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

Hani, Zaher

doi:10.2140/apde.2012.5.339

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

Zaher Hani

Vol. 5 (2012), No. 2, 339–363

DOI: 10.2140/apde.2012.5.339

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 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

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

Vol. 5, No. 2, 2012