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

Abstract

We prove a bilinear $L^2({\mathbb{R}}^d)\times L^2({\mathbb{R}}^d)\to L^2({\mathbb{R}}^{d+1})$ 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 on ${\mathbb{R}}^d$, 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 Schrödinger equation on closed surfaces.

arXiv comment25 pages, Version 2 (revised version incorporating referee's remarks), to appear in Analysis and PDE