## Journal of the Mathematical Society of Japan

### On sharp bilinear Strichartz estimates of Ozawa–Tsutsumi type

#### 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

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

#### 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

#### References

• K. Atkinson and W. Han,
• Spherical harmonics and approximations on the unit sphere: an introduction, Lecture Notes in Math., Springer, Heidelberg, 2012.
• J. Bennett, N. Bez, A. Carbery and D. Hundertmark, Heat-flow monotonicity of Strichartz norms, Anal. PDE, 2 (2009), 147–158.
• J. Bennett, N. Bez and M. Iliopoulou, Flow monotonicity and Strichartz inequalities, Int. Math. Res. Not., 19 (2015), 9415–9437.
• N. Bez, C. Jeavons and T. Ozawa, Some sharp bilinear space-time estimates for the wave equation, Mathematika, 62 (2016), 719–737.
• E. Carneiro, A sharp inequality for the Strichartz norm, Int. Math. Res. Not., 16 (2009), 3127–3145.
• J. Dou and M. Zhu, Reversed Hardy–Littlewood–Sobolev inequality, Int. Math. Res. Not., 19 (2015), 9696–9726.
• T. Duyckaerts, F. Merle and S. Roudenko, Maximizers for the Strichartz norm for small solutions of mass-critical NLS, Ann. Sc. Norm. Super. Pisa Cl. Sci., 10 (2011), 427–476.
• D. Foschi, Maximizers for the Strichartz inequality, J. Eur. Math. Soc., 9 (2007), 739–774.
• D. Foschi and S. Klainerman, Bilinear space-time estimates for homogeneous wave equations, Ann. Sci. École Norm. Sup., 33 (2000), 211–274.
• D. Hundertmark and V. Zharnitsky, On sharp Strichartz inequalities in low dimensions, Int. Math. Res. Not., (2006), Art. ID 34080, 18 pp.
• C. Jeavons, A sharp bilinear estimate for the Klein–Gordon equation in arbitrary space-time dimensions, Differential Integral Equations, 27 (2014), 137–156.
• J. Jiang and S. Shao, On characterisation of the sharp Strichartz inequality for the Schrödinger equation, Anal. PDE, 9 (2016), 353–361.
• S. Klainerman and M. Machedon, Remark on Strichartz-type inequalities, Int. Math. Res. Not., 5 (1996), 201–220.
• P. L. Lions, Compactness in Boltzmann's equation via Fourier integral operators and applications, I, J. Math. Kyoto Univ., 34 (1994), 391–427.
• T. Ozawa and Y. Tsutsumi, Space-time estimates for null gauge forms and nonlinear Schrödinger equations, Differential Integral Equations, 11 (1998), 201–222.
• B. Perthame, Introduction to the collision models in Boltzmann's theory, Modeling of Collisions, 33 (ed. P.-A. Raviart), Masson, Paris, 1997.
• F. Planchon and L. Vega, Bilinear virial identities and applications, Ann. Sci. Éc. Norm. Supér. (4), 42 (2009), 261–290.
• J. J. Rotman, An Introduction to Algebraic Topology, New York: Springer-Verlag, 1988.
• L. Vega, Bilinear virial identities and oscillatory integrals, Harmonic analysis and partial differential equations, Contemp. Math., 505, Amer. Math. Soc., Providence, RI, 2010, 219–232.
• C. Villani, Entropy methods for the Boltzmann equation, Lecture Notes in Math., 1916, Springer, Berlin, 2008, 1–70.