A sharp inequality for the Strichartz norm

TL;DR

This paper establishes a sharp inequality for the Strichartz norm of solutions to the linear Schrödinger equation, identifying Gaussian functions as maximizers and deriving sharp forms of classical inequalities and restriction estimates.

Contribution

It introduces a new sharp inequality for the Strichartz norm with Gaussian maximizers and extends classical results to broader restriction/extension settings.

Findings

Gaussian functions are the unique maximizers.

Derived sharp forms of classical Strichartz inequalities.

Extended inequalities to restriction/extension estimates for paraboloid and cone.

Abstract

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.