Conditional existence of maximizers for the Tomas-Stein inequality for the sphere

The Tomas-Stein inequality for a compact subset $\Gamma$ of the sphere $S^d$ states that the mapping $f\mapsto \widehat{f\sigma}$ is bounded from $L^2(\Gamma,\sigma)$ to $L^{2+4/d}(\R^{d+1})$. Then conditional on a strict comparison between the best constants for the sphere and for the Strichartz inequality for the Schr\"odinger equations, we prove that there exist functions which extremize this inequality, and any extremising sequence has a subsequence which converges to an extremizer. The method is based on the refined Tomas-Stein inequality for the sphere and the profile decompositions. The key ingredient to establish orthogonality in profile decompositions is that we use Tao's sharp bilinear restriction theorem for the paraboloids beyond the Tomas-Stein range. Similar results have been previously established by Frank, Lieb and Sabin \cite{Frank-Lieb-Sabin:2007:maxi-sphere-2d}, where they used the method of the missing mass.