5
$\begingroup$

Can anyone suggest good (a) uses/applications or (b) construction of micro-functions (introduced by Mikio Sato in 1971) in analysis?

I am trying to understand the subject better. Suggestions of literature are very welcome, but also, how would one present the basic concept as concisely as possible if one had to?

BTW, the best I've found so far for (b) is Masaki Kashiwara's 1978 lecture titled Micro-Local Analysis at the International Congress of Mathematicians. (Paper was published in 1980.)

$\endgroup$
  • $\begingroup$ You propably already came across this, but if not this book might be of help: "An Introduction to Sato's hyperfunctions" by Mitsuo Morimoto. (Note that I have no idea of the topic of micro-functions, but I once read the appendix in the book on inductive limits of locally convex spaces, and this part was very readable). $\endgroup$ – Sebastian Nov 3 '13 at 23:26
  • $\begingroup$ That book was helpful for certain things, yes, I would recommend it, but I was thinking more of papers that I do not know about, because the books on this topic are limited in number. $\endgroup$ – user103756 Nov 4 '13 at 12:55
  • 1
    $\begingroup$ Read the original papers. I tried studying algebraic analysis from Kashiwara, Schapira etc´s, but everything is poorly written. However, the papers by Mikio Sato are way more understandable and very easy to understand if you read everything $\endgroup$ – user40276 Sep 5 '14 at 17:50
  • $\begingroup$ I agree with the above comment. This is what I ended up anyway doing in the months since I posted the question. I had to learn some new notational conventions used in the 1971 paper by Sato, but it was worth it. (It's productive notation!) $\endgroup$ – user103756 Sep 26 '14 at 4:00
  • $\begingroup$ @user103756 I think a good intuition for microfunctions is that they are distilled singularities of distributions (hyperfunctions to be precise), if you take a D-module which whose characteristic variety is supported on the zero-section then when you microlocalize it you won't get any microfunction solutions for it except 0. You can use this for example to easily prove a very strong regularity statement for elliptic D-modules (those with 0 char variety). Namely that the sheaf of hyperfunction solutions is quasi isomorphic to the sheaf of analytic solutions. $\endgroup$ – Saal Hardali Nov 1 '17 at 11:14

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy