Suppose you start from partial differential equations and functional analysis (on $\mathbb R^n$ and on real manifolds). Which prominent example problems lead you to work with pseudo-differential operators?
I would appreciate any good examples, as well as some historical outlines on the topic's development. (Shubin's classical book spends a few lines on history and motivation in the preface, but no "natural" examples. I am not aware of any historical outlines in the literature.)
Best Answer
I don't know the history at all, but I have to imagine that the language was invented to provide a context for talking about solution operators for differential equations. Consider, for example, the PDE $D f = f_0$ where $D$ is a nice differential operator. Taking Fourier transforms, this says that $P(\xi)\hat{f} = \hat{f}_0$, where $P$ is the principal symbol of $D$ (a polynomial). Everyone in the world just wants to write $\hat{f} = \frac{1}{P(\xi)}\hat{f}_0$ and take inverse Fourier transforms. In other words, solving the PDE is the same thing as finding an operator $S$ whose Fourier multiplier is $\frac{1}{P(\xi)}$. This most likely fails to be a polynomial, so $S$ is evidently not a differential operator. As far as I can tell, many of those big fat books on pseudodifferential operator theory are all about how to invert as many operators as possible in this sense while salvaging as much regularity as you can. It gets extremely subtle, but I think the motivation is fairly close to the surface.
Aside from that, you might also be led to invent pseudodifferential operators if you cared deeply about the spectral theory of differential operators. The spectral theorem for an operator $T$ is more or less equivalent to the existence of a "functional calculus", i.e. a sensible way to form operators $f(T)$ out of various classes of functions $f$ on the spectrum of $T$. For differential operators (especially on non-compact domains where there need not be a nice eigenspace decomposition), the functional calculus is often obtained via the Fourier transform, and the pseudodifferential calculus manifests itself.