Resolvent estimates for normally hyperbolic trapped sets

TL;DR

This paper establishes resolvent estimates and pole-free regions for semiclassical operators with normally hyperbolic trapped sets, with applications to wave equations in black hole physics.

Contribution

It provides new resolvent estimates for operators with normally hyperbolic trapped sets, relevant to wave equations in Kerr black hole spacetimes.

Findings

Pole-free strips for resolvents are established.

Local smoothing effects with epsilon derivative loss are demonstrated.

Local energy decay results for wave equations are derived.

Abstract

We give pole free strips and estimates for resolvents of semiclassical operators which, on the level of the classical flow, have normally hyperbolic smooth trapped sets of codimension two in phase space. Such trapped sets are structurally stable and our motivation comes partly from considering the wave equation for Kerr black holes and their perturbations, whose trapped sets have precisely this structure. We give applications including local smoothing effects with epsilon derivative loss for the Schr\"odinger propagator as well as local energy decay results for the wave equation.