Sharp L2 estimates of the Schrödinger maximal function in higher dimensions

By Xiumin Du, Ruixiang Zhang. Annals of Mathematics, 2018

214 citations recorded by the source corpus.

We show that, for $n\geq 3$, $\lim_{t \to 0} e^{it\Delta}f(x) = f(x)$ holds almost everywhere for all $f \in H^s (\mathbb{R}^n)$ provided that $s>\frac{n}{2(n+1)}$. 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 divergence set of Schr\"odinger 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.

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/9c157c03a65f096e2babd9f679c86c633927834c

Citations in the corpus

• On Falconer’s distance set problem in the plane (2018, unknown): “The estimates for d ě 4 were further improved by Du and Zhang in [8].”