A sharp inequality for the Strichartz norm

Details A sharp inequality for the Strichartz norm A sharp inequality for the Strichartz norm

2008-09-23

arxiv.org/abs/0809.4054v1

International Mathematics Research Notices, p. 3127-3145, 2009

Mathematics

Analysis of PDEs

15 pages. Submitted

Scientific paper

10.1093/imrn/rnp045

Let $u:\R \times \R^n \to \C$ be the solution of the linear Schr\"odinger equation $iu_t + \Delta u =0$ with initial data $u(0,x) = f(x)$. In the first part of this paper we obtain a sharp inequality for the Strichartz norm $\|u(t,x)\|_{L^{2k}_tL^{2k}_x(\R \times\R^n)}$, where $k\in \Z$, $k \geq 2$ and $(n,k) \neq (1,2)$, that admits only Gaussian maximizers. As corollaries we obtain sharp forms of the classical Strichartz inequalities in low dimensions (works of Foschi and Hundertmark - Zharnitsky) and also sharp forms of some Sobolev-Strichartz inequalities. In the second part of the paper we express Foschi's sharp inequalities for the Schr\"odinger and wave equations in the broader setting of sharp restriction/extension estimates for the paraboloid and the cone.

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