# Global solution of the $3D$ relativistic Vlasov-Maxwell system for the large
radial data

Xuecheng Wang  YMSC, Tsinghua University, Beijing, China, 100084
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

We prove global existence of the $3D$ relativistic Vlasov-Maxwell system for a
class of arbitrary large regular initial data with spherical symmetry, in
which the initial distribution function of particles is assumed to decay fast
but polynomially towards infinity.

##  1\. Introduction

The $3D$ relativistic Vlasov-Maxwell system is one of the fundamental models
in the collisionless plasma physics. It reads as follows,

$(\textup{RVM})\qquad\left\\{\begin{array}[]{c}\partial_{t}f+\hat{v}\cdot\nabla_{x}f+(E+\hat{v}\times
B)\cdot\nabla_{v}f=0,\\\ \nabla\cdot
E=4\pi\displaystyle{\int_{\mathbb{R}^{3}}f(t,x,v)dv},\qquad\nabla\cdot B=0,\\\
\partial_{t}E=\nabla\times
B-4\pi\displaystyle{\int_{\mathbb{R}^{3}}f(t,x,v)\hat{v}dv},\quad\partial_{t}B=-\nabla\times
E,\\\ f(0,x,v)=f_{0}(x,v),\quad E(0,x)=E_{0}(x),\quad
B(0,x)=B_{0}(x).\end{array}\right.$ (1.1)

where
$f:\mathbb{R}_{t}\times\mathbb{R}_{x}^{3}\times\mathbb{R}_{v}^{3}\longrightarrow\mathbb{R}_{+}$
denotes the distribution function of particles,
$E,B:\mathbb{R}_{t}\times\mathbb{R}_{x}^{3}\longrightarrow\mathbb{R}^{3}$
denote the electromagnetic field, $\hat{v}:=v/\sqrt{1+|v|^{2}}$. As a result
of direct computations, we can reduce the Maxwell system into the standard
wave equations as follows,

$\partial_{t}^{2}B-\Delta
B={-4\pi\int_{\mathbb{R}^{3}}\hat{v}\times\nabla_{x}f(t,x,v)dv},\quad\partial_{t}^{2}E-\Delta
E=-4\pi\int_{\mathbb{R}^{3}}\hat{v}\partial_{t}f(t,x,v)dv-4\pi\int_{\mathbb{R}^{3}}\nabla_{x}f(t,x,v)dv.$
(1.2)

The following conservation law holds for the RVM system,

$\mathcal{H}(t):=\int_{\mathbb{R}^{3}}|E^{2}(t,x)|+|B^{2}(t,x)|dx+8\pi\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}\sqrt{1+|v|^{2}}f(t,x,v)dv=\mathcal{H}(0)$
(1.3)

Moreover for any $p\in[1,\infty]$, we have

$\|f(t,x,v)\|_{L^{p}_{x,v}}=\|f(0,x,v)\|_{L^{p}_{x,v}}.$ (1.4)

Note that the following system of equations satisfied by the backward
characteristics,

$\left\\{\begin{array}[]{l}\displaystyle{\frac{d}{ds}X(s;t,{x}_{0},{v}_{0})=\widehat{V}(s;t,{x}_{0},{v}_{0})}\\\
\displaystyle{\frac{d}{ds}V(s;t,{x}_{0},{v}_{0})=K(s,X(s;t,{x}_{0},{v}_{0}),V(s;t,{x}_{0},{v}_{0}))}\\\
X(t;t,{x}_{0},{v}_{0})={x}_{0},V(t;t,{x}_{0},{v}_{0})={v}_{0},\\\
\end{array}\right.$ (1.5)

where $K(s,x,v):=E(s,x)+\hat{v}\times B(s,x)$. Due to the transport nature of
the Vlasov equation, we have
$f(t,x_{0},v_{0})=f(s,X(s;t,{x}_{0},{v}_{0}),V(s;t,{x}_{0},{v}_{0}))$ for any
$s\in(-T,T)$, where $T$ denotes the maximal time of existence.

There is a large literature in the study of the Vlasov-Maxwell system. Large
data global solutions for the RVM have been constructed in two dimensions and
two and half dimensions, we refer readers to [4, 13] and reference therein for
more comprehensive introduction and more details. Moreover, global solutions
for the RVM in the three dimensions for small data have also been constructed,
we refer readers to [6, 19, 21] for more details.

In this paper, we are mainly interested in the large data Cauchy problem of
RVM in the $3D$ case. The result of Glassey-Strauss [5] says that the
classical solution can be globally extended as long as the particle density
has compact support in $v$ for all the time. A new proof of this result based
on Fourier analysis was given by Klainerman-Staffilani [9], which adds a new
perspective to the study of $3D$ RVM system, see also [3, 14, 20]. An
interesting line of research is the continuation criterion for the global
existence of the Vlasov-Maxwell system. In [7], Glassey-Strauss showed that
the lifespan of the solution of the relativistic Vlasov-Maxwell system can be
continued if the initial data decay at rate $|v|^{-7}$ as
$|v|\rightarrow\infty$ and $\|(1+|v|)f(t,x,v)\|_{L^{\infty}_{x}L^{1}_{v}}$
remains bounded for all time. An improvement of this result and a new
continuation criterion was given by Luk-Strain [13], which says that a regular
solution can be extended as long as
$\|(1+|v|^{2})^{\theta/2}f(t,x,v)\|_{L^{q}_{x}L_{v}^{1}}$ remains bounded for
$\theta>2/q,2<q\leq+\infty$, see also Kunze[10], Pallard[15], and Patel[16]
for the recent improvements on the continuation criterion.

We assume that the initial data $(f_{0}(x,v),E_{0}(x),B_{0}(x))$ are
spherically symmetric in the following sense,

$f_{0}(Rx,Rv)=f_{0}(x,v),\quad E_{0}(Rx)=RE_{0}(x),\quad
B_{0}(Rx)=RB_{0}(x),\quad\forall R\in SO(3).$ (1.6)

As a result of direct computation and the uniqueness of solution, we know that
the above radial symmetry property can be propagated from the initial data.
Our main result in this paper is summarized as follows.

######  Theorem 1.1.

Assume that initial data $(f_{0},E_{0},B_{0})$ are radial (in the sense of
(1.6))$,f_{0}(x,v)\in H^{s}(\mathbb{R}_{x}^{3}\times\mathbb{R}_{v}^{3})$,
$E_{0},B_{0}\in H^{s}(\mathbb{R}^{3})$, $s\in\mathbb{Z}_{+},s\geq 6$.
Moreover, for $N_{0}:=10^{8}$, we assume that the following estimate holds for
the initial distribution function $f_{0}(x,v),$

$\sum_{\alpha\in\mathbb{Z}_{+}^{6},|\alpha|\leq
s}\|(1+|x|+|v|)^{N_{0}}\nabla_{x,v}^{\alpha}f_{0}(x,v)\|_{L^{2}_{x,v}}<+\infty.$
(1.7)

Then the relativistic Vlasov-Maxwell system (1.1) admits global solution
$(f(t),E(t),B(t))$ in $H^{s}(\mathbb{R}_{x}^{3}\times\mathbb{R}_{v}^{3})\times
H^{s}(\mathbb{R}^{3}_{x})\times H^{s}(\mathbb{R}^{3}_{x})$.

###  1.1. Local theory and the main idea of proof

Since the assumption imposed on the initial data in (1.7) is stronger than the
assumptions required for initial data in the work of Luk-Strain [13], the
continuation criteria obtained there can be applied directly in this paper.
From the work of Luk-Strain [13], we know that the $3D$ relativistic Vlasov-
Maxwell system is local well-posed and the regularity can be propagated within
the time interval of existence $[0,T)$, where $T$ denotes the maximal time of
existence. Moreover, from the work of Luk-Strain [13], we know that the
lifespan can be extended to $[0,T+\epsilon]$ for some positive number
$\epsilon$ if the following assumption holds,

$\sup_{t\in[0,T)}\sup_{x_{0},v_{0}\in\mathbb{R}^{3}}\int_{0}^{T}\big{(}|E(s,X(s;t,x_{0},v_{0}))|+|B(s,X(s;t,x_{0},v_{0}))|\big{)}ds<+\infty.$
(1.8)

Readers are refereed to [13][Theorem 5.7] for more details.

The main goal of this paper is to show that the acceleration accumulated along
characteristics is indeed uniformly bounded for all time. Hence finishing the
proof of global regularity for the $3D$ relativistic RVM system.

To this end, we propagate a high order moment of the distribution function of
particles. By using the Glassey-Strauss decomposition and the spherical
symmetry of the solution, we show that the quantity in (1.8) is controlled
from the above by the moment of the distribution function, see the estimate
(4.17) in Lemma 4.4.

Actually, we not only show that the boundedness of the high order moment but
also show that it grows at most polynomially over time, see the estimates
(3.28) and (3.29). To this end, as summarized in the Proposition 3.1, the main
observation is that the majority of particles, which are localized around zero
due to the polynomial decay assumption on the initial data, will not be
accelerated much after the speed of particles reaches a certain level.

Intuitively speaking, particles will travel toward infinity when the speed of
particles reaches a certain level. Due to the spherical symmetry, the
electromagnetic field is localized around zero and it becomes weaker as the
radius becomes larger. As a result, the accumulate acceleration caused by the
electromagnetic field is not strong. To get more transparent intuition, we
refer readers to [22] for a similar result in the $3D$ relativistic Vlasov-
Poisson system, which is a simpler model of RVM.

To prove Proposition 3.1, we measure carefully the gain and the loss of using
the smoothing effects. One type of the smoothing effects comes from the
oscillation in time for the electromagnetic field itself. The other type of
the smoothing effect, as pointed out by Klainerman-Staffilani [9], is that the
integration of electromagnetic field along the characteristic is smoother.

This paper is organized as follows.

  1. $\bullet$

In section 2, we introduce the notation and prove two basic lemmas used in
this paper.

  2. $\bullet$

In section 3, we introduce the set-up of propagation of moment and prove the
Theorem 1.1 under the assumption that we have a good control of the
electromagnetic field and the increment of characteristics in terms of the
moment of the distribution function.

  3. $\bullet$

In section 4, we give a rough control of the electromagnetic field based on
the Glassey-Strauss decomposition.

  4. $\bullet$

In section 5, by exploiting the smoothing effect, we use a Fourier method to
control the increment of the magnitudes of the spatial characteristic
$X(s;t,x_{0},v_{0})$ and the velocity characteristic $V(s;t,x_{0},v_{0})$ over
time.

Acknowledgment The author is supported by NSFC-11801299.

##  2\. Preliminary

For any two numbers $A$ and $B$, we use $A\lesssim B$ and $B\gtrsim A$ to
denote $A\leq CB$, where $C$ is an absolute constant. We use the convention
that all constants which only depend on the initial data, e.g., the conserved
quantities ($\|f(t,x,v)\|_{L^{p}_{x,v}},$ $p\in[1,\infty]$,
$\|(1+|v|^{2})f(t,x,v)\|_{L^{1}_{x,v}},$ $\|E(t)\|_{L^{2}},\|B(t)\|_{L^{2}}$),
will be treated as absolute constants.

For any two vectors $v,u\in\mathbb{R}^{3}$, we use $\angle(v,u)$ to denote the
angle between $v$ and $u$ and use the convention that $\angle(v,u)\in[0,\pi]$.
For any $r\in\mathbb{R}_{+}$, we use $r_{-}$ to denote $\min\\{r,1\\}.$ For
any vector $v\in\mathbb{R}^{3}/\\{0\\}$, we use
$\tilde{v}:=v/|v|\in\mathbb{S}^{2}$ to denotes the direction of $v$. Note
that, for any $v\in\mathbb{R}^{3}/\\{0\\}$, we define
$S_{v}:=\\{\omega,\omega\in\mathbb{S}^{2},\omega\cdot\tilde{v}=0\\}$ to be the
great circle on sphere that is orthogonal to the direction $\tilde{v}$.

We fix an even smooth function $\tilde{\psi}:\mathbb{R}\rightarrow[0,1]$,
which is supported in $[-3/2,3/2]$ and equals to “$1$” in $[-5/4,5/4]$. For
any $k\in\mathbb{Z}$, we define the cutoff functions $\psi_{k},\psi_{\leq
k},\psi_{\geq k}:\cup_{n=1,3}\mathbb{R}^{n}\longrightarrow\mathbb{R}$ as
follows,

$\psi_{k}(x):=\tilde{\psi}(|x|/2^{k})-\tilde{\psi}(|x|/2^{k-1}),\quad\psi_{\leq
k}(x):=\tilde{\psi}(|x|/2^{k})=\sum_{l\leq k}\psi_{l}(x),\quad\psi_{\geq
k}(x):=1-\psi_{\leq k-1}(x).$

Moreover, for any $l,n\in\mathbb{Z},l\geq n$, we define the cutoff function
$\varphi_{l;n}(\cdot)$ with threshold $n$ as follows,

$\varphi_{l;n}(x)=\left\\{\begin{array}[]{ll}\psi_{\leq
n}(x)&\textup{if\,\,}l=n\\\ \psi_{l}(x)&\textup{if\,\,}l>n.\\\
\end{array}\right.$ (2.1)

If $n=0$, then we use the convention that $\varphi_{j}(\cdot)$ denotes
$\varphi_{j;0}(\cdot).$

We first record the classic Kirchhoff’s formula, which allows us to represent
the solution of linear wave in physical space.

######  Lemma 2.1.

For any $t\in\mathbb{R},x\in\mathbb{R}^{3}$, the following equality holds,

$|\nabla|^{-1}\sin(t|\nabla|)h(x)=\frac{1}{4\pi}t\int_{\mathbb{S}^{2}}h(x+t\theta)d\theta,$
(2.2)
$|\nabla|^{-1}\cos(t|\nabla|)h(x)=\frac{1}{4\pi}\int_{\mathbb{S}^{2}}|\nabla|^{-1}h(x+t\theta)d\theta+\frac{1}{4\pi}\int_{\mathbb{S}^{2}}t\theta\cdot\frac{\nabla}{|\nabla|}h(x+t\theta)d\theta.$
(2.3)

###### Proof.

Note that

$\int_{\mathbb{R}^{3}}e^{-ix\cdot\xi}\int_{\mathbb{S}^{2}}h(x+t\theta)d\theta
dx=\int_{\mathbb{S}^{2}}e^{it\xi\cdot\theta}\hat{h}(\xi)d\theta=2\pi\hat{h}(\xi)\int_{0}^{\pi}e^{it|\xi|\cos(\phi)}\sin(\phi)d\phi=\frac{4\pi\sin(t|\xi|)}{t|\xi|}\hat{h}(\xi).$
(2.4)

Hence finishing the proof of the desired formula (2.2). Our desired equality
(2.3) holds after taking derivative with respect to “$t$” for the equality
(2.2). ∎

Recall the equations satisfied by the electromagnetic field in (1.2) and the
Kirchhoff’s formula in (2.2). From the Duhamel’s formula, the following
decomposition holds after we do dyadic decomposition for the velocity
variable,

$K(t)=K_{free}(t)+\sum_{j\in\mathbb{Z}_{+}}K_{j}(t),\quad K\in\\{E,B\\}.$
(2.5)

where $K_{free}(t),K\in\\{E,B\\},$ denote the linear wave solution determined
by the initial data of RVM (1.1),

$K_{free}(t)=\cos(t|\nabla|)K_{0}+\sin(t|\nabla|)|\nabla|^{-1}(\partial_{t}K)\big{|}_{t=0},\quad
K\in\\{E,B\\},$ (2.6)
$E_{j}(t):=-\int_{0}^{t}\int_{\mathbb{R}^{3}}\int_{\mathbb{S}^{2}}(t-s)\big{(}\hat{v}\partial_{t}f(s,x+(t-s)\theta,v)+\nabla_{x}f(s,x+(t-s)\theta,v)\big{)}\varphi_{j}(v)d\theta
dvds,$ (2.7)
$B_{j}(t):=-\int_{0}^{t}\int_{\mathbb{S}^{2}}\int_{\mathbb{R}^{3}}(t-s)\hat{v}\times\nabla_{x}f(s,x+(t-s)\theta,v)\varphi_{j}(v)d\theta
dvds.$ (2.8)

The benefit of radial symmetry is mainly exploited in the following Lemma.
Essentially speaking, it says that the average on sphere for an integrable
radial function is well controlled if the center of sphere is far away from
zero. Also, it’s very natural that the radial symmetry won’t provide any gain
when the center is close to zero, which explains why the estimates (2.9) and
(2.10) behaves badly when $r$ approaches to zero.

######  Lemma 2.2.

For any fixed $\omega_{0}\in\mathbb{S}^{2}$, any radial function
$h:\mathbb{R}^{3}\longrightarrow\mathbb{C}$, the following estimate holds for
any
$l\in\mathbb{Z}\cap(-\infty,2],x\in\mathbb{R}^{3}/\\{0\\},s\in\mathbb{R}_{+}/\\{|x|\\}$,

$\big{|}\int_{\mathbb{S}^{2}}h(x+s\omega)\psi_{\leq
l}(\angle(\omega,\omega_{0}))d\omega\big{|}\lesssim\frac{2^{l}}{(2^{l}+|\tilde{x}\times\omega_{0}|)rs}\min\\{\frac{1}{|r-s|}\|h\|_{L^{1}_{x}},\sup_{z\in\mathbb{R}^{3},|z|\in[|r-s|,r+s]}s|z||h(z)|\\},$
(2.9)

where $\tilde{x}:=x/|x|$ and $r:=|x|$. Moreover, for any fixed
$a,b\in\mathbb{R},$ and any radial function
$f:\mathbb{R}_{x}^{3}\times\mathbb{R}_{v}^{3}\longrightarrow\mathbb{R}_{+}$ in
the sense of (1.6), the following estimate holds,

$\big{|}\int_{\mathbb{R}^{3}}\int_{\mathbb{S}^{2}}\int_{S_{v}}f(x-a\tilde{v}-b\omega_{v}+s\omega,v)\psi_{\leq
l}(\angle(\omega,\omega_{0}))d\omega_{v}d\omega
dv\big{|}\lesssim\frac{2^{l}\|f(x,v)\|_{L^{1}_{x,v}}}{(2^{l}+|\tilde{x}\times\omega_{0}|)rs|r-s|}.$
(2.10)

where for any fixed
$v\in\mathbb{R}^{3}/\\{0\\},S_{v}:=\\{\theta\in\mathbb{S}^{2},\theta\cdot\tilde{v}=0\\}$.

###### Proof.

Since $h$ is radial and fixed $\omega_{0}$ is arbitrary, without loss of
generality, we assume that $x=(r,0,0),r:=|x|$. Let
$\omega:=(\cos\theta,\sin\theta\cos\phi,\sin\theta\sin\phi)$ and
$\omega_{0}:=(\cos\theta_{0},\sin\theta_{0}\cos\phi_{0},\sin\theta_{0}\sin\phi_{0})$.
Due to the cutoff function $\psi_{\leq l}(\angle(\omega,\omega_{0}))$, we know
that

$|supp_{\phi}(\psi_{\leq
l}(\angle(\omega,\omega_{0})))|\lesssim\frac{2^{l}}{2^{l}+\sin\theta_{0}}\sim\frac{2^{l}}{2^{l}+|\tilde{x}\times\omega_{0}|}=:C(x,\omega_{0}).$

Therefore, from the above estimate and the radial symmetry of $h$, we have

$\big{|}\int_{\mathbb{S}^{2}}h(x+s\omega)\psi_{\leq
l}(\angle(\omega,\omega_{0}))d\omega\big{|}\lesssim
C(x,\omega_{0})\int_{0}^{\pi}|h((\sqrt{r^{2}+s^{2}+2rs\cos\theta},0,0))|\sin\theta
d\theta$
$\lesssim\frac{C(x,\omega_{0})}{rs}\int_{|r-s|}^{r+s}|h((z,0,0))|zdz\lesssim\frac{C(x,\omega_{0})}{rs}\min\\{\frac{1}{|r-s|}\int_{|r-s|}^{r+s}|h((z,0,0))|z^{2}dz,\sup_{y\in\mathbb{R}^{3},|y|\in[|r-s|,r+s]}s|y||h(y)|\\}.$
(2.11)

Hence finishing the proof of our desired estimate (2.9). In the above
estimate, we used the change coordinates $\theta\longrightarrow
z:=\sqrt{r^{2}+s^{2}+2rs\cos\theta}$. For fixed $a,b\in\mathbb{R}$, we define

$\rho(x;a,b):=\int_{\mathbb{R}^{3}}\int_{S_{v}}f(x-a\tilde{v}-b\omega_{v},v)d\omega_{v}dv.$

Note that, $\forall R\in SO(3),$ the following equality holds from the radial
symmetry of $f(x,v)$ in (1.6),

$\rho(Rx;a,b)=\int_{\mathbb{R}^{3}}\int_{S_{v}}f(Rx-a\tilde{v}-b\omega_{v},v)d\omega_{v}dv=\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}\int_{S_{u}}f(Rx-
aR\tilde{u}-bR\omega_{u},Ru)d\omega_{u}dv$
$=\int_{\mathbb{R}^{3}}\int_{S_{u}}f(x-a\tilde{u}-b\omega_{u},u)d\omega_{u}dv=\rho(x;a,b).$

Therefore, our desired estimate (2.10) holds directly after we applying the
estimate (2.9) to the radial function $\rho(x;a,b)$.

