A sharp k-plane Strichartz inequality for the Schrödinger equation
Research output: Contribution to journal › Article › peer-review
Colleges, School and Institutes
We prove that ‖X(|u|2)‖L3t,ℓ≤ C‖f‖2L2(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.
|Number of pages||17|
|Journal||Transactions of the American Mathematical Society|
|Publication status||Published - 20 Mar 2018|