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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:Thesis
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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