2025/09/06 by Cristian González-Riquelme, González-Riquelme, Cristian, Tolibjon Ismoilov +1
Mathematics · #05B25 #12E20 #42B05 #Advanced Harmonic Analysis Research #Classical Analysis and ODEs (math.CA) #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.2509.05600
openalex publication_date 2025/09/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Sharp Fourier restriction theory and finite field extension theory have both been topics of interest in the last decades. Very recently, in \citeGonzalezOliveira, 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∈ \mathbbFq4∖\\boldsymbol0\ : η12+η22+η32=η42\, the Fourier extension map from L2→ L4 is maximized by constant functions when q=3 \pmod4. In this manuscript, we advance this line of inquiry by establishing sharp inequalities for the L2→ L4 extension inequalities applicable for all remaining cones Γ3⊂ \mathbbFq4. These cones include the (2,2)-cone Γ(2,2)3:=\\boldsymbolη∈ \mathbbFq4∖\\boldsymbol0\ : η12+η22=η32+η42\ for general q=pn and the (3,1)-cone when q=1 \pmod4. 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.