∎

##  3\. Propagation of moments and the proof of Theorem

Let $N_{1}=N_{0}\times 10^{-3}=10^{5}$. Define

$M_{N_{1}}(t):=\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}(1+|v|)^{N_{1}}f(t,x,v)dxdv,\quad\tilde{M}_{N_{1}}(t):=(1+t)^{N_{1}^{2}}+\sup_{s\in[0,t]}M_{N_{1}}(s).$
(3.1)

Note that for any $t\in[0,T),j\in\mathbb{Z}_{+}$, we have

$\int_{\mathbb{R}^{3}}f(t,x,v)\psi_{j}(v)dv\leq\min\\{2^{3j},\int_{\mathbb{R}^{3}}f(t,x,v)\psi_{j}(v)dv\\}\lesssim
2^{3j(1-1/p)}\big{(}\int_{\mathbb{R}^{3}}f(t,x,v)\psi_{j}(v)dv\big{)}^{1/p}$

Hence, we obtain the following basic estimate for any $p\in[1,\infty],$

$\|f(t,x,v)\psi_{j}(v)\|_{L^{p}_{x}L^{1}_{v}}\lesssim
2^{3j(1-1/p)}\big{(}\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}f(t,x,v)\psi_{j}(v)dv\big{)}^{1/p}\lesssim
2^{3j(1-1/p)}\big{(}\min\\{2^{-j},2^{-N_{1}j}M_{N_{1}}(t)\\}\big{)}^{1/p}.$
(3.2)

Let $M_{t}$ denotes the minimum integer such that
$2^{M_{t}}\geq(\tilde{M}_{N_{1}}(t))^{1/{N_{1}}}$. Hence
$2^{M_{t}}\sim(\tilde{M}_{N_{1}}(t))^{1/{N_{1}}}$. We define a set of
majorities of particles at time $s$ as follows,

$R(t,s):=\\{(X(s;t,x_{0},v_{0}),V(s;t,x_{0},v_{0})):|X(0;t,x_{0},v_{0})|+|V(0;t,x_{0},v_{0}))|\leq
2^{\beta M_{t}},x_{0},v_{0}\in\mathbb{R}^{3}\\},\quad\beta:=1/300.$ (3.3)

Let $t\in[0,T)$ and the initial data $(x_{0},v_{0})\in R(t,0)$ of
characteristics in (1.5) be fixed. For the simplicity of notation, we will
omit the dependence of characteristics with respect to the initial data and
view the spatial characteristic $X(s)$ and velocity characteristic $V(s)$ as
regular functions with respect to time $s$.

We will use a standard bootstrap argument to show that the size of any
velocity characteristic $V(s)$, which starts from the major set $R(t,0)$, is
uniformly bounded by $2^{(1-\beta)M}$, $\beta:=1/300$, for all $s\in[0,t]$.
More precisely, we have

######  Proposition 3.1.

For any $t\in[0,T)$, the following relation holds for some sufficiently large
absolute constant $C$,

$R(t,t)\subset B(0,C2^{\beta{M_{t}}})\times B(0,C2^{(1-\beta)M_{t}}).$ (3.4)

###### Proof.

Recall (3.3) and the system of equations satisfied by characteristics in
(1.5). First of all, for any $t_{1},t_{2}\in[0,t]$, the following rough
estimate holds,

$||X(t_{2})|-|X(t_{1})||\leq\int_{t_{1}}^{t_{2}}|\tilde{X}(s)\cdot\hat{V}(s)|ds\leq|t_{2}-t_{1}|,\quad\Longrightarrow\sup_{s\in[0,t]}|X(s)|\leq
2^{\beta M_{t}}+|t|<2^{\beta M_{t}+1}.$ (3.5)

Hence finishing the proof of first part of (3.4).

From the continuity of characteristics and the size of initial data in
$R(t,0)$, we know that one of the following two scenarios holds.

  1. (i)

There exists a maximal time $T^{\ast}\in(0,t)$ and a maximal number
$T^{\ast\ast}>T^{\ast}$ such that the following estimates holds,

$\sup_{s\in[0,T^{\ast}]}|V(s)|\leq
2^{(1-\beta)M_{t}-1},\quad|V(T^{\ast})|=2^{(1-\beta)M_{t}-1},\quad$ (3.6)
$2^{(1-\beta)M_{t}-2}\leq\inf_{s\in[T^{\ast},T^{\ast\ast}]}|V(s)|\leq\sup_{s\in[T^{\ast},T^{\ast\ast}]}|V(s)|\leq
2^{(1-\beta)M_{t}},\quad[T^{\ast},T^{\ast\ast}]\subset[0,t].$ (3.7)

  2. (ii)

The following estimate holds,

$\sup_{s\in[0,t]}|V(s)|\leq 2^{(1-\beta)M_{t}-1}.$ (3.8)

If the second scenario happens or $\tilde{M}_{t}\lesssim 1$, then there is
nothing left to be proved, we restrict ourself to the first scenario under the
assumption that $\tilde{M}_{t}\gg 1$. We will show that the estimate (3.7) can
be improved hence close the bootstrap argument. To obtain an improved
estimate, we control the increment of velocity between any two time in
$[T^{\ast},T^{\ast\ast}]\subset[0,t]$, which is also the main mission of this
paper.

As a result of direct computations, we have

$\frac{d}{ds}|X(s)|^{2}=2X(s)\cdot\hat{V}(s),\quad\frac{d}{ds}|V(s)|=2\tilde{V}(s)\cdot
E(s,X(s)).$ (3.9)

To better see the dynamics of the magnitude of spatial characteristic, we also
study the second order derivative of $|X(s)|^{2}$. As a result, we have

$\frac{d}{ds}\big{(}X(s)\cdot\hat{V}(s)\big{)}=|\hat{V}(s)|+\frac{X(s)\cdot\big{(}E(s,X(s))+\hat{V}(s)\times
B(s,X(s))\big{)}}{{\sqrt{1+|V(s)|^{2}}}}-\frac{X(s)\cdot{V}(s)}{\big{(}{1+|V(s)|^{2}}\big{)}^{3/2}}\big{(}{V}(s)\cdot
E(s,X(s))\big{)}$ $=|\hat{V}(s)|+C_{1}(X(s),V(s))\cdot
E(s,X(s))+C_{2}(X(s),V(s))\cdot B(s,X(s))$ (3.10)

where

$C_{1}(X(s),V(s)):=\frac{X(s)}{\sqrt{1+|V(s)|^{2}}}-\frac{X(s)\cdot
V(s)}{(1+|V(s)|^{2})^{3/2}}V(s),\quad
C_{2}(X(s),V(s)):=\frac{X(s)\times\hat{V}(s)}{\sqrt{1+|V(s)|^{2}}}.$ (3.11)

From the estimate (5.2) in Proposition 5.1, the following rough estimate holds
for any $t_{1},t_{2}\in[T^{\ast},T^{\ast\ast}]\subset[0,t]$, s.t., $t_{1}\leq
t_{2},$

$X(t_{2})\cdot\hat{V}(t_{2})-X(t_{1})\cdot\hat{V}(t_{1})\geq\frac{4}{5}(t_{2}-t_{1})-\big{|}\int_{t_{1}}^{t_{2}}C_{1}(X(s),V(s))\cdot
E(s,X(s))ds\big{|}$ $-\big{|}\int_{t_{1}}^{t_{2}}(C_{2}(X(s),V(s))\cdot
B(s,X(s))ds\big{|}\geq\frac{2}{3}(t_{2}-t_{1})-2^{-2M_{t}+16\alpha
M_{t}},\quad\alpha:=1/100=3\beta.$ (3.12)

Case $1$:  If there exists a time $\tau_{\star}\in[T^{\ast},T^{\ast\ast}]$
s.t., $X(\tau_{\star})\cdot\hat{V}(\tau_{\star})=0$.

From the estimate (3.12), the following estimate holds for $s_{1},s_{2}\geq 0$
s.t, $s_{1}+\tau_{\star},-s_{2}+\tau_{\star}\in[T^{\ast},T^{\ast\ast}]$,

$|X(\tau_{\star})|^{2}-|X(\tau_{\star}-s_{2})|^{2}=\int_{0}^{s_{2}}2X(\tau_{\star}-s_{2}+\tau)\cdot\hat{V}(\tau_{\star}-s_{2}+\tau)d\tau\leq\int_{0}^{s_{2}}-\frac{4}{3}(s_{2}-\tau)+2^{-2M_{t}+16\alpha
M_{t}}d\tau$
$\Longrightarrow|X(\tau_{\star}-s_{2})|^{2}\geq|X(\tau_{\star})|^{2}+\frac{2}{3}s_{2}^{2}-2^{-2M_{t}+16\alpha
M_{t}}s_{2}.$ (3.13)
$|X(\tau_{\star}+s_{1})|^{2}-|X(\tau_{\star})|^{2}\geq\int_{0}^{s_{1}}2\big{(}\frac{2}{3}\tau-2^{-2M_{t}+7\alpha
M_{t}}\big{)}d\tau\gtrsim s_{1}^{2}-2^{-2M_{t}+16\alpha M_{t}}s_{1}.$ (3.14)

Case $2$:  If there doesn’t exist a time
$\tau_{\star}\in[T^{\ast},T^{\ast\ast}]$ s.t.,
$X(\tau_{\star})\cdot\hat{V}(\tau_{\star})=0$.

For this case we know that $|X(s)|^{2}$ is a monotonic function with respect
to $s$, which implies that either $|X(T^{\ast})|^{2}$ or
$|X(T^{\ast\ast})|^{2}$ is the minimum. If $|X(T^{\ast\ast})|^{2}$ is the
minimum , i.e., $X(s)\cdot\hat{V}(s)\leq 0$, then the following estimate holds
from (3.12),

$-X(T^{\ast\ast}-s)\cdot\hat{V}(T^{\ast\ast}-s)\geq
X(T^{\ast\ast})\cdot\hat{V}(T^{\ast\ast})-X(T^{\ast\ast}-s)\cdot\hat{V}(T^{\ast\ast}-s)\geq\frac{2}{3}s-2^{-2M_{t}+16\alpha
M_{t}}.$
$\Longrightarrow|X(T^{\ast\ast})|^{2}-|X(T^{\ast\ast}-\tau)|^{2}=\int_{0}^{\tau}2X(T^{\ast\ast}-s)\cdot\hat{V}(T^{\ast\ast}-s)ds\leq-\frac{2}{3}\tau^{2}+2^{-2M_{t}+16\alpha
M_{t}+1}\tau$
$\Longrightarrow|X(T^{\ast\ast}-\tau)|^{2}\geq|X(T^{\ast\ast})|^{2}+\frac{2}{3}\tau^{2}-2^{-2M_{t}+16\alpha
M_{t}+1}\tau,\quad\tau\in[0,T^{\ast\ast}-T^{\ast}].$ (3.15)

If $|X(T^{\ast})|^{2}$ is the minimum , i.e., $X(s)\cdot\hat{V}(s)\geq 0$,
then the following estimate holds from (3.12),

$|X(T^{\ast}+\tau)|^{2}-|X(T^{\ast})|^{2}=\int_{0}^{\tau}2X(T^{\ast}+s)\cdot\hat{V}(T^{\ast}+s)ds\geq\frac{2}{3}\tau^{2}-2^{-2M_{t}+16\alpha
M_{t}+1}\tau,\quad\tau\in[0,T^{\ast\ast}-T^{\ast}].$ (3.16)

To sum up, from the estimates (3.13–3.16), we have a good control of the
magnitude of characteristic for all time except a small neighborhood of the
fixed local (global) minimum.

Since the Case $2$ is of the same type as in the Case $1$, without loss of
generality, we restrict ourself to the Case $1$. Let
$\tau_{0}:=2^{-2M_{t}+20\alpha M_{t}}$. Based on the possible size of
$|X(\tau_{\star})|$, we separate into two cases as follows.

$\oplus$ If $|X(\tau_{\star})|\geq 2^{-5M_{t}/3}$.

From the rough estimate of increment of magnitude of spatial characteristic in
(3.5), we have

$\sup_{s\in[\tau_{\star}-\tau_{0},\tau_{\star}+\tau_{0}]\cap[T^{\ast},T^{\ast\ast}]}|X(s)|\geq
2^{-5M_{t}/3}-2^{-2M_{t}+20\alpha M_{t}}\gtrsim 2^{-5M_{t}/3}.$ (3.17)

From the above estimate, the estimate (5.1) in Proposition 5.1 and the rough
estimate of the electromagnetic field (4.6) in Lemma 4.2, the following
estimate holds for any $\kappa\in[-2\tau_{0},2\tau_{0}]$, s.t.,
$\tau_{\star}+\kappa\in[T^{\ast},T^{\ast\ast}],$

$\big{|}|V(\tau_{\star}+\kappa)|-|V(\tau_{\star})|\big{|}\lesssim
2^{M_{t}/3+7\alpha
M_{t}}+\big{|}\int_{0}^{\kappa}\frac{2^{(1-\alpha+\epsilon)M_{t}}}{|X(\tau_{\star}+s)|_{-}}ds\big{|}\lesssim
2^{2M_{t}/3+20\alpha M_{t}}.$ (3.18)

For any $\kappa\in[-\tau_{0},\tau_{0}]^{c}$, s.t.,
$\tau_{\star}+\kappa\in[T^{\ast},T^{\ast\ast}],$ from the estimates (3.13) and
(3.14), we have

$|X(\tau_{\star}+\kappa)|\gtrsim|X(\tau_{\star})|+|\kappa|$

From the above estimate, the rough estimate of the electromagnetic field (4.6)
in Lemma 4.2 and the estimate (5.1) in Proposition 5.1, the following estimate
holds for $\kappa\in[-2\tau_{0},2\tau_{0}]^{c}$, s.t.,
$\tau_{\star}+\kappa\in[T^{\ast},T^{\ast\ast}],$

$\big{|}|V(\tau_{\star}+\kappa)|-|V(\tau_{\star}+sign(\kappa)\tau_{0})|\big{|}\lesssim
2^{M_{t}/3+7\alpha
M_{t}}+\big{|}\int_{\tau_{\star}+sign(\kappa)\tau_{0}}^{\tau_{\star}+\kappa}\frac{2^{(1-\alpha+\epsilon)M_{t}}}{\min\\{|X(\tau_{\star})|+|s-\tau_{\star}|,1\\}}ds\big{|}\lesssim
2^{(1-\alpha)M_{t}+2\epsilon M_{t}}.$ (3.19)

where $\epsilon:=60/N_{1}=6\times 10^{-4}.$

$\oplus$ If $|X(\tau_{\star})|\leq 2^{-5M_{t}/3}$.

From the rough estimate of increment of magnitude of spatial characteristic in
(3.5), the following estimate holds for $s\in[-\tau_{0},\tau_{0}]$ s.t.,
$\tau_{\star}+s\in[T^{\ast},T^{\ast\ast}],$

$|X(\tau_{\star}+s)|\leq 2^{-5M_{t}/3}+s\lesssim 2^{-5M_{t}/3}.$

Since now $|X(\tau_{\star}+s)|,s\in[-\tau_{0},\tau_{0}]$, is small, from the
estimate (5.2) in Proposition 5.1, the following improved estimate holds, for
any
$t_{1},t_{2}\in[\tau_{\star}-\tau_{0},\tau_{\star}+\tau_{0}]\cap[T^{\ast},T^{\ast\ast}]$,
s.t., $t_{1}\leq t_{2}$,

$X(t_{2})\cdot\hat{V}(t_{2})-X(t_{1})\cdot\hat{V}(t_{1})\geq\frac{2}{3}(t_{2}-t_{1})-2^{-5M_{t}/3-2M_{t}/3+9\alpha
M_{t}}.$

With the above improved estimate, the following improved estimate holds for
any $s\in[-\tau_{0},\tau_{0}]$, s.t.,
$\tau_{\star}+s\in[T^{\ast},T^{\ast\ast}],$

$\Longrightarrow|X(\tau_{\star}+s)|^{2}\geq|X(\tau_{\star})|^{2}+\frac{2}{3}s^{2}-2^{-7M_{t}/3+9\alpha
M_{t}}|s|.$ (3.20)

Let $\tau_{1}:=2^{-7M_{t}/3+10\alpha M_{t}}<\tau_{0}$. Recall the
decomposition of electromagnetic field in (2.5). From the rough estimate of
electromagnetic field (4.13) in Lemma 4.3, which is used for the case
$j\leq(1+\epsilon)M_{t}$, and the second estimate in (4.17) in Lemma 4.4,
which is used for the the case $j\geq(1+\epsilon)M_{t}$, the following
estimate holds for any $\kappa\in[-2\tau_{1},2\tau_{1}]$, s.t.,
$\tau_{\star}+\kappa\in[T^{\ast},T^{\ast\ast}],$

$\big{|}|V(\tau_{\star}+\kappa)|-|V(\tau_{\star})|\big{|}\lesssim\sum_{j\in\mathbb{Z}_{+}}\sum_{K\in\\{E,B\\}}\int_{\tau_{\star}}^{\tau_{\star}+\kappa}|K_{j}(s,X(s))|ds\lesssim
2^{-7M_{t}/3+10\alpha M_{t}}2^{3(1+3\epsilon)M_{t}}\lesssim
2^{2M_{t}/3+11\alpha M_{t}}.$ (3.21)

For any $\kappa\in[-\tau_{1},\tau_{1}]^{c}$, s.t.,
$\tau_{\star}+\kappa\in[T^{\ast},T^{\ast\ast}],$ from the estimates (3.14),
(3.13), which are used for $\tau\in[-\tau_{0},\tau_{0}]^{c}$ and the estimate
(3.20), which is used when
$\tau\in[-\tau_{1},\tau_{1}]^{c}\cap[-\tau_{0},\tau_{0}]$, we have

$|X(\tau_{\star}+\kappa)|\gtrsim|X(\tau_{\star})|+|\kappa|.$

From the above estimate, the rough estimate of the electromagnetic field (4.6)
in Lemma 4.2 , the estimate (5.1) in Proposition 5.1, the following estimate
holds for any $\kappa\in[-2\tau_{1},2\tau_{1}]^{c}$, s.t.,
$\tau_{\star}+\kappa\in[T^{\ast},T^{\ast\ast}],$

$\big{|}|V(\tau_{\star}+\kappa)|-|V(\tau_{\star}+sign(\kappa)\tau_{1})|\big{|}\lesssim
2^{M_{t}/3+7\alpha
M_{t}}+\big{|}\int_{\tau_{\star}+sign(\kappa)\tau_{1}}^{\tau_{\star}+\kappa}\frac{2^{(1-\alpha+\epsilon)M_{t}}}{\min\\{|X(\tau_{\star})|+|s-\tau_{\star}|,1\\}}ds\big{|}\lesssim
2^{(1-\alpha+2\epsilon)M_{t}}.$ (3.22)

To sum up, in whichever case, from the estimates (3.18), (3.19), (3.21), and
(3.22), the following estimate holds for any
$t_{1},t_{2}\in[T^{\ast},T^{\ast\ast}]\subset[0,t]$,

$\big{|}|V(t_{1})|-|V(t_{2})|\big{|}\lesssim 2^{(1-\alpha)M_{t}+2\epsilon
M_{t}}<2^{(1-2\beta)M_{t}},$ (3.23)
$\Longrightarrow\frac{3}{4}2^{(1-\beta)M_{t}-1}\leq\inf_{s\in[T^{\ast},T^{\ast}+\epsilon]}|V(s)|\leq\sup_{s\in[T^{\ast},T^{\ast}+\epsilon]}|V(s)|\leq\frac{5}{4}2^{(1-\beta)M_{t}-1}$
(3.24)

Hence improving the bootstrap assumption in (3.7). Therefore, we can extend
the size of $\epsilon$ such that $T^{\ast\ast}=t$. To sum up, in whichever
case, we have $\sup_{s\in[0,t]}|V(s)|<2^{(1-\beta)M_{t}}.$ Hence finishing the
proof of (3.4).

∎

Proof of Theorem 1.1 : Recall (3.1). From the conservation law (1.3), we have

$\big{|}\int_{\mathbb{R}^{3}}\int_{|v|\leq
2^{(1-\beta+\epsilon)M_{t}}}(1+|v|)^{N_{1}}f(t,x,v)dxdv\big{|}\lesssim
2^{(N_{1}-1)(1-\beta+\epsilon)M_{t}}\leq(\tilde{M}_{N_{1}}(t))^{(1-\beta+\epsilon)}.$
(3.25)

Recall the definition of the majority set $R(t,s)$ in (3.3). From the relation
(3.4) in Proposition 3.1, we have $|X(0;t,x,v)|+|V(0;t,x,v)|\gtrsim 2^{\beta
M_{t}}$ if $|x|\gtrsim 2^{\beta M_{t}}$ or $|v|\gtrsim 2^{(1-\beta)M_{t}}$.
Therefore, the polynomial decay of the initial data in (1.7) implies that the
following estimate holds if $|v|\gtrsim 2^{(1-\beta)M_{t}}$,

$|f(t,x,v)|=|f_{0}(X(0;t,x,v),V(0;t,x,v))|\lesssim(1+|X(0;t,x,v)|+|V(0;t,x,v)|)^{-N_{0}}\lesssim
2^{-10N_{1}M_{t}/3}.$

Moreover, if $|x|\geq 2^{2\epsilon M_{t}}$, then from the estimate (3.5), we
have

$|X(t,0,x,v)|\geq|x|-t\geq|x|-2^{\epsilon M_{t}}\gtrsim(1+|x|),$

If $|v|\geq 2^{3(1+4\epsilon)M_{t}}$, then the following estimate holds from
the equation (3.9) and the first estimate in (4.17) in Lemma 4.4,

$|V(t,0,x,v)|\geq|v|-\int_{0}^{t}\big{(}|E(s,X(s)|+|B(s,X(s)|\big{)}ds\gtrsim(1+|v|).$

Therefore, from the above three estimates and the assumption of initial data
in (1.7), the following estimate holds if $|v|\gtrsim 2^{(1-\beta)M_{t}}$
regardless the size of $|x|$,

$|f(t,x,v)|=|f_{0}(X(t,0,x,v),V(t,0,x,v))|\lesssim(1+|x|)^{-4}(1+|v|)^{-N_{1}-4}.$
(3.26)

Therefore, from the above estimate, we have

$\big{|}\int_{\mathbb{R}^{3}}\int_{|v|\geq
2^{(1-\beta+\epsilon)M_{t}}}(1+|v|)^{N_{1}}f(t,x,v)dxdv\big{|}\lesssim 1.$
(3.27)

To sum up, from the estimates (3.25) and (3.27), we have

$M_{N_{1}}(t)\lesssim\big{(}\tilde{M}_{N_{1}}(t)\big{)}^{1-\beta+\epsilon}.$

Since the above estimate holds for any $t\in[0,T)$ and $\tilde{M}_{n}(t)$ is
an increasing function with respect to $t$, the following estimate holds for
any $s\in[0,t]$,

$M_{N_{1}}(s)\lesssim\big{(}\tilde{M}_{N_{1}}(s)\big{)}^{1-\beta+\epsilon}\leq\big{(}\tilde{M}_{N_{1}}(t)\big{)}^{1-\beta+\epsilon},$

Hence

$\tilde{M}_{N_{1}}(t)=\sup_{s\in[0,t]}M_{N_{1}}(s)+(1+t)^{N_{1}^{2}}\lesssim\big{(}\tilde{M}_{N_{1}}(t)\big{)}^{1-\beta+\epsilon}+(1+t)^{N_{1}^{2}},\quad\Longrightarrow\tilde{M}_{N_{1}}(t)\lesssim(1+t)^{N_{1}^{2}}.$
(3.28)

Therefore, from the above estimate and the first estimate in (4.17) in Lemma
4.4, we have

$\sup_{x_{0},v_{0}\in\mathbb{R}^{3}}\sup_{t\in[0,T)}\sum_{j\in\mathbb{Z}_{+}}\int_{0}^{T}|K_{j}(s,X(s;t,x_{0},v_{0}))|ds\lesssim(1+T)^{4{N_{1}}}.$
(3.29)

Hence finishing the proof of our desired estimate (1.8) and the theorem from
the the decomposition of the electromagnetic field in (2.5). ∎

##  4\. Rough estimates of the electromagnetic field

In this section, we provide several rough estimates for the electromagnetic
field in terms of the moment of the distribution function, which will be used
as basic tools in the next section for more sophisticated analysis of the
increment of the sizes of characteristics. In particular, as stated in (4.6)
and (4.17), we give a point-wise estimate for the electromagnetic field and
give an upper bound for the targeted quantity in (1.8) respectively.

