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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author = {Bourgain, Jean},
title = {Some new estimates on oscillatory integrals},
booktitle = {Essays on {F}ourier Analysis in Honor of {E}lias {M}. {S}tein},
venue = {{P}rinceton, {NJ},
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series = {Princeton Math. Ser.},
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author = {Bourgain, Jean},
title = {On the {S}chrödinger maximal function in higher dimension},
journal = {Tr. Mat. Inst. Steklova},
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number = {Ortogonal\cprime nye Ryady, Teoriya Priblizheniĭi Smezhnye Voprosy},
pages = {53--66},
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}