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

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

2010-08-17

arxiv.org/abs/1008.2827v3

Mathematics

Analysis of PDEs

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

Scientific paper

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.

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