A sharp k-plane Strichartz inequality for the Schrödinger equation

We prove that ‖X(|u|²)‖_L3t,ℓ≤ C‖f‖²_L2(ℝ2), 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 obtainsharp L²(µ) bounds on Fourier extension operators associated with certain high-dimensional spheres, involving measures µ supported on natural “co-k-planarity” sets.