Abstract
We show that $\mathrm{lim}_{t\to 0} e^{it\Delta}f(x) = f(x)$ almost everywhere for all $f \in H^s(\mathbb{R}^2)$ provided that $s>1/3$. This result is sharp up to the endpoint. The proof uses polynomial partitioning and decoupling.
Citation
Xiumin Du. Larry Guth. Xiaochun Li. "A sharp Schrödinger maximal estimate in $\mathbb{R}^2$." Ann. of Math. (2) 186 (2) 607 - 640, September 2017. https://doi.org/10.4007/annals.2017.186.2.5
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