This talk follows up on the seminar by Andrew Hassell on May 8. However, at the start of my talk I will recall the contents of his seminar, so that the talk is accessible without prior knowledge of the subject.
It was proved by Seeger, Sogge and Stein that Fourier Integral operators (FIOs) in $n$ dimensions map
the Hardy space $H^1$ into $L^1$ provided they have sufficiently negative order, that is, no bigger than $-(n-1)/2$.
Can one use pseudo-differential calculus in situations where no Fourier transform is available? On a manifold, one can use the euclidean pseudo-differential calculus locally, but can one find a global analogue?