On the extremizers of an adjoint Fourier restriction inequality

The adjoint Fourier restriction inequality for the sphere $S^2$ states that if $f\in\lt(S^2,\sigma)$ then $\widehat{f\sigma}\in L^4(\reals^3)$. We prove that all critical points $f$ of the functional $\norm{\widehat{f\sigma}}_{L^4}/\norm{f}_{\lt}$ are smooth; that any complex-valued extremizer for the inequality is a nonnegative extremizer multiplied by the character $e^{ix\cdot\xi}$ for some $\xi$; and that complex-valued extremizing sequences for the inequality are precompact modulo multiplication by characters.