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)
- 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.