A Sharp Inequality for the Strichartz Norm

Let
$u:\mathbb R\times\mathbb R^n\rightarrow\mathbb C$
be the solution of the linear Schrödinger equation
$$\begin{cases}iu_t +\Delta u=0\\\ \ \ \, u(0,x)=f(x)\end{cases}$$
In the first part of this paper, we obtain a sharp inequality for the Strichartz norm
$\parallel u(t,x)\parallel_{L^{2k}_t\ L^{2k}_x(\mathbb R\times\mathbb R^n)}$
, where
$k\in \mathbb Z, k\geq 2$
and
$(n,k)\not=(1,2)$
, that admits only Gaussian maximizers. As corollaries, we obtain sharp forms of the classical Strichartz inequalities in low dimensions (works of Foschi [4] and Hundertmark–Zharnitsky [6]) and also sharp forms of some Sobolev–Strichartz inequalities. In the second part of the paper, we express Foschi's [4] sharp inequalities for the Schrödinger and wave equations in the broader setting of sharp restriction/extension estimates for the paraboloid and the cone.