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May I humblely ask what is a good reference for conormal distributions (for student with some rudimentary pseudo-differential operator background)? I heard from my advisor that it is useful in index theory, but most of the lecture notes I read are quite opaque (like Simanca's) by using baroque notations, and it is difficult to see the theory in transparent manner without going over all the technical details. For a concrete example, it is not clear to me how to compute the principal symbol of the product of two conormal distributions.

I assume this is a subject well known to experts(like Richard Melrose, Rafe Mezzeo, etc), but I found it difficult to find a good reference that does not assume I know most of the basic theory already.

Update: I know Hormander's book is the Bible in our field. It is just I do not know if I will have enough time to digest it properly. Is there any alternatives?

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  • $\begingroup$ It seems to me that it's hard to learn this stuff without grinding through a lot of details. A reasonable preliminary thing to read is Guillemin and Sternberg's Geometric Asymptotics. $\endgroup$
    – Deane Yang
    Commented Dec 7, 2014 at 22:42
  • $\begingroup$ @Deane Yang: I have no objection for details, I am just afraid that flipping a page or less a day made me unable to see the forest through the trees. $\endgroup$
    – Bombyx mori
    Commented Dec 7, 2014 at 22:50

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I think the third volume of Hormander's The Analysis of Linear Partial Differrential Operators is a good reference. chapter 18 is an introduction to PsDOs, especially, in section 18.2, it's devoted to conormal distributions, which I think is very useful for people with PsDO background. He starts with the fact that the wave front set of the kernel of an PsDO is contained in the Conormal distribution, and further discussed some regularity properties of such distributions, which in turn gives an invariant definition on manifolds. In addition, you will see what conormal distributions look like locally in this section as well.

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  • $\begingroup$ However, the natural topology of the space of conormal distributions is not deeply studied in those references. Do you know a reference where the topology of the space of conormal distributions is studied? $\endgroup$
    – Jesús A. Álvarez López
    Commented Aug 7, 2022 at 6:11

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