log^{p}(2+|t|)\|\psi\|_{C^{N}}.$

The proof is completed by letting
$C=C(r)=\tilde{c}_{r}\left(\frac{s}{d}\right)^{|\alpha|+\beta}$. ∎

######  Lemma 2.6.

Suppose $m>n$, $h_{1}:\mathbb{R}^{n}\rightarrow\mathbb{R}$ and
$h_{2}:\mathbb{R}^{m}\rightarrow\mathbb{R}$ satisfy

$h_{2}(x_{1},\cdots,x_{m})=h_{1}(x_{1},\cdots,x_{n})+x_{n+1}^{2}+\cdots+x_{m}^{2}.$

Then $M(h_{2})\curlyeqprec M(h_{1})+\left(-\frac{m-n}{2},0\right).$

###### Proof.

Suppose that $h_{1}:{\mathbb{R}}^{n}\rightarrow{\mathbb{R}}$,
$M(h_{1})\curlyeqprec(\beta,p)$. Without loss of generality, it suffices to
establish the uniform estimate for the integral

(19)
$\int_{{\mathbb{R}}^{n}}e^{ith_{1}(x)}\left(\int_{{\mathbb{R}}^{m}}e^{it\left(c_{1}y_{1}^{2}+\cdots+c_{m}y_{m}^{2}+P(x,y)\right)}\psi(x,y)\,dy\right)dx,$

where $c_{j}=\pm 1$ and $P\in\mathcal{H}_{r}(\epsilon)$ with $\epsilon$ to be
determined later (only depends on $r$). Let

$Z(x,y):=c_{1}y_{1}^{2}+\cdots+c_{m}y_{m}^{2}+P(x,y).$

Applying the Cauchy inequality and the contraction mapping principle, if
$\epsilon$ is sufficiently small, for all $x\in
B_{{\mathbb{C}}^{n}}(0,\frac{r}{4})$, there exists a unique $y=y(x)\in
B_{{\mathbb{C}}^{m}}(0,\frac{r}{4})$ such that $\nabla_{y}Z(x,y(x))=0$.
Moreover, by the implicit function theorem (see [19]), we know that $y=y(x)$
is holomorphic in $B_{{\mathbb{C}}^{n}}(0,\frac{r}{4})$. Therefore, if we use
the stationary phase method ([20, Theorem 7.7.5]), we can get an expansion for
the inner integral

(20) $\int_{{\mathbb{R}}^{m}}e^{itZ(x,y)}\psi(x,y)\,dy,\ \mbox{with the first
term}\
\frac{c_{m}t^{-\frac{m}{2}}}{\sqrt{\mbox{detHess}Z(x,y(x))}}e^{itZ(x,y(x))}.$

Our choice of the regions in $x,y$ ensures that
$Z(x,y(x))\in\mathcal{H}_{\frac{r}{4}}(\epsilon^{\prime})$ with an appropriate
$\epsilon^{\prime}$, and the derivatives of $y(x)$ in $x$ have uniform bounds.
Therefore, the main term of (19) is

$I_{0}={c_{m}t^{-\frac{m}{2}}}\int_{{\mathbb{R}}^{m}}e^{it(h_{1}(x)+Z(x,y(x)))}\tilde{\psi}(x)\,dx.$

And we have

$|I_{0}|\leq
C_{r}(1+|t|)^{\beta-\frac{m}{2}}\log^{p}(2+|t|)\,||\psi||_{C^{N}}$

for some $N\in{\mathbb{N}}$ independent of $r$. Similar argument can be
applied to other terms in the expansion of (20). Thus we finish the proof. ∎

Several useful examples are given below. They will play important roles in the
proof of the main result. The proof of Proposition 2.7 is postponed to Section
4.2.

######  Proposition 2.7.

The following assertions hold:

$\displaystyle(a)\;M(x^{k+1})\curlyeqprec\left(-\tfrac{1}{k+1},0\right),\forall\,k\in{\mathbb{N}},\;(b)\,M(x_{1}^{2}x_{2}-x_{2}^{3})\curlyeqprec\left(-\tfrac{2}{3},0\right),\;(c)\,M(x_{1}x_{2}x_{3})\curlyeqprec(-1,1).$

###  2.3. Analysis of $I(v,t)$

To establish (6) or (3). we give a detailed analysis of (5) based on known
results in [38]. First, we write $I(v,t)$ as an integral on ${\mathbb{R}}^{d}$
by introducing a nonnegative function $\eta\in
C_{0}^{\infty}({\mathbb{R}}^{d})$ supported in $(-2\pi,2\pi)^{d}$. Besides, it
should be non-vanishing on some neighborhood of $\mathbb{T}^{d}$ such that

$\sum_{j\in{\mathbb{Z}}^{d}}\eta(x+2\pi j)=1,\;\forall\,x\in{\mathbb{R}}^{d}.$

Noticing that $\omega$ is periodic, we have

(21)
$I(v,t)=\frac{1}{(2\pi)^{d}}\sum_{j}\int_{\mathbb{T}^{d}}e^{it\phi(v,\xi)}\,\frac{\eta(\xi+2\pi
j)}{\omega(\xi)}d\xi=\frac{1}{(2\pi)^{d}}\int_{{\mathbb{R}}^{d}}e^{it\phi(v,\xi)}\frac{\eta(\xi)}{\omega(\xi)}d\xi.$

####  2.3.1. Classification of critical points

We note that the phase and amplitude of (21) have singularity at 0. Also, a
direct calculation gives

(22)
$\nabla\omega(\xi)=\left(\frac{\sin\xi_{1}}{\omega(\xi)},\cdots,\frac{\sin\xi_{d}}{\omega(\xi)}\right),\
\xi\in\mathbb{T}^{d}\backslash\\{0\\}.$

For any $v\in{\mathbb{R}}^{d}$, we investigate the critical point set of
$\phi$, which is denoted by

(23)
$\mathcal{C}_{v}:=\\{\xi\in\mathbb{T}^{d}\backslash\\{0\\}\,:\,\nabla_{\xi}\phi(v,\xi)=v-\nabla\omega(\xi)=0\\}.$

If $\xi\in\mathcal{C}_{v}$, by (22) it is clear that

(24)
$|v|^{2}=\frac{\sum_{j=1}^{d}\sin^{2}\xi_{j}}{\sum_{j=1}^{d}(2-2\cos\xi_{j})}<1.$

Therefore, $|v|>1$ implies $\mathcal{C}_{v}=\emptyset$. For the case in the
vicinity of the light cone, see Lemma 2.10 below. On the other hand, it
follows from [37, Proposition 5.6] that $\mathcal{C}_{v}\neq\emptyset$ for all
$v\in B_{{\mathbb{R}}^{d}}(0,1)$. So we have to classify the critical points
and velocities in this case. Since
$\mbox{Hess}_{\xi}\phi(v,\xi)=-\mbox{Hess}_{\xi}\omega(\xi)$, we set

(25)
$\Sigma_{k}=\Sigma_{k}(v,d):=\\{\xi\in{\mathbb{R}}^{d}:\mbox{corank\,Hess}_{\xi}\omega(\xi)=k\\}$

for $k=0,1,\cdots,d$. For simplicity we do not specify the dependence of these
notations on $d$ and $v$. Then, the set of degenerate critical points $\Sigma$
and the corresponding velocity set $\Omega$ can be formulated as

(26)
$\Sigma:=\bigcup_{k=1}^{d}\Sigma_{k},\;\;\mbox{and}\;\;\;\Omega:=\bigcup_{k=1}^{d}\nabla\omega(\Sigma_{k}),$

where we denote by $\nabla\omega(U)$ (resp. $(\nabla\omega)^{-1}(U)$) the
image (resp. preimage) of $U$ under the map $\nabla\omega$ (resp.
$(\nabla\omega)^{-1}$), cf. (23). We also set
$\Omega_{k}=\nabla\omega(\Sigma_{k})$ for $k=0,1,\cdots,d$. Thus $\Omega_{0}$
consists of velocity $v$ such that $\mathcal{C}_{v}$ has only nondegenerate
critical points. Meanwhile, $\Omega_{0}\cup\Omega=B_{{\mathbb{R}}^{d}}(0,1)$.

Now we give the characterization of $\Sigma$. The following lemma can be
proved directly by [38, Proposition 2.4] with a careful analysis to
$\mbox{Hess}_{\xi}\omega$. Only the first quadrant $[0,\pi]^{d}$ will be
considered by symmetry.

######  Lemma 2.8.

Suppose that $d\geq 2$ and $\xi=(\xi_{1},\cdots,\xi_{d})\in[0,\pi]^{d}$. Then
we have

  * (a)

$\Sigma_{1}=\Sigma_{1}^{\prime}\cup\Sigma_{1}^{\prime\prime}$, where
$\Sigma_{1}^{\prime\prime}$ is the set of $\xi$ with exactly two components
equal to $\frac{\pi}{2}$, and

$\displaystyle\Sigma_{1}^{\prime}=\left\\{\xi:\sum_{j=1}^{d}\left(\cos\xi_{j}+\sec\xi_{j}\right)=2d,\
\xi\neq 0\right\\}.$

  * (b)

If $d\geq 3$, $\Sigma_{j}$ consists of $\xi$ with exactly ($j+1$) components
equal to $\frac{\pi}{2}$ for $j=2,\cdots,d-1$.

  * (c)

$\Sigma_{d}=\emptyset$.

Therefore, if $d\geq 3$, the most degenerate case is

$\Sigma_{d-1}=\left\\{\left(\tfrac{\pi}{2},\cdots,\tfrac{\pi}{2}\right)\right\\},\quad\mbox{and}\quad\Omega_{d-1}=\nabla\omega(\Sigma_{d-1})=\left\\{\left(\tfrac{1}{\sqrt{2d}},\cdots,\tfrac{1}{\sqrt{2d}}\right)\right\\}.$

By a direct computation, one can verify that the following assertion holds.

######  Corollary 2.9.

If $d\geq 3$, then $(\nabla\omega)^{-1}(\Omega_{d-1})=\Sigma_{d-1}$ and
$\Omega_{i}\cap\Omega_{j}=\emptyset$, $\forall\,i,j\geq 2,$ $i\neq j$.
Moreover, there is a constant $b_{0}=b_{0}(d)\in(0,1)$ such that
$\Omega\subset B_{{\mathbb{R}}^{d}}(0,b_{0})$.

