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.
Categories: math.CA