  We apply the Bennett-Carbery-Tao multilinear restriction estimate in order to
bound restriction operators and more general oscillatory integral operators. We
get improved L^p estimates in the Stein restriction problem for dimension at
least 5 and a small improvement in dimension 3. We prove similar estimates for
Hormander-type oscillatory integral operators when the quadratic term in the
phase function is positive definite, getting improvements in dimension at least
5. We also prove estimates for Hormander-type oscillatory integral operators in
even dimensions. These last oscillatory estimates are related to improved
bounds on the dimensions of curved Kakeya sets in even dimensions.