To prove our desired rough estimates, for simplicity, we don’t distinguish the
frequencies of the electromagnetic field. The Glassey-Strauss decomposition as
stated in the following Lemma is very convenient and useful.

######  Lemma 4.1.

For any $j\in\mathbb{Z}_{+}$, the following decomposition holds

$K_{j}(t,x)=\sum_{-j\leq l\leq
2,l\in\mathbb{Z}}\int_{0}^{t}K_{S;j}^{l}(t,s,x)+K_{T;j}^{l}(t,s,x)ds,\quad
K\in\\{E,B\\},$ (4.1)

where

$E_{T;j}^{l}(t,s,x):=\int_{\mathbb{R}^{3}}\int_{\mathbb{S}^{2}}\frac{(1-|\hat{v}|^{2})(\hat{v}+\theta)}{(1+\hat{v}\cdot\theta)^{2}}f(s,x+(t-s)\theta,v)\varphi_{j}(v)\varphi_{l;-j}(\angle(v,-\theta))d\theta
dv,$ (4.2)
$B_{T;j,l}^{l}(t,s,x):=\int_{\mathbb{R}^{3}}\int_{\mathbb{S}^{2}}\frac{(1-|\hat{v}|^{2})(\theta\times\hat{v})}{(1+\hat{v}\cdot\theta)^{2}}f(s,x+(t-s)\theta,v)\varphi_{j}(v)\varphi_{l;-j}(\angle(v,-\theta))d\theta
dv,$ (4.3)
$E_{S;j}^{l}(t,s,x):=\int_{\mathbb{R}^{3}}\int_{\mathbb{S}^{2}}(t-s)K(s,x+(t-s)\theta,v)\cdot\nabla_{v}\big{(}\frac{\varphi_{j}(v)(\hat{v}+\theta)}{1+\hat{v}\cdot\theta}\big{)}\varphi_{l;-j}(\angle(v,-\theta))f(s,x+(t-s)\theta,v)d\theta
dv,$ (4.4)
$B_{S;j}^{l}(t,s,x)=\int_{\mathbb{R}^{3}}\int_{\mathbb{S}^{2}}(t-s)K(s,x+(t-s)\theta,v)\cdot\nabla_{v}\big{(}\frac{\varphi_{j}(v)(\theta\times\hat{v})}{(1+\hat{v}\cdot\theta)}\big{)}\varphi_{l;-j}(\angle(v,-\theta))f(s,x+(t-s)\theta,v)d\theta
dv.$ (4.5)

###### Proof.

The desired decomposition (4.1) follows from redoing the Glassey-Strauss
decomposition first for $K_{j},K\in\\{E,B\\},$ and then used a dyadic
decomposition with threshold $-j$ for the angular between $v$ and $-\theta$.
See [6][Theorem 3] for more details. ∎

######  Lemma 4.2.

For any $x\in\mathbb{R}^{3}/\\{0\\},s\in[0,t]$, the following estimate holds

$\sum_{j\in\mathbb{Z}_{+}}\sum_{K\in\\{E,B\\}}|K_{j}(s,x)|\lesssim\frac{2^{(1+\epsilon)M_{t}}}{r_{-}},\quad
r_{-}:=\min\\{1,r\\},\quad\epsilon:=60/{N_{1}}=6\times 10^{-4}.$ (4.6)

Moreover, we have the following estimate as a byproduct,

$\sum_{j\in\mathbb{Z}_{+},j\notin[(1-\alpha)M_{t},(1+\epsilon)M_{t}]}\sum_{K\in\\{E,B\\}}K_{j}(s)\lesssim\frac{2^{(1-\alpha+\epsilon)M_{t}}}{r_{-}},\quad\alpha:=1/100=3\beta.$
(4.7)

###### Proof.

Recall the decomposition of electromagnetic field in (2.5) and the Glassey-
Strauss decomposition in (4.1).

We first estimate the “T” part. Recall (4.2) and (4.3). Note that, from the
estimate (2.10) in Lemma 2.2, the estimate (3.2), and the volume of support of
$v$ and $\theta$, we have

$|E_{T;j}^{l}(s,\tau,x)|+|B_{T;j}^{l}(s,\tau,x)|\lesssim
2^{-j-2l}\min\\{\frac{1}{r|s-\tau||r-(s-\tau)|}\min\\{2^{-j},2^{-{N_{1}}j+{N_{1}}M_{t}}\\},2^{3j+2l}\\}.$
(4.8)

Now we estimate the “S” part. Recall (4.4) and (4.5). From the Cauchy-Schwarz
inequality and the estimates in Lemma 2.2, the estimate (3.2), and the volume
of support of $v$, we have

$|E_{S;j}^{l}(s,\tau,x)|+|B_{S;j}^{l}(s,\tau,x)|\lesssim
2^{-j-2l}|s-\tau|\big{(}\int_{\mathbb{R}^{3}}\int_{\mathbb{S}^{2}}f(\tau,x+(s-\tau)\theta,v)\varphi_{j}(v)\varphi_{l;-j}(\angle(v,-\theta))dvd\theta\big{)}^{1/2}$
$\times\big{(}\int_{\mathbb{R}^{3}}\int_{\mathbb{S}^{2}}(|E(\tau,x+(s-\tau)\theta)|^{2}+|B(\tau,x+(s-\tau)\theta)|^{2})\varphi_{j}(v)\varphi_{l;-j}(\angle(v,-\theta))dvd\theta\big{)}^{1/2}$
$\lesssim
2^{-j-2l}|s-\tau|\big{(}\frac{2^{3j+2l}}{r|s-\tau||r-(s-\tau)|}\big{)}^{1/2}\big{[}\min\\{\frac{\min\\{2^{-j},2^{-{N_{1}}j+{N_{1}}M_{t}}\\}}{r|s-\tau||r-(s-\tau)|},2^{3j+2l}\\}\big{]}^{1/2}.$
(4.9)

Based on the possible size of $j$, we separate into two cases as follow.

Case $1$: If $j\geq(1+\epsilon/2)M_{t}.$ From the estimate (4.8) and the
estimate (4.9), we have

$\sum_{j\in\mathbb{Z}_{+},j\geq(1+\epsilon/2)M_{t}}\sum_{l\in[-j,2]\cap\mathbb{Z}}\sum_{K\in\\{E,B\\}}\int_{0}^{s}|K_{T;j}^{l}(s,\tau,x)|+|K_{S;j}^{l}(s,\tau,x)|d\tau$
$\lesssim\sum_{j\in\mathbb{Z}_{+},j\geq(1+\epsilon/2)M_{t}}(1+j)\big{[}\int_{0}^{s}\big{(}2^{2j}\big{)}^{9/10}\big{(}\frac{2^{j}2^{-{N_{1}}j+{N_{1}}M_{t}}}{r|s-\tau||r-(s-\tau)|}\big{)}^{1/10}d\tau$
$+\int_{0}^{s}(s-\tau)^{1/4}2^{3j/2}\big{(}\frac{1}{r|r-(s-\tau)|}\big{)}^{1/2}\big{(}\frac{2^{-{N_{1}}j+{N_{1}}M_{t}}}{r|r-(s-\tau)|}\big{)}^{1/4}\big{(}2^{3j+2l}\big{)}^{1/4}ds\lesssim\frac{1}{r_{-}^{3/4}}.$
(4.10)

Case $2$: If $j\leq(1+\epsilon/2)M_{t}.$ Let
$\delta:=2^{-(1+\epsilon/2)M_{t}}$. From the estimate (4.8), we have

$\sum_{j\in\mathbb{Z}_{+},j\leq(1+\epsilon/2)M_{t}}\sum_{l\in[-j,2]\cap\mathbb{Z}}\sum_{K\in\\{E,B\\}}\int_{0}^{s}|K_{T;j}^{l}(s,\tau,x)|d\tau\lesssim\sum_{j\in\mathbb{Z}_{+},j\leq(1+\epsilon/2)M_{t}}\sum_{l\in[-j,2]\cap\mathbb{Z}}\int_{[s-\delta,s]\cup[s-r-\delta,s-r+\delta]}2^{2j}d\tau$
$+\int_{[0,s-\delta]\cap[s-r-\delta,s-r+\delta]^{c}}\frac{2^{-2j-2l}}{r|s-\tau||r-(s-\tau)|}d\tau\lesssim\frac{(M_{t})^{2}}{r\delta}+M_{t}2^{(2+\epsilon)M_{t}}\delta\lesssim\frac{2^{(1+\epsilon)M_{t}}}{r_{-}}.$
(4.11)

Let $\tilde{\delta}:=2^{-nM_{t}}$. From the estimate (4.9), we have

$\sum_{j\in\mathbb{Z}_{+},j\leq(1+\epsilon/2)M_{t}}\sum_{l\in[-j,2]\cap\mathbb{Z}}\sum_{K\in\\{E,B\\}}\int_{0}^{s}|K_{S;j}^{l}(s,\tau,x)|d\tau$
$\lesssim\sum_{j\in\mathbb{Z}_{+},j\leq(1+\epsilon/2)M_{t}}\sum_{l\in[-j,2]\cap\mathbb{Z}}\int_{[0,s]\cap[s-r-\tilde{\delta},s-r+\tilde{\delta}]^{c}}\frac{2^{-l}}{r|r-(s-\tau)|}d\tau$
$+\int_{[0,s]\cap[s-r-\tilde{\delta},s-r+\tilde{\delta}]}\frac{(1+j)2^{2j}(s-\tau)^{1/2}}{(r|r-(s-\tau)|)^{1/2}}d\tau\lesssim\frac{2^{(1+\epsilon/2)M_{t}}M_{t}}{r}+\frac{1}{r^{1/2}}\lesssim\frac{2^{(1+\epsilon)M_{t}}}{r_{-}}.$
(4.12)

To sum up, our desire estimate (4.6) holds from the estimates (4.10), (4.11),
and (4.12).

By using the same strategy used in above estimates and letting
$\delta:=(\tilde{M}_{n}(t))^{-(1-\epsilon/2)/(n-1)}$ and
$\tilde{\delta}:=(\tilde{M}_{n}(t))^{-1}$, we obtain our desired estimates
(4.7) after combining the estimate (4.10) and changing the range of the
summation with respect to $j$ in (4.12) from
$j\in\mathbb{Z}_{+},{j}\leq(1+\epsilon/2)M_{t}$ to
$j\in\mathbb{Z}_{+},j\leq(1-\alpha)M_{t}$.

∎

######  Lemma 4.3.

The following estimate holds for any
$s\in[0,t],j\in\mathbb{Z}_{+},x\in\mathbb{R}^{3},$ s.t.,
$j\leq(1+\epsilon)M_{t}$,

$|E_{j}(s,x)|+|B_{j}(s,x)|\lesssim 2^{3(1+2\epsilon)M_{t}}.$ (4.13)

###### Proof.

Note that our desired estimate (4.13) is a trivial consequence of (4.6) in
Lemma 4.2 if $|x|\geq 1$. It would be sufficient to consider the case $|x|\leq
1$. Recall the Glassey-Strauss decomposition in (4.1). From the obtained
estimate (4.8), we have

$\sum_{l\in[-j,2]\cap\mathbb{Z}}\int_{0}^{s}|E_{T;j}^{l}(s,\tau,x)|+|B_{T;j}^{l}(s,\tau,x)|d\tau\lesssim(1+j)|t|2^{2j}\lesssim
2^{2(1+2\epsilon)M_{t}}.$ (4.14)

Hence, it remains to estimate the “$S$” part. Recall (4.4) and (4.5). From the
rough estimate of the electromagnetic field (4.6) in Lemma 4.2 and the
obtained estimate (4.9), we know that the following estimate holds if
$\tau\neq s-r$,

$\sum_{K\in\\{E,B\\}}|K_{S;j}^{l}(s,\tau,x)|\lesssim\min\\{\frac{(s-\tau)2^{(1+\epsilon)M_{t}}}{|r-(s-\tau)|}2^{-j-2l}2^{3j+2l},2^{2(1+\epsilon)M_{t}}\big{(}\frac{s-\tau}{r|r-(s-\tau)|}\big{)}^{1/2}\\}.$
(4.15)

From the estimate (4.15) , we have

$\sum_{K\in\\{E,B\\}}|K_{S;j}(s,x)|\lesssim\int_{s-r-r/2}^{s-r+r/2}2^{2(1+\epsilon)M_{t}}\big{(}\frac{s-\tau}{r|r-(s-\tau)|}\big{)}^{1/2}d\tau+\int_{[0,s]\cap[s-r-r/2,s-r+r/2]^{c}}\frac{(s-\tau)2^{3(1+\epsilon)M}}{|r-(s-\tau)|}d\tau$
$\lesssim
2^{2(1+\epsilon)M_{t}}r^{1/2}+2^{3(1+\epsilon)M_{t}}\big{(}1+|t\ln(t)|+r\ln(1/r)\big{)}\lesssim
2^{3(1+2\epsilon)M_{t}}.$ (4.16)

To sum up, our desired estimate (4.13) holds from the estimates (4.14) and
(4.16). ∎

######  Lemma 4.4.

For any $t\in[0,T)$, the following estimates hold for any initial data
${x}_{0},{v}_{0}\in\mathbb{R}^{3}$ of characteristics,

$\sum_{j\in\mathbb{Z}_{+}}\sum_{K\in\\{E,B\\}}\int_{0}^{t}|K_{j}(s,X(s))|ds\lesssim
2^{3(1+3\epsilon)M_{t}},\quad\sum_{j\in\mathbb{Z},j\geq(1+\epsilon)M_{t}}\sum_{K\in\\{E,B\\}}\int_{0}^{t}|K_{j}(s,X(s))|ds\lesssim
1.$ (4.17)

###### Proof.

Recall the decomposition of electromagnetic field in (2.5) and the Glassey-
Strauss decomposition in (4.1). Note that, the following estimate holds from
the estimate (4.6) in Lemma 4.2, the estimate (4.10), the estimate (4.13) in
Lemma4.3,

$\sum_{K\in\\{E,B\\}}|K(t,x)|\lesssim\big{(}\frac{1}{r^{3/4}}+2^{3(1+2\epsilon)M_{t}}\big{)}\psi_{\leq
2}(r)+2^{(1+2\epsilon)M_{t}}\psi_{\geq 2}(r).$ (4.18)

Moreover, the following rough estimates hold straightforwardly,

$\sum_{K\in\\{E,B\\}}\int_{0}^{t}\int_{0}^{s}|K_{T;j}^{l}(s,\tau,X(s))|d\tau
ds$ $\lesssim
2^{j}\int_{0}^{t}\int_{\tau}^{t}\int_{\mathbb{R}^{3}}\int_{0}^{2\pi}\int_{0}^{\pi}f(\tau,X(s)+(s-\tau)\omega,v)\varphi_{j}(v)\varphi_{l;-j}(\angle(v,-\omega))\sin\theta
d\theta d\phi dvdsd\tau,$ (4.19)
$\sum_{K\in\\{E,B\\}}\int_{0}^{t}\int_{0}^{s}|K_{S;j}^{l}(s,\tau,X(s))|d\tau
ds\lesssim\sum_{K\in\\{E,B\\}}2^{j}\int_{0}^{t}\int_{\tau}^{t}\int_{\mathbb{R}^{3}}\int_{0}^{2\pi}\int_{0}^{\pi}|K(\tau,X(s)+(s-\tau)\omega)|$
$\times\varphi_{j}(v)\varphi_{l;-j}(\angle(v,-\omega))(s-\tau)f(\tau,X(s)+(s-\tau)\omega,v)\sin\theta
d\theta d\phi dvdsd\tau,$ (4.20)

As observed by Pallard [14], for fixed $\tau$, we can do change of coordinates
$(s,\theta,\phi)\longrightarrow(X(s)+(s-\tau)\omega)$. As a result of direct
computation (see also [14]), the Jacobian of the transformation is
$(X^{\prime}(s)\cdot\omega+1)(s-\tau)^{2}\sin\theta$. From the Hölder
inequality, we have

$\big{|}\int_{\tau}^{t}\int_{0}^{2\pi}\int_{0}^{\pi}h(\tau,X(s)\omega)\sin\theta
d\theta d\phi
ds\big{|}\lesssim\big{|}\int_{\tau}^{t}\int_{0}^{2\pi}\int_{0}^{\pi}|h(\tau,X(s)+(s-\tau)\omega)|^{p}(X^{\prime}(s)\cdot\omega+1)(s-\tau)^{2}\sin\theta
d\theta d\phi ds\big{|}^{1/p}$
$\times\big{|}\int_{\tau}^{t}\int_{0}^{2\pi}\int_{0}^{\pi}\big{(}(X^{\prime}(s)\cdot\omega+1)(s-\tau)^{2}\big{)}^{-q/p}(\sin\theta)^{(1-1/p)q}d\theta
d\phi ds\big{|}^{1/q},\quad\frac{1}{p}+\frac{1}{q}=1.$ (4.21)

Note that the following estimate holds for any $q\in(1,3/2)$,

$\int_{\tau}^{t}\int_{0}^{2\pi}\int_{0}^{\pi}\big{(}(X^{\prime}(s)\cdot\omega+1)(s-\tau)^{2}\big{)}^{-q+1}\sin\theta
d\theta d\phi ds\lesssim(1+t)^{2}.$

Let $p=3/(1-1/n)$ and $q=13$. From the estimates (4.18–4.21) and the estimate
(3.2), we have

$\sum_{j\geq(1+\epsilon)M_{t}}\sum_{K\in\\{E,B\\}}\sum_{-j\leq l\leq
2,l\in\mathbb{Z}}\int_{0}^{t}\int_{0}^{s}|K_{S;j}^{l}(s,\tau,X(s))|d\tau
ds+\int_{0}^{t}\int_{0}^{s}|K_{T;j}^{l}(s,\tau,X(s))|d\tau ds$
$\lesssim\sum_{j\geq(1+\epsilon)M_{t}}\sum_{K\in\\{E,B\\}}2^{j}(1+t)^{2}\big{[}t\int_{0}^{t}\|\int_{\mathbb{R}^{3}}|K(\tau,x)|f(\tau,x,v)\varphi_{j}(v)dv\|_{L^{p}_{x}}d\tau+\int_{0}^{t}\|\int_{\mathbb{R}^{3}}f(\tau,x,v)\varphi_{j}(v)dv\|_{L^{p}_{x}}d\tau\big{]}$
(4.22)
$\lesssim\sum_{j\geq(1+\epsilon)M_{t}}\sup_{\tau\in[0,t]}2^{j}(1+t)^{4}\big{[}2^{3(1+2\epsilon)M_{t}}\|\int_{\mathbb{R}^{3}}f(\tau,x,v)\varphi_{j}(v)dv\|_{L^{p}_{x}}+\|\int_{\mathbb{R}^{3}}r^{-3/4}\psi_{\leq
2}(r)f(\tau,x,v)\varphi_{j}(v)dv\|_{L^{p}_{x}}\big{]}$
$\lesssim\sum_{j\geq(1+\epsilon)M_{t}}2^{j}\big{[}2^{3(1+3\epsilon)M_{t}}2^{3j(1-1/p)}\big{(}2^{-{N_{1}}j+{N_{1}}M_{t}}\big{)}^{1/p}+2^{3j(1-1/q)}\big{(}2^{-{N_{1}}j+{N_{1}}M_{t}}\big{)}^{1/q}\big{]}\lesssim
1.$ (4.23)

Our desired estimates in (4.17) hold from the above estimate and the estimate
(4.13) in Lemma (4.3). ∎

##  5\. Increment of the sizes of characteristics over time

Recall the equations satisfied by the magnitudes of characteristics in (3.9).
As stated in the following Proposition, our main goal in this section is to
quantitatively control the increment of the size of characteristics over time.

######  Proposition 5.1.

Under the assumptions in (3.6) and (3.7), the following estimate holds for any
$t_{1},t_{2}\in[T^{\ast},T^{\ast\ast}]\subset[0,t]$,

$\big{|}\int_{t_{1}}^{t_{2}}\tilde{V}(s)\cdot
E(s,X(s))ds\big{|}\lesssim\int_{t_{1}}^{t_{2}}|K(s,X(s))|\min\\{\frac{2^{-3M_{t}/2+24\alpha
M_{t}}}{|X(s)|_{-}},2^{M_{t}/2+3\alpha M_{t}}|X(s)|+2^{-2M_{t}/3+8\alpha
M_{t}}\\}ds$ $+2^{M_{t}/3+7\alpha
M_{t}}+\int_{t_{1}}^{t_{2}}\frac{2^{(1-\alpha+\epsilon)M_{t}}}{|X(s)|_{-}}ds,\quad\alpha:=1/100,\quad\epsilon:=60/N_{1}=6\times
10^{-4},$ (5.1) $\big{|}\int_{t_{1}}^{t_{2}}C_{1}(X(s),V(s))\cdot
E(s,X(s))ds\big{|}+\big{|}\int_{t_{1}}^{t_{2}}C_{2}(X(s),V(s))\cdot
B(s,X(s))ds\big{|}\lesssim 2^{-(\alpha-2\beta-3\epsilon)M_{t}}|t_{2}-t_{1}|$
$+\sum_{i=1,2}|X(t_{i})|\min\\{\frac{2^{-2M_{t}+15\alpha
M_{t}}}{|X(t_{i})|_{-}},2^{M_{t}/2+3\alpha
M_{t}}|X(t_{i})|+2^{-2M_{t}/3+8\alpha M_{t}}\\},$ (5.2)

where $C_{1}(\cdot,\cdot)$ and $C_{2}(\cdot,\cdot)$ are defined in (3.11).

