A parametrix for the fundamental solution of the Klein-Gordon equation on asymptotically de Sitter spaces

TL;DR

This paper constructs a parametrix for the fundamental solution of the Klein-Gordon equation on asymptotically de Sitter spaces, enabling asymptotic analysis and uniform estimates for solutions without caustics.

Contribution

It introduces a new parametrix construction for the Klein-Gordon equation on asymptotically de Sitter spaces, facilitating asymptotic expansions and L^p estimates.

Findings

Constructed a parametrix for the fundamental solution

Derived asymptotic expansions for solutions

Established uniform L^p estimates for functions at infinity

Abstract

In this paper we construct a parametrix for the forward fundamental solution of the wave and Klein-Gordon equations on asymptotically de Sitter spaces without caustics. We use this parametrix to obtain asymptotic expansions for solutions of the inhomogeneous equation and to obtain a uniform L^p estimate for a family of bump functions traveling to infinity.