Abstract
We provide a comprehensive analysis of sharp bilinear estimates of Ozawa-Tsutsumi type for solutions u of the free Schr$\"o$dinger 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.
Original language | English |
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Pages (from-to) | 459-476 |
Journal | Journal of the Mathematical Society of Japan |
Volume | 69 |
Issue number | 2 |
Early online date | 12 May 2017 |
DOIs | |
Publication status | Published - 2017 |
Bibliographical note
17 pages, various improvements to the exposition and resultsKeywords
- bilinear estimates
- Schrödinger equation
- sharp constants