  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.
