## Abstract

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 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 obtainsharp L^{2}(µ) bounds on Fourier extension operators associated with certain high-dimensional spheres, involving measures µ supported on natural “co-k-planarity” sets.Original language | English |
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Pages (from-to) | 5617-5633 |

Number of pages | 17 |

Journal | Transactions of the American Mathematical Society |

Volume | 370 |

Issue number | 8 |

DOIs | |

Publication status | Published - 20 Mar 2018 |

## Keywords

- math.CA

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