####  2.3.2. Decay estimates

We consider several cases separately. As we discussed before, outside the
light cone there is no critical points of $\phi$, this case is simple. In
fact, integrating by parts gives that the decay of the Green’s function
$G(tv,t)$ is of order $\mathcal{O}(|t|^{-N})$ for any $N\in\mathbb{N}$.
Moreover, it holds uniformly in $|v|>1$, i.e. the constant in the
$\mathcal{O}$’s term is independent of $v$, see [37, Proposition 5.3].
However, in the vicinity of the light cone, we need more analysis. The
following result handles these two regions (cf. (7)) in all dimensions $d\geq
2$.

######  Lemma 2.10.

Suppose that $d\geqslant 2$, then there exist $C=C(d)>0$ and
$\mathbf{c}=\mathbf{c}(d)\in(b_{0},1)$ such that

$|G(tv,t)|\leqslant C(1+|t|)^{-\frac{d}{2}}$

holds uniformly in $|v|\geq\mathbf{c}$ and $t\in{\mathbb{R}}$.

This lemma follows from [38, Proposition 2.1, Proposition 2.2, Proposition
3.10], relying on the method of stationary phase, the properties of Airy
function and the Green’s function in the continuous setting. With Lemma 2.10
in hand, it suffices to consider the estimate of $I(v,t)$ for $v\in
B_{{\mathbb{R}}^{d}}(0,\mathbf{c})$. In view of the singularity of
$\frac{1}{\omega}$, we introduce a cutoff function $\chi\in
C_{0}^{\infty}({\mathbb{R}}^{d})$ supported near the origin and decompose (21)
into two parts:

(27)
$\displaystyle\begin{split}I(v,t)&=\frac{1}{(2\pi)^{d}}\int_{{\mathbb{R}}^{d}}e^{it\phi(v,\xi)}\frac{\eta(\xi)}{\omega(\xi)}\chi(\xi)\,d\xi+\frac{1}{(2\pi)^{d}}\int_{{\mathbb{R}}^{d}}e^{it\phi(v,\xi)}\frac{\eta(\xi)}{\omega(\xi)}\left(1-\chi(\xi)\right)d\xi\\\
&=:I_{1}(v,t)+I_{2}(v,t).\end{split}$

$I_{1}$ is handled by [38, Proposition 2.3] with a decay rate
$\mathcal{O}(|t|^{-d+1})$ as $t\rightarrow\infty$, uniformly in $v\in
B_{{\mathbb{R}}^{d}}(0,\mathbf{c})$. This bound is better than that of Lemma
2.10.

Now we analyze $I_{2}$, if $\mathcal{C}_{v}=\emptyset$, then Lemma 2.3
applies. If $v\in\Omega_{0}$, i.e. all critical points are nondegenerate, in
the spirit of the stationary phase method we have

######  Lemma 2.11.

For any $v_{0}\in\Omega_{0}$, there exist a neighbourhood $V$ of $v_{0}$ and
$C=C(v_{0},d)>0$ such that

$|I_{2}(v,t)|\leq C(1+|t|)^{-\frac{d}{2}}$

holds uniformly in $(v,t)\in V\times{\mathbb{R}}$.

The proof follows from Lemma 2.6 and a partition of unity in the space of wave
number $\xi$. In what follows, we only need to consider $I_{2}(v,t)$ in the
degenerate case, i.e. $v\in\Omega$. In view of Lemma 2.8, without loss of
generality, we can assume

(28)
$\displaystyle\begin{split}&\Sigma_{1}=\Sigma_{1}^{\prime}\cup\left\\{\left(\tfrac{\pi}{2},\tfrac{\pi}{2},\xi_{*},\eta_{*}\right):~{}\xi_{*},\eta_{*}\neq\tfrac{\pi}{2}\right\\},\\\
&\Sigma_{2}=\left\\{\left(\tfrac{\pi}{2},\tfrac{\pi}{2},\tfrac{\pi}{2},\xi_{*}\right):~{}\xi_{*}\neq\tfrac{\pi}{2}\right\\}\;\mbox{and}\;\;\Sigma_{3}=\left(\tfrac{\pi}{2},\tfrac{\pi}{2},\tfrac{\pi}{2},\tfrac{\pi}{2}\right)\end{split}$

when $d=4$. We will see that the type of singularity depends on the number of
components equal to $\frac{\pi}{2}$. In view of (26), we have to consider
every possibility in $\Omega_{k}$, $k=1,2,3$ separately to get the decay
estimates. We have the following result.

######  Theorem 2.12.

Suppose that $d=4$ and
$v_{0}\in\Omega=\Omega_{1}\cup\Omega_{2}\cup\Omega_{3}$. Then there exist a
neighbourhood $V$ of $v_{0}$ and $C=C(v_{0},d)>0$ such that

(29) $|I_{2}(v,t)|\leq
C(1+|t|)^{\beta}\log^{p}(2+|t|),\;\mbox{with}\;\;(\beta,p)=\begin{cases}(-\frac{3}{2},1),&\text{if
$~{}v_{0}\in\Omega_{3}$};\\\ (-\frac{5}{3},0),&\text{if
$~{}v_{0}\in\Omega_{2}$};\\\ (-\frac{3}{2},0),&\text{if
$~{}v_{0}\in\Omega_{1}\backslash(\Omega_{2}\cup\Omega_{3})$},\end{cases}$

holds uniformly in $(v,t)\in V\times{\mathbb{R}}$.

The proof of Theorem 2.12 can be found in Section 3. Combining Lemma 2.11 and
Theorem 2.12, we finish the estimates for $I_{2}$ by a finite covering on the
velocity space $B_{{\mathbb{R}}^{d}}(0,\mathbf{c})$. Thus we complete the
proof of Theorem 1.1 by the discussion in this section.

The index $(-\frac{3}{2},0)$ in (29) is not sharp, but it is sufficient for
our purposes. All cases in $d=2,3,4$ and the corresponding decay rates are
exhibited in Table 1. In Section 4.2 we will discuss these results by the
method of Newton polyhedra.

######  Remark 2.13.

The decay rate is determined by the vicinity of the light cone when $d=2$ (see
[38]), and the set $\Omega_{d-1}\times\Sigma_{d-1}$ when $d=3,4$. We believe
that this still works for dimensions $d>4$.

Table 1. Disperive estimates for the DW | $\Sigma_{1}^{\prime}$ | $\Sigma_{1}^{\prime\prime}$ | $\Sigma_{2}$ | $\Sigma_{3}$
---|---|---|---|---
dim 2 | $|t|^{-5/6}$ | $|t|^{-3/4}$ |  |
dim 3 | $|t|^{-4/3}$ | $|t|^{-5/4}$ | $|t|^{-7/6}$ |
dim 4 | $|t|^{-3/2}$ | $|t|^{-3/2}$ | $|t|^{-5/3}$ | $|t|^{-3/2}\log(|t|)$

##  3\. Proof of Theorem 1.1

By the argument in Section 2.3.2, we have reduced Theorem 1.1 to Theorem 2.12.

###  3.1. Simplification

The following discussion is not restricted in the case $d=4$. Let us fix some
$v_{0}\in\Omega$, then $(\nabla\omega)^{-1}(v_{0})\cap\Sigma\neq\emptyset$. We
start with the integral $I_{2}(v,t)$ where $v$ is close to $v_{0}$. We
naturally classify all the $\xi^{\prime}s$ in the support of $\eta(1-\chi)$,
denoted by $\mathcal{U}$, into three parts: regular points (i.e.
$\nabla_{\xi}\phi(v_{0},\xi)\neq 0$), nondegenerate critical points and
degenerate critical points of $\phi(v_{0},\cdot)$. For any
$\xi\in\mathcal{U}$, suppose that

(30) $M(\phi(v_{0},\cdot),\xi)\curlyeqprec(\beta_{\xi},p_{\xi})$

for some $(\beta_{\xi},p_{\xi})$. By Definition 2.2, there exist
$\epsilon_{\xi}>0$ and neighborhood $\mathbf{m}_{\xi}$ of $\xi$ such that (16)
holds. Therefore, $\mathcal{U}\subset\cup_{\xi\in\mathcal{U}}\mathbf{m}_{\xi}$
and we can choose finite sets $\\{\mathbf{m}_{\xi_{j}},j=1,2,\cdots,N_{0}\\}$
to cover it. By a partition of unity, there are nonnegative functions
$\\{\varphi_{j},\,j=1,2,\cdots,N_{0}\\}$ such that

$\varphi_{j}\in
C_{0}^{\infty}(\mathbf{m}_{\xi_{j}}),\quad\text{and}\;\;\sum_{j=1}^{N_{0}}\varphi_{j}\equiv
1\;\;\mbox{on}\;\;\mathcal{U}.$

We insert these functions into $I_{2}$:

$I_{2}(v,t)=\int
e^{it\phi}\;\frac{\eta(1-\chi)}{\omega}=\sum_{j=1}^{N_{0}}\int
e^{it\phi}\;\frac{\eta(1-\chi)}{\omega}\varphi_{j}=:\sum_{j=1}^{N_{0}}I_{2}^{j}(v,t).$

Assume that (30) holds, then for any $j$,
$M(\phi(v_{0}),\xi_{j})\curlyeqprec(\beta_{\xi_{j}},p_{\xi_{j}})$. Thus

(31) $|I_{2}^{j}(v,t)|\leqslant
C_{j}(1+|t|)^{\beta_{\xi_{j}}}\log^{p_{\xi_{j}}}(2+|t|)$

holds uniformly in $t\in{\mathbb{R}}$ and all $v$ such that
$|v-v_{0}|<\epsilon_{j}$ , where $C_{j}>0$ is a constant. Therefore, adding
these inequalities together gives estimate for $I_{2}$ in similar manner, with
index $(\beta,p)=\max_{j}\\{(\beta_{\xi_{j}},p_{\xi_{j}})\\}$ in the
lexicographic order.

In what follows, we prove (30) for all $\xi\in\mathcal{U}$. If
$\nabla_{\xi}\phi(v_{0},\xi)\neq 0$, then Lemma 2.3 applies. If $\xi$ is a
nondegenerate critical point of $\phi(v_{0},\cdot)$, we can simply use Lemma
2.11. Finally, we need to establish (30) for all
$\xi_{0}\in\Sigma=\cup_{j=1}^{d-1}\Sigma_{j}$. Thus we shall consider
$\xi_{0}\in\Sigma_{j}$ and correspondingly $v_{0}\in\Omega_{j}$,
$j=1,\cdots,d-1$. In view of Corollary 2.9, when $d=4$ it suffices to prove
(30) with index equals to that of Theorem 2.12. Now we first give some
notations:

  * (N1)

Let $\mu:=v-v_{0}$ and $\phi_{ori}(\mu,\xi):=\phi(\mu+v_{0},\xi+\xi_{0}).$

  * (N2)

For $x\in{\mathbb{R}}^{d}$, let $x^{\prime}=(x_{1},\cdots,x_{d-1})$ and
$Q(x^{\prime}):=\left(\sum_{j=1}^{d-1}x_{j}\right)^{3}-\sum_{j=1}^{d-1}x_{j}^{3}.$
Let $\boldsymbol{e_{j}}$ $(j=1,\cdots,d)$ be the standard coordinate vector in
${\mathbb{R}}^{d}$.

