Extremizers for adjoint Fourier restriction on hyperboloids: the higher dimensional case

TL;DR

This paper proves the existence of maximizers for the Lorentz-invariant adjoint Fourier restriction inequality on hyperboloids in dimensions three and higher, extending previous results from lower dimensions using bilinear restriction techniques.

Contribution

It generalizes the existence of maximizers for the adjoint Fourier restriction inequality to higher dimensions ($d geq 2$) by employing bilinear restriction methods.

Findings

Maximizers exist for the inequality in dimensions $d geq 3$.

Extension of previous results from $d=1,2$ to higher dimensions.

Use of bilinear restriction theory to achieve the generalization.

Abstract

We prove that in dimensions $d \geq 3$, the non-endpoint, Lorentz-invariant $L^2 \to L^p$ adjoint Fourier restriction inequality on the $d$-dimensional hyperboloid $\mathbb{H}^d \subseteq \mathbb{R}^{d+1}$ possesses maximizers. The analogous result had been previously established in dimensions $d=1,2$ using the convolution structure of the inequality at the lower endpoint (an even integer); we obtain the generalization by using tools from bilinear restriction theory.