 we introduce more notations. As in
{\cite{Guo5}}, for  $l\geq0$, $\alpha$ and $\beta$, we define the velocity
weight function $w_l(\alpha,\beta)$ by
\begin{equation}
\label{def.wab}
w_l(\alpha,\beta)(v)={w}^{l+K-|\alpha|-|\beta|}(v)=\langle v\rangle^{(-\gamma)(l+K-|\alpha|-|\beta|)},
\end{equation}
where $K$ is an integer. Corresponding to given $f=f(t,x,v)$, we define the instant energy
functional $\mathcal {E}_{l}(t)$ and the instant high-order energy
functional $\mathcal {E}^h_{l}(t)$, for $l\geq 0$, to be functionals satisfying the equivalent relations
\begin{eqnarray}
\mathcal {E}_{l}(t)&\sim& \sum\limits_{|\alpha|\leq
K}\|\partial^{\alpha}E(t)\|^2+\sum\limits_{|\alpha|+|\beta|\leq
K}\left\|w_l(\alpha,\beta)\partial_{\beta}^{\alpha}f(t)\right\|^{2},\label{def e}\\
\mathcal {E}^{h}_{l}(t)&\sim& \sum\limits_{|\alpha|\leq
K}\|\partial^{\alpha}E(t)\|^2+\sum\limits_{1\leq|\alpha|\leq
K}\|\partial^{\alpha}{\bf P}f\|^2+\sum\limits_{|\alpha|+|\beta|\leq
K}\left\|w_l(\alpha,\beta)\partial_{\beta}^{\alpha}({\bf I-P})f\right\|^{2},\label{def he}
\end{eqnarray}
and define the dissipation rate functional $\mathcal {D}_{l}(t)$ by
\begin{equation}\label{def d}
\mathcal {D}_{l}(t)=\sum\limits_{|\alpha|\leq
K-1}\|\partial^{\alpha}E(t)\|^2+\sum\limits_{1\leq|\alpha|\leq
K}\|\partial^{\alpha}{\bf P}f(t)\|^2+\sum\limits_{|\alpha|+|\beta|\leq
K}\left\|w_l(\alpha,\beta)\partial_{\beta}^{\alpha}({\bf
I-P})f(t)\right\|^{2}_{N^{s}_{\gamma}}.
\end{equation}
Here, $E=E(t,x)$ is understood to be determined by $f(t,x,v)$ in terms of
\begin{equation*}
E=-\nabla_x\phi, \quad \phi=\frac{1}{4\pi |x|} \ast_x\rho_f, \quad \rho_f=\int_{{\bf
R}^3}(f_+-f_-)\sqrt{\mu(v)}\,dv.
\end{equation*}
The main result of the paper is stated as follows.

\begin{Theorem}\label{thm.gl}
Let $-3<\gamma<-2s$, $1/2\leq s<1$. Fix $l_0\geq 0$, $1/2<p<1$, the integer $K\geq 8$, and define
\begin{equation*}
l_1=\frac{5}{4(1-p)}\frac{\gamma+2s}{\gamma}.
\end{equation*}
Let $f_0(x,v)=[f_{0,+}(x,v),f_{0,-}(x,v)]$ satisfy
$F_{\pm}(0,x,v)=\mu(v)+\sqrt{\mu(v)}f_{0,{\pm}}(x,v)\geq 0$, and
\begin{equation}\label{thm.gl.a1}
\int_{{\bf R}^3}\rho_{f_0}\,dx=0,\ \ \int_{{\bf R}^3}(1+|x|)|\rho_{f_0}|\,dx<\infty.
\end{equation}
Then, there are functionals $\mathcal{E}_{l}(t)$ and $\mathcal{E}_l^{h}(t)$ in the sense of \eqref{def e} and \eqref{def he} such that the following thing holds true.  If
\begin{equation}\label{thm.gl.a2}
\epsilon_{0}=\sqrt{\mathcal{E}_{l_0+l_1}(0)}+\left\|w^{l_2}f_0\right\|_{Z_1}+\|(1+|x|)\rho_0\|_{L^1},
\end{equation}
is sufficiently small, where $l_2>\frac{5(\gamma+2s)}{4\gamma}$ is a constant, then there exists a unique global solution $f(t,x,v)$ to the Cauchy
problem (\ref{perturbed eqn})-(\ref{perturbed data}) of the Vlasov-Poisson-Boltzmann system such that
$F_{\pm}(t,x,v)=\mu(v)+\sqrt{\mu(v)}f_{\pm}(t,x,v)\geq 0$ and
\begin{eqnarray}
&&\mathcal{E}_{l_0+l_1}(t)  \lesssim \epsilon_0^2,\label{thm.gl.c1}\\
&&\mathcal{E}_{l_0}(t) \lesssim\epsilon_0^2(1+t)^{-\frac{3}{2}},\label{thm.gl.c2}\\
&&\mathcal{E}_{l_0}^{h}(t) \lesssim \epsilon_0^2(1+t)^{-\frac{3}{2}-p},\label{thm.gl.c3}
\end{eqnarray}
for any $t\geq 0$.
\end{Theorem}

