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

TL;DR

This paper establishes a bilinear estimate for oscillatory integral operators and applies it to refine Strichartz estimates on closed manifolds, aiding in proving well-posedness for nonlinear Schrödinger equations.

Contribution

It introduces a new bilinear estimate for oscillatory integrals with different asymptotic parameters and uses it to improve Strichartz estimates on closed manifolds.

Findings

Proved a bilinear $L^2 imes L^2 o L^2$ estimate for oscillatory integrals.

Derived bilinear refinements to Strichartz estimates on closed manifolds.

Enabled progress towards global well-posedness results for nonlinear Schrödinger equations.

Abstract

We prove a bilinear $L^2(\R^d) \times L^2(\R^d) \to L^2(\R^{d+1})$ 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 $\R^d$, 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.