  * (N3)

In the rank one case, i.e. $(v_{0},\xi_{0})\in\Omega_{d-1}\times\Sigma_{d-1}$,
the weight is taken as
$\alpha:=(\frac{1}{3},\cdots,\frac{1}{3},\frac{1}{2})\in{\mathbb{R}}^{d}$ (see
Definition 2.4). This choice essentially comes from the principle face of the
Newton polyhedra, see Section 4.

###  3.2. Corank 3

We begin with a decomposition for the phase $\phi$ in all dimensions $d\geq
3$, which will be also used in Theorem 4.5.

######  Lemma 3.1.

Suppose that $d\geq 3$, $(v_{0},\xi_{0})\in\Omega_{d-1}\times\Sigma_{d-1}$,
then there exist an invertible linear transform $\Phi$ on ${\mathbb{R}}^{d}$,
a constant $c$ and $R\in H_{\alpha,d}$ such that the equality

(32)
$\left(\phi_{ori}(\mu,\cdot)\circ\Phi\right)(y)=c+\Phi(y)\mu+y_{d}^{2}+Q(y^{\prime})+R(y)$

holds for $y$ near the origin.

###### Proof.

Near the origin (of $\xi$), we have

$\phi_{ori}(\mu,\xi)=-a_{1}+\mu\cdot\xi+a_{2}\left(\sum_{j=1}^{d}\xi_{j}\right)^{2}-a_{3}\left(\sum_{j=1}^{d}\xi_{j}\right)^{3}+a_{4}\sum_{j=1}^{d}\xi_{j}^{3}+W(\xi)$

by the Taylor’s formula. Here $a_{j}$ are positive constants and $W$ collects
terms with degree greater than three. Moreover, one can easily check that
every monomial in $W$ consists of factors with the form
$\sum_{j=1}^{d}\xi_{j}^{\,q}$ where $q\geq 1$ is odd. For instance, the term
in $W$ of degree 5 (i.e. 5-jet) has the form

$c_{1}\left(\sum_{j=1}^{d}\xi_{j}\right)^{5}+c_{2}\left(\sum_{j=1}^{d}\xi_{j}^{3}\right)\left(\sum_{j=1}^{d}\xi_{j}\right)^{2}+c_{3}\sum_{j=1}^{d}\xi_{j}^{5}$

with constants $c_{j}$, $j=1,2,3$. Let $z=\Phi_{1}(\xi)$, where
$z_{j}=\xi_{j}$, $j=1,\cdots,d-1$ and $z_{d}=\sum_{j=1}^{d}\xi_{j}$. Then

(33)
$\displaystyle(\phi_{ori}(\mu,\cdot)\circ\Phi_{1}^{-1})(z)=-a_{1}+\Phi_{1}^{-1}(z)\cdot\mu+a_{2}z_{d}^{2}+a_{4}Q(z^{\prime})+\tilde{W}(z),$

where

$\tilde{W}(z)=-a_{3}z_{d}^{3}-3a_{4}z_{d}^{2}\sum_{j=1}^{d-1}z_{j}+3a_{4}z_{d}\sum_{j=1}^{d-1}z_{j}^{2}+W\circ\Phi_{1}^{-1}(z).$

It is clear that $W\circ\Phi_{1}^{-1}\in H_{\alpha,d}$ by the choice of
$\alpha$. Therefore, $\tilde{W}\in H_{\alpha,d}$. Then we get the conclusion
by absorbing the coefficients in (33). ∎

######  Remark 3.2.

(a) The perturbation part $\Phi(y)\mu$ in (32) is linear in $y$. Thus for any
given $\epsilon>0$ and $r>0$, we know that
$\Phi(\cdot)\mu\in\mathcal{H}_{r}(\epsilon)$ when $|\mu|$ is sufficiently
small. Moreover, in the proof of Theorem 1.1 we can always make sure that
perturbation is linear. (b) By the splitting lemma (cf. e.g. [34, Theorem
4.13]), we may obtain a result similar to Lemma 3.1, but the perturbation part
is not linear any more.

In the case $d=4$. In view of (28),
$\xi_{0}=(\frac{\pi}{2},\frac{\pi}{2},\frac{\pi}{2},\frac{\pi}{2})$. By Lemma
3.1, we can locally write $\phi_{ori}$ as

$\displaystyle\phi_{ori}=c+y_{4}^{2}+\Phi(y)\mu+Q(y^{\prime})+R(y).$

Notice that

$Q(y^{\prime})=(y_{1}+y_{2}+y_{3})^{3}-y_{1}^{3}-y_{2}^{3}-y_{3}^{3}=3(y_{1}+y_{2})(y_{1}+y_{3})(y_{2}+y_{3}).$

Under the coordinate changes $z_{1}=y_{1}+y_{2}$, $z_{2}=y_{1}+y_{3}$ and
$z_{3}=y_{2}+y_{3}$, $\phi_{ori}$ becomes

$\displaystyle
c+z_{4}^{2}+3z_{1}z_{2}z_{3}+\tilde{\Phi}(z)\cdot\mu+\tilde{R}(z).$

By Lemma 2.5 and 2.6, we have

$\displaystyle M$
$\displaystyle\left(z_{4}^{2}+3z_{1}z_{2}z_{3}+\tilde{R}\right)\curlyeqprec
M\left(z_{1}z_{2}z_{3}\right)+\left(-\frac{1}{2},0\right).$

Then (30) in this case follows from Remark 3.2 (a) and Proposition 2.7 (c).

###  3.3. Corank 2

In this case
$\xi_{0}=\left(\frac{\pi}{2},\frac{\pi}{2},\frac{\pi}{2},\xi_{*}\right)^{T}$,
$\xi_{*}\neq\frac{\pi}{2}$. We will find a proper coordinate transform to
change the phase to “good” form. We calculate the Hessian of
$\phi(v_{0},\cdot)$ at $\xi_{0}$, that is,

$\mbox{Hess}_{\xi}\phi(v_{0},\xi_{0})=-\mbox{Hess}\,\omega(\xi_{0})=c_{1}\begin{pmatrix}{\huge{\text{$\mathbbm{1}$}_{3}}}\begin{matrix}\quad\quad\quad
a_{1}\\\ \quad\quad\quad a_{1}\\\ \quad\quad\quad a_{1}\end{matrix}&\\\
\begin{matrix}a_{1}&a_{1}&a_{1}&a_{1}^{2}+b_{1}\end{matrix}\end{pmatrix}$

for some nonzero constants $a_{1},b_{1},c_{1}$ depending on $\xi_{*}$. Here
$\mathbbm{1}_{3}$ is the $3\times 3$ matrix with all elements $1$. The matrix
$\mbox{Hess}_{\xi}\phi(v_{0},\xi_{0})$ has zero eigenvalue of multiplicity 2
whose eigenvectors can be chosen as $\gamma_{1}=(1,-1,0,0)^{T}$ and
$\gamma_{2}=(1,1,-2,0)^{T}$. Then we turn $\phi_{ori}$ into a new phase

(34) $\displaystyle
c+\mu\boldsymbol{E}x+L(x_{3},x_{4})+\frac{\sqrt{2}}{2}\omega(\xi_{0})^{-1}(x_{2}^{3}-x_{1}^{2}x_{2})+\mathcal{V}(x)$

by a linear transform $\xi=\boldsymbol{E}x$, where the matrix
$\boldsymbol{E}=(\gamma_{1},\gamma_{2},\boldsymbol{e_{3}},\boldsymbol{e_{4}}).$
In the expansion we also have $\mathcal{V}\in H_{\alpha,4}$ with
$\alpha=\left(\frac{1}{3},\frac{1}{3},\frac{1}{2},\frac{1}{2}\right)$ and

$L(x_{3},x_{4})=-\frac{\sqrt{2}}{8}\omega(\xi_{0})^{-3}\Big{(}x_{3}^{2}+2x_{3}x_{4}\sin\xi_{*}-\left(2\omega(\xi_{0})^{2}\cos\xi_{*}-\sin^{2}\xi_{*}\right)x_{4}^{2}\Big{)}.$

We then apply a rotation in $x_{3},x_{4}$ transforming (34) into the form

$c+\mu{\boldsymbol{E_{1}}}y+(a_{3}y_{3}^{2}+a_{4}y_{4}^{2}-y_{1}^{2}y_{2}+y_{2}^{3})+\mathcal{V}_{1}(y)$

with non-vanishing constants $a_{3},a_{4}$. The polynomial in the bracket is
the normal form of $D_{4}^{-}$ singularity, which is one of the stable
singularities and has been fully investigated. Moreover, we have

$\displaystyle
M\left(a_{3}y_{3}^{2}+a_{4}y_{4}^{2}-y_{1}^{2}y_{2}+y_{2}^{3}\right)$
$\displaystyle\curlyeqprec M\left(-y_{1}^{2}y_{2}+y_{2}^{3}\right)+(-1,0)$

by Lemma 2.6. Therefore, we complete the proof by Proposition 2.7 (b) since
$|\mu|$ is sufficiently small.

###  3.4. Corank 1

Since a combination of the splitting lemma [34, Theorem 4.13] and Lemma 2.6
gives the uniform estimates with index pair $(-\frac{3}{2},0)$, which is
better than the case Corank 3, the proof is completed.

