# Sharp resolvent bounds and resonance-free regions

## TL;DR

This paper establishes a link between polynomial bounds on the resolvent in a strip and logarithmic bounds, leading to improved resonance-free region estimates in semiclassical scattering on certain manifolds.

## Contribution

It provides a new method to derive logarithmic resolvent bounds from polynomial estimates, enhancing understanding of resonance-free regions in semiclassical scattering.

## Key findings

- Polynomial resolvent estimates imply logarithmic bounds in the same strip.
- Improved resonance-free region estimates for convex co-compact surfaces.
- Enhanced bounds on the resolvent near the real axis.

## Abstract

In this note, we consider semiclassical scattering on a manifold which is Euclidean near infinity or asymptotically hyperbolic. We show that, if the cut-off resolvent satisfies polynomial estimates in a strip of size $O(h |\log h|^{-\alpha})$ below the real axis, for some $\alpha\geq 0$, then the cut-off resolvent is actually bounded by $O(|\log h|^{\alpha+1} h^{-1})$ in this strip. As an application, we improve slightly the estimates on the real axis given by Bourgain and Dyatlov in the case of convex co-compact surfaces.