Exponentials rarely maximize Fourier extension inequalities for cones

Abstract We prove the existence of maximizers and the precompactness of ‐normalized maximizing sequences modulo symmetries for all valid scale‐invariant Fourier extension inequalities on the cone in . In the range for which such inequalities are conjectural, our result is conditional on the boundedness of the extension operator. Global maximizers for the Fourier extension inequality on the cone in have been characterized in the lowest dimensional cases . We further prove that these functions are critical points for the to Fourier extension inequality if and only if .