Maximizers of the $L^2\to L^4$ Fourier extension inequality for cones in finite fields

Abstract: Sharp Fourier restriction theory and finite field extension theory have both been topics of interest in the last decades. Very recently, in \cite{GonzalezOliveira}, the research into the intersection of these two topics started. There it was established that, for the $(3,1)$-cone $\Gamma_{(3,1)}^3:={\boldsymbol{\eta}\in \mathbb{F}q^4\setminus{\boldsymbol{0}} : \eta_1^2+\eta_2^2+\eta_3^2=\eta_4^2},$ the Fourier extension map from $L^2\to L^{4}$ is maximized by constant functions when $q=3\, \pmod{4}$. In this manuscript, we advance this line of inquiry by establishing sharp inequalities for the $L^{2}\to L^{4}$ extension inequalities applicable for all remaining cones $\Gamma^3\subset \mathbb{F}_q^4$. These cones include the $(2,2)$-cone $\Gamma{(2,2)}^3:={\boldsymbol{\eta}\in \mathbb{F}_q^4\setminus{\boldsymbol{0}} : \eta_1^2+\eta_2^2=\eta_3^2+\eta_4^2}$ for general $q=p^n$ and the $(3,1)$-cone when $q=1\, \pmod{4}$. Moreover, we classify all the extremizers in each case. We note that the analogous problem for the (2, 2)-cone in the euclidean setting remains open.