###### Proof.

To improve presentation, we postpone the proof of the above two estimates to
the end of this subsection. ∎

Recall the decomposition of electromagnetic field in (2.5). Because the rough
estimate of $K_{j},K\in\\{E,B\\}$, in (4.7) in Lemma 4.2 is sufficient for our
purpose, we only need to consider the uncovered case
$j\in[(1-\alpha)M_{t},(1+\epsilon)M_{t}]$. In the rest of this section, we use
a Fourier method to prove our desired estimates for the uncovered case.

First of all, we introduce the set-up of different frameworks for the
electromagnetic field in different scenarios. Recall the detailed formulas of
$E_{j}(t,x)$ and $B_{j}(t,x)$ in (2.7) and (2.8). After represented them on
Fourier side and using the Vlasov equation to substitute $\partial_{t}f$, the
following decomposition holds after doing dyadic decompositions for both the
frequency and the angle between $\xi$ and $v$,

$K_{j}(t,x)=\sum_{k\in\mathbb{Z}}\sum_{l\in[-j,2]\cap\mathbb{Z}}\frac{-i}{2}\big{(}K_{k;j,l}(t,x)-\overline{K_{k;j,l}(t,x)}\big{)},\quad
K\in\\{E,B\\},$ (5.3)

where

$B_{k;j,l}(t,x)=\int_{0}^{t}V_{k;j,l}^{b}(t,s,x)ds,\quad
E_{k;j,l}(t,x)=\int_{0}^{t}V_{k;j,l}^{e}(t,s,x)+High_{k;j,l}^{0}(t,s,x)ds,$
(5.4)
$V_{k;j,l}^{u}(t,s,x)=\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}e^{ix\cdot\xi+it|\xi|-is|\xi|}m_{u}(\xi,v)\widehat{f}(s,\xi,v)\varphi_{k;j,l}(\xi,v)d\xi
dvds,\quad u\in\\{e,b\\}$ (5.5)
$High_{k;j,l}^{0}(t,s,x):=\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}e^{ix\cdot\xi+it|\xi|-is|\xi|}\frac{1}{|\xi|}\nabla_{v}\hat{v}\cdot\big{(}\widehat{E}(s,\xi-\eta)+\hat{v}\times\widehat{B}(s,\xi-\eta)\big{)}\widehat{f}(s,\eta,v)\varphi_{k;j,l}(\xi,v)d\xi
d\eta dv,$ (5.6)

where the cutoff function $\varphi_{k;j,l}(\xi,v)$ and the symbols
$m_{u}(\xi,v),u\in\\{e,b\\},$ are defined as follows,

$\varphi_{k;j,l}(\xi,v):=\psi_{k}(\xi)\varphi_{j}(v)\varphi_{l;-j}(\angle(\xi,-v)),\quad
m_{e}(\xi,v):=i\frac{(\hat{v}\cdot\xi)\hat{v}-\xi}{|\xi|},\quad
m_{b}(\xi,v):=-i\frac{\hat{v}\times\xi}{|\xi|}.$ (5.7)

where the cutoff functions $\varphi_{j}(\cdot)$ and $\varphi_{l;-j}(\cdot)$
are defined in (2.1).

We remind readers that the angular cutoff function in (5.7), which measure the
angle between frequency $\xi$ and $v$, different from the one we used in the
Glassey-Strauss decomposition (4.1), which measure the angle in physical
space.

Recall (1.1). To take account of the linear effect, we study the profile
$g(t,x,v):=f(t,x+t\hat{v},v)$ of the distribution function $f(t,x,v)$ as
follows,

$\partial_{t}g(t,x,v)=(E(t,x+t\hat{v})+\hat{v}\times
B(t,x+t\hat{v}))\cdot\nabla_{v}f(t,x+t\hat{v},v),\quad
f(t,x,v)=g(t,x-t\hat{v},v).$ (5.8)

To exploit the oscillation with respect to time in some cases, we will use the
normal form transformation for $V_{k;j,l}^{u}(t,s,x)$ in (5.5). More
precisely, after doing integration by parts with respect to $s$, the following
equality holds,

$\int_{0}^{t}V_{k;j,l}^{u}(t,s,x)ds=\int_{0}^{t}\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}e^{ix\cdot\xi+it|\xi|-is|\xi|-is\hat{v}\cdot\xi}m_{u}(\xi,v)\widehat{g}(s,\xi,v)\varphi_{k;j,l}(\xi,v)d\xi
dvds$
$=End_{k;j,l}^{u}(t,t,x)-End_{k;j,l}^{u}(t,0,x)+\int_{0}^{t}High^{u}_{k;j,l}(t,s,x)ds,\quad
u\in\\{e,b\\},$ (5.9)

where

$End_{k;j,l}^{u}(t,t,x)=\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}e^{ix\cdot\xi-
it\hat{v}\cdot\xi}\frac{m_{u}(\xi,v)\varphi_{k;j,l}(\xi,v)}{|\xi|(|\xi|+\hat{v}\cdot\xi)}\widehat{g}(t,\xi,v)d\xi
dv$
$=\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}e^{ix\cdot\xi}\frac{m_{u}(\xi,v)\varphi_{k;j,l}(\xi,v)}{|\xi|(|\xi|+\hat{v}\cdot\xi)}\widehat{f}(t,\xi,v)d\xi
dv,$ (5.10)
$End_{k;j,l}^{u}(t,0,x)=\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}e^{ix\cdot\xi+it|\xi|}\frac{m_{u}(\xi,v)\varphi_{k;j,l}(\xi,v)}{|\xi|(|\xi|+\hat{v}\cdot\xi)}\widehat{f}(0,\xi,v)d\xi
dv,$ (5.11)
$High_{k;j,l}^{u}(t,s,x):=\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}e^{ix\cdot\xi+it|\xi|-is|\xi|-is\hat{v}\cdot\xi}\frac{m_{u}(\xi,v)}{|\xi|(|\xi|+\hat{v}\cdot\xi)}\partial_{t}\widehat{g}(s,\xi,v)\varphi_{k;j,l}(\xi,v)d\xi
dv$ (5.12)
$=\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}e^{ix\cdot\xi+it|\xi|-is|\xi|}\nabla_{v}\big{(}\frac{m_{u}(\xi,v)\varphi_{k;j,l}(\xi,v)}{|\xi|(|\xi|+\hat{v}\cdot\xi)}\big{)}\cdot\big{(}\widehat{E}(s,\xi-\eta)+\hat{v}\times\widehat{B}(s,\xi-\eta)\big{)}\hat{f}(s,\eta,v)d\eta
d\xi dv.$ (5.13)

Since $End_{k;j,l}^{u}(t,0,x)$ only depends on the initial data, which is
regular, it is uniformly bounded over time.

For simplicity of notation, we formulate
$High_{k;j,l}^{m}(t,s,x),m\in\\{0,e,b\\},$ in (5.6) and (5.13), uniformly as
follows,

$High_{k;j,l}^{m}(t,s,x)=\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}e^{ix\cdot\xi+it|\xi|-is|\xi|}\tilde{m}_{k,j,l;m}(\xi,v)\big{(}\widehat{E}(s,\xi-\eta)+\hat{v}\times\widehat{B}(s,\xi-\eta)\big{)}\hat{f}(s,\eta,v)d\eta
dvd\xi$ (5.14)

where

$\tilde{m}_{k,j,l;0}(\xi,v)=\varphi_{k;j,l}(\xi,v)(|\xi|)^{-1}\nabla_{v}\hat{v},\quad\quad\tilde{m}_{k,j,l;u}(\xi,v)=\nabla_{v}\big{(}\frac{m_{u}(\xi,v)\varphi_{k;j,l}(\xi,v)}{|\xi|(|\xi|+\hat{v}\cdot\xi)}\big{)},\quad
u\in\\{e,b\\}.$ (5.15)

As pointed out by Klainerman-Staffilani [9](see also the smoothing effect
pointed out by Bouchut-Golse-Pallard [1]), a very important observation, which
will also be used here, is that the integration of electromagnetic field along
the characteristic is smoother than the electromagnetic field itself.
Intuitively speaking, the smoothing effect comes from the different speeds of
wave and particles.

However, we need to be very careful about the gain and the loss of using the
smoothing effect, which, in practice, is doing integration by parts with
respect to “$s$”. This process will be carried out in (5.20).

As a matter of fact, the loss of doing integration over time along the spatial
characteristic $X(s)$ depends on the angle between the frequency and the
velocity characteristics $V(s)$, see the resulted symbol in (5.24).
Presumably, the loss can be very large because another electromagnetic field,
which is the main unknown, will be introduced when “$\partial_{s}$” hits the
velocity characteristics, see (5.23) and the system of equations satisfied by
characteristics in (1.5).

To better measure the loss, we dyadically localize the angle between the
frequency variable and the velocity characteristics $V(s)$, which is fixed at
any fixed time $s$. Motivated from the above discussion and the general form
of nonlinearities in (5.14), we study the following bilinear form,

$T_{k;j}^{l,n}(K,f)(s,\tau,x):=\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}e^{ix\cdot\xi+i(s-\tau)|\xi|}\varphi_{k;j,l}(\xi,v)m_{1}(\xi,v)m_{2}(\xi,\omega_{0})\varphi_{n;l}(\angle(\xi,\omega_{0}))\widehat{K}(\tau,\xi-\eta)\hat{f}(\tau,\eta,v)d\eta
dvd\xi,$ (5.16)

where $\omega_{0}\in\mathbb{S}^{2}$ is fixed,
$k\in\mathbb{Z},j\in\mathbb{Z}_{+}$, $l\in[-j,2]\cap\mathbb{Z}$,
$n\in[l,2]\cap\mathbb{Z}$ , $K\in\\{E,B\\}$, the cutoff function
$\varphi_{k;j,l}(\xi)$ was defined in (5.7), and the symbols $m_{1}(\xi,v)$
and $m_{2}(\xi,\omega_{0})$ satisfies the following assumption for any
$k\in\mathbb{Z},j\in\mathbb{Z}_{+}$,

$\sup_{|a|\leq
N_{1}^{2}}\sup_{\xi,v\in\mathbb{R}^{3}}\big{[}2^{ak}\big{|}\big{(}\tilde{v}\cdot\nabla_{\xi}\big{)}^{a}m_{1}(\xi,v)\big{|}+2^{a(k+l)}\big{|}\big{(}\tilde{v}\times\nabla_{\xi}\big{)}^{a}m_{1}(\xi,v)\big{|}\big{]}\varphi_{k;j,l}(\xi,v)\lesssim\|m_{1}\|_{\mathcal{S}^{\infty}_{k,j,l}},$
(5.17) $\sup_{|a|\leq
N_{1}^{2}}\sup_{\xi\in\mathbb{R}^{3},\omega_{0}\in\mathbb{S}^{2}}\big{[}2^{ak}\big{|}\big{(}\omega_{0}\cdot\nabla_{\xi}\big{)}^{a}m_{2}(\xi,\omega_{0})\big{|}+2^{a(k+n)}\big{|}\big{(}\omega_{0}\times\nabla_{\xi}\big{)}^{a}m_{2}(\xi,\omega_{0})\big{|}\big{]}\psi_{[k-4,k+4]}(\xi)\varphi_{n;l}(\angle(\xi,\omega_{0}))\lesssim\|m_{2}\|_{\mathcal{S}^{\infty}_{k;n,l}}.$
(5.18)

We use $\|m_{1}\|_{\mathcal{S}^{\infty}_{k,j,l}}$ and
$\|m_{2}\|_{\mathcal{S}^{\infty}_{k}}$, which are not norms, simply for
notational convenience.

In applications, $\omega_{0}$ will be the direction (module the sign) of
velocity characteristics $\tilde{V}(s)$, the symbol $m_{1}(\xi,v)$ will be the
price of using the normal form transformation or one (if not using the normal
form transformation), and the symbol $m_{2}(\xi,\omega_{0})$ will be the price
of doing integration over time along the spatial characteristic $X(s)$ or one.
More precisely, recall (5.14), after doing dyadic decomposition for the angle
between $\xi$ and $-V(s)$, we have

$High_{k;j,l}^{m}(s,\tau,x)=\sum_{n\in[l,2]\cap\mathbb{Z}}H_{k;j,l}^{m;n}(s,\tau,x),\quad
H_{k;j,l}^{m;n}(s,\tau,x):=\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}e^{ix\cdot\xi+i(s-\tau)|\xi|}\tilde{m}_{k,j,l;m}(\xi,v)$
$\times\varphi_{n;l}(\angle(\xi,-\tilde{V}(s)))\big{(}\widehat{E}(\tau,\xi-\eta)+\hat{v}\times\widehat{B}(\tau,\xi-\eta)\big{)}\hat{f}(\tau,\eta,v)d\eta
dvd\xi.$ (5.19)

Let $C:\mathbb{R}_{x}^{3}\times R_{v}^{3}\longrightarrow\mathbb{R}_{x}^{3}$ be
a differentiable function. To control the increment of magnitude of
characteristics between any two time
$t_{1},t_{2}\in[T^{\ast},T^{\ast\ast}]\subset[0,t]$, we estimate the following
general form, which is applicable for quantities in (3.9) and (3.10).
Moreover, recall (5.19), the following equality holds after doing integration
by parts with respect to “$s$”,

$\int_{t_{1}}^{t_{2}}\int_{0}^{s}C(X(s),V(s))\cdot
H_{k;j,l}^{m;n}(s,\tau,X(s))d\tau
ds=\sum_{i=1,2,3}G_{m}^{n;i}(C)(t_{1},t_{2}),$ (5.20)

where $m\in\\{0,e,b\\}$ and the detailed formulas of
$G_{m}^{n;i}(C)(t_{1},t_{2}),i\in\\{1,2,3\\},$ are given as follow,

$G_{m}^{n;1}(C)(t_{1},t_{2})=-\int_{t_{1}}^{t_{2}}\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}e^{iX(s)\cdot\xi}\mathcal{M}_{m}(C)(s,\xi,v)\cdot\big{[}\big{(}\widehat{E}(s,\xi-\eta)+\hat{v}\times\widehat{B}(s,\xi-\eta)\big{)}\widehat{f}(s,\eta,v)\big{]}dvd\xi
d\eta ds,$ (5.21)
$G_{m}^{n;2}(C)(t_{1},t_{2})=\sum_{i=1,2}(-1)^{i}\int_{0}^{t_{i}}\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}e^{iX(t_{i})\cdot\xi+it_{i}|\xi|-i\tau|\xi|}\mathcal{M}_{m}^{n}(C)(t_{i},\xi,v)\cdot\big{[}\big{(}\widehat{E}(\tau,\xi-\eta)$
$+\hat{v}\times\widehat{B}(\tau,\xi-\eta)\big{)}\widehat{f}(\tau,\eta,v)\big{]}dvd\tau
d\xi d\eta,$ (5.22)
$G_{m}^{n;3}(C)(t_{1},t_{2})=-\int_{t_{1}}^{t_{2}}\int_{0}^{s}\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}\big{(}e^{iX(s)\cdot\xi+is|\xi|-i\tau|\xi|}\big{)}\partial_{s}\mathcal{M}_{m}^{n}(C)(s,\xi,v)\cdot\big{[}\big{(}\widehat{E}(\tau,\xi-\eta)$
$+\hat{v}\times\widehat{B}(\tau,\xi-\eta)\big{)}\widehat{f}(\tau,\eta,v)\big{]}dvd\tau
dsd\xi,$ (5.23)

where

$\mathcal{M}_{m}^{n}(C)(s,\xi,v):=\big{(}\
\tilde{m}_{k,j,l;m}(\xi,v)\big{)}^{\top}C(X(s),V(s))\big{(}{(|\xi|+\hat{V}(s)\cdot\xi)}\big{)}^{-1}\varphi_{n;l}(\angle(\xi,-\tilde{V}(s))).$
(5.24)

Recall that $j\in[(1-\alpha)M_{t},(1+\epsilon)M_{t}]$. For the sake of
readers, based on the possible size of $k$ (corresponds to the size of output
frequency $\xi$), $l$ (corresponds to the size of angle between $\xi$ and
$-v$) and $n$(corresponds to the size of angle between $\xi$ and velocity
characteristic $-V(s)$), we summarize our main strategies as follows.

  1. $\bullet$

The case $k\leq 40M_{t}$ and $k\leq-l+j+(1-3\alpha/2)M_{t}$. For this case, we
work with the formulation (5.4) directly. We don’t use the normal form
transformation and don’t use the integration by parts in characteristic time.

  2. $\bullet$

The case $k\leq 40M_{t}$ and $k\geq-l+j+(1-3\alpha/2)M_{t}$. For this case, we
always use the normal form transformation and work with the formulation (5.9).
Moreover, we don’t use the integration by parts in characteristic time if $l$
is large or $n$ is small and do use it if $l$ is small and $n$ is large.

  3. $\bullet$

The case $k\geq 40M_{t}$. For the very large frequency case, we use both the
normal form transformation and the integration by parts in characteristic
time.

######  Lemma 5.1.

For any fixed $k\in\mathbb{Z},j\in\mathbb{Z}_{+}$, $l\in[-j,2]\cap\mathbb{Z}$,
s.t., $k\leq 40M_{t},j\in[(1-\alpha)M_{t},(1+\epsilon)M_{t}]$, and
$k\leq-l+j+(1-3\alpha/2)M_{t}$, the following estimate holds for any
$s\in[0,t]$,

$\sum_{K\in\\{E,B\\}}\big{|}K_{k;j,l}(s,x)\big{|}\lesssim\min\\{\frac{2^{(1-\alpha)M}}{r_{-}},2^{3k+3l-j}+2^{2k+2l}\\},\quad
r:=|x|.$ (5.25)

###### Proof.

Recall the decomposition (5.4) and the detailed formulas of
$V_{k;j,l}^{u}(t,s,x),u\in\\{e,b\\}$, and $High_{k;j,l}^{0}(t,s,x)$ in
(5.5–5.6). From the volume of support of $\xi$ and the estimate (3.2), we have

$\sum_{u\in\\{e,b\\}}|V_{k;j,l}^{u}(s,\tau,x)|\lesssim
2^{3k+3l}\|\widehat{f}(\tau,\xi,v)\varphi_{j}(v)\|_{L^{1}_{v}L^{\infty}_{\xi}}\lesssim
2^{3k+3l-j},$
$|High_{k;j,l}^{0}(s,\tau,x)|\lesssim\sum_{K\in\\{E,B\\}}2^{2k+2l-j}\||K(\tau,x)f(\tau,x,v)\varphi_{j}(v)\|_{L^{1}_{x,v}}\lesssim
2^{2k+2l}.$ (5.26)

From the above two estimates, we know that the second part of our desired
estimate (5.25) holds. Moreover, it allows us to focus on the case
$k\geq(1-\alpha)M_{t}/2$ for obtaining first part of the desired estimate in
(5.25).

From the Kirchhoff’s formulas (2.2) and (2.3) in Lemma 2.1, we have

$V_{k;j,l}^{u}(s,\tau,x)=I_{u}^{1}(s,\tau,x)+I_{u}^{2}(s,\tau,x),\quad
I_{u}^{1}(s,\tau,x):=\int_{\mathbb{S}^{2}}\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}(s-\tau)K_{k,l}^{u}(y,\theta,v)f(\tau,x-y-(s-\tau)\theta,v)\psi_{j}(v)dydvd\theta$
(5.27)
$I_{u}^{2}(s,\tau,x):=\int_{\mathbb{S}^{2}}\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}E_{k,l}^{u}(y,v)f(\tau,x-y-(s-\tau)\theta,v)\psi_{j}(v)dydvd\theta,\quad
u\in\\{e,b\\},$ (5.28)
$High_{k;j,l}^{0}(s,\tau,x)=J_{u}^{1}(s,\tau,x)+J_{u}^{2}(s,\tau,x),\quad
J_{u}^{1}(t,s,x):=\int_{\mathbb{S}^{2}}\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}(s-\tau)K_{k,l}^{high;0}(y,\theta,v)\big{(}E(\tau,x-y-(s-\tau)\theta)$
$+\hat{v}\times
B(\tau,x-y-(s-\tau)\theta)\big{)}\cdot\nabla_{v}\hat{v}f(\tau,x-y-(s-\tau)\theta,v)\psi_{j}(v)dydvd\theta,$
(5.29)
$J_{u}^{2}(s,\tau,x):=\int_{\mathbb{S}^{2}}\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}E_{k,l}^{high;0}(y,v)\big{(}E(\tau,x-y-(s-\tau)\theta)+\hat{v}\times
B(\tau,x-y-(s-\tau)\theta)\big{)}\cdot\nabla_{v}\hat{v}$ $\times
f(\tau,x-y-(s-\tau)\theta,v)\psi_{j}(v)dydvd\theta,$ (5.30)

where

$K_{k,l}^{u}(y,\theta,v):=\int_{\mathbb{R}^{3}}e^{iy\cdot\xi}im_{u}(\xi,v)(|\xi|+\theta\cdot\xi)\varphi_{l;-j}(\angle(\xi,-v))\psi_{k}(\xi)d\xi,\quad
u\in\\{e,b\\},$ (5.31)
$K_{k,l}^{high;0}(y,\theta,v):=\int_{\mathbb{R}^{3}}e^{iy\cdot\xi}(1+\theta\cdot\xi/|\xi|)\varphi_{l;-j}(\angle(\xi,-v))\psi_{k}(\xi)d\xi.$
(5.32)
$E_{k,l}^{u}(y,v):=\int_{\mathbb{R}^{3}}e^{iy\cdot\xi}m_{u}(\xi,v)\varphi_{l;-j}(\angle(\xi,-v))\psi_{k}(\xi)d\xi,\quad
u\in\\{e,b\\}.$ (5.33)
$E_{k,l}^{high;0}(y,v)=\int_{\mathbb{R}^{3}}e^{iy\cdot\xi}|\xi|^{-1}\varphi_{l;-j}(\angle(\xi,-v))\psi_{k}(\xi)d\xi$
(5.34)

After doing integration by parts in $\tilde{v}$ direction and
$\tilde{v}^{\bot}$ directions, the following estimate holds for any
$u\in\\{e,b\\},$

$\sup_{\theta\in\mathbb{S}^{2}}2^{-k-l}|K_{k,l}^{u}(y,\theta,v)|+|K_{k,l}^{high;0}(y,\theta,v)|+2^{-l}|E_{k,l}^{u}(y,v)|+2^{k}|E_{k,l}^{high;0}(y,v)|$
$\lesssim
2^{3k+2l}(1+2^{k}|y\cdot\tilde{v}|)^{-10}(1+2^{k+l}|y\times\tilde{v}|)^{-10}.$
(5.35)