##  4\. Newton Polyhedra

###  4.1. Basic concepts

In this section we first recall some basics of Newton polyhedra, see [42, 15,
23]. In our setting, it is convenient to consider the real-analytic cases. For
the concepts in convex analysis and polytopes we refer to [4, 6]. Let $S$ be a
real-analytic function defined in a neighborhood of the origin. Without loss
of generality, in the sequel we always assume that the phase $S$ satisfies

(35) $S(0)=0,\quad\nabla S(0)=0.$

Consider the associated Taylor series

(36) $S(x)=\sum_{m\in\mathbb{N}^{d}}s_{m}x^{m}.$

The set $\mathcal{T}(S):=\\{m\in\mathbb{N}^{d}:s_{m}\neq 0\\}$ is called the
Taylor support of $S$. The Newton polyhedron of $S$, denoted by
$\mathcal{N}(S)$, is the convex hull of the set

$\bigcup_{m\in\mathcal{T}(S)}\left(m+\mathbb{R}^{d}_{+}\right),$

where ${\mathbb{R}}^{d}_{+}=\\{x\in{\mathbb{R}}^{d}:x_{j}\geq 0,\,1\leq j\leq
d\\}$. For every compact face $\mathcal{P}$ of the polyhedron
$\mathcal{N}(S)$, we call

$S_{\mathcal{P}}(x)=\sum_{m\in\mathcal{P}}s_{m}x^{m}$

the $\mathcal{P}$-part of the series (36). We say that $S$ is
$\mathbb{R}$-nondegenerate if $\nabla S_{\mathcal{P}}(x)$ is nonvanishing on
$(\mathbb{R}\backslash\\{0\\})^{d}$ for every compact face $\mathcal{P}$, that
is,

(37)
$\bigcup_{\mathcal{P}}\left(\bigcap_{j=1}^{d}\left\\{x:{\partial_{j}S_{\mathcal{P}}}(x)=0\right\\}\right)\subset\;\bigcup_{j=1}^{d}\\{x:x_{j}=0\\}.$

Note that an edge or a vertex is also a face. Assume that $S$ is not
identically zero, then the Newton distance $d_{S}$ of $\mathcal{N}(S)$ is
defined by

$d_{S}=\inf\\{\rho>0:(\rho,\rho,\cdots,\rho)\in\mathcal{N}(S)\\}.$

The principle face $\pi(S)$ of $\mathcal{N}(S)$ is the face of minimal
dimension containing the point
$\boldsymbol{d_{S}}=(d_{S},d_{S},\cdots,d_{S}).$ The following result is a
consequence of the main theorem in Varchenko [42], who proved that the leading
term of the asymptotic expansion of the oscillatory integral in (38) can be
read by the Newton polyhedron under certain conditions. See also Gilula [15,
Theorem 2.3].

######  Theorem 4.1.

Suppose $S$ is $\mathbb{R}$-nondegenerate and
$\psi:\mathbb{R}^{d}\rightarrow\mathbb{R}$ is smoothly supported close enough
to the origin. Let $k_{0}\in{\mathbb{N}}$ be the greatest codimension over all
faces of $\mathcal{N}(S)$ containing the point $\boldsymbol{d_{S}}$, then we
have

(38) $\left|\int_{{\mathbb{R}}^{d}}e^{itS(x)}\psi(x)\,dx\right|\leqslant
C(1+|t|)^{-\frac{1}{d_{S}}}\log^{k_{0}-1}(2+|t|).$

Note also that the Newton distance depends on the chosen coordinate system
defined near the origin in which $S$ is expressed. The height of $S$ is
defined by

$ht_{S}:=\sup\\{d_{S,x}\\},$

where the supremum is taken over all local analytic coordinate systems which
preserves 0 and $d_{S,x}$ is the Newton distance in coordinates $x$. A given
coordinate system $x_{0}$ is said to be adapted to $S$ if
$d_{S,x_{0}}=ht_{S}$.

In the case $d=2$, Varchenko found sufficient conditions (see [42, Proposition
0.7, 0.8]) to determine whether a given coordinate system is adapted, which
turned out to be also necessary in [23]. Furthermore, the following result on
stability of oscillatory integrals will also be used, it can be obtained from
[27, Theorem 2.1].

######  Theorem 4.2.

Assume that $S:\mathbb{R}^{2}\rightarrow\mathbb{R}$ satisfying (35) and $\psi$
is smoothly supported near the origin. Moreover, if the inequality

$\left|\int_{\mathbb{R}^{2}}e^{itS(x)}\psi(x)dx\right|\leqslant
C(1+|t|)^{\beta_{S}}\log^{p_{S}}(2+|t|)$

holds uniformly in $t\in{\mathbb{R}}$ with some exponent pair
$(\beta_{S},p_{S})$, then $M(S)\curlyeqprec(\beta_{S},p_{S})$.

###  4.2. Results in $d=2,3,4$

In this part we shall see that the dispersive estimates for the DW on
${\mathbb{Z}}^{d}$ with $d=2,3,4$ can also be interpreted via the method of
Newton polyhedra.

More precisely, following from [38] and the proof of Theorem 2.12, we see that
the phase can be expressed in the form $S=S_{1}+q_{S}$ after omitting the
coefficients, perturbations and high order terms. Here $q_{S}$ is the
quadratic term separating from other variables and will be put into the
bracket in Table 3.

Thanks to Lemma 2.6, it suffices to consider the stability of oscillatory
integral with phase $S_{1}$, and we reduce the problem to dimensions no more
than 2 except the last case. Moreover, by the following results in [42] for
$d=2$, we know that in all cases the phase $S_{1}$ is expressed in adapted
coordinate system except $x_{1}x_{2}x_{3}$.

######  Proposition 4.3.

If the point $\boldsymbol{d_{S}}=(d_{S},d_{S})$ lies on a compact face
$\Gamma$ of the Newton polyhedron. Let $a_{1}x_{1}+x_{2}=a_{2}$ be the
equation of the straight line on which $\Gamma$ lies, where $a_{j}$, $j=1,2$
are natural numbers. Then the corresponding coordinate system is adapted if
the polynomial $S_{\Gamma}(\cdot\,,1)$ does not have a real root of
multiplicity larger than $\frac{a_{2}}{1+a_{1}}$.

Indeed, it suffices to consider $x_{1}^{2}+x_{1}x^{2}_{2}$ (equivalent to
$x_{1}^{2}-x_{2}^{4}$) and $x_{1}^{2}x_{2}-x_{2}^{3}$. The support hyperplanes
are $2x_{1}+x_{2}=4$ and $x_{1}+x_{2}=3$, respectively. In each case
$S_{\Gamma}(\cdot\,,1)$ has root of multiplicity $1$. The corresponding Newton
polyhedron $\mathcal{N}(S_{1})$ and the Newton distance $d_{S_{1}}$ are listed
in Table 3. Then an application of [42, Theorem 0.6], Theorem 4.2 and Lemma
2.6 gives the desired exponent pairs $(\beta_{S},p_{S})$ for the uniform
estimates.

Table 2. Cases in dimensions 2,3,4 |  | $S=S_{1}(+q_{S})$ | $\mathcal{N}(S_{1})$ | $d_{S_{1}}$ | $(\beta_{S},p_{S})$
---|---|---|---|---|---
dim 2 | $\Sigma_{1}^{\prime}$ | $x_{1}^{3}(+x_{2}^{2})$ | $[3,\infty)$ | 3 | $(5/6,0)$
dim 3 | $\Sigma_{1}^{\prime}$ | $x_{1}^{3}(+x_{2}^{2}+x_{3}^{2})$ | $(4/3,0)$
dim 2 | $\Sigma_{1}^{\prime\prime}$ | $x_{1}^{2}+x_{1}x_{2}^{2}$ | $\\{(\lambda+1,2-2\lambda):\lambda\in[0,1]\\}+{\mathbb{R}}^{2}_{+}$ | $\dfrac{4}{3}$ | $(3/4,0)$
dim 3 | $\Sigma_{1}^{\prime\prime}$ | $x_{1}^{2}+x_{1}x_{2}^{2}(+x_{3}^{2})$ | $(5/4,0)$
dim 3 | $\Sigma_{2}$ | $x_{1}^{2}x_{2}-x_{2}^{3}(+x_{3}^{2})$ | $\\{(2\lambda,3-2\lambda):\lambda\in[0,1]\\}+{\mathbb{R}}^{2}_{+}$ | $\dfrac{3}{2}$ | $(7/6,0)$
dim 4 | $\Sigma_{2}$ | $x_{1}^{2}x_{2}-x_{2}^{3}(+x_{3}^{2}+x_{4}^{2})$ | $(5/3,0)$
dim 4 | $\Sigma_{3}$ | $x_{1}x_{2}x_{3}(+x_{4}^{2})$ | $(1,1,1)+{\mathbb{R}}^{3}_{+}$ | 1 | $(3/2,1)$

Finally, we consider the phase $x_{1}x_{2}x_{3}$. We obtain an upper bound
$\mathcal{O}(t^{-1}\log^{2}t)$ directly by Theorem 4.1. But it is not sharp.
To see this, we use a change of variables

$x_{1}=z_{1},\,x_{2}=z_{2}-z_{3},\,x_{3}=z_{2}+z_{3}$

to obtain a new phase $z_{1}(z_{2}^{2}-z_{3}^{2})$. Then we get the decay rate
$\mathcal{O}(t^{-1}\log t)$ by Theorem 4.1 again. This estimate is sharp, see
Remark 1.2.

Next, we give the proof of Proposition 2.7.

###### Proof of Proposition 2.7.

The assertion $(a)$ can be obtained by van der Corput lemma, cf. e.g. [40,
Chapter 8], while $(b)$ has been proved above, see also [12, Table 4.3.2]. Now
we prove $(c)$. We make use of the notations in [26] and let
$S(x)=x_{1}x_{2}x_{3}$. Set $\alpha=(1/3,1/3,1/3)$ and
$\kappa=S|_{\mathbb{S}^{2}}$. A direct computation shows that if
$\theta\in\mathbb{S}^{2}$ satisfies
$\kappa(\theta)=\mbox{d}\kappa|_{\theta}=0$, then $\theta$ is a nondegenerate
critical point, here $d\kappa$ is the differential of $\kappa$. Noting that
Definition 2.2 carries over to real-analytic manifolds, by Lemma 2.6 we have
$M(\kappa,\theta)\curlyeqprec(-1,0).$ Then by [26, Theorem 1] and [26, Theorem
2 (1a)], it suffices to prove the following estimate

(39) $M(h)\curlyeqprec(-1,1)$

holds for $h$ in the set
$\big{\\{}x_{3}^{2}+\mbox{sym}^{2}\mathbb{Z}^{2},x_{3}^{2}+\mbox{sym}^{3}\mathbb{Z}^{2},x_{3}^{2}-x_{1}^{2}x_{2}^{2}\big{\\}},$
where $\mbox{sym}^{2}\mathbb{Z}^{2}$ and $\mbox{sym}^{3}\mathbb{Z}^{2}$ denote
the set of nonzero binary quadratic forms and binary cubic forms (not
equivalent to $x_{1}^{3}$) in $(x_{1},x_{2})$ respectively.

