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arXiv · 1404.2466

On sharp bilinear Strichartz estimates of Ozawa-Tsutsumi type

Jonathan Bennett·Neal Bez·Chris Jeavons·Nikolaos Pattakos
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Abstract

We provide a comprehensive analysis of sharp bilinear estimates of Ozawa-Tsutsumi type for solutions u of the free Schr\"odinger equation, which give sharp control on $|u|^2$ in classical Sobolev spaces. In particular, we provide a generalization of their estimates in such a way that provides a unification with some sharp bilinear estimates proved by Carneiro and Planchon-Vega, via entirely different methods, by seeing them all as special cases of a one parameter family of sharp estimates. We show that the extremal functions are solutions of the Maxwell-Boltzmann functional equation and provide a new proof that this equation admits only Gaussian solutions. We also make a connection to certain sharp estimates on $u^2$ involving certain dispersive Sobolev norms.

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Jonathan Bennett, Neal Bez, Chris Jeavons, Nikolaos Pattakos. 2014-04-09. On sharp bilinear Strichartz estimates of Ozawa-Tsutsumi type. https://doi.org/10.2969/jmsj%2F06920459

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