# Accessible reference for (scattering) $\Psi DO$'s on manifolds

I am currently trying to understand Hassell, Tao, and Wunsch's paper on Strichartz estimates on non-trapping asymptotically conic manifolds, however, my understanding of pseudodifferential operators on non-compact Riemannian manifolds is fairly weak.

I'm familiar with the theory of $\Psi DO$'s as presented in Abel's book Pseudo-Differential and Singular Integral Operators, which covers uniform symbol classes $S_{1,0}^m(\mathbb R^n\times\mathbb R^N)$, with a very brief excursion into $\Psi DO$'s on compact manifolds, but nothing about non-compact manifolds. The only references I can find are either very encyclopedic, such as Hörmander's books, or near undecipherable, like Melrose's notes. Are there any books or lecture notes that can provide a gentle introduction to pseudodifferential operators on manifolds without immediately pursuing the utmost generality?

## 1 Answer

I take a very brief look at the paper and I did not see $\Psi DO$ on manifold with boundary being used heavily anywhere (no conormal distribution, multiple blow-ups, heavy handed symbol estimates, etc). The gist of the paper is not the non-compact setting but dispersive PDE.

I think the following notes I took may be helpful, but there are plenty of mistakes here and there. There are also notes by many other experts. Personally, I suggest against going into the field as an undergraduate. I think the right strategy should be to isolate pieces of mathematics you cannot understand in that paper, and try to prove it on your own by learning the machinery from books/notes, or from references papers cited whenever possible. The "bottom up" approach simply takes too much time, and you may be aware that there are many approaches for dealing with non-compact manifolds. So learning one may not be enough. The goal of your paper reading should be focusing on the content (dispersive PDE), not the form (the myriad array of techniques or technical detail presented in it). This is the advice offered to me by Atiyah.