Propagation of Singularity with Normally Hyperbolic Trapping

TL;DR

This paper establishes microlocal estimates for wave equations with normally hyperbolic trapping, introducing a novel symbol class and demonstrating minimal loss in estimates on Kerr(-de Sitter) spacetimes.

Contribution

It introduces a new symbol class for microlocal analysis and proves estimates with minimal loss in the presence of normally hyperbolic trapping.

Findings

Microlocal estimates with arbitrarily small loss on Kerr(-de Sitter) spacetimes.

Introduction of a new symbol class constructed via blow-up techniques.

Enhanced understanding of wave propagation in trapped geometries.

Abstract

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.