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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.

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    $\begingroup$ An approximate inverse (called a parametric) of a differential operator with variable coefficients. $\endgroup$
    – Deane Yang
    Sep 7, 2015 at 6:14
  • $\begingroup$ Look for the section discussing elliptic PDEs. $\endgroup$
    – Deane Yang
    Sep 7, 2015 at 6:15
  • $\begingroup$ Another natural place psido appear is the Dirichlet to Neumann map. I'm not sure of the right reference but you can Google to find things about this. $\endgroup$ Sep 7, 2015 at 6:42
  • $\begingroup$ @DeaneYang What "section" are you referencing? Also, can you give an explicit example? Maybe a worked parametrix with specific coefficients? $\endgroup$
    – Patch
    Sep 7, 2015 at 6:47
  • $\begingroup$ What are you reading to learn about pseudodifferential operators? Every exposition I've ever read discusses how to prove that weak solutions to an elliptic PDE with smooth coefficients are always smooth using pseudodifferential operators. Part of this is constructing a parametrix to the elliptic PDO. This involves solving for an infinite asymptotic expansion of the parametrix symbol. If you use only finitely many (say, just 1 term) for an explicitly defined elliptic PDO, then you get an explicitly defined pseudo-DO of negative order. $\endgroup$
    – Deane Yang
    Sep 7, 2015 at 16:15

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