Some remarks are given as follows. Notice that $l_0$ can take zero while the choice of  other parameters $l_1$, $l_2$, $K$ and $p$ could not be optimal in order for initial data to have the weaker regularity and velocity moments. The restriction of those parameters is related to obtaining the following closed a priori estimate
$$
X(t)\lesssim \epsilon_0^2+ X^{\frac{3}{2}}(t) +X^2(t)
$$
with respect to the time-weighted energy norm $X(t)$ defined by
\begin{equation}\label{key.X}
\begin{split}
X(t)=\sup\limits_{0\leq \tau\leq t}\mathcal {E}_{l_0+l_1}(\tau)+\sup\limits_{0\leq\tau\leq t}(1+\tau)^{\frac{3}{2}}\mathcal {E}_{l_0}(\tau)+\sup\limits_{0\leq\tau\leq t}(1+\tau)^{\frac{3}{2}+p}\mathcal {E}^h_{l_0}(\tau).
\end{split}
\end{equation}
Here, the parameter $p$ is introduced to take care the time-decay of the high-order energy functional $\mathcal{E}_{l_0}^{h}(t)$.  {The time decay rate $(1+t)^{-\frac{3}{2}-p}$ for $\mathcal{E}_{l_0}^{h}(t)$ in \eqref{thm.gl.c3} is not optimal compared to the linearized system; see Theorem \ref{basic decay lemma}  given later on. It is then of interest to upgrade it to $(1+t)^{-5/2}$, that is to prove \eqref{thm.gl.c3} in the case of $p=1$, and we will make a further discussion of possibility at the end of the paper.
Moreover, similar results in the} hard potential case $\gamma+2s\geq 0$ with $0<s<1$ could be considered in the simpler way, but for the soft potential case $-3<\gamma<-2s$ in Theorem \ref{thm.gl}, the restriction $s\geq 1/2$ is necessary in our proof due to the technique of the approach; we will clarify this point later. The assumptions concerning \eqref{thm.gl.a1} and \eqref{thm.gl.a2} arise from the linearized analysis of the Vlasov-Poisson-Boltzmann system in order to assure the strong enough time-decay rate of the linearized solution operator; see \eqref{est.rho} in the proof of Theorem \ref{basic decay lemma}.

In what follows let us point out several key technical points in the proof of Theorem \ref{thm.gl}. First of all, we emphasize the role of the velocity weight $w_l(\alpha,\beta)$ in  {\eqref{def.wab}.
Such weight was firstly introduced in \cite{SG1,SG2} to deal with the time decay of the Boltzmann equation for soft potentials on torus, and it was also used recently in \cite{Guo5} to investigate the Vlasov-Poisson-Landau system for Coulomb potentials.}
In fact, the linearized non cutoff Boltzmann operator  enjoys the anisotropic dissipation norm
\begin{equation}
\label{diss.norm}
\|f\|_{N