Extremizers for Fourier restriction on hyperboloids

Abstract

The $L^2\to L^p$ adjoint Fourier restriction inequality on the $d$-dimensional hyperboloid ${\mathbb{H}}^d\subset {\mathbb{R}}^{d+1}$ holds provided $6\le p<\infty$, if $d=1$, and $2(d+2)/d\le p\le 2(d+1)/(d-1)$, if $d\ge 2$. Quilodrán [35] recently found the values of the optimal constants in the endpoint cases $(d,p)\in \{(2,4),(2,6),(3,4)\}$ and showed that the inequality does not have extremizers in these cases. In this paper we answer two questions posed in [35], namely: (i) we find the explicit value of the optimal constant in the endpoint case $(d,p)=(1,6)$ (the remaining endpoint for which $p$ is an even integer) and show that there are no extremizers in this case; and (ii) we establish the existence of extremizers in all non-endpoint cases in dimensions $d\in \{1,2\}$. This completes the qualitative description of this problem in low dimensions.