Applications of pseudodifferential operators to PDE I am planning to build a PDE course centred around pseudodifferential operators. I know some important applications of pseudodifferential operators to PDEs, but I don't know enough to get the whole picture. So my question is, after setting up the stage and covering the basic stuff on pseudodifferential operators, including the algebra and mapping properties, what are my choices on the "main part" of the course? One application per answer, and please include references if possible.
To get started, I have the following examples (please expand these if you want)


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*Pseudodifferential operators can be used to construct parametrices for (properly) elliptic operators on closed manifolds, hence proving the Fredholm property of elliptic operators.

*Ellipitic boundary value problems can be reduced to pseudodifferential equations on the boundary of the domain.

*For a class of hyperbolic systems, one can construct microlocal symmetrizers by using pseudodifferntial operators. This class significantly extends symmetric hyperbolic systems.

 A: After establishing the basic calculus and the construction and use of parametrices for elliptic operators, I suggest to go for Hörmander's theorem on propagation of singularities. This result is basic to understanding why high-frequency waves propagate along (geometro-optical) rays. Its proof (and, of course, its statement using wavefront sets) is a core example of microlocal analysis, and it shows that more useful things can be done with symbols than only taking their reciprocals. I find Hörmander's original proof in the ICM Nice 1970 proceedings very readable: Construct an operator which commutes with the given operator (the d'Alembertian, say) and (micro-)localizes to a neighbourhood of a given bicharacteristic, and then apply the $C^\infty$-wellposedness (established differently). Other proofs use microlocal energy estimates ("positive commutator method") or Egorov's Theorem (FIO calculus). Proofs of the theorem can be found, e.g., in the books by Grigis&Sjöstrand (CUP 1994), by Taylor (PUP 1981), by Unterberger (Aarhus Univ. 1976), and by Eskin (AMS 2011).
A: In my humble opinion, Pseudodifferential operators (PO) are usefull for (elliptic and hypoelliptic) PDE. For hyperbolic PDE, Fourier Intergral Operators (FIO) are more suitable for building parametrices. (example : the wave equation). 
Garding Inequality ans its generalizations are important tools in the framework of PO and are.used to solve the Dirichlet Problem.
