Local smoothing for scattering manifolds with hyperbolic trapped sets

TL;DR

This paper establishes resolvent estimates for the Laplace-Beltrami operator on scattering manifolds with hyperbolic trapped sets, leading to local smoothing results by combining microlocal analysis and existing estimates.

Contribution

It introduces a novel approach to combine estimates near trapped regions and infinity on scattering manifolds using microlocal calculus.

Findings

Proves resolvent estimates for hyperbolic trapped sets

Derives local smoothing as a corollary of resolvent bounds

Integrates existing results to handle different regions on the manifold

Abstract

We prove a resolvent estimate for the Laplace-Beltrami operator on a scattering manifold with a hyperbolic trapped set, and as a corollary deduce local smoothing. We use a result of Nonnenmacher-Zworski to provide an estimate near the trapped region, a result of Burq and Cardoso-Vodev to provide an estimate near infinity, and the microlocal calculus on scattering manifolds to combine the two.