# 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)

• 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.
• I'm not 100% sure, but I think you can do things like Schauder estimates and a lot of the other parts of "elliptic theory" with psiDO's. I've never read it, but I once glanced at Taylor's book "Pseudodifferential Operators and Nonlinear PDE": unc.edu/math/Faculty/met/nonlin.html which seems to have a bit of this. Anyways, it might also have some other topics you'd be interested in. – Otis Chodosh Apr 18 '13 at 16:21
• Stein's Harmonic Analysis has a chapter or two on pseudo-differential operators and related topics; he covers such things as the algebra and mapping properties in the main text, so you might find useful 'subsequent topics' in the Notes and References section of these chapters. If you want the students to do some of the talking you could also hand out topics from the Notes and References for them to prepare talks on. – Jens Apr 18 '13 at 18:24
• Bazin (mathoverflow.net/users/21907), Motivation for and history of pseudo-differential operators, mathoverflow.net/questions/97604 (version: 2012-05-23) – Bazin Apr 18 '13 at 18:30

## 2 Answers

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).

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.