A sharp bilinear estimate for the Klein-Gordon equation in arbitrary space-time dimensions. (English) Zbl 1313.35036

Summary: We prove a sharp bilinear inequality for the Klein-Gordon equation on \(\mathbb {R}^{d+1}\) for any \(d\geq 2\). This extends work of Ozawa-Rogers and Quilodrán for the Klein-Gordon equation and generalizes 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 maximizers for this estimate do not exist and that any maximizing sequence of initial data concentrates at spatial infinity.


35B45 A priori estimates in context of PDEs
35L10 Second-order hyperbolic equations
Full Text: arXiv