For these two-dimensional cases, it is convenient to combine [42, Theorem 0.6]
with Theorem 4.2 to get the decay rates. An alternative way for the case
$\mbox{sym}^{3}\mathbb{Z}^{2}$ is reducing it to $x_{1}^{2}x_{2}$ or
$D_{4}^{\pm}$ singularities by the arguments in [34, Page 85]. Then we get
that

$M(x_{1}^{2}x_{2})\curlyeqprec\left(-\frac{1}{2},0\right),\;\;\mbox{and}\;\;M(x_{1}^{2}x_{2}^{2})\curlyeqprec\left(-\frac{1}{2},1\right)$

by [24, Theorem 1] and [36, Theorem 1.6], respectively. Consequently, we have
proved (39). Applying [26, Theorem 1] with $(\beta_{1},p_{1})=(-1,1)$,
$(\beta_{2},p_{2})=(-1,0)$ and $|\alpha|=1$, we finish the proof of
Proposition 2.7. ∎

######  Remark 4.4.

As we mentioned before, in the framework of the DW, the perturbation is linear
and the result [23, Theorem 1.1] can also be applied to get the desired
exponents.

###  4.3. Results for all odd $d\geq 3$

In this part we verify the assertion in the discussion after Conjecture 1.5.

######  Theorem 4.5.

For all odd $d\in{\mathbb{N}}$, $d\geq 3$, the inequality

(40) $|I(v_{0},t)|\leqslant C_{d}(1+|t|)^{-\frac{2d+1}{6}}$

holds, where $C_{d}$ only depends on $d$ and
$v_{0}=(\frac{1}{\sqrt{2d}},\cdots,\frac{1}{\sqrt{2d}})\in{\mathbb{R}}^{d}$,
i.e. $\\{v_{0}\\}=\Omega_{d-1}$.

###### Proof.

It follows from Corollary 2.9 that $\phi(v_{0},\cdot)$ has only one critical
point $\xi_{0}=(\frac{\pi}{2},\cdots,\frac{\pi}{2})$. Then from the argument
in Section 2.3.2, it suffices to consider the integral

$\widetilde{I}(v,t)=\int_{{\mathbb{R}}^{d}}e^{it\phi_{ori}(\mu,\xi)}\varphi(\xi)\,d\xi,$

where $\phi_{ori}(\mu,\xi)=\phi(\mu+v_{0},\xi+\xi_{0})$ and $\varphi$ is a
smooth function having support near the origin with $\varphi(0)\neq 0$. We
will take $\mu=0$ and show that (40) holds for $\widetilde{I}(v_{0},t)$.

Step 1. Note that rankHess${}_{\xi}\phi_{ori}(0,0)=1$. By Lemma 3.1, there
exist a linear transform $\Phi$ and $R\in H_{\alpha,d}$ such that

$(\phi_{ori}(\mu,\cdot)\circ\Phi)(y)=c+\Phi(y)\mu+y_{d}^{2}+Q(y^{\prime})+R(y),$

where $c$ is a constant,
$\alpha=(\frac{1}{3},\cdots,\frac{1}{3},\frac{1}{2})$. From the argument in
Step 2 below, we shall see that $Q$ is not ${\mathbb{R}}$-nondegenerate, which
makes it a little complicated. Therefore, we further introduce an invertible
linear transform as follows. Suppose that $d=2m_{0}+1$ for some $m_{0}\geq 1$.
Let $z=\Psi(y)$, where

(41) $\begin{cases}z_{i}=\dfrac{y_{i}+y_{i+1}}{2},\\\
z_{i+1}=\dfrac{y_{i}-y_{i+1}}{2},\;\;i~{}\mbox{odd},\;1\leq i\leq 2m_{0}-1;\\\
z_{2m_{0}+1}=y_{2m_{0}+1}.\end{cases}$

Under this change of coordinates, we get

(42)
$(\phi_{ori}(\mu,\cdot)\circ\Phi\circ\Psi^{-1})(z)=c+z_{2m_{0}+1}^{2}+(\Phi\circ\Psi^{-1})(z)\mu+2Y(z^{\prime})+R\circ\Psi^{-1}(z),$

where $z^{\prime}=(z_{1},\cdots,z_{2m_{0}})$ and

(43)
$Y(z^{\prime})=4\left(\sum_{j=1}^{m_{0}}z_{2j-1}\right)^{3}-\left(\sum_{j=1}^{m_{0}}z_{2j-1}^{3}\right)-3\sum_{j=1}^{m_{0}}z_{2j-1}z_{2j}^{2}.$

Now we take $\mu=0$ and set

$\tilde{\phi}(z)=z_{2m_{0}+1}^{2}+2Y(z^{\prime})+R\circ\Psi^{-1}(z)=:\mathbf{S}(z)+R\circ\Psi^{-1}(z).$

We shall show that $\tilde{\phi}$ is ${\mathbb{R}}$-nondegenerate.

Step 2. In the sequel we will consider $\mathcal{N}(\mathbf{S})$. First note
that $\mathcal{T}(\mathbf{S})$ lies on the simplex

$\Gamma_{\mathbf{S}}=\bigg{\\{}x\in{\mathbb{R}}_{+}^{2m_{0}+1}:\sum_{j=1}^{2m_{0}}\frac{x_{j}}{3}+\frac{x_{2m_{0}+1}}{2}=1\bigg{\\}}.$

We shall show that $\mathbf{S}$ is ${\mathbb{R}}$-nondegenerate. And for every
compact face $\mathcal{P}$ on $\mathcal{N}(\tilde{\phi})$, the
$\mathcal{P}$-part $\tilde{\phi}_{\mathcal{P}}=\mathbf{S}_{\mathcal{P}}$. Thus
$\tilde{\phi}$ is ${\mathbb{R}}$-nondegenerate.

Note that the index of every even number only appears once in (43). Therefore,
for any compact face $\Gamma$ on $\mathcal{N}(\mathbf{S})$, the $\Gamma$-part
$\mathbf{S}_{\Gamma}$ containing the term like $z_{2j-1}z_{2j}^{2}$ must be
${\mathbb{R}}$-nondegenerate. This is because the $2j$-th partial derivative
is $\partial_{2j}S_{\Gamma}(z)=2z_{2j-1}z_{2j}$, then (37) is fulfilled. By
the same consideration, we can also deal with the term $z_{2m_{0}+1}^{2}$.
Therefore, we reduce the problem to consider the left terms in $Y$, which is
(by a permutation)
$Y_{1}(y):=4\left(\sum_{j=1}^{m_{0}}y_{j}\right)^{3}-\left(\sum_{j=1}^{m_{0}}y_{j}^{3}\right).$
We will show that $Y_{1}$ is ${\mathbb{R}}$-nondegenerate in
${\mathbb{R}}^{m_{0}}$. Firstly, we know that $Y_{1}$ itself corresponds to a
compact face. Moreover, $\nabla Y_{1}(y)=0$ if and only if

(44)
$4\left(\sum_{j=1}^{m_{0}}y_{j}\right)^{2}=y_{1}^{2}=y_{2}^{2}=\cdots=y_{m_{0}}^{2}.$

We claim that $\nabla Y_{1}(y)=0$ has only zero solution. Suppose for
contradiction that, by a scaling, $y_{1}^{2}=1$. Then
$\sum_{j=1}^{m_{0}}y_{j}$ must be an integer, which contradicts (44). By a
similar argument, one can also check that the polynomial $Q$ is not
$\mathbb{R}$-nondenegerate.

Therefore, it suffices to consider other compact faces of
$\mathcal{N}(Y_{1})$, the dimensions of which are less than $m_{0}-1$. For
$k\in{\mathbb{N}}$, define the $(k-1)$-simplex

$\Theta_{k-1}=\left\\{x\in{\mathbb{R}}^{k}_{+}:\sum_{j=1}^{k}x_{j}=3,\;j=1,\cdots,k\right\\}.$

Its vertices are $\\{3e_{j}\\}_{j=1}^{k}$, where
$e_{j}=(0,\cdots,0,1,0,\cdots,0)$ is a standard coordinate vertor for
$j=1,\cdots,k$. By the representation theorem of polyhedra (cf.[4, Page 68]),
we have

(45) $\mathcal{N}(Y_{1})=\Theta_{m_{0}-1}+{\mathbb{R}}_{+}^{m_{0}}.$

And compact faces of $\mathcal{N}(Y_{1})$ are exactly those of
$\Theta_{m_{0}-1}$. By [6, Theorem 12.1], any proper face $\Pi$ of
$\Theta_{m_{0}-1}$ is also a simplex having vertices which are exactly those
vertices of $\Theta_{m_{0}-1}$ in $\Pi$. Therefore, without loss of generality
we assume that the vertices set of $\Pi$ is $\\{3e_{j}\\}_{j=1}^{m}$ with some
$n$ satisfying $1\leq n<m_{0}$. Then it is obvious that the $\Pi$-part
$Y_{1,\Pi}$ has the similar expression as $Y_{1}$, which can be written as
$Y_{1,\Pi}(y)=4\left(\sum_{j=1}^{n}y_{j}\right)^{3}-\left(\sum_{j=1}^{n}y_{j}^{3}\right).$
Therefore, by induction we show that $Y_{1}$ is ${\mathbb{R}}$-nondegenerate,
and so is $\mathbf{S}$.

