A SHARP k-PLANE STRICHARTZ INEQUALITY FOR THE SCHRÖDINGER EQUATION

We prove that || X ( | u | 2 ) | | L i , l 3 ≤ C || f | | L 2 ( R 2 ) 2 , where u(x, t) is the solution to the linear time-dependent Schrödinger equation on R2 with initial datum f and X is the (spatial) X-ray transform on R2. In particular, we identify the best constant C and show that a datum f is an extremiser if and only if it is a gaussian. We also establish bounds of this type in higher dimensions d, where the X-ray transform is replaced by the k-plane transform for any 1 ≤ k ≤ d − 1. In the process we obtain sharp L2(μ) bounds on Fourier extension operators associated with certain high-dimensional spheres involving measures μ supported on natural “co-k-planarity” sets.