## Decay rate of nonlocal differential operator?

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.

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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 $1 < p < \infty$.

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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 $\Vert a_{jk}(x,D)\Vert_{L^2\rightarrow L^2}\le C_N(1+ \vert j-k\vert)^{-N}.$

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What if anything does $N$ depend on in the displayed inequality? – Deane Yang Sep 5 at 13:14
@Deane, I believe @Bazin means for any $N$ (of course the constant $C$ depends on $N$ as indicated), so it says that the $L^2\to L^2$ norm decays "rapidly" in $|j-k|$. – Otis Chodosh Sep 5 at 14:12
Otis, agreed. Just wanted it stated explicitly. On the other hand, it would be best if the original poster just worked out the proof and figured this out independently. – Deane Yang Sep 5 at 15:14