Hello,
I am wondering whether anyone know some good references for the theory of wave front set, microlocal analysis? I have some basic knowledge of distribution theory at the level of the Rudin's functional analysis (the first part). As for PDE theory, I learned this topic mainly by Folland's "Introduction to Partial Differential Equations".
When I learned the distribution theory, the book by Strichartz gave me many intuition and helped me
a lot. I am wondering whether there is a similar book to introduce the theory of wave front set, microlocal analysis?
Thank you very much!
Best Answer
There are many references at various levels of difficulty; it also depends on what aspects are you interested in. I cite out of memory, so beware of inaccuracies (which can be corrected according to your needs).
A very good reference is Hormander I (The theory of linear partial diff. op), chapter VIII. The emphasis there is on the $C^\infty$ theory, with Hormander's own definition of WF as a limit. You cutoff the function near a point, Fourier transform it, then examine in which directions the Fourier transform decays fast and in which ones it does not. These last directions stabilize as the cutoff support tends to the point, and what remains is the WF set at the point. Further results, written in an even denser style, are contained in Michael Taylor's book on Pseudodifferential Operators. More recently, the concept has been generalized to include directions of Sobolev regularity and has found applications in nonlinear equations; there results are scattered in a number of papers (JM Bony wrote some papers on this).
There are other points of view; an important one is the analytic wave front set. Here the accent is more on the algebraic aspects, since the set of all solutions to an equation or system of PDEs is studied as a whole. Here the best references are japanese, a good starting point being Akira Kaneko's book on hyperfunction theory, and continuing with the works of Kashiwara (including a book), Sato, Schapira.
EDIT: I understand that an easier introduction would be helpful. You should try with section II.B of Alinhac-Gerard Operateurs pseudodifferentiels et Theoreme de Nash-Moser. It's very readable (assuming you read french :)