Resolvent and spectral measure on non-trapping asymptotically hyperbolic manifolds I: Resolvent construction at high energy

TL;DR

This paper constructs the high-energy resolvent on non-trapping asymptotically hyperbolic manifolds using advanced microlocal analysis techniques, laying groundwork for further spectral and geometric analysis.

Contribution

It introduces a novel resolvent construction at high energy for a broad class of asymptotically hyperbolic manifolds, extending previous methods.

Findings

Successful construction of high-energy resolvent using semiclassical techniques

Generalization of previous resolvent constructions to non-trapping asymptotically hyperbolic manifolds

Foundation for applications to restriction theorems, spectral multipliers, and Strichartz estimates

Abstract

This is the first in a series of papers in which we investigate the resolvent and spectral measure on non-trapping asymptotically hyperbolic manifolds with applications to the restriction theorem, spectral multiplier results and Strichartz estimates. In this first paper, we use semiclassical Lagrangian distributions and semiclassical intersecting Lagrangian distributions, along with Mazzeo-Melrose 0-calculus, to construct the high energy resolvent on general non- trapping asymptotically hyperbolic manifolds, generalizing the work due to Melrose, Sa Barreto and Vasy. We note that there is an independent work by Y. Wang which also constructs the high-energy resolvent.