Strichartz estimates on asymptotically de Sitter spaces

TL;DR

This paper establishes local weighted Strichartz estimates with derivative losses for the Klein-Gordon equation on asymptotically de Sitter spaces, highlighting the challenges in obtaining global dispersive estimates and applying results to small-data global existence.

Contribution

It introduces a new family of local weighted Strichartz estimates for Klein-Gordon equations on asymptotically de Sitter spaces, with insights into their limitations and applications.

Findings

Proved local weighted Strichartz estimates with derivative losses.

Provided heuristic argument against global dispersive estimates.

Applied estimates to small-data global existence for semilinear equations.

Abstract

In this article we prove a family of local (in time) weighted Strichartz estimates with derivative losses for the Klein-Gordon equation on asymptotically de Sitter spaces and provide a heuristic argument for the non-existence of a global dispersive estimate on these spaces. The weights in the estimates depend on the mass parameter and disappear in the "large mass" regime. We also provide an application of these estimates to establish small-data global existence for a class of semilinear equations on these spaces.