On the extremizers of an adjoint Fourier restriction inequality

The adjoint Fourier restriction inequality for the sphere S2 states that if f∈L2(S2,σ) then fσ̂∈L4(R3). We prove that all critical points f of the functional ‖fσ̂‖L4/‖f‖L2 are smooth, any complex-valued extremizer for the inequality is a nonnegative extremizer multiplied by the character eix⋅ξ for some ξ, and complex-valued extremizing sequences for the inequality are precompact modulo multiplication by characters.