Resonance expansions for tensor-valued waves on asymptotically Kerr-de Sitter spaces

TL;DR

This paper establishes exponential decay estimates for tensor-valued waves on perturbations of Schwarzschild-de Sitter spaces, using novel pseudodifferential inner products to handle vector bundles without positive definite metrics.

Contribution

It introduces pseudodifferential inner products to analyze wave decay on tensor bundles in black hole spacetimes, extending previous low energy results to high energy estimates.

Findings

Proves exponential decay of tensor-valued waves on perturbations of Schwarzschild-de Sitter spaces.

Develops pseudodifferential inner products for vector bundles without positive definite metrics.

Establishes high energy estimates crucial for understanding wave asymptotics in black hole spacetimes.

Abstract

In recent joint work with Vasy, we analyze the low energy behavior of differential form-valued waves on black hole spacetimes. In order to deduce asymptotics and decay from this, one in addition needs high energy estimates for the wave operator acting on sections of the form bundle. The present paper provides these on perturbations of Schwarzschild-de Sitter spaces in all spacetime dimensions $n\geq 4$. In fact, we prove exponential decay, up to a finite-dimensional space of resonances, of waves valued in any finite rank subbundle of the tensor bundle, which in particular includes differential forms and symmetric tensors. As the main technical tool for working on vector bundles that do not have a natural positive definite inner product, we introduce pseudodifferential inner products, which are inner products depending on the position in phase space.