I'm reading up on $\Psi DO$'s and trying to find some examples of symbols that are not quite so trivial.
Obviously, the first example of a symbol that most people talk about is just a polynomial in $\xi$ with coefficients in $C^{\infty}$ that depend on $x$: $$P(x,\xi) = \sum_{|\alpha|<m}a_{\alpha}(x)\xi^{\alpha}, \quad \text{with $\alpha$ a multi-index.}$$
To me, however, these are fairly uninteresting. Particularly because in many cases they correspond exactly to regular old differentiation in the $x$ variables.
So I tried thinking of my own examples in $\mathbb{R}^1$, just to get a good handle on what's really going on (and in the hopes that I might be able to use Mathematica to visualize some of the pseudoderivaties.)
The first example of an order-$1$ symbol I thought up was $$P(x,\xi) = \cos(x)\log(1+\xi^2).$$
Indeed, we have that $$\left| \left(\frac{\partial}{\partial \xi}\right)^{\alpha} \left(\frac{\partial}{\partial x}\right)^{\beta} (\cos(x)\log(1+\xi^2)) \right| \leq C_{\alpha,\beta}(1+|\xi|)^{1-|\alpha|}.$$
However I soon realized that even this example was a little too simple, since the $x$ "portion" of $P(x,\xi)$ just factors out of the integral, $$\int_{\mathbb{R}}\int_{\mathbb{R}}e^{i\langle x-y,\xi\rangle}P(x,\xi)u(y)d\xi dy = \cos(x)\int_{\mathbb{R}}\int_{\mathbb{R}}e^{i\langle x-y,\xi \rangle}\log(1+\xi^2)u(y)d\xi dy,$$ making it essentially a Fourier multiplier issue. Somehow the nuance of allowing $P(x,\xi)$ to be a function of both $x$ and $\xi$ is lost. This is still a problem in $\mathbb{R}^n$, of course, if we simply have $P(x,\xi)=P_1(x)P_2(\xi)$.
The last example I could come up with then was $$P(x,\xi) = \exp\left({\frac{-(1+\xi^2)}{1+x^2}}\right), \quad \text{for $|x|<1$ ($\equiv 0$ for $|x|\geq1$)}.$$ But again I am not quite satisfied, seeing as $P(x,\xi)\in C_c^{\infty}(\mathbb{R})\times\mathcal{S}(\mathbb{R})$, and so $P\in \bigcap_{m}S^m = S^{-\infty}$, meaning it is just a smoothing operator.
Can anyone offer some interesting examples of symbols/operators of order $m\in \mathbb{Z}$, that are not just products of functions nor smoothing operators? Examples in $\mathbb{R}^1$ would be preferable, but anything else is good too.