Now we prove that for every compact face $\mathcal{P}$ on
$\mathcal{N}(\tilde{\phi})$, it holds that
$\tilde{\phi}_{\mathcal{P}}=\mathbf{S}_{\mathcal{P}}$. By (42) and the
structure of $R\circ\Psi^{-1}$ as we showed in the proof of Lemma 3.1, it
suffices to consider the term of the following form
$T_{1}=z_{2m_{0}+1}\left(\sum_{j=1}^{m_{0}}z_{2j-1}\right)^{2}$ as well as the
higher-order terms

$\displaystyle T_{2N+1}$
$\displaystyle=\sum_{j=1}^{m_{0}}\Big{(}(z_{2j-1}-z_{2j})^{2N+1}+(z_{2j-1}+z_{2j})^{2N+1}\Big{)}$
$\displaystyle=2\sum_{j=1}^{m_{0}}\left(\sum_{m=1}^{N+1}\tbinom{2N+1}{2m-1}\;z_{2j-1}^{2m-1}\,z_{2j}^{2(N-m+1)}\right),\,N\in{\mathbb{N}},N\geq
2.$

One easily sees that every monomial in $T_{2N+1}$ can be dominated by some
terms in (43), thus is governed by $\mathbf{S}$. As for $T_{1}$, we consider
the unbounded face of $\mathcal{N}(\mathbf{S})$,

$\displaystyle U_{\mathbf{S}}$
$\displaystyle=\\{x\in{\mathbb{R}}_{+}^{2m_{0}+1}:x_{2j}=0,\,1\leq j\leq
m_{0}\\}\cap\mathcal{N}(\mathbf{S})$
$\displaystyle=\bigg{\\{}x\in{\mathbb{R}}_{+}^{2m_{0}+1}:\sum_{j=1}^{m_{0}}\frac{x_{2j-1}}{3}+\frac{x_{2m_{0}+1}}{2}\geq
1,x_{2i}=0,\,1\leq i\leq m_{0}\bigg{\\}}.$

Then $\mathcal{T}(T_{1})\subset\mbox{ri}(U_{\mathbf{S}})$, i.e. the relative
interior of $U_{\mathbf{S}}$. Therefore, the high-degree terms
$R\circ\Psi^{-1}$ are negligible. In conclusion, we have proved that
$\tilde{\phi}$ is ${\mathbb{R}}$-nondegenerate and
$\mathcal{N}(\mathbf{S})=\mathcal{N}(\tilde{\phi})$.

Step 3. Now we compute the Newton distance $t_{0}$ of
$\mathcal{N}(\mathbf{S})$ and the maximal codimension $M_{t_{0}}$ over all
faces containing the point
$\boldsymbol{t_{0}}=(t_{0},\cdots,t_{0})\in{\mathbb{R}}^{d}$. We shall prove
that $t_{0}=\frac{6}{4m_{0}+3}$ and $M_{t_{0}}=1$. We first observe that terms
possessing even number indices as $v_{2j-1}v_{2j}^{2}$ in $\mathbf{S}$ must be
used to obtain $\boldsymbol{t_{0}}$. The points in $\mathcal{T}(\mathbf{S})$
corresponding to these terms are
$A_{2j}=(0,\cdots,0,1,2,0\cdots,0)\in{\mathbb{R}}^{2m_{0}+1}$ for
$j=1,\cdots,m_{0}$, where the two non-zero elements are the ($2j-1$)-th and
$2j$-th components. Furthermore, the point related to the only quadratic term
$v_{2m_{0}+1}^{2}$ is $A_{2m_{0}+1}=(0,\cdots,0,2)$. Now we compute $t_{0}$.
Suppose that there is a point $\boldsymbol{\lambda}=(\lambda,\cdots,\lambda)$
of same components lying on $l_{bi}\cap\mathcal{N}(\mathbf{S})$, where
$l_{bi}=\\{x\in{\mathbb{R}}^{2m_{0}+1}:\,x_{1}=\cdots=x_{2m_{0}+1}\\}$ is the
bisectrix. We hope to find such $\lambda$ in the convex combination of points
in $\mathcal{T}(\mathbf{S})$. Hence we need to give weight $\frac{\lambda}{2}$
to the points $A_{2j}$, $j=1,\cdots,m_{0}$ and $A_{2m_{0}+1}$. We get

(46)
$\frac{\lambda}{2}(A_{2}+\cdots+A_{2m_{0}}+A_{2m_{0}+1})=\left(\frac{\lambda}{2},\>\lambda,\>\frac{\lambda}{2},\>\lambda,\>\cdots,\>\frac{\lambda}{2},\>\lambda,\>\lambda\right)=:\boldsymbol{\lambda_{1}}.$

Now we need to search points in $\mathcal{T}(Y_{1})$ to obtain
$\boldsymbol{\lambda}-\boldsymbol{\lambda_{1}}$. Clearly, the vertices set
$B=\\{3e_{2j-1}\\}_{j=1}^{m_{0}}$ is enough. Indeed, we give weight
$\frac{\lambda}{6}$ to every point in $B$ and add them together as in (46),
then we get $\boldsymbol{\lambda}$. To calculate $\lambda$, note that

$(m_{0}+1)\,\frac{\lambda}{2}+m_{0}\,\frac{\lambda}{6}=1$

gives $\lambda=\frac{6}{4m_{0}+3}=\frac{6}{2d+1}$. Thus by the definition,
$t_{0}\leq\frac{6}{4m_{0}+3}$. However, since all points of
$\mathcal{T}(\mathbf{S})$ lie on $\Gamma_{\mathbf{S}}$, we know
$\mathcal{N}(\mathbf{S})\subset\Gamma_{\mathbf{S}}+{\mathbb{R}}_{+}^{2m_{0}+1}$.
Note that $l_{bi}\cap\Gamma_{S}$ is exactly $\boldsymbol{\lambda}$. We get
$t_{0}\geq\lambda$, and thus $t_{0}=\frac{6}{4m_{0}+3}$.

Step 4. Now we prove that $M_{t_{0}}=1$. Notice that
$\mathcal{A}=\Gamma_{S}\cap\mathcal{N}(\mathbf{S})$ is a compact face of
$\mathcal{N}(\mathbf{S})$. By the computation of $t_{0}$:

(47)
$\boldsymbol{t_{0}}=\frac{t_{0}}{6}(B_{1}+B_{2}+\cdots+B_{m_{0}})+\frac{t_{0}}{2}(A_{2}+\cdots+A_{2m_{0}}+A_{2m_{0}+1}),$

we know that $\boldsymbol{t_{0}}$ is in the convex combination of the set

$A\cup
B:=\\{A_{j}:j=2,4,\cdots,2m_{0},2m_{0}+1\\}\cup\\{B_{j}=3e_{2j-1}:j=1,2,\cdots,m_{0}\\}$

with strictly positive coefficients. The set $A\cup B$ lies on the face
$\mathcal{A}$. Furthermore, these $d$ points of $A\cup B$ are linearly
independent in ${\mathbb{R}}^{d}$, so that are affinely independent. Then the
codimension of the face $\mathcal{A}$ is equal to 1. Thus, by (47) and [6,
Theorem 3.5, Exercises 3.1], $\boldsymbol{t_{0}}$ is in the relative interior
of $\mathcal{A}$ , that is, $\boldsymbol{t_{0}}\in\mbox{ri}\,(\mathcal{A})$.
Then we have $M_{t_{0}}=1$. Thanks to Theorem 4.1, we finish the proof of
Theorem 4.5. ∎

######  Remark 4.6.

It is not clear whether Theorem 4.5 is sharp or not. From the proof, we have
reduced Conjecture 1.5 to the uniform estimates of oscillatory integral with
phase (43). Moreover, similar procedure can also be applied to the setting of
the DKG.

##  5\. Space-time Estimates and nonlinear equation

For two functions $h_{1}$ and $h_{2}$, we write $h_{1}\lesssim h_{2}$ if there
exists $C>0$ independent of time $t$ such that $h_{1}\leq Ch_{2}$. Sometimes
for a function $h$ with variables in both space and time, we write
$h(t)=h(\cdot,t)$ for simplicity.

###  5.1. $l^{p}\rightarrow l^{q}$ estimates

In this part we first give the proof of Theorem 1.3.

###### Proof of Theorem 1.3.

Since the solution is given by $u(t)=G(t)*f$, it suffices to estimate the
Green’s function. In the case $d=4$, by Theorem 1.1 we have

$|G(t)|_{{\infty}}\leqslant C(1+|t|)^{-(\frac{3}{2})^{-}}.$

Recall that the meaning of “$a^{-}$” has been explained before Theorem 1.3.
The other endpoint is

$|G(t)|_{2}\leqslant C,$

which can be obtained directly by the Plancherel theorem. By interpolation we
can deduce that for $2\leqslant p\leqslant\infty$, there exists $C_{p}>0$ such
that

$|G(t)|_{p}\leqslant C_{p}(1+|t|)^{-\zeta_{p}}.$

By Young’s inequality, for $r\geq 2$ satisfying
$1+\frac{1}{q}=\frac{1}{r}+\frac{1}{p}$,

$\displaystyle|u(t)|_{q}=|G(t)*f|_{q}$
$\displaystyle\leqslant|G(t)|_{r}|f|_{p}\leqslant
C_{r}(1+|t|)^{-\zeta_{r}}|f|_{p}$
$\displaystyle=C_{r}(1+|t|)^{-(\frac{3}{2})^{-}\left(1-2(1+\frac{1}{q}-\frac{1}{p})\right)}|f|_{p}$
$\displaystyle=C_{r}(1+|t|)^{-\beta_{p,q}}|f|_{p}.$

To sharpen this estimate, we use the Sobolev embedding on ${\mathbb{Z}}^{4}$,
see e.g. [21, Theorem 3.6]. For any $g\in l^{2}$, we have

(48) $\displaystyle|G(t)*g|_{4}$
$\displaystyle\lesssim\Big{|}|\nabla(G*g)|\Big{|}_{2}=\Big{|}\widehat{|\nabla(G*g)|}\Big{|}_{2}=\Big{|}\omega\cdot\widehat{G*g}\Big{|}_{2}$
$\displaystyle=\Big{|}\omega\cdot\widehat{G}\cdot\widehat{g}\Big{|}_{2}=\Big{|}\omega\cdot\frac{\sin
t\omega}{\omega}\cdot\widehat{g}\Big{|}_{2}$ $\displaystyle=\Big{|}\sin
t\omega\cdot\widehat{g}\Big{|}_{2}\leqslant|\widehat{g}|_{2}=|g|_{2}.$

Moreover, we have $G(x-y,t)=\overline{G(y-x,t)}=G(y-x,t)$ by (15), and hence
$\langle G*f,g\rangle=\langle f,G*g\rangle.$ Combining this with (48), we
obtain

$\displaystyle|G*f|_{2}=\sup_{g\in l^{2},|g|_{2}=1}{|\langle G*f,g\rangle|}$
$\displaystyle=\sup_{g\in l^{2},|g|_{2}=1}{|\langle f,G*g\rangle|}$
$\displaystyle\leqslant\sup_{g\in
l^{2},|g|_{2}=1}{|f|_{\frac{4}{3}}|G*g|_{4}}\lesssim|f|_{\frac{4}{3}}.$

By Young’s inequality we also have

$|G*f|_{\infty}\leqslant(1+|t|)^{-(\frac{3}{2})^{-}}|f|_{1}.$

For $2<q<\infty$, the Riesz-Thorin interpolation theorem yields the desired
result

$|u(t)|_{q}=|G(t)*f|_{q}\leqslant C_{q}(1+|t|)^{-\zeta_{q}}|f|_{p}.$

∎

######  Remark 5.1.

In Theorem 1.3, we only sharpen the estimate on a specific segment, one may
conclude more assertions about $l^{p}\rightarrow l^{q}$ estimates. For
example, a sharper estimate can be proved on the segment with vertices
$(\frac{1}{2},\frac{1}{4})$ and $(1,0)$ by (48) and the interpolation
argument. In this paper we will not pursue the optimal estimate.

