In this paper we establish sharp estimates (up to logarithmic losses) for the multilinear oscillatory integral operator studied by Phong, Stein, and Sturm [17] and Carbery and Wright [3] on any product d j=1 L pj (R) with each p j ≥ 2, extending the known results outside the previously-studied range Our theorem assumes a second-order nondegeneracy condition of Varčenko type, and as a corollary reproduces a variant of Varčenko's theorem and implies Fourier decay estimates for measures of smooth density on degenerate hypersurfaces in R d .