A sharp bilinear estimate for the Klein-Gordon equation in arbitrary space-time dimensions

TL;DR

This paper establishes a precise bilinear inequality for the Klein-Gordon equation across multiple dimensions, leading to a sharp Strichartz estimate in five dimensions and insights into the behavior of maximising sequences.

Contribution

It extends bilinear estimates for the Klein-Gordon equation to arbitrary space-time dimensions and analyzes the non-existence of maximisers for the associated Strichartz estimate.

Findings

Proves a sharp bilinear inequality for Klein-Gordon in any dimension.

Derives a sharp Strichartz estimate in 5D for data in H^1.

Shows maximisers do not exist; sequences concentrate at infinity.

Abstract

We prove a sharp bilinear inequality for the Klein-Gordon equation on $\sr^{d+1}$, for any $d \geq 2$. This extends work of Ozawa-Rogers and Quilodr\'an for the Klein-Gordon equation and generalises work of Bez-Rogers for the wave equation. As a consequence we obtain a sharp Strichartz estimate for the solution of the Klein-Gordon equation in five spatial dimensions for data belonging to $H^1$. We show that maximisers for this estimate do not exist and that any maximising sequence of initial data concentrates at spatial infinity.