We prove microlocal estimates at the trapped set of asymptotically Kerr spacetimes:
these are spacetimes whose metrics decay inverse polynomially in time to a stationary
subextremal Kerr metric. This combines two independent results. The first
one is purely dynamical: we show that the stable and unstable manifolds of
a decaying perturbation of a time-translation-invariant dynamical system
with normally hyperbolic trapping are smooth and decay to their stationary
counterparts. The second, independent, result provides microlocal estimates
for operators whose null-bicharacteristic flow has a normally hyperbolic
invariant manifold, under suitable nondegeneracy conditions on the stable
and unstable manifolds; this includes operators on closed manifolds, as well
as operators on spacetimes for which the invariant manifold lies at future
infinity.
Keywords
normally hyperbolic trapping, propagation of singularities,
wave equations, black holes
