Decay rate of nonlocal differential operator?

Question

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.

Answer 1

You have an explicit formula: the symbol of $\chi_jm(D)\chi_k$ is $$ a_{jk}(x,\xi)=\chi_j(x)\iint e^{-2\pi iy\eta}m(\xi+\eta)\chi_k(y+x) dyd\eta.\tag C $$ It is difficult to handle this with $\chi_j$ characteristic functions, because of lack of regularity. Essentially the same service will be given with $\chi_j(x)=\chi_0(x-j)$ where $\chi_0\in C^\infty_c(\mathbb R^n)$. In that case, integrations by parts in $(C)$ yield $$ \vert \partial_x^\alpha\partial_\xi^\beta a_{jk}(x,\xi)\vert \le C_{N\alpha\beta} (1+ \vert j-k\vert)^{-N}\vert\xi\vert^{-\vert\beta\vert} $$ and thus $ \Vert a_{jk}(x,D)\Vert_{L^2\rightarrow L^2}\le C_N(1+ \vert j-k\vert)^{-N}. $

Answer 2

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 $1 < p < \infty$.