Note that, for $y\in\mathbb{R}^{3}$ in (5.27–5.30), we can represent it in
terms of the cylinder coordinate system as follows,

$y=\big{(}\tilde{v}\cdot y\big{)}\tilde{v}+r_{y}\omega_{y},\quad
r_{y}\in\mathbb{R}_{+},\omega_{y}\in S_{v},\quad
S_{v}:=\\{\omega\in\mathbb{S}^{2}:\omega\cdot v=0\\}.$ (5.36)

Therefore, from the estimate of kernel in (5.35), the estimate (2.10) in Lemma
2.2, and the estimate (3.2), we have

$\big{|}I_{u}^{1}(s,\tau,x)\big{|}\lesssim\int_{\mathbb{S}^{2}}\int_{\mathbb{R}^{3}}\int_{S_{v}}\int_{\mathbb{R}}\int_{\mathbb{R}}(s-\tau)2^{4k+3l}(1+a2^{k})^{-10}(1+r2^{k+l})^{-10}f(\tau,x-a\tilde{v}-r\omega_{v}-(s-\tau)\theta,v)\psi_{j}(v)rdrdad\omega_{v}dvd\theta$
$\lesssim\frac{2^{k+l}\|\int_{\mathbb{R}^{3}}f(\tau,x,v)\varphi_{j}(v)dv\|_{L^{1}_{x}}}{r|r-(t-s)|}\lesssim\frac{2^{k+l-j}}{r|r-(t-s)|}.$
(5.37)

Following the same strategy, we have

$\big{|}I_{u}^{2}(s,\tau,x)\big{|}\lesssim\frac{2^{l-j}}{r(t-s)|r-(t-s)|}.$
(5.38)
$|J_{u}^{1}(s,\tau,x)|\lesssim\sum_{K\in\\{E,B\\}}\frac{2^{-j}}{r|r-(t-s)|}\|\int_{\mathbb{R}^{3}}|K(\tau,x)|f(\tau,x,v)\varphi_{j}(v)dv\|_{L^{1}_{x}}\lesssim\frac{1}{r|r-(t-s)|}$
(5.39)
$|J_{u}^{2}(s,\tau,x)|\lesssim\sum_{K\in\\{E,B\\}}\frac{2^{-k-j}}{r(t-s)|r-(t-s)|}\|\int_{\mathbb{R}^{3}}|K(\tau,x)|f(\tau,x,v)\varphi_{j}(v)dv\|_{L^{1}_{x}}\lesssim\frac{2^{-k}}{r(t-s)|r-(t-s)|}.$
(5.40)

Moreover, from the estimate of kernels in (5.35), Cauchy-Schwarz inequality
and the volume of support of $v$, we have

$\big{|}I_{u}^{1}(s,\tau,x)\big{|}\lesssim 2^{k+l+3j+\epsilon
M_{t}},\quad\big{|}I_{u}^{2}(s,\tau,x)\big{|}\lesssim
2^{l+3j},\quad\big{|}J_{u}^{1}(s,\tau,x)\big{|}\lesssim 2^{3k/2+l+2j+\epsilon
M_{t}},\big{|}J_{u}^{2}(s,\tau,x)\big{|}\lesssim 2^{k/2+l+2j}.$ (5.41)

Let $\delta=2^{-10M_{t}}$. From the above estimate (5.41) and the estimates
(5.37–5.40), the following estimate holds for any $u\in\\{e,b\\}$,

$\int_{0}^{s}\big{(}\big{|}I_{u}^{1}(s,\tau,x)\big{|}+\big{|}J_{u}^{1}(s,\tau,x)\big{|}\big{)}d\tau$
$\lesssim\int_{[0,s]\cap[s-r-\delta,s-r+\delta]}2^{\epsilon
M_{t}}(2^{k+l+3j}+2^{3k/2+l+2j})d\tau+\int_{[0,s]\cap[s-r-\delta,s-r+\delta]^{c}}\frac{2^{k+l-j}+1}{r|r-(s-\tau)|}d\tau$
$\lesssim\delta 2^{\epsilon
M_{t}}(2^{k+l+3j}+2^{3k/2+l+2j})+\ln(1/\delta)\frac{1}{r}(2^{k+l-j}+1)\lesssim\frac{2^{(1-\alpha)M_{t}}}{r_{-}},$
(5.42)
$\int_{0}^{s}\big{(}\big{|}I_{u}^{2}(s,\tau,x)\big{|}+\big{|}J_{u}^{2}(s,\tau,x)\big{|}\big{)}d\tau$
$\lesssim\int_{[0,s-{\delta}]\cap[s-r-\delta,s-r+\delta]^{c}}\frac{2^{-j}+2^{-k}}{r(s-\tau)|r-(s-\tau)|}d\tau+\int_{([0,s-{\delta}]\cap[s-r-\delta,s-r+\delta])\cup[s-{\delta},t]}2^{3j}d\tau$
$\lesssim{\delta}2^{3j}+\frac{(2^{-j}+2^{-k})\ln(1/{\delta})}{r^{2}}\lesssim\frac{2^{-(1-\alpha-\epsilon)M_{t}/2}}{r^{2}_{-}}.$
(5.43)

Note that the above estimate (5.43) is sufficient for the estimate of
$I_{u}^{2}(s,\tau,x)$ and $J_{u}^{2}(s,\tau,x)$ if $r\geq
2^{-3(1-\alpha-\epsilon)M_{t}/2}$. If $r\leq
2^{-3(1-\alpha-\epsilon)M_{t}/2}$, then from the estimates (5.38), (5.40), and
(5.41), the following estimate holds,

$\int_{0}^{s}\big{(}\big{|}I_{u}^{2}(s,\tau,x)\big{|}+\big{|}J_{u}^{2}(s,\tau,x)\big{|}\big{)}d\tau\lesssim\int_{s-2^{-M_{t}}}^{s}(2^{l+3j}+2^{k/2+l+2j})d\tau+\int_{0}^{\max\\{s-2^{-M_{t}},0\\}}\frac{2^{-j}+2^{-k}}{r(s-\tau)|r-(s-\tau)|}d\tau$
$\lesssim
2^{2(1+2\epsilon)M_{t}}+\frac{2^{M_{t}}(2^{-j}+2^{-k})}{r}\lesssim\frac{2^{(1-\alpha)M_{t}}}{r}$
(5.44)

Recall the decomposition (5.4) and the decompositions in (5.27) and (5.29). To
sum up, our desired estimate (5.25) holds from the estimates (5.42), (5.43),
and (5.44).

∎

Now, we proceeds to consider the case when $k\geq-l+j+(1-3\alpha/2)M_{t}$. As
we explained at the beginning, we will use normal form transformation for this
case. Recall (5.9). The estimate of endpoint case is summarized in the
following Lemma.

######  Lemma 5.2.

For any fixed $k\in\mathbb{Z},j\in\mathbb{Z}_{+}$, $l\in[-j,2]\cap\mathbb{Z}$,
s.t., $j\in[(1-\alpha)M_{t},(1+\epsilon)M_{t}],$
$k\geq-l+j+(1-3\alpha/2)M_{t}$, the following estimate holds for any
$s\in[0,t]$,

$|End_{k;j,l}^{e}(s,s,x)|+|End_{k;j,l}^{b}(s,s,x)|\lesssim\min\\{\frac{2^{(1-\alpha)M_{t}}}{r_{-}},2^{-k-l+3j}\\}.$
(5.45)

###### Proof.

Recall the detailed formula of $End_{k;j,l}^{u}(s,s,x),u\in\\{e,b\\}$, in
(5.10). In terms of kernel, we have

$End_{k;j,l}^{u}(s,s,x)=\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}\widetilde{K_{k,l}^{u}}(y,v)f(s,x-y,v)\varphi_{j}(v)dydv,\quad
u\in\\{e,b\\},$

where

$\widetilde{K_{k,l}^{u}}(y,v)=\int_{\mathbb{R}^{3}}e^{iy\cdot\xi}\frac{m_{u}(\xi,v)\psi_{k}(\xi)}{|\xi|(|\xi|+\hat{v}\cdot\xi)}\varphi_{l;-j}(\angle(v,-\xi))d\xi.$

Recall the detailed formulas of symbol $m_{u}(\xi,v)$ in (5.7). After doing
integration by parts in $\tilde{v}$ direction and $\tilde{v}^{\bot}$
directions, the following estimate holds for the kernels,

$|\widetilde{K_{k,l}^{u}}(y,v)|\lesssim
2^{2k+l}(1+2^{k}|y\cdot\tilde{v}|)^{-100/\epsilon}(1+2^{k+l}|y\times\tilde{v}|)^{-100/\epsilon}.$
(5.46)

From the above estimate of kernels and the volume of support of $v$, we have

$|End_{k;j,l}^{u}(s,s,x)\lesssim 2^{-k-l+3j}\lesssim 2^{(1+3\alpha)M_{t}}$
(5.47)

In particular, from the above estimate, we have the following estimate if
$|x|\leq 2^{-4\alpha M_{t}}$,

$|End_{k;j,l}^{e}(s,s,x)|+|End_{k;j,l}^{b}(s,s,x)|\lesssim{r}^{-1}2^{(1-\alpha)M_{t}}.$
(5.48)

Now, we restrict ourself to the case $|x|\geq 2^{-4\alpha M_{t}}$. Due to the
fast decay rate of kernels in the estimate (5.46), the following estimate
holds if $|y|\geq 2^{-k-l+\epsilon M_{t}}$,

$\sum_{u\in\\{e,b\\}}\big{|}\int_{\mathbb{R}^{3}}\int_{|y|\geq
2^{-k-l+\epsilon
M_{t}}}\widetilde{K_{k,l}^{u}}(y,v)f(s,x-y,v)\varphi_{j}(v)dydv\big{|}\lesssim
2^{-k-l-50j}\lesssim r^{-1}.$ (5.49)

Note that $|x-y|\sim|x|$ if $|y|\leq 2^{-k-l+\epsilon M_{t}}$ and $|x|\geq
2^{-4\alpha M_{t}}$. From the radial symmetry of $f(t,\cdot,\cdot)$ and the
estimate of kernel in (5.46), the following estimate holds if $|x|\geq
2^{-4\alpha M_{t}}$,

$\sum_{u\in\\{e,b\\}}\big{|}\int_{\mathbb{R}^{3}}\int_{|y|\leq
2^{-k-l+\epsilon
M_{t}}}\widetilde{K_{k,l}^{u}}(y,v)f(s,x-y,v)\varphi_{j}(v)dydv\big{|}\lesssim
2^{2k+l}\frac{2^{-2k-2l+2\epsilon
M_{t}}}{r^{2}}2^{-j}\lesssim\frac{2^{2\epsilon
M_{t}}}{r^{2}}\lesssim\frac{2^{5\alpha M_{t}}}{r}$ (5.50)

To sum up, our desired estimate (5.45) holds from the estimates (5.47),
(5.48), (5.49), and (5.50). ∎

Recall the decompositions in (5.4) and (5.9). Now our mission is reduced to
estimate the quadratic terms, $High^{m}_{k;j,l}(t,s,x),$ $m\in\\{0,e,b\\}$ in
(5.14) for the case $k\geq-l+j+(1-3\alpha/2)M_{t}$.

Recall the decomposition of $High^{m}_{k;j,l}(t,s,x)$,$m\in\\{0,e,b\\}$, in
(5.19). The key ingredients of estimating $H_{k;j,l}^{m;n}(t,s,x)$, are two
point-wise bilinear estimates in Lemma 5.7, which are applicable regardless
whether using the integration by parts in characteristic time. We will use
them as black boxes for the estimate of $H_{k;j,l}^{m;n}(t,s,x)$,
$m\in\\{0,e,b\\}$, in the next two Lemmas. For the sake of clarity, we
postpone and and elaborate the proof of Lemma 5.7 to the next subsection.

######  Lemma 5.3.

Under the assumptions (3.6) and (3.7), for any fixed
$s\in[T^{\ast},T^{\ast\ast}]\subset[0,t]$,
$k\in\mathbb{Z},j\in\mathbb{Z}_{+}$, $l\in[-j,2]\cap\mathbb{Z}$, s.t., $k\leq
40M_{t},j\in[(1-\alpha)M_{t},(1+\epsilon)M_{t}]$, and
$k\geq-l+j+(1-3\alpha/2)M_{t}$, if moreover we have
$l\geq-2(1-7\alpha)M_{t}/3$ or $n\leq-M_{t}/2-7\alpha M_{t}$, then the
following estimate holds,

$\big{|}\int_{0}^{s}C_{1}(X(s),V(s))\cdot\big{(}H_{k;j,l}^{0;n}(s,\tau,X(s))+H_{k;j,l}^{e;n}(s,\tau,X(s))\big{)}d\tau\big{|}+\big{|}\int_{0}^{s}C_{2}(X(s),V(s))\cdot
H_{k;j,l}^{b;n}(s,\tau,X(s))d\tau\big{|}\lesssim 2^{-\alpha M_{t}},$ (5.51)
$\big{|}\int_{0}^{s}\tilde{V}(s)\cdot\big{(}H_{k;j,l}^{0;n}(s,\tau,X(s))+H_{k;j,l}^{e;n}(s,\tau,X(s))\big{)}d\tau\big{|}\lesssim\min\\{\frac{2^{(1-\alpha)M_{t}}}{|X(s)|_{-}},2^{k/2+3(1+3\epsilon)M_{t}}|X(s)|+2^{3(1+3\epsilon)M_{t}}\\}.$
(5.52)

###### Proof.

Recall the detailed formulas of symbols $\tilde{m}_{k,j,l;m}(\xi,v)$ in (5.15)
and smooth coefficients in (3.11). Note that, the following decomposition
holds for any $u$,

$u\cdot\nabla_{v}=u\cdot\tilde{v}(\tilde{v}\cdot\nabla_{v})+\sum_{i=1,2,3}u\cdot(\tilde{v}\times\mathbf{e}_{i})\big{(}(\tilde{v}\times\mathbf{e}_{i})\cdot\nabla_{v}\big{)},$
(5.53)

where
$\mathbf{e}_{1}:=(1,0,0),\mathbf{e}_{2}:=(0,1,0),\mathbf{e}_{3}:=(0,0,1).$
Recall (5.17). With the above equality, as a result of direct computations, we
have

$\|\tilde{m}_{k,j,l;0}(\xi,v)\|_{\mathcal{S}^{\infty}_{k,j,l}}\lesssim
2^{-k-j},\quad\|\tilde{m}_{k,j,l;e}(\xi,v)\|_{\mathcal{S}^{\infty}_{k,j,l}}+\|\tilde{m}_{k,j,l;b}(\xi,v)\|_{\mathcal{S}^{\infty}_{k,j,l}}\lesssim
2^{-k-j-2l},$ (5.54)
$\|\tilde{V}(s)\cdot\tilde{m}_{k,j,l;e}(\xi,v)\varphi_{n;l}(\angle(\xi,-\tilde{V}(t)))\|_{\mathcal{S}^{\infty}_{k,j,l}}\lesssim
2^{-k-j-2l+n},$ (5.55)
$\|C_{1}(X(s),V(s))\tilde{m}_{k,j,l;e}(\xi,v)\varphi_{n;l}(\angle(\xi,-\tilde{V}(t)))\|_{\mathcal{S}^{\infty}_{k,j,l}}+\|C_{2}(X(s),V(s))\tilde{m}_{k,j,l;b}(\xi,v)\varphi_{n;l}(\angle(\xi,-\tilde{V}(t)))\|_{\mathcal{S}^{\infty}_{k,j,l}}$
$\lesssim|X(s)|\big{(}|\tilde{X}(s)\times\tilde{V}(s)|+2^{n}\big{)}2^{-(1-\beta)M_{t}-k-j-2l}.$
(5.56)

Therefore, from the above estimates of symbols, the estimate (3.5), and the
estimates (5.96) and (5.97) in Lemma 5.7, we have

$\big{|}\int_{0}^{s}C_{1}(X(s),V(s))\cdot\big{(}H_{k;j,l}^{0;n}(s,\tau,X(s))+H_{k;j,l}^{e;n}(s,\tau,X(s))\big{)}d\tau\big{|}+\big{|}\int_{0}^{s}C_{2}(X(s),V(s))\cdot
H_{k;j,l}^{b;n}(s,\tau,X(ts))d\tau\big{|}$
$\lesssim|X(s)|2^{-(1-\beta)M_{t}-k-j-2l}\frac{2^{k+l/2+n+(1+3\alpha)M_{t}}}{|X(s)|_{-}}\lesssim
2^{-\alpha M_{t}}.$
$\big{|}\int_{0}^{s}\tilde{V}(s)\cdot\big{(}H_{k;j,l}^{0;n}(s,\tau,X(s))+H_{k;j,l}^{e;n}(s,\tau,X(s))\big{)}d\tau\big{|}\lesssim\min\\{\frac{2^{-3l/2+n+4\alpha
M_{t}}}{|X(s)|_{-}},2^{k/2+3(1+3\epsilon)M_{t}}|X(s)|+2^{3(1+3\epsilon)M_{t}}\\}.$

Hence finishing the proof of our desired estimates (5.51) and (5.52). ∎

######  Lemma 5.4.

Under the assumptions (3.6) and (3.7), for any fixed
$t_{1},t_{2}\in[T^{\ast},T^{\ast\ast}]\subset[0,t]$,
$k\in\mathbb{Z},j\in\mathbb{Z}_{+}$, $l\in[-j,2]\cap\mathbb{Z}$, s.t., $k\leq
40M_{t},j\in[(1-\alpha)M_{t},(1+\epsilon)M_{t}]$, and
$k\geq-l+j+(1-3\alpha/2)M_{t}$, if moreover we have
$l\leq-2(1-7\alpha)M_{t}/3$ and $n\geq-M_{t}/2-7\alpha M_{t}$, then the
following estimate holds,

$\big{|}\int_{t_{1}}^{t_{2}}\int_{0}^{s}C_{1}(X(s),V(s))\cdot\big{(}H_{k;j,l}^{0;n}(s,\tau,X(s))+H_{k;j,l}^{e;n}(s,\tau,X(s))\big{)}d\tau
ds\big{|}+\big{|}\int_{t_{1}}^{t_{2}}\int_{0}^{s}C_{2}(X(s),V(s))\cdot
H_{k;j,l}^{b;n}(s,\tau,X(s))d\tau ds\big{|}$
$\lesssim\sum_{i=1,2}|X(t_{i})|\min\\{\frac{2^{-2M_{t}+15\alpha
M_{t}}}{|X(t_{i})|_{-}},2^{M_{t}/2+3\alpha
M_{t}}|X(t_{i})|+2^{-2M_{t}/3+8\alpha M_{t}}\\}+2^{-M_{t}/6+16\alpha
M_{t}}(t_{2}-t_{1}).$ (5.57)
$\big{|}\int_{t_{1}}^{t_{2}}\int_{0}^{s}\tilde{V}\cdot\big{(}H_{k;j,l}^{0;n}(s,\tau,X(s))+H_{k;j,l}^{e;n}(s,\tau,X(s))\big{)}d\tau
ds\big{|}\lesssim 2^{M_{t}/3+7\alpha
M_{t}}+\int_{t_{1}}^{t_{2}}\min\\{\frac{2^{-M_{t}/2+10\alpha
M_{t}}}{|X(s)|_{-}},2^{3M_{t}/2+4\alpha M_{t}}\\}ds$
$+\int_{t_{1}}^{t_{2}}|K(s,X(s))|\min\\{\frac{2^{-3M_{t}/2+24\alpha
M_{t}}}{|X(s)|_{-}},2^{M_{t}/2+3\alpha M_{t}}|X(s)|+2^{-2M_{t}/3+8\alpha
M_{t}}\\}ds.$ (5.58)

###### Proof.

For this case, we do integration by parts in characteristic time. Recall the
decomposition (5.20) and the associated symbol in (5.24). Recall (5.18). Based
on possible destination of $\partial_{s}\mathcal{M}_{m}^{n}(C)(s,\xi,v)$, we
separate $G_{m}^{n;3}(C)(t_{1},t_{2})$ further into three parts as follows,

$G_{m}^{n;3}(C)(t_{1},t_{2})=G_{m;1}^{n;3}(C)(t_{1},t_{2})+G_{m;2}^{n;3}(C)(t_{1},t_{2})+G_{m;3}^{n;3}(C)(t_{1},t_{2})$
(5.59)
$G_{m;1}^{n;3}(C)(t_{1},t_{2})=-\int_{t_{1}}^{t_{2}}\int_{0}^{s}\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}\big{(}e^{iX(s)\cdot\xi+is|\xi|-i\tau|\xi|}\big{)}\hat{V}(s)\cdot\nabla_{x}C(X(s),V(s))\tilde{m}_{k,j,l;m}(\xi,v)\cdot\big{[}\big{(}\widehat{E}(\tau,\xi-\eta)$
$+\hat{v}\times\widehat{B}(\tau,\xi-\eta)\big{)}\widehat{f}(\tau,\eta,v)\big{]}\big{(}{(|\xi|+\hat{V}(s)\cdot\xi)}\big{)}^{-1}\varphi_{n;l}(\angle(\xi,\tilde{V}(s)))dvd\tau
dsd\xi,$ (5.60)
$G_{m;2}^{n;3}(C)(t_{1},t_{2})=-\int_{t_{1}}^{t_{2}}\int_{0}^{s}\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}\big{(}e^{iX(s)\cdot\xi+is|\xi|-i\tau|\xi|}\big{)}\widetilde{\mathcal{M}_{m}^{n}}(C)(s,\xi)\cdot\big{[}\
\tilde{m}_{k,j,l;m}(\xi,v)\big{(}\widehat{E}(\tau,\xi-\eta)$
$+\hat{v}\times\widehat{B}(\tau,\xi-\eta)\big{)}\widehat{f}(\tau,\eta,v)\big{]}\varphi_{n;l}(\angle(\xi,\tilde{V}(s)))dvd\tau
dsd\xi,$ (5.61)
$G_{m;3}^{n;3}(C)(t_{1},t_{2})=-\int_{t_{1}}^{t_{2}}\int_{0}^{s}\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}\big{(}e^{iX(s)\cdot\xi+is|\xi|-i\tau|\xi|}\big{)}C(X(s),V(s))\
\tilde{m}_{k,j,l;m}(\xi,v)\cdot\big{[}\big{(}\widehat{E}(\tau,\xi-\eta)$
$+\hat{v}\times\widehat{B}(\tau,\xi-\eta)\big{)}\widehat{f}(\tau,\eta,v)\big{]}\partial_{s}\big{(}(|\xi|+\hat{V}(s)\cdot\xi)^{-1}\varphi_{n;l}(\angle(\xi,\tilde{V}(s)))\big{)}dvd\tau
dsd\xi,\quad a\in\\{1,2\\},$ (5.62)

where

$\widetilde{\mathcal{M}_{m}^{n}}(C)(s,\xi)=(K(s,X(s),V(s))\cdot\nabla_{v}C(X(s),V(s))(|\xi|+\hat{V}(s)\cdot\xi)^{-1}.$
(5.63)

