Smoothing estimates for non commutative spaces

"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_\taⁿ."

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2018

Author dc:creator

• Zhao, Mingyu

Contributors dc:contributor

• Junge, Marius
• Ruan, Zhong-Jin
• Boca, Florin
• Oikhberg, Timur

• harmonic analysis, Hardy-Littlewood-Sobolev inequalities, Functional analysis, Operator space, Operator algebras, non-commutaive $L_p$ spaces

Statement dc:rights

• Copyright 2018 by Mingyu Zhao. All rights reserved.

Language dc:language

en

Handle dc:identifier

http://hdl.handle.net/2142/101570

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/101570