Sharp Fourier extension for functions with localized support on the circle

Abstract

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.