A sharp Schrödinger maximal estimate in R2

Abstract

We study the almost everywhere pointwise convergence of the solutions to Schrödinger equations in $\mathbb{R}^2$. It is shown 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. It comes from the following Schrödinger maximal estimate:

$$ \left\| \sup_{0 < t \leq 1} | e^{it \Delta} f| \right\|_{L^3(B(0,1))} \leq C_s \| f \|_{H^s(\mathbb{R}^2)}\,, $$

for any $s > 1/3$ and any function $f \in H^s(\mathbb{R}^2)$. The proof uses polynomial partitioning and decoupling.