Maximizers of the $L^2\to L^4$ Fourier extension inequality for cones in finite fields
Cristian González-Riquelme, Tolibjon Ismoilov
TL;DR
The paper establishes a sharp L^2→L^4 Fourier extension inequality for the 2+2 cone \Gamma^{3}_{(2,2)} in field_q^4 and provides a complete extremizer classification, extending prior results for the (3,1)-cone and covering the q\equiv1\pmod{4} case. The authors introduce a Segre-embedding parametrization of the cone, analyze the combinatorial structure of Σ_ξ and associated sets, and implement a symmetrization and mass-transport framework to reduce the problem to plane-based configurations. Through detailed decomposition into punctured planes and careful AM-GM estimates, they prove the sharp constant and show that extremizers are precisely complex exponentials restricted to the cone. When q≡1 mod 4, the same sharp constant and extremizers hold for the (3,1)-cone, completing the sharp-constant picture for four-dimensional finite-field cones. These results illuminate the discrete geometry behind Euclidean cone restriction and provide explicit extremizers via additive characters.
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 $Γ_{(3,1)}^3:=\{\boldsymbolη\in \mathbb{F}_q^4\setminus\{\boldsymbol{0}\} : η_1^2+η_2^2+η_3^2=η_4^2\},$ 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 $Γ^3\subset \mathbb{F}_q^4$. These cones include the $(2,2)$-cone $Γ_{(2,2)}^3:=\{\boldsymbolη\in \mathbb{F}_q^4\setminus\{\boldsymbol{0}\} : η_1^2+η_2^2=η_3^2+η_4^2\}$ 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.