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 Title: A sharp Schrödinger maximal estimate in R2 Author(s): Du, Xiumin Director of Research: Li, Xiaochun Doctoral Committee Chair(s): Erdogan, Burak Doctoral Committee Member(s): Boca, Florin; Laugesen, Richard Department / Program: Mathematics Discipline: Mathematics Degree Granting Institution: University of Illinois at Urbana-Champaign Degree: Ph.D. Genre: Dissertation Subject(s): Schrödinger maximal extimate Variations on the Strichartz inequality 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. Issue Date: 2017-06-20 Type: Text URI: http://hdl.handle.net/2142/98116 Rights Information: Copyright 2017 Xiumin Du Date Available in IDEALS: 2017-09-29 Date Deposited: 2017-08
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