Strichartz estimates for the Klein--Gordon equation in $\mathbb{R}^{3+1}$

TL;DR

This paper establishes both standard and reversed Strichartz estimates for the Klein--Gordon equation in four-dimensional space-time using fundamental solutions and fractional integrations, and applies these results to semilinear equations.

Contribution

It introduces a novel approach to derive Strichartz estimates without Fourier analysis, utilizing fundamental solutions and fractional integrations.

Findings

Proved standard and reversed Strichartz estimates for Klein--Gordon in 3+1 dimensions.

Applied estimates to analyze semilinear Klein--Gordon equations.

Provided a Fourier-free methodology for dispersive estimates.

Abstract

In this paper we prove standard and reversed Strichartz estimates for the Klein--Gordon equation in $\mathbb{R}^{3+1}$. Instead of the Fourier theory, our analysis is based on fundamental solutions of the free equations and fractional integrations. In the final part of this paper, we apply Strichartz estimates in the study of a semilinear Klein--Gordon equation.