Sharp endpoint multilinear estimates for oscillatory integrals and spectral clusters

We prove sharp $k$-linear $L^p$ estimates for Carleson–Sjölin oscillatory integral operators with arbitrary separated frequency scales for all $k\ge 2$ and $1\le p\le \mathrm{\infty }$. The estimates are sharp, including the endpoint logarithmic behavior for general Carleson–Sjölin phases. Moreover, we obtain log-free endpoint bilinear spectral cluster estimates on every closed three-dimensional Riemannian manifold, resolving a problem of Burq–Gérard–Tzvetkov. As a consequence, we establish sharp $k$-linear $L^p$ spectral cluster estimates for all $k\ge 2$ and $1\le p\le \mathrm{\infty }$.