As a consequence of Theorem 1.3, for $1\leqslant p<r\leqslant\infty,$ we have

$\displaystyle\|u\|_{L^{q}_{t}l^{r}}^{q}$
$\displaystyle=\|G(t)*f\|_{L^{q}_{t}l^{r}}^{q}=\int_{-\infty}^{+\infty}|G(t)*f|_{r}^{q}dt\lesssim\,|f|_{p}^{q}\int_{0}^{+\infty}(1+|t|)^{-q\beta_{p,r}}\,dt.$

The integral on the right hand side converges if and only if

$3(1/p-1/r-1/2)q>1.$

In addition, if $\frac{1}{r}=(-2)\frac{1}{p}+2$ and $1\leqslant
p\leqslant\frac{4}{3}$, the index $\beta_{p,r}$ can be replaced by
$\zeta_{r}$. In summary, we have the following

######  Theorem 5.2.

Suppose $u$ is the solution to (1) on ${\mathbb{Z}}^{4}$ with $g,F\equiv 0$.
Then for

(49) $1\leqslant
p<r\leqslant\infty,~{}0<q<\infty,~{}\frac{1}{p}>\frac{1}{2}+\frac{1}{r}+\frac{1}{3q},$

it holds that

$\|u\|_{L^{q}_{t}l^{r}}\leqslant C_{p,q,r}|f|_{p}.$

Furthermore, if $(\frac{1}{p},\frac{1}{r})$ lies on the segment with vertices
$(\frac{3}{4},\frac{1}{2})$ and $(1,0)$, the same conclusion holds for
$\frac{1}{r}+\frac{1}{3q}<\frac{1}{2}$ and $(q,r,p)=(\infty,2,\frac{4}{3})$.

This is the first version of Strichartz estimates. In next section, we prove
the second version by a different method.

###  5.2. Strichartz estimates

We first recall a classical result in the abstract setting.

######  Theorem 5.3 (Theorem 1.2 in [29]).

Let $(X,dx)$ be a measure space and $H$ be a Hilbert space. Suppose that for
each time $t\in{\mathbb{R}}$ we have an operator $U(t):H\rightarrow L^{2}(X)$
which obeys the energy estimate

$\|U(t)f\|_{L^{2}_{x}}\lesssim\|f\|_{H}$

and the truncated decay estimate for some $\sigma>0$

$\|U(t)(U(s))^{*}g\|_{L^{\infty}_{x}}\lesssim(1+|t-s|)^{-\sigma}\|g\|_{L^{1}_{x}},$

where $(U(t))^{*}$ denotes the adjoint of $U(t)$. Then the estimates

$\|U(t)f\|_{L^{q}_{t}L^{r}_{x}}\lesssim\|f\|_{H},$
$\left\|\int_{s<t}U(t)(U(s))^{*}F(s)ds\right\|_{L^{q}_{t}L^{r}_{x}}\lesssim\|F\|_{L^{\tilde{q}^{\prime}}_{t}L^{\tilde{r}^{\prime}}_{x}}$

hold for all $\sigma$-admissible exponent pairs $(q,r)$,
$(\tilde{q},\tilde{r})$, i.e. $q,r\geqslant 2$, $(q,r,\sigma)\neq(2,\infty,1)$
and $\frac{1}{q}\leqslant\sigma(\frac{1}{2}-\frac{1}{r})$.

The standard argument of Keel-Tao can be used to obtain the second version of
Strichartz estimates, i.e. Theorem 1.4.

###### Proof of Theorem 1.4.

Our proof relies on the representation of the solution, which is given by
Duhamel’s formula,

(50)
$u(x,t)=\cos(tD)g(x)+\frac{\sin(tD)}{D}f(x)+\int_{0}^{t}\frac{\sin(t-s)D}{D}F(x,s)ds.$

Here we define

(51) $h(D)f:=\mathcal{F}^{-1}(h(\omega))*f$

for any proper function $h$. Note that the definition of $D$ coincides with
the notion $\sqrt{-\Delta}$ in operator theory, see (14). Let
$H=l^{2}({\mathbb{Z}}^{4})$, $X={\mathbb{Z}}^{4}$ and the truncated operators
$U_{\pm}(t)=\chi_{\\{t\geqslant 0\\}}e^{\pm itD}$ for any fixed
$t\in{\mathbb{R}}$ in Theorem 5.3. One easily checks that $U_{\pm}(t)$ satisfy
the following properties:

  * (P1)

$U_{\pm}(t):l^{2}\rightarrow l^{2}$ and $|U_{\pm}(t)f|_{2}\leqslant|f|_{2}$;

  * (P2)

$(U_{+}(t))^{*}=U_{-}(t)$;

  * (P3)

$e^{itD}e^{isD}f=e^{i(t+s)D}f$.

By (P2), (P3) and Young’s inequality, we deduce that

$\displaystyle|U_{+}(t)(U_{+}(s))^{*}g|_{\infty}$
$\displaystyle=|U_{+}(t)U_{-}(s)g|_{\infty}\leqslant|e^{itD}e^{-isD}g|_{\infty}$
$\displaystyle=|\mathcal{F}(e^{i(t-s)\omega})*g|_{\infty}\leqslant|\mathcal{F}(e^{i(t-s)\omega})|_{\infty}|g|_{1}$
$\displaystyle\lesssim(1+|t-s|)^{-\frac{3}{2}}\log(2+|t-s|)|g|_{1},$

which is the truncated decay estimate. In the last line we have used the fact
that

(52)
$|\mathcal{F}(e^{it\omega})|_{\infty}\lesssim(1+|t|)^{-\frac{3}{2}}\log(2+|t|).$

The proof of (52) is similar to that of Theorem 1.1, see also the remark in
[38, Page 692]. The energy estimate is contained in (P1), so Theorem 5.3
implies

(53) $\|e^{itD}f\|_{L^{q}_{t}l^{r}}\lesssim|f|_{2}$

and

(54)
$\left\|\int_{0}^{t}e^{i(t-s)D}F(x,s)ds\right\|_{L^{q}_{t}l^{r}}\lesssim\|F\|_{L_{t}^{\tilde{q}^{\prime}}l^{\tilde{r}^{\prime}}}$

for $t,s\in{\mathbb{R}}$ as long as the index pairs $(q,r)$ and
$(\tilde{q},\tilde{r})$ are $(\frac{3}{2})^{-}$-admissible, i.e. $q,r\geqslant
2$ and

$\frac{1}{q}\leqslant\left(\frac{3}{2}\right)^{-}\left(\frac{1}{2}-\frac{1}{r}\right)<\frac{3}{2}\left(\frac{1}{2}-\frac{1}{r}\right).$

With (50), (53) and (54) in hands, by Euler’s formula, we have

(55) $\displaystyle\|u\|_{L^{q}_{t}l^{r}}$
$\displaystyle\leqslant\|\cos(tD)g\|_{L^{q}_{t}l^{r}}+\left\|\frac{\sin(tD)}{D}f\right\|_{L^{q}_{t}l^{r}}+\;\left\|\int_{0}^{t}\frac{\sin(t-s)D}{D}F(x,s)ds\right\|_{L^{q}_{t}l^{r}}$
$\displaystyle\lesssim|g|_{2}+\left|\frac{1}{D}f\right|_{2}+\left\|\frac{1}{D}F\right\|_{L_{t}^{\tilde{q}^{\prime}}l^{\tilde{r}^{\prime}}}.$

The boundedness of the operator $\frac{1}{D}$ is considered in Proposition 5.4
below. Apply it to the final line of (55), with $b=2$ for the second term and
$b=\tilde{r}^{\prime}$ for the third term. We finish the proof after figuring
out the index relation as (11) and (12). ∎

######  Proposition 5.4.

Suppose that $b>\frac{d}{d-1}$, then
$\frac{1}{D}:l^{a}({\mathbb{Z}}^{d})\rightarrow l^{b}({\mathbb{Z}}^{d})$ is
bounded for $a\in[1,\frac{bd}{b+d}]$.

To see Proposition 5.4 is valid, we first need the following lemma, see also
[32, 33]. Here we give an elementary proof.

######  Lemma 5.5.

For $d\geq 2$, there exist constants $\widetilde{C_{d}},\widetilde{c_{d}}>0$
depending on $d$ such that

$\widetilde{c_{d}}(1+|x|)^{-d-\delta}\leq\;\left|\int_{\mathbb{T}^{d}}e^{ix\cdot\xi}\,\omega(\xi)^{\delta}\,d\xi\right|\;\,\leq\widetilde{C_{d}}(1+|x|)^{-d-\delta}$

holds for $|x|$ large enough and $\delta=\pm 1$.

###### Proof.

We only prove the case $\delta=-1$, the other case is similar. By the
distribution theory,

(56) $\mathcal{F}({|\cdot|^{w}})=C_{d,w}|\cdot|^{-d-w}$

for all $w\in{\mathbb{C}}$ with $-d<\Re w<0$, where $\Re w$ is the real part
of $w$, and $\mathcal{F}$ is the Fourier transform of a distribution in
${\mathbb{R}}^{d}$. We then have

$\int_{\mathbb{T}^{d}}e^{ix\cdot\xi}\frac{1}{\omega(\xi)}d\xi=\int_{\mathbb{R}^{d}}\frac{\eta(\xi)}{w(\xi)}e^{ix\cdot\xi}d\xi=\mathcal{F}{\left(\frac{\eta}{w}\right)}(x)$

with the same cut-off $\eta$ as in (21). Since $\omega^{-1}\in
L^{1}({\mathbb{T}^{d}})$ for $d\geq 2$, the notation $\mathcal{F}$ here also
denotes the Fourier transform in ${\mathbb{R}}^{d}$.
$\mathcal{F}{\left(\frac{\eta}{\omega}\right)}$ can be decomposed into two
parts as below.

