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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 topics development (Shubins 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 literature.).

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.)