Semiclassical resolvent estimates at trapped sets

TL;DR

This paper extends semiclassical resolvent estimates to include trapped sets, providing intermediate bounds that bridge previous trapping and non-trapping results, enhancing understanding of wave behavior in complex geometries.

Contribution

It introduces new resolvent estimates that incorporate trapping effects by allowing one cutoff to be supported at the trapped set, advancing prior microlocal analysis techniques.

Findings

Derived resolvent estimates at trapped sets.

Bridged gap between trapping and non-trapping estimates.

Extended applicability of semiclassical analysis in trapping scenarios.

Abstract

We extend our recent results on propagation of semiclassical resolvent estimates through trapped sets when a priori polynomial resolvent bounds hold. Previously we obtained non-trapping estimates in trapping situations when the resolvent was sandwiched between cutoffs microlocally supported away from the trapping, a microlocal version of a result of Burq and Cardoso-Vodev. We now allow one of the two cutoffs to be supported at the trapped set, giving estimates which are intermediate between the trapping and non-trapping ones.