Sharp Fourier extension for functions with localized support on the circle

A well-known conjecture states that constant functions are extremizers of the L^{2} \to L^{6} Tomas–Stein extension inequality for the circle. We prove that functions supported in a \sqrt{6}/80 -neighborhood of a pair of antipodal points on S^{1} satisfy the conjectured sharp inequality. In the process, we make progress on a program formulated by Carneiro, Foschi, Oliveira e Silva and Thiele to prove the sharp inequality for all functions.