A sharp Schrödinger 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.

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Authors

Xiumin Du

University of Illinois at Urbana-Champaign, Urbana IL

Current address:

Institute for Advanced Study, Princeton, NJ Larry Guth

Massachusetts Institute of Technology, Cambridge, MA

Xiaochun Li

University of Illinois at Urbana-Champaign, Urbana IL