Semiclassical resolvent estimates at trapped sets

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 χ microlocally supported away from the trapping: ∥χR h (E+i0)χ∥=𝒪(h -1 ), a microlocal version of a result of Burq and Cardoso-Vodev. We now allow one of the two cutoffs, χ ˜, to be supported at the trapped set, giving ∥χR h (E+i0)χ ˜∥=𝒪(a(h)h -1 ) when the a priori bound is ∥χ ˜R h (E+i0)χ ˜∥=𝒪(a(h)h -1 ).