  This paper is concerned with the localized behaviors of the solution $u$ to
the Navier-Stokes equations near the potential singular points. We establish
the concentration rate for the $L^{p,\infty}$ norm of $u$ with $3\leq
p\leq\infty$. Namely, we show that if $z_0=(t_0,x_0)$ is a singular point, then
for any $r>0$, it holds \begin{align} \limsup_{t\to
t_0^-}||u(t,x)-u(t)_{x_0,r}||_{L^{3,\infty}(B_r(x_0))}>\delta^*,\notag
\end{align} and \begin{align} \limsup_{t\to
t_0^-}(t_0-t)^{\frac{1}{\mu}}r^{\frac{2}{\nu}-\frac{3}{p}}||u(t)||_{L^{p,\infty}(B_r(x_0))}>\delta^*\notag
for~3<p\leq\infty,
~\frac{1}{\mu}+\frac{1}{\nu}=\frac{1}{2}~and~2\leq\nu\leq\frac{2}{3}p,\notag
\end{align}where $\delta^*$ is a positive constant independent of $p$ and
$\nu$. Our main tools are some $\varepsilon$-regularity criteria in
$L^{p,\infty}$ spaces and an embedding theorem from $L^{p,\infty}$ space into a
Morrey type space. These are of independent interests.
