Thanks for Bazin's answer and those comments. Just as Otis Chodosh  said, the norm of $\|\chi_jm(D)\chi_k\|$ will decrease rapidly if $\chi_j, \chi_k$ are test functions. This is may not true for characteristic functions. If $m \in S^0$, then the kernel $K(x-y)$ of the operator $m(D)$ belongs to $C^\infty(R^n\times R^n\setminus \{0\})$ and 
$$
|D^\alpha K(x)| \leq C_\alpha |x|^{-(n+\alpha)}.
$$
The kernel of $\chi_jm(D)\chi_k$ is given by
$$
K'(x,y) = \chi_jK(x-y)\chi_k.
$$
It's easy to check that
$$
\|\chi_jm(D)\chi_k\|_{L^p, L^p} \leq C|j-k|^{-n}
$$
for $1 < p < \infty$.