Journal of the Mathematical Society of Japan
- J. Math. Soc. Japan
- Volume 69, Number 2 (2017), 459-476.
On sharp bilinear Strichartz estimates of Ozawa–Tsutsumi type
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.
J. Math. Soc. Japan, Volume 69, Number 2 (2017), 459-476.
First available in Project Euclid: 20 April 2017
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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