Propagation of Singularities with Normally Hyperbolic Trapping

We prove a new microlocal estimate with normally hyperbolic trapping, which can be applied to Kerr and Kerr-de Sitter spacetimes. We use a new type of symbol class, and corresponding operator class, which is constructed by blowing up the intersection of the unstable manifold and fiber infinity. For scalar wave equations on Kerr and Kerr-de Sitter spacetimes, the extra loss of the microlocal estimates compared with the standard propagation of singularities without trapping is arbitrarily small.