A sharp Schrodinger maximal estimate in ${\mathbb{R}}^2$

Abstract

We show that ${\lim }_{t\to 0}{e}^{it\Delta }f(x)=f(x)$ 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.

29 pages, 4 figures, accepted for publication in the Annals of Math