# A good reference for the wave front set

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!

• The standard reference is, of course, Lars Hormander, Analysis of Linear Partial Differential Operators, vols 1-4. Unfortunately I wouldn't say it is a "similar book" to the book of Strichartz. – Willie Wong Dec 3 '10 at 14:27
• I'm really out of date with this stuff, but in my day the books I looked at were Treves (Introduction to Pseudodifferential and Fourier Integral Operators) and stuff written by Michael Taylor. Also really nice is Geometric Asymptotics by Guillemin and Sternberg. – Deane Yang Dec 3 '10 at 14:44
• Another book that I used was by Chazarain and Piriou: books.google.com/… – Deane Yang Dec 3 '10 at 15:00
• Actually, I think vol 1 of Hormander is about the most lucid book I have ever seen (and I am no analyst). – Igor Rivin Dec 3 '10 at 15:11
• The most elementary introduction that I know is the book "Elementary Introduction to the Theory of Pseudodifferential Operaotors" by Xavier Saint Raymond, it has only 100 pages and assumes basic knowledge of real analysis only. – Tim van Beek Dec 5 '10 at 9:13

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).