TY - JOUR

T1 - Sharp L2 estimates of the Schrödinger maximal function in higher dimensions

AU - Du, Xiumin

AU - Zhang, Ruixiang

N1 - Funding Information:
The material is based upon work supported by the National Science Foundation under Grant No. DMS-1638352, the Shiing-Shen Chern Fund and the James D. Wolfensohn Fund while the authors were in residence at the Institute for Advanced Study during the academic year 2017–2018.
Publisher Copyright:
© 2019 Annals of Mathematics.

PY - 2019/5/1

Y1 - 2019/5/1

N2 - 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.

AB - 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.

KW - Decoupling

KW - Fourier restriction

KW - Refined Strichartz

KW - Schrödinger equation

KW - Schrödinger maximal function

KW - Weighted restriction

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U2 - 10.4007/annals.2019.189.3.4

DO - 10.4007/annals.2019.189.3.4

M3 - Article

AN - SCOPUS:85066924677

VL - 189

SP - 837

EP - 861

JO - Annals of Mathematics

JF - Annals of Mathematics

SN - 0003-486X

IS - 3

ER -