On sharp bilinear Strichartz estimates of Ozawa-Tsutsumi type

Jonathan Bennett, Neal Bez, Chris Jeavons, Nikolaos Pattakos

Research output: Contribution to journalArticlepeer-review

6 Citations (Scopus)
106 Downloads (Pure)

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 languageEnglish
Pages (from-to)459-476
JournalJournal of the Mathematical Society of Japan
Volume69
Issue number2
Early online date12 May 2017
DOIs
Publication statusPublished - 2017

Keywords

  • bilinear estimates
  • Schrödinger equation
  • sharp constants

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