#### Vol. 5, No. 2, 2012

 Recent Issues
 The Journal About the Journal Editorial Board Subscriptions Editors’ Interests Scientific Advantages Submission Guidelines Submission Form Editorial Login Ethics Statement ISSN: 1948-206X (e-only) ISSN: 2157-5045 (print) Author Index To Appear Other MSP Journals
A bilinear oscillatory integral estimate and bilinear refinements to Strichartz estimates on closed manifolds

### Zaher Hani

Vol. 5 (2012), No. 2, 339–363
##### Abstract

We prove a bilinear ${L}^{2}\left({ℝ}^{d}\right)×{L}^{2}\left({ℝ}^{d}\right)\to {L}^{2}\left({ℝ}^{d+1}\right)$ 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.

##### Keywords
bilinear oscillatory integrals, bilinear Strichartz estimates, transversality, semiclassical time scale, nonlinear Schrödinger equation on compact manifolds
##### Mathematical Subject Classification 2000
Primary: 35B45, 42B20, 58J40
Secondary: 35A17, 35S30