2026-09-22T03:29:03Z https://www.ideals.illinois.edu/oai-pmh

oai:www.ideals.illinois.edu:2142/101570 2023-07-11 col_2142_5131 col_2142_16340 com_2142_5130 com_2142_16339 com_2142_8903

Junge, Marius Ruan, Zhong-Jin Boca, Florin Oikhberg, Timur Zhao, Mingyu 2018-09-27T16:17:49Z 2018-09-27T16:17:49Z 2018-07-12 2018-08 "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 $\rx_{\ta}^n$." Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2018-09-27 without embargo terms The student, Mingyu Zhao, accepted the attached license on 2018-07-12 at 03:40. The student, Mingyu Zhao, submitted this Dissertation for approval on 2018-07-12 at 03:52. This Dissertation was approved for publication on 2018-07-12 at 11:13. DSpace SAF Submission Ingestion Package generated from Vireo submission #12846 on 2018-09-27 at 10:47:59 Made available in DSpace on 2018-09-27T16:17:49Z (GMT). No. of bitstreams: 2 ZHAO-DISSERTATION-2018.pdf: 612055 bytes, checksum: 44bd1eb629eeb4e5abb3e18e7b4463a5 (MD5) LICENSE.txt: 4208 bytes, checksum: 255a166d1be3e7404111465d78d7dc04 (MD5) Previous issue date: 2018-07-12 application/pdf http://hdl.handle.net/2142/101570 en Copyright 2018 by Mingyu Zhao. All rights reserved. harmonic analysis, Hardy-Littlewood-Sobolev inequalities, Functional analysis, Operator space, Operator algebras, non-commutaive $L_p$ spaces Smoothing estimates for non commutative spaces text text Mathematics Mathematics University of Illinois at Urbana-Champaign Dissertation Ph.D.