Recall (5.18) and (3.11). As a result of direct computations, we have

$\|(|\xi|+\hat{V}(s)\cdot\xi)^{-1}\|_{\mathcal{S}^{\infty}_{k;n,l}}\lesssim
2^{-k-2n},$ (5.64)
$\|\partial_{s}\big{(}(|\xi|+\hat{V}(s)\cdot\xi)^{-1}\varphi_{n;l}(\angle(\xi,\tilde{V}(s)))\big{)}\|_{\mathcal{S}^{\infty}_{k;n,l}}\lesssim\sum_{K\in\\{E,B\\}}|K(s,X(s))|2^{-(1-\beta)M_{t}-k-3n},$
(5.65)
$\|\hat{V}(s)\cdot\nabla_{x}C_{1}(X(s),V(s))\tilde{m}_{k,j,l;e}(\xi,v)\|_{\mathcal{S}^{\infty}_{k,j,l}}\lesssim
2^{-(1-\beta)M_{t}-k-j-2l+n},\quad\hat{V}(s)\cdot\nabla_{x}C_{2}(X(s),V(s))=0,$
(5.66)
$\|\widetilde{\mathcal{M}_{m}^{n}}(C_{1})(s,\xi)\|_{\mathcal{S}^{\infty}_{k}}+\|\widetilde{\mathcal{M}_{m}^{n}}(C_{2})(s,\xi)\|_{\mathcal{S}^{\infty}_{k;n,l}}\lesssim\sum_{K\in\\{E,B\\}}|X(s)||K(s,X(s))|2^{-k-2n-2(1-\beta)M_{t}}.$
(5.67)
$\|\widetilde{\mathcal{M}_{m}^{n}}(\tilde{V})(s,\xi)\|_{\mathcal{S}^{\infty}_{k;n,l}}\lesssim\sum_{K\in\\{E,B\\}}|K(s,X(s))|2^{-k-2n-(1-\beta)M_{t}}.$
(5.68)

With the above preparation, we are ready to estimate the terms in
decompositions (5.20) and (5.59) one by one. Recall (5.21). From the estimate
(5.77) in Lemma 5.5, we have

$|G_{e}^{n;1}(C_{1})(t_{1},t_{2})|+|G_{0}^{n;1}(C_{1})(t_{1},t_{2})|+|G_{b}^{n;1}(C_{2})(t_{1},t_{2})|\lesssim
2^{-M_{t}/2+10\alpha M_{t}}(t_{2}-t_{1}),$ (5.69)
$|G_{e}^{n;1}(\tilde{V})(t_{1},t_{2})|+|G_{0}^{n;1}(\tilde{V})(t_{1},t_{2})|\lesssim\int_{t_{1}}^{t_{2}}\min\\{\frac{2^{-M_{t}/2+10\alpha
M_{t}}}{|X(s)|_{-}},2^{3M_{t}/2+4\alpha M_{t}}\\}ds.$ (5.70)

Recall (5.22). From the estimates (5.56) and (5.64), and the estimates (5.96)
and (5.97) in Lemma 5.7, we have

$|G_{e}^{n;2}(C_{1})(t_{1},t_{2})|+|G_{0}^{n;2}(C_{1})(t_{1},t_{2})|+|G_{b}^{n;2}(C_{2})(t_{1},t_{2})|\lesssim\sum_{i=1,2}2^{-k-2n}2^{-(1-\beta)M_{t}-k-j-2l}|X(t_{i})|$
$\times\min\\{\frac{2^{k+l/2+n+(1+3\alpha)M_{t}}}{|X(t_{i})|_{-}},2^{3k/2+3j+2n+l+\epsilon
M_{t}}|X(t_{i})|+2^{k+3j+2n+2l+(1+3\epsilon)M_{t}}\\}.$
$\lesssim\sum_{i=1,2}|X(t_{i})|\min\\{\frac{2^{-2M_{t}+15\alpha
M_{t}}}{|X(t_{i})|_{-}},2^{M_{t}/2+2\alpha
M_{t}}|X(t_{i})|+2^{-2M_{t}/3+8\alpha M_{t}}\\}$ (5.71)

From the estimates (5.55) and (5.64) and the estimates (5.96) and (5.97) in
Lemma 5.7, we have

$|G_{e}^{n;2}(\tilde{V})(t_{1},t_{2})|+|G_{0}^{n;2}(\tilde{V})(t_{1},t_{2})|$
$\lesssim\sum_{i=1,2}2^{-k-j-2l+n}2^{-k-2n}\min\\{\frac{2^{k+l/2+(1+3\alpha)M_{t}}}{|X(t_{i})|_{-}},2^{3k/2+3j+2n+l+\epsilon
M_{t}}|X(t_{i})|+2^{k+3j+2n+2l+(1+3\epsilon)M_{t}}\\}$
$\lesssim\sum_{i=1,2}\min\\{\frac{2^{-M_{t}+15\alpha
M_{t}}}{|X(t_{i})|_{-}},2^{3M_{t}/2+2\alpha
M_{t}}|X(t_{i})|+2^{M_{t}/3+7\alpha M_{t}}\\}\lesssim 2^{M_{t}/3+7\alpha
M_{t}}.$ (5.72)

Recall (5.60). From the estimates (5.66) and (5.54) and the estimates (5.96)
and (5.97) in Lemma 5.7, we have

$|G_{e;1}^{n;3}(C_{1})(t_{1},t_{2})|+|G_{0;1}^{n;3}(C_{1})(t_{1},t_{2})|+|G_{b;1}^{n;3}(C_{2})(t_{1},t_{2})|$
$\lesssim\int_{t_{1}}^{t_{2}}2^{-(1-\beta)M_{t}-k-j-2l+n}2^{-k-2n}\min\\{\frac{2^{k+l/2+(1+3\alpha)M_{t}}}{|X(s)|_{-}},2^{3k/2+3j+2n+l+\epsilon
M_{t}}|X(s)|+2^{k+3j+2n+2l+(1+3\epsilon)M_{t}}\\}ds$
$\lesssim\int_{t_{1}}^{t_{2}}\min\\{\frac{2^{-2M_{t}+16\alpha
M_{t}}}{|X(s)|_{-}},2^{M_{t}/2+3\alpha M_{t}}|X(s)|+2^{-2M_{t}/3+8\alpha
M_{t}}\\}ds\lesssim 2^{-2M_{t}/3+8\alpha M_{t}}(t_{2}-t_{1}).$ (5.73)

Since the function $\tilde{V}:(X(s),V(s))\longrightarrow\tilde{V}(s)$ doesn’t
depend on $X(s)$, we have

$|G_{e;1}^{n;2}(\tilde{V})(t_{1},t_{2})|+|G_{0;1}^{n;2}(\tilde{V})(t_{1},t_{2})|=0.$
(5.74)

Recall (5.61) and (5.62). From the estimates of symbols in (5.54), (5.56),
(5.64), (5.65), and (5.67), the estimates (5.96) and (5.97) in Lemma 5.7, and
the rough estimate of the electromagnetic field (4.6) in Lemma 4.2, we have

$\sum_{a=2,3}|G_{e;a}^{n;3}(C_{1})(t_{1},t_{2})|+|G_{0;a}^{n;3}(C_{1})(t_{1},t_{2})|+|G_{b;a}^{n;3}(C_{2})(t_{1},t_{2})|$
$\lesssim\int_{t_{1}}^{t_{2}}|X(s)||K(s,X(s))|\min\\{2^{-k-2(1-\beta)M_{t}-3n}2^{-k-j-2l}\frac{2^{k+l/2+n+(1+3\alpha)M_{t}}}{|X(s)|_{-}},$
$2^{-k-2(1-\beta)M_{t}-3n}2^{-k-j-2l}\big{[}2^{3k/2+3j+2n+l+\epsilon
M_{t}}|X(s)|+2^{k+3j+2n+2l+(1+3\epsilon)M_{t}}\big{]}\\}ds$
$\lesssim\int_{t_{1}}^{t_{2}}\min\\{2^{(1+20\alpha)M_{t}}|X(s)|+2^{-M/6+16\alpha
M},\frac{2^{-3M/2+24\alpha M}}{|X(s)|_{-}}\\}ds\lesssim 2^{-M/6+16\alpha
M}(t_{2}-t_{1}).$ (5.75)

From the estimates of symbols in (5.55), (5.64), and (5.65), and the estimates
(5.96) and (5.97) in Lemma 5.7, we have

$\sum_{a=2,3}|G_{e;a}^{n;3}(\tilde{V})(t_{1},t_{2})|+|G_{0;a}^{n;3}(\tilde{V})(t_{1},t_{2})|\lesssim\int_{t_{1}}^{t_{2}}|K(s,X(s))|2^{-k-(1-\beta)M_{t}-2n}$
$\times
2^{-k-j-2l}\min\\{\frac{2^{k+l/2+(1+3\alpha)M_{t}}}{|X(s)|_{-}},2^{3k/2+3j+2n+l+\epsilon
M_{t}}|X(s)|+2^{k+3j+2n+2l+(1+3\epsilon)M_{t}}\\}ds$
$\lesssim\int_{t_{1}}^{t_{2}}|K(s,X(s))|\min\\{\frac{2^{-3M_{t}/2+24\alpha
M_{t}}}{|X(s)|_{-}},2^{M_{t}/2+3\alpha M_{t}}|X(s)|+2^{-2M_{t}/3+8\alpha
M_{t}}\\}ds.$ (5.76)

To sum up, recall the decompositions (5.20) and (5.59), our desired estimate
(5.57) holds from the estimates (5.69), (5.71), (5.73), and (5.75). The
desired estimate (5.58) holds from the estimates (5.70), (5.72), (5.74), and
(5.76).

∎

######  Lemma 5.5.

Under the assumptions (3.6) and (3.7), for any
$t_{1},t_{2}\in[T^{\ast},T^{\ast\ast}]\subset[0,t],$ any differentiable
function $C:\mathbb{R}_{x}^{3}\times R_{v}^{3}\longrightarrow\mathbb{R}^{3}$,
any fixed $k\in\mathbb{Z},j\in\mathbb{Z}_{+}$,
$l\in[-j,2]\cap\mathbb{Z},n\in[l,2]$, s.t., $k\leq
40M_{t},j\in[(1-\alpha)M_{t},(1+\epsilon)M_{t}]$, and
$k\geq-l+j+(1-3\alpha/2)M_{t}$, the following estimate holds for any
$m\in\\{0,u,b\\}$,

$\big{|}G_{m}^{n;1}(C)(t_{1},t_{2})\big{|}\lesssim\int_{t_{1}}^{t_{2}}|C(X(\tau),V(\tau))|\min\\{(|X(\tau)|_{-})^{-1}2^{-3k/2+2j-2l+2\alpha
M_{t}},2^{-k/2+2j-l}\\}d\tau.$ (5.77)

###### Proof.

Recall (5.21) and (5.24). In terms of kernel, we have

$|G_{m}^{n;1}(C)(t_{1},t_{2})|=\int_{t_{1}}^{t_{2}}\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}K^{1}_{k;n}(\tau,y)K_{k;l,m}(z,v)\big{(}E(\tau,X(\tau)-y-z)+\hat{v}\times
B(\tau,X(\tau)-y-z)\big{)}$ $\times
f(\tau,X(\tau)-y-z,v)\varphi_{j}(v)\psi_{\leq
n+4}(\angle(v,\tilde{V}(\tau)))dvdydzd\tau,$ (5.78)

where the kernel $K_{k;l,m}(z,v)$ and the kernel $K^{1}_{k;n}(y,v)$ are
defined as follows,

$K_{k;l,m}(z,v)=\int_{\mathbb{R}^{3}}e^{iz\cdot\xi}\tilde{m}_{k,j,l;m}(\xi,v)\varphi_{l;-j}(\angle(\xi,-v))\psi_{[k-4,k+4]}(\xi)d\xi,$
(5.79)
$K^{1}_{k;n}(\tau,y):=\int_{\mathbb{R}^{3}}e^{iy\cdot\xi}C(X(\tau),V(\tau))\big{(}{(|\xi|+\hat{V}(\tau)\cdot\xi)}\big{)}^{-1}\psi_{k}(\xi)\varphi_{n;l}(\angle(\xi,-\tilde{V}(\tau)))d\xi.$

In the equality (5.78), we used the fact that the angle between $v$ and
$\tilde{V}(\tau)$ is less than $2^{n+2}$ because the angle between $-v$ and
$\xi$ is less than $2^{l}$ and the angle between between $\xi$ and
$-\tilde{V}(\tau)$, which is a fixed vector, is less than $2^{n}$.

Recall the detailed formulas of symbols $\tilde{m}_{k,j,l;m}(\xi,v)$ in
(5.15). After doing integration by parts in $\xi$ in directions $\tilde{v}$
and $\tilde{v}^{\bot}$, the following estimate holds for any
$m\in\\{0,e,b\\},$

$|K_{k;l,m}(z,v)|\lesssim
2^{2k-j}(1+2^{k}|z\cdot\tilde{v}|)^{-1000/\epsilon}(1+2^{k+l}|z\times\tilde{v}|)^{-1000/\epsilon}.$
(5.80)

After doing integration by parts in $\xi$ in directions $\tilde{V}(\tau)$ and
$\tilde{V}(\tau)^{\bot}$, the following estimate holds,

$\big{|}K^{1}_{k;n}(\tau,y)\big{|}\lesssim
2^{2k}|C(X(\tau),V(\tau))|(1+2^{k}|y\cdot\tilde{V}(\tau)|)^{-100/\epsilon}(1+2^{k+n}|y\times\tilde{V}(\tau)|)^{-100/\epsilon}.$
(5.81)

From the Cauchy-Schwarz inequality for the integration with respect to $z$,
the conservation laws in (1.3), the boundedness of $L^{\infty}_{x,v}-$norm of
$f$ and the volume of support of $v$. As a result, the following estimate
holds from the estimates of kernels in (5.80) and (5.81),

$|G_{m}^{n;1}(C)(t_{1},t_{2})|$
$\lesssim\sum_{K\in\\{E,B\\}}\int_{t_{1}}^{t_{2}}\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}|K^{1}_{k;n}(\tau,y)|\psi_{\leq
n+4}(\angle(v,\tilde{V}(\tau)))\|K(\tau,X(\tau)-y-z)\|_{L^{2}_{z}}\|K_{k;l,m}(z,v)\|_{L^{2}_{z}}\varphi_{j}(v)dydvd\tau$
$\lesssim\int_{t_{1}}^{t_{2}}|C(X(\tau),V(\tau))|2^{-k/2-j-2n-l}2^{3j+2n}d\tau\lesssim\int_{t_{1}}^{t_{2}}|C(X(\tau),V(\tau))|2^{-k/2+2j-l}d\tau.$
(5.82)

For fixed $\tau$, based on the possible size of $|X(\tau)|$, we separate into
two cases as follow.

Case $1$: If $|X(\tau)|\leq 2^{-k-l+2\epsilon M_{t}}$. From the estimate
(5.82), we have

$|G_{m}^{n;1}(C)(t_{1},t_{2})|\lesssim\int_{t_{1}}^{t_{2}}|C(X(\tau),V(\tau))|(|X(\tau)|)^{-1}{2^{-3k/2+2j-2l+2\epsilon
M_{t}}}d\tau.$ (5.83)

Case $2$: If $|X(\tau)|\geq 2^{-k-l+2\epsilon M_{t}}$. For this case we have
$|X(\tau)-y-z|\sim|X(\tau)|$ if $|y|+|z|\leq 2^{-k-l+\epsilon M_{t}}$, which
is the main subcase we only have to consider. If $|y|+|z|\geq 2^{-k-l+\epsilon
M_{t}}$, then from the estimates of kernels in (5.81) and (5.80), we know that
the size of kernels is very small. From the rough estimate of electromagnetic
field (4.6) in Lemma 4.2, the estimates of kernels and the volume of support
of $v$, we have

$|G_{m}^{n;1}(C)(t_{1},t_{2})|\lesssim\int_{t_{1}}^{t_{2}}|C(X(\tau),V(\tau))|\big{[}2^{-2k-j-2n-2l}(|X(\tau)|_{-})^{-1}2^{(1+\epsilon)M_{t}}2^{3j+2n}+2^{-500M_{t}}\big{]}d\tau$
(5.84)

To sum up, our desired estimate (5.77) holds from the estimates (5.82), (5.83)
and (5.84). ∎

######  Lemma 5.6.

Under the assumptions (3.6) and (3.7), for any
$t_{1},t_{2}\in[T^{\ast},T^{\ast\ast}]\subset[0,t],m\in\\{0,e,b\\}$, the
following estimate holds for any differentiable function
$C:\mathbb{R}_{x}^{3}\times\mathbb{R}_{v}^{3}\longrightarrow\mathbb{R}^{3}$
s.t., $\|C(x,v)\|_{L^{\infty}_{x,v}}\lesssim 1$,

$\big{|}\int_{t_{1}}^{t_{2}}\int_{0}^{s}C(X(s),V(s))High_{k;j,l}^{m}(s,\tau,X(s))d\tau
ds\big{|}\lesssim
2^{-2k/7+5M_{t}+4j}\big{(}\min\\{2^{-j},2^{-N_{1}j+N_{1}M_{t}}\\}\big{)}^{1/14}.$
(5.85)

###### Proof.

For this case, we do integration by parts in characteristic time. Recall the
equality in (5.20) and the corresponding symbols in (5.24), and (5.15). Note
that from the $L^{\infty}_{x}\longrightarrow L^{7/4}$ type Sobolev embedding,
the estimate of velocity of characteristics in (3.6), and the basic estimate
in (3.2), the following estimate holds for any $m\in\\{0,e,b\\},$

$\sum_{i=1,2}\sum_{n\in[l,2]\cap\mathbb{Z}}|G_{m}^{n;i}(C)(t_{1},t_{2})|\lesssim
2^{-j-2l}2^{12k/7}2^{-2k}2^{2(1-\beta)M_{t}}\int_{0}^{t}\|\int_{\mathbb{R}^{3}}|K(s,x)|f(s,x,v)\varphi_{j}(v)dv\|_{L^{7/4}_{x}}ds$
$\lesssim
2^{-j-2l}2^{-2k/7+2M_{t}}\int_{0}^{t}\|\int_{\mathbb{R}^{3}}f(s,x,v)\varphi_{j}(v)dv\|_{L^{14}_{x}}ds\lesssim
2^{-2k/7+2M_{t}+4j}\big{(}\min\\{2^{-j},2^{-N_{1}j+N_{1}M_{t}}\\}\big{)}^{1/14}$
(5.86)

Recall the detailed formula of $G_{m}^{n;3}(C)(t_{1},t_{2})$ in (5.23).
Similarly, by using the same strategy used in the above estimate, the
following estimate holds for $G_{m}^{n;3}(C)(t_{1},t_{2})$ from the first
estimate (4.17) in Lemma 4.4,

$\sum_{i=1,2}\sum_{n\in[l,2]\cap\mathbb{Z}}|G_{m}^{n;3}(C)(t_{1},t_{2})|\lesssim\sum_{K_{1},K_{2}\in\\{E,B\\}}2^{-j-2l}2^{12k/7}2^{-2k+2(1-\beta)M_{t}}\int_{t_{1}}^{t_{2}}|K_{1}(s,X(s))|$
$\times\int_{0}^{s}\|\int_{\mathbb{R}^{3}}|K_{2}(\tau,x)|f(\tau,x,v)\varphi_{j}(v)dv\|_{L^{7/4}_{x}}d\tau
ds\lesssim
2^{-2k/7+5M_{t}}2^{4j}\big{(}\min\\{2^{-j},2^{-nj+nM_{t}}\\}\big{)}^{1/14}.$
(5.87)

Hence, our desired estimate (5.85) holds straightforwardly from the estimate
(5.86) and the estimate (5.87). ∎

Proof of Proposition 5.1.

Recall the decomposition of the electromagnetic field in (2.5) and the
decompositions of $K_{j}$ in (5.3), (5.4) and (5.9).

The desired estimate (5.1) follows from the second estimate in (4.17) in Lemma
4.4, which is used for the case $j\geq(1+\epsilon)M_{t}$, the estimate (4.7)
in Lemma 4.2 and the estimate (4.13) in Lemma 4.3, which are used for the case
$j\leq(1-\alpha)M_{t}$, the estimate (5.25) in Lemma 5.1, which is used for
the case
$j\in[(1-\alpha)M_{t},(1+\epsilon)M_{t}],k\leq-l+j+(1-3\alpha/2)M_{t}$, the
first estimate in (5.45) in Lemma 5.2, the estimate (5.52) in Lemma 5.3, the
estimate (5.58) in Lemma 5.4, which are used for the case
$j\in[(1-\alpha)M_{t},(1+\epsilon)M_{t}],-l+j+(1-3\alpha/2)M_{t}\leq k\leq
40M_{t}$, and the second estimate in (5.45) in Lemma 5.2 and the estimate
(5.85) in Lemma 5.6, which are used for the case when $k\geq 40M_{t}$.

Similarly, from the estimate (4.7) in Lemma 4.2, the estimate (5.25) in Lemma
5.1, the estimate (5.45) in Lemma 5.2, the estimate (5.51) in Lemma 5.3, the
estimate (5.57) in Lemma 5.4, the estimate (5.85) in Lemma 5.6, we know that
our desired estimate (5.2) holds.

∎

###  5.1. A pointwise bilinear estimate

As summarized in the Lemma 5.7, our main goal of this subsection is to prove
two point-wise estimates for the bilinear form $T_{k;j}^{l,n}(K,f)(t,s,x)$,
which have been used as black boxes previously in the proof of Lemma 5.3 and
Lemma 5.4. These two estimates schematically give their dependence to the
localized angles and the symbols of the bilinear form.

