Propagation of Singularity with Normally Hyperbolic Trapping

We prove microlocal estimates with normally hyperbolic trapping. We use a new type of symbol class which is constructed by blowing up the intersection of the unstable manifold and the fiber infinity. For scalar wave equations on Kerr(-de Sitter) spacetimes, the extra loss of the microlocal estimates compared with the standard propagation of singularities is arbitrarily small.