Sharp L2 estimates of the Schrödinger maximal function in higher dimensions

Xiumin Du, Ruixiang Zhang

Research output: Contribution to journalArticlepeer-review

84 Scopus citations


We show that, for n≥3, limt→0eitΔf(x)=f(x) holds almost everywhere for all f∈Hs(ℝn) 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 L2 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 languageEnglish (US)
Pages (from-to)837-861
Number of pages25
JournalAnnals of Mathematics
Issue number3
StatePublished - May 1 2019
Externally publishedYes


  • 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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