## 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!

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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 2010 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I'm really out of date with this stuff, but in my day the books I looked at were Treves (Introduction to Pseudodifferential and Fourier Integral Operators) and stuff written by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 2010 at 14:44
Another book that I used was by Chazarain and Piriou: books.google.com/… – Deane Yang Dec 3 2010 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 2010 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 2010 at 9:13
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## 1 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 :)

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@Prof. Piero D'Ancona, thank you very much. These references are very useful. I will have a look. :-) – Anand Dec 3 2010 at 16:16
Alinhac-Gerard has been translated in GSM 82. – timur Dec 3 2010 at 18:57
@Timur, Thank you. :-) I will have a look of the translated version. – Anand Dec 3 2010 at 22:20
For convenience of references, here is the book by Alinhac-Gerard: books.google.ch/… – Anand Dec 3 2010 at 22:29