A sharp k-plane Strichartz inequality for the Schrödinger equation

Research output: Contribution to journalArticle

Colleges, School and Institutes

Abstract

We prove that kX(|u| 2 )kL3 t,ℓ ≤ Ckfk 2 L2(R2), where u(x, t) is the solution to the linear time-dependent Schr¨odinger equation on R 2 with initial datum f, and X is the (spatial) X-ray transform on R 2 . 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 L 2 (µ) bounds on Fourier extension operators associated with certain high-dimensional spheres, involving measures µ supported on natural “co-k-planarity” sets.

Bibliographic note

18 pages. Author accepted version. To appear in Trans. Amer. Math. Soc

Details

Original languageEnglish
Pages (from-to)5617-5633
Number of pages17
JournalTransactions of the American Mathematical Society
Volume370
Early online date20 Mar 2018
Publication statusE-pub ahead of print - 20 Mar 2018

Keywords

  • math.CA