On existence of extremizers for the Tomas-Stein inequality for S1

The Tomas-Stein inequality or the adjoint Fourier restriction inequality for the sphere S1 states that the mapping f↦fσˆ is bounded from L2(S1) to L6(R2). We prove that there exists an extremizer for this inequality. We also prove that any extremizer satisfies |f(−x)|=|f(x)| for almost every x∈S1.