$\mathcal{F}{\left(\frac{\eta}{\omega}\right)}=\mathcal{F}{\left(\left(\frac{\eta(\cdot)|\cdot|}{\omega(\cdot)}-1\right)|\cdot|^{-1}\right)}+\mathcal{F}{\left(|\cdot|^{-1}\right)}=:\mathcal{F}(g_{1})+\mathcal{F}(g_{2}).$

Inserting a smooth cut-off $\psi$ with $\psi=1$ near the origin, we write

$\mathcal{F}{\left(\frac{\eta}{\omega}\right)}=\mathcal{F}(g_{1}\psi)+\mathcal{F}(g_{1}(1-\psi))+\mathcal{F}(g_{2})=:I_{1}+I_{2}+I_{3}.$

By (56) with $w=-1$, we get that $I_{3}$ behaves like $|x|^{-d+1}$ as
$|x|\rightarrow\infty$. Moreover, we claim that

$I_{1}=\mathcal{O}(|x|^{-d}),~{}~{}I_{2}=\mathcal{O}(|x|^{-N})$

for all large $N$ as $|x|\rightarrow\infty$. For $I_{1}$, a direct computation
and an induction argument lead to the derivative estimates

$\partial^{\beta}(g_{1}(x)\psi(x))=\mathcal{O}(|x|^{1-|\beta|})$

for all $\beta\in\mathbb{N}^{d}$. When $|\beta|=d$, we have
$\partial^{\beta}(g_{1}\psi)\in L^{1}(\mathbb{R}^{d})$. Then we get the
desired estimate for $I_{1}$. As for $I_{2}$, a similar argument shows that

$\partial^{N}(g_{1}(x)(1-\psi(x)))\in L^{1}(\mathbb{R}^{d})$

for all $N\in\mathbb{N}$ large enough. The proof is completed. ∎

###### Proof of Proposition 5.4.

In view of the natural embedding of $l^{p}$ spaces, it suffices to prove this
result for the endpoint $a=\frac{bd}{b+d}$. By the asymptotic estimate in
Lemma 5.5, we have

(57)
$\bigg{|}\frac{1}{D}f\bigg{|}_{b}\lesssim\big{|}(1+|\cdot|)^{-d+1}*|f|\big{|}_{b}=\sup_{h\in
l^{b^{\prime}},|h|_{b^{\prime}}=1}{\big{|}\langle(1+|\cdot|)^{-d+1}*|f|,h\rangle\big{|}}.$

The discrete Hardy-Littlewood-Sobolev (DHLS for abbreviation) inequality, [22,
Section 1], and Hölder inequality yield

(58)
$\displaystyle\big{|}\left\langle(1+|\cdot|)^{-d+1}*|f|,h\right\rangle\big{|}$
$\displaystyle\leq\sum_{x,y\in{\mathbb{Z}}^{d},x\neq
y}\frac{|f(x)h(y)|}{(1+|x-y|)^{d-1}}+\sum_{w\in{\mathbb{Z}}^{d}}|f(w)h(w)|$
$\displaystyle\;\lesssim|f|_{\frac{bd}{b+d}}+|f|_{b}\lesssim|f|_{\frac{bd}{b+d}}.$

The last inequality follows from the natural embedding
$l^{\frac{bd}{b+d}}\subset l^{b}$ since $\frac{bd}{b+d}<b$. The restriction on
the index in the DHLS gives $b>\frac{d}{d-1}$. Then we insert (58) into (57)
and finish the proof. ∎

A direct corollary of Theorem 1.4 is as follows.

######  Corollary 5.6.

Suppose that $u$ is the solution to (1) on ${\mathbb{Z}}^{4}$ with $g,F\equiv
0$, then we have

(59) $\|u\|_{L^{q}_{t}l^{r}}\leqslant C|f|_{{\frac{4}{3}}}$

for $(q,r)$ satisfying

$q,r\geqslant
2,~{}\frac{1}{q}<\frac{3}{2}\left(\frac{1}{2}-\frac{1}{r}\right).$

######  Remark 5.7.

Theorem 5.2 and Corollary 5.6 are both Strichartz type estimates, and have
advantages of their own. For instance, when $p=\frac{4}{3}$, Corollary 5.6
admits a wider range of $(q,r)$. However, if we fix an index pair $(p,r)$,
Theorem 5.2 may provide a wider range of $q$ than Corollary 5.6. Moreover,
Theorem 5.2 enables us to deal with initial data arbitrarily close to $l^{2}$,
which may be useful in taking limit processes.

Strichartz estimates on ${\mathbb{Z}}^{2}$ and ${\mathbb{Z}}^{3}$ can be
obtained similarly by the two methods above except for one case. In fact,
there are some difficulties if we apply Theorem 5.3 when $d=2$. Here we state
the results on ${\mathbb{Z}}^{3}$ analogue to Theorem 1.4.

######  Theorem 5.8.

Suppose that $u$ is the solution to (1) on ${\mathbb{Z}}^{3}$. Then

$\|u\|_{L^{q}_{t}l^{r}}\leqslant
C_{q,r,\tilde{p},\tilde{q}}\left(|g|_{2}+|f|_{{\frac{6}{5}}}+||F||_{L_{t}^{\tilde{q}^{\prime}}l^{\tilde{p}}}\right)$

for

$q,r\geqslant
2,~{}\frac{1}{q}\leqslant\frac{7}{6}\left(\frac{1}{2}-\frac{1}{r}\right)$

and

$\tilde{q}\geqslant
2,~{}1<\tilde{p}\leqslant\frac{6}{5},~{}\frac{1}{\tilde{q}}\leqslant\frac{7}{6}\left(\frac{1}{\tilde{p}}-\frac{5}{6}\right).$

###  5.3. Nonlinear equations

We use space-time estimates to obtain global well-posedness for small initial
data. In this section we suppose $g\equiv 0$ in (1) without loss of
generality.

######  Definition 5.9.

We say $F(u)$ has the $k$-th order nonlinearity if there is a constant $C$
such that

$|F(u)|_{p}\leqslant C|u|^{k}_{kp}$

for all $1\leqslant p\leqslant\infty$. A simple example is $F(u)=|u|^{k-1}u$.

######  Theorem 5.10.

Suppose that $F$ has the order of nonlinearity $k\geqslant 4$. If $f$ has
sufficiently small norm in $l^{1}$, then the solution to (1) exists in
$l^{p}~{}(2\leqslant p\leqslant\infty)$ for all time and the decay estimate in
$l^{p}$ holds

$|u(t)|_{p}\leqslant C_{p}(1+t)^{-\zeta_{p}}.$

###### Proof.

We will use a standard contraction mapping argument. Recall that $\zeta_{p}$
has been defined in Theorem 1.3. For $p=k$, consider the metric space

$\mathcal{M}=\left\\{h:|h|_{\mathcal{M}}=\sup_{t\geqslant
0}(1+t)^{\zeta_{k}}|h(\cdot\,,t)|_{l^{k}}\leqslant 2C_{0}|f|_{l^{1}}\right\\}$

with $C_{0}$ to be determined later. We define the map $\Lambda$ on
$\mathcal{M}$,

(60) $\Lambda u(t)=G(t)*f+\int_{0}^{t}G(t-s)*F(u(s))ds.$

We first prove that $\Lambda u\in\mathcal{M}$. Taking norms in $l^{k}$ to
(60), it holds that

$|\Lambda
u(t)|_{k}\leqslant(1+t)^{-\zeta_{k}}|f|_{1}+\int_{0}^{t}(1+t-s)^{-\zeta_{k}}|u(s)|^{k}_{k}ds,$

in which we have used Theorem 1.3 with $\beta_{1,k}=\zeta_{k}$ and the fact
$|F(u)|\leqslant|u|_{k}^{k}$. Note that

$1+t\leqslant(1+t-s)(1+s)$

for $0\leqslant s\leqslant t$, we obtain

$(1+t)^{\zeta_{k}}|\Lambda
u(t)|_{k}\leqslant|f|_{1}+|u|_{\mathcal{M}}^{k}\int_{0}^{t}(1+s)^{\zeta_{k}-k\zeta_{k}}ds=:|f|_{1}+|u|_{\mathcal{M}}^{k}I_{k}.$

To make sure the integrability of $I_{k}$, we need $\zeta_{k}-k\zeta_{k}<-1$,
implying $k>3$. Thus

$|\Lambda u|_{\mathcal{M}}\leqslant|f|_{1}+I_{k}|u|_{\mathcal{M}}^{k}\leqslant
C_{0}(|f|_{1}+|u|_{\mathcal{M}}^{k}).$

If $|f|_{1}$ is sufficiently small such that
$|u|_{\mathcal{M}}^{q}\leqslant|f|_{1}$, we have

$|\Lambda u|_{\mathcal{M}}\leqslant
C_{0}(|f|_{1}+|u|_{\mathcal{M}}^{k})\leqslant 2C_{0}|f|_{1}.$

Thus $\Lambda u\in\mathcal{M}$. Similarly, one can show that $\Lambda$ is a
contraction with

$\displaystyle(1+t)^{\zeta_{k}}|\Lambda u_{1}-\Lambda u_{2}|_{k}$
$\displaystyle\leqslant
CI_{k}(|u_{1}|_{\mathcal{M}}^{k-1}+|u_{2}|_{\mathcal{M}}^{k-1})|u_{1}-u_{2}|_{\mathcal{M}}$
$\displaystyle\leqslant
2CI_{k}(2C_{0}|f|_{1})^{k-1}|u_{1}-u_{2}|_{\mathcal{M}}$

by adjusting $C_{0}$ properly. Therefore, $\Lambda$ admits a fixed point in
$\mathcal{M}$, which is the solution to (1). As for general $2\leqslant
p\leqslant\infty$, we repeat the same argument with $\zeta_{p}$. ∎

We can also prove the following result similarly.

######  Theorem 5.11.

Suppose that $F(u)=|u|^{k-1}u$ with $k\geqslant 3$. Let $p=\frac{2k+1}{2k}$
and $q=\frac{2k+1}{2}$. If $f$ has sufficiently small $l^{p}$ norm, then the
solution to (1) exists in $l^{q}$ for all time and

$|u(t)|_{q}\leqslant C_{k}(1+t)^{-\frac{3}{2}\left(1-\frac{4}{2k+1}\right)}.$

## Acknowledgement

The authors wish to express gratitude to Prof. J.-C. Cuenin, I. A. Ikromov and
D. Müller for their helpful explanations. C.Bi is indebted to Prof. H.-Q. Li
for many useful suggestions. B.H. is supported by NSFC, No. 11831004, and by
Shanghai Science and Technology Program [Project No. 22JC1400100].

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