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Description
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 |
This item appears in the following Collection(s)
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Dissertations and Theses - Mathematics
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Graduate Dissertations and Theses at Illinois
Graduate Theses and Dissertations at Illinois