Sharp extension inequalities on finite fields

Sharp restriction theory and the finite field extension problem have both received a great deal of attention in the last two decades, but so far they have not intersected. In this paper, we initiate the study of sharp restriction theory on finite fields. We prove that constant functions maximize the Fourier extension inequality from the parabola P¹⊂ F²q and the paraboloid P²⊂ Fq³ at the euclidean Stein-Tomas endpoint; here, Fqᵈ denotes the (dual) d-dimensional vector space over the finite field Fq with q=pⁿ elements, where p is a prime number greater than $3$ or $2$, respectively. We fully characterize the maximizers for the L²→ L⁴ extension inequality from P² whenever q≡ 1(mod\, 4). Our methods lead to analogous results on the hyperbolic paraboloid, whose corresponding euclidean problem remains open. We further establish that constants maximize the L²→ L⁴ extension inequality from the cone Γ³:=\(ξ,τ, σ)∈ F⁴q: τσ=ξ²\ \ 0\ whenever q≡ 3(mod\, 4). By contrast, we prove that constant functions fail to be critical points for the corresponding inequality on Γ³∪ \ 0\ over Fₚ⁴. While some inspiration is drawn from the euclidean setting, entirely new phenomena emerge which are related to the underlying arithmetic and discrete structures.