On the extremizers of an adjoint Fourier restriction inequality (Q438175)

Let \(S^2\) denote the unit sphere in \(\mathbb R^3\), equipped with surface measure \(\sigma\). The adjoint Fourier restriction inequlity states that there exists \(C < \infty\) such that NEWLINE\[NEWLINE \| \widehat{f \sigma}\|_{L^4(\mathbb R^3)} \leq C \| f \|_{L^2(S^2, \sigma)} NEWLINE\]NEWLINE for all \(f \in L^2(S^2)\).NEWLINENEWLINEIn this paper, the authors prove that all critical points of the functional \(\| \widehat{f \sigma}\|_{L^4(\mathbb R^3)} / \| f \|_{L^2(S^2, \sigma)}\) are smooth. The authors also characterize general complex-valued extremizers in terms of positive ones, and show that the precompactness does continue to hold for complex-valued extremizing sequences, modulo the action of a natural noncompact symmetry group of the inequality.