Hi, Moers. Let $m(\xi) \in S^0$, that is, $$ |D^\alpha m(\xi)| \leq C<\xi>^{-|\alpha|}, \quad \forall \xi \in R^n. $$ It's well known that $m(D)$ is bounded in $L^p$ for $1 < p < \infty$. Let $j, k \in Z^n$ and $\chi_j, \chi_k$ denotes the characteristic functions of unit cubic with center $j, k$ respectively. If $m(D)$ is local(i.e., $supp M(D)\varphi \subset supp \varphi$), then $\chi_jm(D)\chi_k =0$. In general, we only have some decay information like $$ \|\chi_jm(D)\chi_k\|_{L^2, L^2} \leq C|j - k|^{-\gamma}, \quad |j - k| \gg 1. $$ My questions then arises: what decay rate $\gamma$ we can expect?

Thanks.