A bilinear oscillatory integral estimate and bilinear refinements to Strichartz estimates on closed manifolds

Abstract

We prove a bilinear $L^2\left({ℝ}^d\right)\times L^2\left({ℝ}^d\right)\to L^2\left({ℝ}^{d+1}\right)$ estimate for a pair of oscillatory integral operators with different asymptotic parameters and phase functions satisfying a transversality condition. This is then used to prove a bilinear refinement to Strichartz estimates on closed manifolds, similar to that derived by Bourgain on ${ℝ}^d$, but at a relevant semiclassical scale. These estimates will be employed elsewhere to prove global well-posedness below $H^1$ for the cubic nonlinear Schrödinger equation on closed surfaces.