Journal of the Mathematical Society of Japan

On sharp bilinear Strichartz estimates of Ozawa–Tsutsumi type

Jonathan BENNETT, Neal BEZ, Chris JEAVONS, and 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 Schrödinger equation, which give sharp control on $|u|^2$ in classical Sobolev spaces. In particular, we generalise 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. The extremal functions are solutions of the Maxwell–Boltzmann functional equation and hence Gaussian. For $u^2$ we argue that the natural analogous results involve certain dispersive Sobolev norms; in particular, despite the validity of the classical Ozawa–Tsutsumi estimates for both $|u|^2$ and $u^2$ in the classical Sobolev spaces, we show that Gaussians are not extremisers in the latter case for spatial dimensions strictly greater than two.

Article information

Source
J. Math. Soc. Japan Volume 69, Number 2 (2017), 459-476.

Dates
First available in Project Euclid: 20 April 2017

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1492653636

Digital Object Identifier
doi:10.2969/jmsj/06920459

Subjects
Primary: 35B45: A priori estimates
Secondary: 35Q40: PDEs in connection with quantum mechanics

Keywords
bilinear estimates Schrödinger equation sharp constants

Citation

BENNETT, Jonathan; BEZ, Neal; JEAVONS, Chris; PATTAKOS, Nikolaos. On sharp bilinear Strichartz estimates of Ozawa–Tsutsumi type. J. Math. Soc. Japan 69 (2017), no. 2, 459--476. doi:10.2969/jmsj/06920459. https://projecteuclid.org/euclid.jmsj/1492653636


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