Recall (5.16). From the Kirchhoff’s formulas in (2.2) and (2.3). We have the
following formulation in terms of kernel,

$T_{k;j}^{l,n}(K,f)(s,\tau,x)=M_{k;j}^{l,n}(K,f)(s,\tau,x)+Err_{k;j}^{l,n}(K,f)(s,\tau,x),$
(5.88)

where

$M_{k;j}^{l,n}(K,f)(s,\tau,x):=\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}\int_{\mathbb{S}^{2}}(s-\tau)K(\tau,x-y-z+(s-\tau)\theta)f(\tau,x-y-z+(s-\tau)\theta,v)$
$\times\mathcal{M}_{k;j}^{l}(y,\theta,v)K_{k;n}(z)\varphi_{j}(v)\psi_{\leq
n+4}(\angle(v,-\omega_{0}))d\theta dydzdv,$ (5.89)
$Err_{k;j}^{l,n}(K,f)(s,\tau,x)=\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}\int_{\mathbb{S}^{2}}K(\tau,x-y-z+(s-\tau)\theta)f(\tau,x-y-z+(s-\tau)\theta,v)$
$\times\mathcal{E}_{k;j}^{l}(y,v)K_{k;n}(z)\varphi_{j}(v)\psi_{\leq
n+4}(\angle(v,-\omega_{0}))d\theta dydzdv,$ (5.90)

where we used the fact that $\angle(v,-\omega_{0})\leq 2^{n+2}$ as
$\angle(\xi,\omega_{0})\leq 2^{n}$ and $\angle(\xi,-v)\leq 2^{l}$ due to the
cutoff functions and the assumption that $n\in[l,2]\cap\mathbb{Z}$, and the
kernels are defined as follows,

$\mathcal{M}_{k;j}^{l}(y,\theta,v):=\int_{\mathbb{R}^{3}}e^{iy\cdot\xi}i(|\xi|+\theta\cdot\xi)m_{1}(\xi,v)\psi_{k}(\xi)\varphi_{l;-j}(\angle(\xi,-v))d\xi,$
(5.91)
$\mathcal{E}_{k;j}^{l}(y,v):=\int_{\mathbb{R}^{3}}e^{iy\cdot\xi}m_{1}(\xi,v)\psi_{k}(\xi)\varphi_{l;-j}(\angle(\xi,-v))d\xi,\quad
K_{k;n}(z):=\int_{\mathbb{R}^{3}}e^{iy\cdot\xi}\varphi_{n;l}(\angle(\xi,\omega_{0}))m_{2}(\xi,\omega_{0})\psi_{[k-4,k+4]}(\xi)d\xi.$
(5.92)

By doing integration by parts in $\tilde{v}$ ($\omega_{0}$) direction and
$\tilde{v}^{\bot}$ ($\omega_{0}^{\bot}$) directions, the following estimate
holds from assumptions of symbols in (5.17) and (5.18),

$\sup_{\theta\in\mathbb{S}^{2}}\mathcal{M}_{k;j}^{l}(y,\theta,v)\lesssim
2^{4k+2l}\|m_{1}\|_{\mathcal{S}^{\infty}_{k,j,l}}(1+2^{k}|y\cdot\tilde{v}|)^{-10^{3}/\epsilon}(1+2^{k+l}|y\times\tilde{v}|)^{-10^{3}/\epsilon}.$
(5.93) $|\mathcal{E}_{k;j}^{l}(y,v)|\lesssim
2^{3k+2l}\|m_{1}\|_{\mathcal{S}^{\infty}_{k,j,l}}(1+2^{k}|y\cdot\tilde{v}|)^{-10^{3}/\epsilon}(1+2^{k+l}|y\times\tilde{v}|)^{-10^{3}/\epsilon}.$
(5.94) $|K_{k;n}(z)|\lesssim
2^{3k+2n}\|m_{2}\|_{\mathcal{S}^{\infty}_{k;n,l}}(1+2^{k}|z\cdot{\omega}_{0}|)^{-10^{3}/\epsilon}(1+2^{k+n}|z\times{\omega}_{0}|)^{-10^{3}/\epsilon}.$
(5.95)

######  Lemma 5.7.

For any fixed $x\in\mathbb{R}^{3}/\\{0\\},$ $s\in[0,t],$
$k\in\mathbb{Z},j\in\mathbb{Z}_{+}$,
$l\in[-j,2]\cap\mathbb{Z},n\in[l,2]\cap\mathbb{Z}$, s.t., $k\leq
40M_{t},j\in[(1-\alpha)M_{t},(1+\epsilon)M_{t}]$,
$k\geq-l+j+(1-3\alpha/2)M_{t}$, the following estimate holds for the bilinear
form defined in (5.16),

$\sum_{K\in\\{E,B\\}}\int_{0}^{s}\big{|}T_{k;j}^{l,n}(K,f)(s,\tau,x)\big{|}d\tau\lesssim\frac{\|m_{1}\|_{\mathcal{S}^{\infty}_{k,j,l}}\|m_{2}\|_{\mathcal{S}^{\infty}_{k;n,l}}}{2^{n}+|\tilde{x}\times\omega_{0}|}\frac{2^{k+l/2+n+(1+3\alpha)M_{t}}}{r_{-}},\quad
r:=|x|.$ (5.96)

Moreover, the following rough estimate also holds,

$\sum_{K\in\\{E,B\\}}\int_{0}^{s}\big{|}T_{k;j}^{l,n}(K,f)(s,\tau,x)\big{|}d\tau\lesssim\|m_{1}\|_{\mathcal{S}^{\infty}_{k,j,l}}\|m_{2}\|_{\mathcal{S}^{\infty}_{k;n,l}}\big{[}2^{3k/2+3j+2n+l+\epsilon
M_{t}}r+2^{k+3j+2n+2l+(1+3\epsilon)M_{t}}\big{]}.$ (5.97)

###### Proof.

Let $\tilde{\delta}:=2^{-k-l+\epsilon M_{t}}$. Based on the possible size of
$t-s$, we separate into two cases as follows.

Case $1$: If $|s-\tau|\leq\tilde{\delta}$, i.e., $\tau\in[s-\tilde{\delta},s]$
. Recall (5.16). Note that after using the Cauchy-Schwarz inequality for the
integration with respect to $\xi$ and the volume of support of $\xi$ and $v$,
the following estimate holds if $r:=|x|\leq\tilde{\delta}2^{\epsilon M_{t}}$,

$|T_{k;j}^{l,n}(K,f)(s,\tau,x)|\lesssim\|m_{1}\|_{\mathcal{S}^{\infty}_{k,j,l}}\|m_{2}\|_{\mathcal{S}^{\infty}_{k;n,l}}2^{(3k+2l)/2}\int_{\mathbb{R}^{3}}\|K(\tau,x)f(\tau,x,v)\|_{L^{2}_{x}}\varphi_{j}(v)\psi_{\leq
n+4}(\angle(-v,\omega_{0}))dv$
$\lesssim\|m_{1}\|_{\mathcal{S}^{\infty}_{k,j,l}}\|m_{2}\|_{\mathcal{S}^{\infty}_{k;n,l}}2^{3k/2+3j+2n+l}\lesssim\|m_{1}\|_{\mathcal{S}^{\infty}_{k,j,l}}\|m_{2}\|_{\mathcal{S}^{\infty}_{k;n,l}}\frac{2^{k/2+3j+2n+2\epsilon
M_{t}}}{r}.$ (5.98)

Note that, $|x-y-z+(s-\tau)\theta|\sim|x|$ if $|y|+|z|\leq\tilde{\delta}$ and
$|x|\geq\tilde{\delta}2^{\epsilon M_{t}}$. Moreover, from the estimates of
kernels in (5.93–5.95), we know that the kernel is extremely small if
$|y|+|z|\geq\tilde{\delta}$. Therefore, from the rough estimate of
electromagnetic field (4.6) in Lemma 4.2 and the volume of support of $v$, the
following estimate holds if $r:=|x|\geq\tilde{\delta}2^{\epsilon M_{t}}$,

$|T_{k;j}^{l,n}(K,f)(s,\tau,x)|\lesssim\|m_{1}\|_{\mathcal{S}^{\infty}_{k,j,l}}\|m_{2}\|_{\mathcal{S}^{\infty}_{k;n,l}}\big{[}(s-\tau)2^{k}\frac{2^{(1+\epsilon)M_{t}}}{r}2^{3j+2n}+2^{-10M_{t}}\|\int_{\mathbb{R}^{3}}|K(\tau,x)|f(\tau,x,v)\varphi_{j}(v)dv\|_{L^{1}_{x}}\big{]}.$
(5.99)

Combining the estimate (5.98) and the estimate (5.99), we have

$\int_{s-\tilde{\delta}}^{s}|T_{k;j}^{l,n}(K,f)(s,\tau,x)|ds\lesssim\|m_{1}\|_{\mathcal{S}^{\infty}_{k,j,l}}\|m_{2}\|_{\mathcal{S}^{\infty}_{k;n,l}}\big{[}\frac{2^{-k/2+3j+2n-l+3\epsilon
M_{t}}}{r}+\frac{2^{-k+3j+2n-2l+(1+6\epsilon)M_{t}}}{r}+2^{-5M_{t}}\big{]}$
$\lesssim\|m_{1}\|_{\mathcal{S}^{\infty}_{k,j,l}}\|m_{2}\|_{\mathcal{S}^{\infty}_{k;n,l}}\frac{2^{k+l/2+2n+(1+10\alpha)M_{t}/2}}{r_{-}}.$
(5.100)

Case $2$: If $|s-\tau|\geq\tilde{\delta}$, i.e.,
$\tau\in[0,s-\tilde{\delta}]$.

To better see the angular relations, we localize the angle of $v$ by using the
following partition of unity. Let
$\eta_{i}(\omega),\omega\in\mathbb{S}^{2},i\in\\{1,\cdots,K\\}$ be a labeled
partition of unit for the unit sphere such that the support of
$\eta_{i}(\cdot)$ is contained in small ball on sphere with radius of size
$2^{l}$ and the supports of $\eta_{i}(\cdot)$ overlaps only finite times. More
precisely, we have

$\forall\omega\in\mathbb{S}^{2},\quad
1=\sum_{i=1,\cdots,L}\eta_{i}(\omega),\quad|supp(\eta_{i}(\cdot))|\lesssim
2^{2l},\quad|L|\sim 2^{-2l},\quad\eta_{i}(\omega)\geq
0,\quad\eta_{i}(\omega)\eta_{j}(\omega)=0\,\,\textup{if}\,\,|i-j|\geq C.$
(5.101)

Moreover, we fix a choice of
$\\{\omega_{i}\\}_{i=1}^{L}\subset\mathbb{S}^{2}$. s.t., $\omega_{i}\in
supp(\eta_{i}(\cdot))$. Therefore, $supp(\eta_{i}(\cdot))\subset
supp(\psi_{\leq l+10}(\cdot-\omega_{i}))$.

Once we localize $\tilde{v}$ inside $supp(\eta_{i}(\cdot))$ , due to the
cutoff function $\varphi_{l;-j}(\angle(-v,\xi))$, we know that $\xi$ is also
localized in a sector of size $2^{l}$ centered at $\omega_{i}$. Moreover, due
to the cutoff function $\psi_{\leq n+4}(\angle(\xi,\omega_{0})$, where
$\omega_{0}\in\mathbb{S}^{2}$ is fixed, we know that there are at most
$2^{2n-2l}$ sectors on sphere to be considered.

Recall the integral (2.4) in the proof of Kirchhoff’s formula. From the
stationary phase point of view, we know that $\theta$ is localized near
$\xi/|\xi|$ and $-\xi/|\xi|$ with radius of size $(|s-\tau||\xi|)^{-1/2}$.
Hence, it is also localized roughly near the fixed direction $\omega_{0}$ with
radius of size $(|s-\tau||\xi|)^{-1/2}+2^{n}$.

Motivated from the above discussion, we define
$\bar{\theta}_{s}^{l}:=(1+|s|2^{k})^{-1/2}2^{\epsilon M_{t}}+2^{l+20}$ and the
following two cutoff functions,

$\varphi_{ess;i}(s,\theta):=\sum_{\mu\in\\{+,-\\}}\psi(\angle(\theta,\mu\omega_{i})/\bar{\theta}_{s}^{l})\psi(\angle(\theta,\mu\omega_{0})/\bar{\theta}_{s}^{n}),\quad\varphi_{gd}(s,\theta)=1-\sum_{i=1,\cdots
L}\varphi_{ess;i}(s,\theta).$ (5.102)

Recall (5.89). After using the partition of unity in (5.101) for the direction
of $v$, $\tilde{v}$, and the partition of unity in (5.102) for $\theta$, we
have

$M_{k;j}^{l,n}(K,f)(s,\tau,x)=M_{k;j}^{l,n;gd}(K,f)(s,\tau,x)+\sum_{i=1,\cdots
K_{n}}M_{k;j}^{l,n;ess,i}(K,f)(s,\tau,x),\quad K_{n}\sim 2^{2n-2l}$ (5.103)

where the angular localized
$M_{k;j}^{l,n;u}(K,f)(s,\tau,x),u\in\\{gd,ess;i\\},$ in physical space are
defined as follows,

$M_{k;j}^{l,n;u}(K,f)(s,\tau,x):=\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}\int_{\mathbb{S}^{2}}(s-\tau)\mathcal{M}_{k;j}^{l}(y,\theta,v)K_{k;n}(z)\psi_{\leq
n+4}(\angle(-v,\omega_{0}))\big{)}\varphi_{j}(v)\varphi_{u}(s-\tau,\theta)\eta_{i}(\tilde{v})$
$\times
K(\tau,x-y-z+(s-\tau)\theta)f(\tau,x-y-z+(s-\tau)\theta,v)dydzdvd\theta,\quad
u\in\\{gd,ess;i\\}.$ (5.104)

Let $\xi$ be fixed. After changing coordinates such that
$\xi\cdot\theta=|\xi|\cos\sigma$, we do integration by parts in $\sigma$ many
times. As a result, the following estimate holds for any $N\in\mathbb{Z}_{+}$
if $|s-\tau|\geq 2^{-k+\epsilon M_{t}}$,

$\big{|}\int_{\mathbb{S}^{2}}e^{i(s-\tau)\xi\cdot\theta}\varphi_{gd}(s-\tau,\theta)d\theta\big{|}\lesssim_{N}2^{-NM_{t}}.$
(5.105)

Therefore, from the above estimate and the volume of support of $\xi$, the
following estimate holds for any $\tau\in[0,s-\tilde{\delta}]$,

$\big{|}M_{k;j}^{l,n;gd}(K,f)(s,\tau,x)\big{|}\lesssim\|m_{1}\|_{\mathcal{S}^{\infty}_{k,j,l}}\|m_{2}\|_{\mathcal{S}^{\infty}_{k;n,l}}2^{-200M_{t}}2^{(3k+2l)/2}\int_{\mathbb{R}^{3}}\||K(\tau,z)|f(\tau,z,v)\|_{L^{2}_{z}}\varphi_{j}(v)dv$
$\lesssim
2^{-10M_{t}}\|m_{1}\|_{\mathcal{S}^{\infty}_{k,j,l}}\|m_{2}\|_{\mathcal{S}^{\infty}_{k;n,l}}.$
(5.106)

Now we focus on the essential part. We first rule out the case
$|y|+|z|\geq\tilde{\delta}2^{-\epsilon M_{t}/2}=2^{-k-l+\epsilon M_{t}/2}$, in
which the kernels provide sufficiently fast decay. Note that, from the rough
estimate of the electromagnetic field (4.6) in Lemma 4.2 and the estimates of
kernels in (5.93) and (5.95), we have

$\big{|}\int_{\mathbb{R}^{3}}\int_{\mathbb{S}^{2}}\int_{|y|+|z|\geq\tilde{\delta}2^{-\epsilon
M_{t}/2}}(s-\tau)\mathcal{M}_{k;j}^{l}(y,\theta,v)K_{k;n}(z)\psi_{\leq
n+4}(\angle(-v,\omega_{0}))\big{)}\varphi_{j}(v)\varphi_{u}(s-\tau,\theta)\eta_{i}(\tilde{v})$
$\times K(\tau,x-y-z+(s-\tau)\theta)f(\tau,x-y-z+(s-\tau)\theta,v)dydzd\theta
dv\big{|}$
$\lesssim\|m_{1}\|_{\mathcal{S}^{\infty}_{k,j,l}}\|m_{2}\|_{\mathcal{S}^{\infty}_{k;n,l}}\int_{\mathbb{R}^{3}}\int_{\mathbb{S}^{2}}\int_{|y|+|z|\geq\tilde{\delta}2^{-\epsilon
M_{t}/2}}(s-\tau)\frac{2^{4k+2l}2^{3k+2n}(1+2^{k+l}|y|)^{-10^{3}/\epsilon}(1+2^{k+n}|z|)^{-10^{3}/\epsilon}}{|x-y-z+(s-\tau)\theta|_{-}}$
$\times
2^{3j+(1+\epsilon)M_{t}}dydzd\theta\lesssim\|m_{1}\|_{\mathcal{S}^{\infty}_{k,j,l}}\|m_{2}\|_{\mathcal{S}^{\infty}_{k;n,l}}.$
(5.107)

Recall the estimate of kernel $K_{k;n}(z)$ in (5.95). Note that, the following
estimate holds from the above estimate (5.107) and the Cauchy-Schwarz
inequality for fixed $z$,

$\sum_{i=1,\cdots,K_{n}}\big{|}M_{k;j}^{l,n;ess,i}(K,f)(s,\tau,x)\big{|}\lesssim\|m_{2}\|_{\mathcal{S}^{\infty}_{k;n,l}}\big{[}\|m_{1}\|_{\mathcal{S}^{\infty}_{k,j,l}}$
$+\int_{|z|\leq\tilde{\delta}2^{-\epsilon
M_{t}/2}}(s-\tau)2^{3k+2n}(1+2^{k}|z\cdot{\omega}_{0}|)^{-10^{3}/\epsilon}(1+2^{k+n}|z\times{\omega}_{0}|)^{-10^{3}/\epsilon}F(s,\tau,x,z)dz\big{]},$
(5.108)

where

$0\leq F(s,\tau,x,z)$
$\lesssim\big{[}\sum_{i=1,\cdots,K_{n}}\int_{|y|\leq\tilde{\delta}2^{-\epsilon
M_{t}/2}}\int_{\mathbb{S}^{2}}\int_{\mathbb{R}^{3}}|\mathcal{M}_{k;j}^{l}(y,\theta,v)||K(\tau,x-y-z+(s-\tau)\theta)|^{2}\eta_{i}(\tilde{v})\varphi_{ess;i}(s-\tau,\theta)\varphi_{j}(v)dvd\theta
dy\big{]}^{1/2}$ $\times\big{[}\sum_{i=1,\cdots
K_{n}}\int_{|y|\leq\tilde{\delta}2^{-\epsilon
M_{t}/2}}\int_{\mathbb{S}^{2}}\int_{\mathbb{R}^{3}}|\mathcal{M}_{k;j}^{l}(y,\theta,v)||f(\tau,x-z-y+(s-\tau)\theta,v)|\eta_{i}(\tilde{v})\varphi_{ess;i}(s-\tau,\theta)\varphi_{j}(v)dvd\theta
dy\big{]}^{1/2},$ (5.109)

Note that,

$\textup{if\,\,}|\tilde{v}-\theta|\geq
2^{5}\bar{\theta}_{s-\tau}^{l}\,\quad\Longrightarrow\quad(\tilde{v},\theta)\notin\cup_{i=1,\cdots,L}supp(\eta_{i}(\tilde{v})\varphi_{ess;i}(t,\theta))\subset\mathbb{S}^{2}\times\mathbb{S}^{2}.$

With the above fact, from the estimate of kernel in (5.93) and the estimate
(2.9) in Lemma 2.2 and the point-wise estimate of electromagnetic field (4.6)
in Lemma 4.2 and the volume of support of $v$, the following estimate holds
for any fixed $z\in\mathbb{R}^{3}$,

$\sum_{i=1,\cdots,K_{n}}\sum_{K\in\\{E,B\\}}\int_{|y|\leq\tilde{\delta}2^{-\epsilon
M_{t}/2}}\int_{\mathbb{S}^{2}}\int_{\mathbb{R}^{3}}|\mathcal{M}_{k;j}^{l}(y,\theta,v)||K(\tau,x-y-z+(s-\tau)\theta)|^{2}\eta_{i}(\tilde{v})\varphi_{ess;i}(s-\tau,\theta)\varphi_{j}(v)dvd\theta
dy$ $\lesssim\int_{|y|\leq\tilde{\delta}2^{-\epsilon
M_{t}/2}}2^{4k+2l}\|m_{1}\|_{\mathcal{S}^{\infty}_{k,j,l}}\frac{\bar{\theta}_{s-\tau}^{n}2^{3j}\big{(}\bar{\theta}_{s-\tau}^{l}\big{)}^{2}\min\\{||x-y-z|-(s-\tau)|^{-1},2^{(1+\epsilon)M_{t}}|s-\tau|\\}}{(\bar{\theta}_{s-\tau}^{n}+|\tilde{x}\times\omega_{0}|)|x-y-z|(s-\tau)}dy.$
(5.110)

Moreover, after changing coordinates for $y$ in terms of the cylinder
coordinates system with $\tilde{v}$ as axis as in (5.36), from the estimate of
kernel in (5.93), the following estimate holds after using the estimate (2.10)
in Lemma 2.2 or using the volume of support of $\theta$ and $v$,

$\sum_{i=1,\cdots K_{n}}\int_{|y|\leq\tilde{\delta}2^{-\epsilon
M_{t}/2}}\int_{\mathbb{S}^{2}}\int_{\mathbb{R}^{3}}|\mathcal{M}_{k;j}^{l}(y,\theta,v)||f(\tau,x-z-y+(s-\tau)\theta,v)|\eta_{i}(\tilde{v})\varphi_{ess;i}(s-\tau,\theta)\varphi_{j}(v)dvd\theta
dy$ $\lesssim\sum_{i=1,\cdots
K_{n}}\int_{\mathbb{R}}\int_{\mathbb{R}}\int_{\mathbb{R}^{3}}\int_{\mathbb{S}_{v}}\int_{\mathbb{S}^{2}}2^{4k+2l}\|m_{1}\|_{\mathcal{S}^{\infty}_{k,j,l}}(1+2^{k}a)^{-10}(1+2^{k+l}r)^{-10}\eta_{i}(\tilde{v})\varphi_{ess;i}(s-\tau,\theta)\varphi_{j}(v)$
$\times f(\tau,x-z-a\tilde{v}-r\omega_{v}+(s-\tau)\theta,v)d\theta
d\omega_{v}dvrdrda$
$\lesssim\|m_{1}\|_{\mathcal{S}^{\infty}_{k,j,l}}\min\\{\frac{2^{k}C_{n}(x,\omega_{0})\bar{\theta}_{s-\tau}^{n}2^{-j}}{|x-z|(s-\tau)||x-z|-(s-\tau)|},|K_{n}|2^{k}2^{3j+2l}\big{(}\bar{\theta}_{s-\tau}^{l}\big{)}^{2}\\},$
(5.111)

where $C_{n}(x,\omega_{0}):=({2^{n}+|\tilde{x}\times\omega_{0}|})^{-1}$.

