A sharp k-plane Strichartz inequality for the Schrödinger equation

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Colleges, School and Institutes

Abstract

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.

Details

Original languageEnglish
Pages (from-to)5617-5633
Number of pages17
JournalTransactions of the American Mathematical Society
Volume370
Issue number8
Publication statusPublished - 20 Mar 2018

Keywords

  • math.CA