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 Title: Smoothing estimates for non commutative spaces Author(s): Zhao, Mingyu Director of Research: Junge, Marius Doctoral Committee Chair(s): Ruan, Zhong-Jin Doctoral Committee Member(s): Boca, Florin; Oikhberg, Timur Department / Program: Mathematics Discipline: Mathematics Degree Granting Institution: University of Illinois at Urbana-Champaign Degree: Ph.D. Genre: Dissertation Subject(s): harmonic analysis, Hardy-Littlewood-Sobolev inequalities, Functional analysis, Operator space, Operator algebras, non-commutaive $L_p$ spaces Abstract: In the first part of this thesis, we follow Varopoulos's perspective to establish the noncommutaive Sobolev inequaties (namely, Hardy-Littlewood-Sobolev inequalites), and extend the Sobolev embedding from noncommutative $L_p$ spaces to general Orlicz function spaces related with Cowling and Meda's work. Also we will show some examples to illustract the relation between the Orlicz function, dispersive estimate on semigroup $T_t$ and general resolvent formula on the generator $A$ of the semigroup (i.e. $Ax= \lim_{t\rightarrow 0} \frac{T_t x - x}{t}$). And we prove a borderline case of noncommutaive Sobolev inequality, namely the noncommutative Trudinger Moser's inequality. The focus of the second part of the thesis is the completely bounded version of noncommutative Sobolev inequalities. We prove a cb version of the Sobolev inequality for noncommutative $L_p$ spaces. As a tool, we further develop a general embedding theory for von Neumann algebra, continuing the work for \cite{junge2010mixed}. Finally we prove the cb version of Varopolous's theorem and provide some examples and applications. The third part of the thesis proves the existence of abstract Strichartz estimates on $\rx_{\ta}$ for operators that satisfies ultracontractivity and energy estimate. And we show the abstract Strichartz estimates are applicable to the Schr\"{o}dinger equation problem on quantum Euclidean spaces $\rx_{\ta}^n$. Issue Date: 2018-07-12 Type: Text URI: http://hdl.handle.net/2142/101570 Rights Information: Copyright 2018 by Mingyu Zhao. All rights reserved. Date Available in IDEALS: 2018-09-27 Date Deposited: 2018-08
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