## Abstract

We show that, for n≥3, lim_{t→0}e^{itΔ}f(x)=f(x) holds almost everywhere for all f∈H^{s}(ℝ^{n}) provided that s > n/2(n+1). Due to a counterexample by Bourgain, up to the endpoint, this result is sharp and fully resolves a problem raised by Carleson. Our main theorem is a fractal L^{2} restriction estimate, which also gives improved results on the size of the divergence set of the Schrödinger solutions, the Falconer distance set problem and the spherical average Fourier decay rates of fractal measures. The key ingredients of the proof include multilinear Kakeya estimates, decoupling and induction on scales.

Original language | English (US) |
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Pages (from-to) | 837-861 |

Number of pages | 25 |

Journal | Annals of Mathematics |

Volume | 189 |

Issue number | 3 |

DOIs | |

State | Published - May 1 2019 |

Externally published | Yes |

## Keywords

- Decoupling
- Fourier restriction
- Refined Strichartz
- Schrödinger equation
- Schrödinger maximal function
- Weighted restriction

## ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty

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