Geodesic
Sharp $L^2$ estimates of the Schrödinger maximal function in higher dimensions
Xiumin Du1 ; Ruixiang Zhang2
1 University of Maryland, College Park, MD
2 University of Wisconsin-Madison, Madison, WI
Annals of mathematics, Tome 189 (2019) no. 3, pp. 837-861

Voir la notice de l'article provenant de la source Annals of Mathematics website

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 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.

DOI : 10.4007/annals.2019.189.3.4

Xiumin Du 1 ; Ruixiang Zhang 2

1 University of Maryland, College Park, MD
2 University of Wisconsin-Madison, Madison, WI 
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Xiumin Du; Ruixiang Zhang. Sharp $L^2$ estimates of the Schrödinger maximal function in higher dimensions. Annals of mathematics, Tome 189 (2019) no. 3, pp. 837-861. doi: 10.4007/annals.2019.189.3.4

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