Pseudodifferential operators and amplitudes

Question

I am studying psudodifferential operators on $\mathbb{R}^n$. Let $U\subset \mathbb{R}^n$ an open subset. A function is $b\in C^\infty(U\times U\times U \times \mathbb{R}^n)$ is an amplitude of order $r$ if for any multiindices $\alpha$, $\beta$, $\gamma$, there is constant $C_{\alpha,\beta,\gamma}$ such that \begin{equation} sup_{x,y\in U} |D^\alpha _xD^\beta _yD^\gamma _\xi b(x,y,\xi)|\leq C_{\alpha,\beta,\gamma} (1+|\xi|)^{r-|\gamma|}. \end{equation} A pseudodifferential operator is given by \begin{equation} B\phi (x)=\frac{1}{(2\pi)^n}\int_{\mathbb{R}^n} \int_U e^{i(x-y)\xi}b(x,y,\xi)\phi(y) dy d\xi. \end{equation} Unfortunately, I don't understand why if I have an amplitude defining a pseudodifferential operator, and the amplitude vanishes in a neighborhood of the diagonal, then it is of order $-\infty$. I thank you in advance for the help.

Answer 1

Because then you can use the identity $$ \frac1{i(x-y)^2}(x-y)\frac\partial{\partial \xi}e^{i(x-y)\xi}=e^{i(x-y)\xi} $$ to integrate by parts as many times as you wish and obtain an ampitude of arbitrary large negative order $r-N$, where $N$ is the number of integration by parts. Specifically, $$ \int e^{i(x-y)\xi}b(x,y,\xi)\phi(y)d\xi dy=\int\left(\left[\frac1{i(x-y)^2}(x-y)\frac\partial{\partial \xi}\right]^N e^{i(x-y)\xi}\right)b(x,y,\xi)\phi(y)d\xi dy\\ =(-1)^N\int e^{i(x-y)\xi}\left(\left[\frac1{i(x-y)^2}(x-y)\frac\partial{\partial \xi}\right]^N b(x,y,\xi)\right)\phi(y)d\xi dy. $$ The amplitude in parentheses is smooth (because $b\equiv0$ in a neighborhood of the diagonal $x=y$) and is of order $r-N$ if $b$ is an amplitude of order $r$.