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

摘要

We prove a bilinear $L^2(\R^d) \times L^2(\R^d) \to L^2(\R^{d+1})$ estimatefor a pair of oscillatory integral operators with different asymptoticparameters and phase functions satisfying a transversality condition. This isthen used to prove a bilinear refinement to Strichartz estimates on closedmanifolds, similar to that on $\R^d$, but at a relevant semi-classical scale.These estimates will be employed elsewhere to prove global well-posedness below$H^1$ for the cubic nonlinear Schr\"odinger equation on closed surfaces.