Maximizers of the L2 → L4 Fourier extension inequality for cones in finite fields

Sharp Fourier restriction inequalities in euclidean spaces and the restriction phenomenon in finite fields have both been topics of interest in the last few decades. Very recently, the research at the intersection of these two topics began. In [8], it was established that, for the (3,1)[jls-end-space/]-cone Γ(3,1)3:={η∈Fq4∖{0}:η12+η22+η32=η42}[jls-end-space/], the L2→L4 Fourier extension inequality is saturated by constant functions when (Formula presented). In this manuscript, we advance this line of inquiry by establishing sharp forms of L2→L4 Fourier extension inequalities for all the remaining cones Γ3⊂Fq4[jls-end-space/]. These cones include the (2,2)[jls-end-space/]-cone Γ(2,2)3:={η∈Fq4∖{0}:η12+η22=η32+η42} for general q=pn and the (3,1)[jls-end-space/]-cone when (Formula presented). Moreover, we classify all the extremizers in both cases. We note that the corresponding problem for the (2,2)[jls-end-space/]-cone in the euclidean setting remains open.