A Sharp Inequality for the Strichartz Norm

Let u : R x R-n -> C be the solution of the linear Schrodinger equation {u((0,x) = f(x).)(iut + Delta u = 0) In the first part of this paper, we obtain a sharp inequality for the Strichartz norm parallel to u(t, x)parallel to (LtLx2k)-L-2k(R x R-n), where k is an element of Z, k >= 2, and (n, k) not equal (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 Schrodinger and wave equations in the broader setting of sharp restriction/extension estimates for the paraboloid and the cone.