The Fourier extension operator on large spheres and related oscillatory   integrals
  We obtain new estimates for a class of oscillatory integral operators with folding canonical relations satisfying a curvature condition. The main lower bounds showing sharpness are proved using Kakeya set constructions. As a special case of the upper bounds we deduce optimal $L^p(mathbb{S}^2)\to L^q(R \mathbb{S}^2)$ estimates for the Fourier extension operator on large spheres in $\mathbb{R}^3$, which are uniform in the radius $R$. Two appendices are included, one concerning an application to Lorentz space bounds for averaging operators along curves in $R^3$, and one on bilinear estimates. 
Jonathan Bennett and Andreas Seeger