Resolvent estimates and local decay of waves on conic manifolds

TL;DR

This paper establishes resolvent estimates and local decay results for waves on conic manifolds with singularities, using advanced propagation of singularities techniques, leading to applications in energy decay and smoothing estimates.

Contribution

It introduces a logarithmic resonance-free region for the resolvent on conic manifolds and applies propagation of singularities to derive decay and smoothing results.

Findings

Existence of a logarithmic resonance-free region for the resolvent.

Exponential local energy decay on conic manifolds.

Lossless local smoothing estimate for Schrödinger equation.

Abstract

We consider manifolds with conic singularites that are isometric to $\mathbb{R}^{n}$ outside a compact set. Under natural geometric assumptions on the cone points, we prove the existence of a logarithmic resonance-free region for the cut-off resolvent. The estimate also applies to the exterior domains of non-trapping polygons via a doubling process. The proof of the resolvent estimate relies on the propagation of singularities theorems of Melrose and the second author to establish a "very weak" Huygens' principle, which may be of independent interest. As applications of the estimate, we obtain a exponential local energy decay and a resonance wave expansion in odd dimensions, as well as a lossless local smoothing estimate for the Schr{\"o}dinger equation.