Fundamental Solutions for the Klein-Gordon Equation in de Sitter Spacetime

TL;DR

This paper constructs fundamental solutions for the Klein-Gordon equation in de Sitter spacetime, enabling representation of solutions and establishing $L^p-L^q$ estimates for both homogeneous and inhomogeneous cases.

Contribution

It introduces explicit fundamental solutions for the Klein-Gordon equation in de Sitter spacetime and derives associated $L^p-L^q$ estimates, advancing mathematical understanding of wave propagation in curved spacetime.

Findings

Explicit fundamental solutions constructed for the Klein-Gordon equation in de Sitter spacetime.

Representation formulas for solutions to the Cauchy problem established.

Proved $L^p-L^q$ estimates for solutions with and without source terms.

Abstract

In this article we construct the fundamental solutions for the Klein-Gordon equation in de Sitter spacetime. We use these fundamental solutions to represent solutions of the Cauchy problem and to prove $L^p-L^q$ estimates for the solutions of the equation with and without a source term.