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 $1 < p < \infty$.

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 $. Hence $ |K'(x,y)| \leq C|j-k|^{-n} $, which in turn implies that $$ \|\chi_jm(D)\chi_k\|_{L^p, L^p} \leq C|j-k|^{-n} $$ for $1 < p < \infty$.