A sharp inequality for the Strichartz norm

Abstract

Let $u:\R \times \R^n \to \C$ be the solution of the linear Schrödinger equation $i{u}_t+\Delta u=0$ with initial data $u(0,x)=f(x)$. In the first part of this paper we obtain a sharp inequality for the Strichartz norm $\Vert u(t,x){\Vert }_{L_t^{2k}{L}_x^{2k}(\mathbb{R}\times {\mathbb{R}}^n)}$, where $k\in \mathbb{Z}$, $k\ge 2$ and $(n,k)\ne (1,2)$, 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 Schrödinger and wave equations in the broader setting of sharp restriction/extension estimates for the paraboloid and the cone.

15 pages. Submitted

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