With the above preparations, based on the possible size of “$r$” and “$\tau$”,
we separate into four sub-cases as follows.

Subcase $1$: If $r\leq\tilde{\delta}2^{\epsilon M_{t}}$ and
$\tau\in[s-r-\tilde{\delta},s-r+\tilde{\delta}]\cap[0,s-\tilde{\delta}]$.
Recall (5.104). Note that we have $|s-\tau|\lesssim\tilde{\delta}2^{\epsilon
M_{t}}$ for the case we are considering. From the Cauchy-Schwarz inequality
for the integration with respect to $y$, the estimate of kernels in (5.93) and
(5.95), and the volume of support of $\theta$ and $v$, we have

$\sum_{i=1,\cdots
K_{n}}\int_{s-r-\tilde{\delta}}^{\min\\{s-\tilde{\delta},s-r+\tilde{\delta}\\}}\big{|}M_{k;j}^{l,n;ess,i}(K,f)(s,\tau,x)\big{|}d\tau$
$\lesssim\|m_{1}\|_{\mathcal{S}^{\infty}_{k,j,l}}\|m_{2}\|_{\mathcal{S}^{\infty}_{k;n,l}}\int_{s-r-\tilde{\delta}}^{s-r+\tilde{\delta}}(s-\tau)|K_{n}|2^{5k/2+l}\big{(}\bar{\theta}_{s-\tau}^{l}\big{)}^{2}2^{3j+2l}d\tau$
$\lesssim\|m_{1}\|_{\mathcal{S}^{\infty}_{k,j,l}}\|m_{2}\|_{\mathcal{S}^{\infty}_{k;n,l}}2^{5k/2+3j+2n+l+6\epsilon
M_{t}}2^{-2k-l}\lesssim\|m_{1}\|_{\mathcal{S}^{\infty}_{k,j,l}}\|m_{2}\|_{\mathcal{S}^{\infty}_{k;n,l}}{2^{k+4\alpha
M_{t}+2n+l/2}}r^{-1}.$ (5.112)

Subcase $2$: If $r\geq\tilde{\delta}2^{\epsilon M_{t}}$ and
$\tau\in[s-r-\tilde{\delta},s-r+\tilde{\delta}]\cap[0,s-\tilde{\delta}]$.

For this case we are considering, we have $|s-\tau|\sim r$ and
$|x-y-z|,|x-z|\sim r$ if $|y|+|z|\leq\tilde{\delta}=2^{-k-l+\epsilon M_{t}}$.
From the estimate (5.109) and the estimates (5.110) and (5.111), the following
estimate holds for fixed $z\in\mathbb{R}^{3}$ s.t., $|z|\leq\tilde{\delta},$

$\int_{s-r-\tilde{\delta}}^{s-r+\tilde{\delta}}(s-\tau)F(s,\tau,x,z)d\tau\lesssim\|m_{1}\|_{\mathcal{S}^{\infty}_{k,j,l}}\int_{s-r-\tilde{\delta}}^{s-r+\tilde{\delta}}(s-\tau)\big{(}\min\\{2^{3j+2n+k}\big{(}\bar{\theta}_{s-\tau}^{l}\big{)}^{2},\frac{C_{n}(x,\omega_{0})2^{k-j}\bar{\theta}_{s-\tau}^{n}}{r^{2}||x-z|-(s-\tau)|}\\}\big{)}^{1/2}$
$\times\big{(}\int_{|y|\leq\tilde{\delta}2^{-\epsilon
M_{t}/2}}\frac{2^{2\epsilon
M_{t}}C_{n}(x,\omega_{0})\bar{\theta}_{s-\tau}^{n}2^{4k+2l+3j}\big{(}\bar{\theta}_{s-\tau}^{l}\big{)}^{2}}{r^{2}||x-y-z|-(s-\tau)|^{1-\epsilon}}dy\big{)}^{1/2}d\tau$
$\lesssim\|m_{1}\|_{\mathcal{S}^{\infty}_{k,j,l}}\big{[}C_{n}(x,\omega_{0}){2^{5k/2+l+3j/2+\epsilon
M_{t}}}\big{(}\int_{s-r-\tilde{\delta}}^{s-r+\tilde{\delta}}\int_{|y|\leq\tilde{\delta}2^{-\epsilon
M_{t}/2}}\frac{1}{||x-y-z|-(s-\tau)|^{1-\epsilon}}dyd\tau\big{)}^{1/2}$
$\times\big{(}\int_{s-r-\tilde{\delta}}^{s-r+\tilde{\delta}}\min\\{2^{2n+3j}\big{(}\bar{\theta}_{s-\tau}^{l}\big{)}^{4}\bar{\theta}_{s-\tau}^{n},\frac{2^{-j}(\bar{\theta}^{l}_{s-\tau})^{2}(\bar{\theta}_{s-\tau}^{n})^{2}}{r^{2}||x-z|-(s-\tau)|}\\}d\tau\big{)}^{1/2}\big{]}\lesssim\|m_{1}\|_{\mathcal{S}^{\infty}_{k,j,l}}C_{n}(x,\omega_{0})2^{k+3j/2-l/2+4\epsilon
M_{t}}$
$\times\big{(}\min\\{\tilde{\delta}2^{3j+2n}(2^{4l+n}+r^{-2}2^{-2k+n}+r^{-5/2}2^{-5k/2}),r^{-2}2^{-j+2l+2n}+r^{-3}2^{-j-k+2n}+r^{-4}2^{-j-2k}\\}\big{)}^{1/2}$
(5.113)
$\lesssim\|m_{1}\|_{\mathcal{S}^{\infty}_{k,j,l}}C_{n}(x,\omega_{0})2^{k+l/2+n+(1+3\alpha)M_{t}}r^{-1}.$
(5.114)

In the above final estimate, we used the second estimate in (5.113) if $r\geq
2^{-k-2l}$ and used the first estimate in (5.113) if
$\tilde{\delta}2^{\epsilon M_{t}}\leq r\leq 2^{-k-2l}$. Therefore, after
combining the estimates (5.108) and (5.114), we have

$\sum_{i=1,\cdots,K_{n}}\int_{s-r-\tilde{\delta}}^{\min\\{s-\tilde{\delta},s-r+\tilde{\delta}\\}}\big{|}M_{k;j}^{l,n;ess,i}(K,f)(s,\tau,x)\big{|}d\tau\lesssim
C_{n}(x,\omega_{0})\|m_{1}\|_{\mathcal{S}^{\infty}_{k,j,l}}\|m_{2}\|_{\mathcal{S}^{\infty}_{k;n,l}}2^{k+l/2+n+(1+3\alpha)M_{t}}r_{-}^{-1}.$
(5.115)

Subcase $3$: If $r\geq\tilde{\delta}2^{-\epsilon M_{t}/3}$ and
$\tau\in[s-r-\tilde{\delta},s-r+\tilde{\delta}]^{c}\cap[0,s-\tilde{\delta}]$.
For this case, we have $|x-y-z|,|x-z|\sim r$ and
$||x-y|-(s-\tau)|\sim||x|-(s-\tau)|$ if $|y|,|z|\leq 2^{-k-l+\epsilon
M_{t}/2}=\tilde{\delta}2^{-\epsilon M_{t}/2}$ . From the estimate (5.109) and
the estimates (5.110) and (5.111), the following estimate holds for fixed
$z\in\mathbb{R}^{3}$ s.t., $|z|\leq\tilde{\delta},$

$\int_{[0,s-\tilde{\delta}]\cap[s-r-\tilde{\delta},s-r+\tilde{\delta}]^{c}}(s-\tau)F(s,\tau,x,z)d\tau\lesssim
C_{n}(x,\omega_{0})\|m_{1}\|_{\mathcal{S}^{\infty}_{k,j,l}}\big{[}\int_{[0,s-\tilde{\delta}]\cap[s-r-\tilde{\delta},s-r+\tilde{\delta}]^{c}}(s-\tau)$
$\times\min\\{\frac{2^{k+j-l/2}(\bar{\theta}_{s-\tau}^{l})\bar{\theta}_{s-\tau}^{n}}{(r(s-\tau)|r-(s-\tau)|)},\frac{2^{k+3j-l/2+n}(\bar{\theta}^{l}_{s-\tau})^{2}(\bar{\theta}_{s-\tau}^{n})^{1/2}}{(r(s-\tau)|r-(s-\tau)|)^{1/2}}\\}ds\big{]}.$
(5.116)

If $|s-\tau|2^{k}\geq 2^{-2l}$, i.e., $\bar{\theta}_{s-\tau}^{l}\leq
2^{l+\epsilon M_{t}}$, then we use the first estimate in (5.116). Meanwhile,
if $|s-\tau|2^{k}\leq 2^{-2l}$, i.e.,
$\bar{\theta}_{s-\tau}^{l}\leq(s-\tau)^{-1/2}2^{-k/2+\epsilon M_{t}}$, then we
use the second estimate in (5.116). As a result, we have

$\int_{[0,s-\tilde{\delta}]\cap[s-r-\tilde{\delta},s-r+\tilde{\delta}]^{c}}(s-\tau)F(s,\tau,x,z)d\tau\lesssim
C_{n}(x,\omega_{0})\|m_{1}\|_{\mathcal{S}^{\infty}_{k,j,l}}2^{k+l/2+n+(1+3\alpha)M_{t}}r_{-}^{-1}.$

After combining it with the estimate (5.108), we have

$\sum_{i=1,\cdots,K_{n}}\int_{[0,s-\tilde{\delta}]\cap[s-r-\tilde{\delta},s-r+\tilde{\delta}]^{c}}\big{|}M_{k;j}^{l,n;ess,i}(K,f)(s,\tau,x)\big{|}d\tau\lesssim
C_{n}(x,\omega_{0})\|m_{1}\|_{\mathcal{S}^{\infty}_{k,j,l}}\|m_{2}\|_{\mathcal{S}^{\infty}_{k;n,l}}2^{k+l/2+n+(1+3\alpha)M_{t}}r_{-}^{-1}.$
(5.117)

Subcase $4$: If $r\leq\tilde{\delta}2^{-\epsilon M_{t}/3}$ and
$\tau\in[s-r-\tilde{\delta},s-r+\tilde{\delta}]^{c}\cap[0,s-\tilde{\delta}]$.
For this case, we have $||x-y-z|-(s-\tau)|\sim|s-\tau|$ if $|y|+|z|\leq
2^{-k-l+\epsilon M_{t}/2}=\tilde{\delta}2^{-\epsilon M_{t}/2}$. Recall
(5.104). From the estimates of kernels in (5.93) and (5.95), the rapid decay
rate of kernel if $|y|\geq 2^{-k-l+\epsilon
M_{t}/2}=\tilde{\delta}2^{-\epsilon M_{t}/2}$, the point-wise estimate of
electromagnetic field (4.6) in Lemma 4.2 and the estimate (5.111), the
following estimate holds,

$\sum_{i=1,\cdots,K_{n}}\int_{[0,s-\tilde{\delta}]\cap[s-r-\tilde{\delta},s-r+\tilde{\delta}]^{c}}\big{|}M_{k;j}^{l,n;ess,i}(K,f)(s,\tau,x)\big{|}d\tau$
$\lesssim\|m_{1}\|_{\mathcal{S}^{\infty}_{k,j,l}}\|m_{2}\|_{\mathcal{S}^{\infty}_{k;n,l}}\big{[}\int_{|z|\leq
2^{-k-n+\epsilon
M/2}}\int_{[0,s-\tilde{\delta}]\cap[s-r-\tilde{\delta},s-r+\tilde{\delta}]^{c}}2^{3k+2n}(1+2^{k}|z\cdot{\omega}_{0}|)^{-100/\epsilon}(1+2^{k+n}|z\times{\omega}_{0}|)^{-100/\epsilon}$
$\times(s-\tau)\frac{2^{(1+\epsilon)M_{t}}}{|s-\tau|}\min\\{\frac{2^{k}C_{n}(x,\omega_{0})\bar{\theta}_{s-\tau}^{n}2^{-j}}{|x-z|(s-\tau)||x-z|-(s-\tau)|},|K_{n}|2^{k}2^{3j+2l}\big{(}\bar{\theta}_{s-\tau}^{l}\big{)}^{2}\\}d\tau
dz+2^{-10M_{t}}\big{]}.$ (5.118)

If $s-\tau\leq 2^{-k-2l+\alpha M_{t}}$ then from the second estimate of
(5.118), we have

$\textup{(\ref{march10eqn71})}\lesssim\|m_{1}\|_{\mathcal{S}^{\infty}_{k,j,l}}\|m_{2}\|_{\mathcal{S}^{\infty}_{k;n,l}}2^{k+4(1+\epsilon)M_{t}+2n}\big{(}2^{-k-2l+\alpha
M_{t}}2^{2l}+2^{-k}\big{)}r^{-1}2^{-k-l+\epsilon M_{t}}\lesssim
r^{-1}2^{k+l+2n+6\alpha M_{t}},$ (5.119)

If $s-\tau\geq 2^{-k-2l+\alpha M_{t}}$,( i.e., $\bar{\theta}_{s-\tau}^{l}\sim
2^{l}$), and $2^{-k-n+2\epsilon M_{t}}\leq r\leq\tilde{\delta}2^{\epsilon
M_{t}}$, then we have $|x-z|\sim r$ if $|z|\leq 2^{-k-n+\epsilon M/2}$. From
the geometric mean of two estimates in (5.118), we have

$\textup{(\ref{march10eqn71})}\lesssim
C_{n}(x,\omega_{0})\|m_{1}\|_{\mathcal{S}^{\infty}_{k,j,l}}\|m_{2}\|_{\mathcal{S}^{\infty}_{k;n,l}}r^{-1/2}2^{k+2(1+\epsilon)M_{t}+l+3n/2}$
$\lesssim
C_{n}(x,\omega_{0})\|m_{1}\|_{\mathcal{S}^{\infty}_{k,j,l}}\|m_{2}\|_{\mathcal{S}^{\infty}_{k;n,l}}r^{-1}2^{k+(1+2\alpha)M_{t}+l+3n/2}.$
(5.120)

If $s-\tau\geq 2^{-k-2l+\alpha M_{t}}$,( i.e., $\bar{\theta}_{s-\tau}^{l}\sim
2^{l}$), and $r\leq 2^{-k-n+2\epsilon M_{t}}$, then from the geometric mean of
two estimates in (5.118), we have

$\textup{(\ref{march10eqn71})}\lesssim
C_{n}(x,\omega_{0})\|m_{1}\|_{\mathcal{S}^{\infty}_{k,j,l}}\|m_{2}\|_{\mathcal{S}^{\infty}_{k;n,l}}\int_{|z|\leq
2^{-k-n+2\epsilon
M_{t}}}2^{3k+2n}|x-z|^{-1/2}2^{k+2(1+\epsilon)M_{t}+l+3n/2}dz\lesssim
C_{n}(x,\omega_{0})$
$\times\|m_{1}\|_{\mathcal{S}^{\infty}_{k,j,l}}\|m_{2}\|_{\mathcal{S}^{\infty}_{k;n,l}}2^{k+2(1+\epsilon)M_{t}+l+3n/2}2^{k/2-n/2+6\epsilon
M_{t}}\lesssim
C_{n}(x,\omega_{0})\|m_{1}\|_{\mathcal{S}^{\infty}_{k,j,l}}\|m_{2}\|_{\mathcal{S}^{\infty}_{k;n,l}}r^{-1}2^{k+(1+2\alpha)M_{t}+3l/2}.$
(5.121)

To sum up, after combining the estimates (5.119–5.121), in whichever case we
have

$\sum_{i=1,\cdots,K_{n}}\int_{[0,s-\tilde{\delta}]\cap[s-r-\tilde{\delta},s-r+\tilde{\delta}]^{c}}\big{|}M_{k;j}^{l,n;ess,i}(K,f)(s,\tau,x)\big{|}d\tau\lesssim
C_{n}(x,\omega_{0})\|m_{1}\|_{\mathcal{S}^{\infty}_{k,j,l}}\|m_{2}\|_{\mathcal{S}^{\infty}_{k;n,l}}r^{-1}2^{k+(1+3\alpha)M_{t}+l/2+n}.$
(5.122)

Recall the decomposition (5.103). To sum up, after combining the estimates
(5.106), (5.112), (5.115), (5.117), and (5.122), we have

$\int_{0}^{s-\tilde{\delta}}\big{|}M_{k;j}^{l,n}(K,f)(s,\tau,x)\big{|}d\tau\lesssim
C_{n}(x,\omega_{0})\|m_{1}\|_{\mathcal{S}^{\infty}_{k,j,l}}\|m_{2}\|_{\mathcal{S}^{\infty}_{k;n,l}}r_{-}^{-1}2^{k+(1+3\alpha)M_{t}+l/2+n}.$
(5.123)

Recall the detailed formulas of $M_{k;j}^{l,n}(K,f)(s,\tau,x)$ in (5.89) and
$Err_{k;j}^{l,n}(K,f)(s,\tau,x)$ in (5.90) and the estimate of corresponding
kernels in (5.93) and (5.94). Note that we have $s-\tau\geq\tilde{\delta}\geq
2^{-k+\epsilon M_{t}}$. The smallness of extra $2^{-k}$ comes from the symbol
compensate the loss of smallness “$s-\tau$” in
$Err_{k;j}^{l,n}(K,f)(s,\tau,x)$. Therefore, with minor modification of the
proof of the estimate (5.123), we have

$\int_{0}^{s-\tilde{\delta}}\big{|}Err_{k;j}^{l,n}(K,f)(s,\tau,x)\big{|}d\tau\lesssim
C_{n}(x,\omega_{0})\|m_{1}\|_{\mathcal{S}^{\infty}_{k,j,l}}\|m_{2}\|_{\mathcal{S}^{\infty}_{k;n,l}}r_{-}^{-1}2^{k+(1+3\alpha)M_{t}+l/2+n}.$
(5.124)

Recall the decomposition (5.88). To sum up, our desired estimate (5.96) holds
from the estimates (5.100), (5.123), and (5.124).

Moreover, we obtain a rough point-wise estimate as a byproduct of the above
argument, which will be used when $r$ is extremely small. Recall (5.16). Note
that after using the Cauchy-Schwarz inequality for the integration with
respect to $\xi$ and the volume of support of $\xi$ and $v$, the following
estimate holds

$|T_{k;j}^{l,n}(K,f)(s,\tau,x)|\lesssim\|m_{1}\|_{\mathcal{S}^{\infty}_{k,j,l}}\|m_{2}\|_{\mathcal{S}^{\infty}_{k;n,l}}2^{(3k+2l)/2}\int_{\mathbb{R}^{3}}\|K(\tau,x)f(\tau,x,v)\|_{L^{2}_{x}}\varphi_{j}(v)\psi_{\leq
n+4}(\angle(-v,\omega_{0}))dv$
$\lesssim\|m_{1}\|_{\mathcal{S}^{\infty}_{k,j,l}}\|m_{2}\|_{\mathcal{S}^{\infty}_{k;n,l}}2^{3k/2+3j+2n+l}.$
(5.125)

Moreover, for any fixed $\tau\in[0,s]$ s.t., $|s-\tau|\geq
10(r+2^{-k-l+\epsilon M_{t}})$, we have $|x-y-z-(s-\tau)|\sim(s-\tau)$ if
$|y|,|z|\leq 2^{-k-l+\epsilon M_{t}}$. Therefore, from the the rapid decay
rate of kernels in (5.93) and (5.95) if $|y|,|z|\geq 2^{-k-l+\epsilon
M_{t}/2}=\tilde{\delta}2^{-\epsilon M_{t}/2}$, the estimate of good part in
(5.106), the point-wise estimate of electromagnetic field (4.6) in Lemma 4.2,
we have

$|T_{k;j}^{l,n}(K,f)(s,\tau,x)|\lesssim\|m_{1}\|_{\mathcal{S}^{\infty}_{k,j,l}}\|m_{2}\|_{\mathcal{S}^{\infty}_{k;n,l}}\big{[}(s-\tau)\frac{2^{(1+\epsilon)M_{t}}}{|s-\tau|}|K_{n}|2^{k}2^{3j+2l}\big{(}\bar{\theta}_{s-\tau}^{l}\big{)}^{2}\big{]}.$
(5.126)

Combining the estimates (5.125) and (5.126), we have

$\int_{0}^{s}|T_{k;j}^{l,n}(K,f)(s,\tau,x)d\tau|\lesssim\|m_{1}\|_{\mathcal{S}^{\infty}_{k,j,l}}\|m_{2}\|_{\mathcal{S}^{\infty}_{k;n,l}}\big{[}\int_{[0,s]\cap[s-10(r+2^{-k-l+\epsilon
M_{t}}),s+10(r+2^{-k-l+\epsilon M_{t}})]}2^{3k/2+3j+2n+l}d\tau$
$+\int_{[0,s]\cap[s-10(r+2^{-k-l+\epsilon M_{t}}),s+10(r+2^{-k-l+\epsilon
M_{t}})]^{c}}(s-\tau)\frac{2^{(1+\epsilon)M_{t}}}{|s-\tau|}|K_{n}|2^{k}2^{3j+2l}\big{(}\bar{\theta}_{s-\tau}^{l}\big{)}^{2}d\tau\big{]}$
$\lesssim\|m_{1}\|_{\mathcal{S}^{\infty}_{k,j,l}}\|m_{2}\|_{\mathcal{S}^{\infty}_{k;n,l}}\big{(}2^{3k/2+3j+2n+l+\epsilon
M_{t}}r+2^{k+3j+2n+2l+(1+3\epsilon)M_{t}}\big{)}.$ (5.127)

Hence finishing the proof of our desired estimate (5.97). ∎

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