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

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

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Xiumin Du, IAS
Fine Hall 314

Joint with Guth and Li, recently we showed that the solution to the free Schrödinger equation converges to its initial data almost everywhere, provided that the initial data is in the Sobolev space $H^s(\mathbb{R}^2)$ with $s>1/3$. This is sharp up to the endpoint, due to a counterexample by Bourgain. This pointwise convergence problem can be approached by estimates of Schrödinger maximal functions, which have some similar flavor as the Fourier restriction estimates. In this talk, I'll first explain quickly how to reduce the original estimate in three dimensions to an essentially two dimensional one, via polynomial partitioning method. Then we'll see that the reduced problem asks how to control the size of the solution on a sparse and spread-out set, and it can be solved by refined Strichartz estimates derived from l^2 decoupling theorem and induction on scales.