Fredholm Theory of Non-Elliptic Operators in the Presence of Normally Hyperbolic Trapping

We present an improved Fredholm theory of non-elliptic operators for when the corresponding classical dynamical system exhibits normally hyperbolic trapping with smooth backward and forward trapped sets. It takes place on coisotropic Sobolev spaces with weak regularity at the backward trapped set ${\mathrm{\Gamma }}_u$, which are roughly speaking made of distributions $v\in H^s$ satisfying ${\mathrm{\Phi }}_{u}v\in H^{s+1}$ for some quantization ${\mathrm{\Phi }}_u$ of the defining function ${\phi }_u$ of ${\mathrm{\Gamma }}_u$. We then apply it to the case of wave operators on spacetimes.