Elliptic capacity propagation and Fourier restriction to the sphere

We prove the bounded-data Fourier restriction conjecture for the sphere in three dimensions: the Fourier extension operator maps $L^{\infty }(S^2)$ boundedly into $L^p({\mathbb{R}}^3)$ for every p > 3. This is the full conjectured open range, and the threshold p = 3 is sharp.