Vol. 5, No. 2, 2012

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ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
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 L2(d) × L2(d) L2(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 derived by Bourgain on d, but at a relevant semiclassical scale. These estimates will be employed elsewhere to prove global well-posedness below H1 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
Milestones
Received: 16 August 2010
Revised: 13 January 2011
Accepted: 13 February 2011
Published: 27 August 2012
Authors
Zaher Hani
Mathematics Department
University of California, Los Angeles
520 Portola Plaza, Math Sciences Building
Los Angeles, CA 90095
United States
http://www.math.ucla.edu/~zhani