{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/113210"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/113210","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Noncommutative Sobolev type inequalities","abstract":"In this thesis, we study Sobolev type inequalities in the noncommutative (quantum) settings. We establish the abstract Bakry-\\'Emery criterion for the operator-valued $f$-Sobolev inequalities on the derivation triple. We recapture the celebrated Bakry-\\'Emery theorem for operator-valued functions on Riemannian manifolds. We discuss examples including noncommutative $f$-Sobolev inequalities on the intervals, Lindblad operators of finite dimensional matrix algebras, and discrete graphs. By generalizing the monotone metrics in the space of quantum states, we develop a deeper understanding of $f$-Sobolev inequalities in the noncommutative setting.","abstract_html":"In this thesis, we study Sobolev type inequalities in the noncommutative (quantum) settings. We establish the abstract Bakry-\\'Emery criterion for the operator-valued $f$-Sobolev inequalities on the derivation triple. We recapture the celebrated Bakry-\\'Emery theorem for operator-valued functions on Riemannian manifolds. We discuss examples including noncommutative $f$-Sobolev inequalities on the intervals, Lindblad operators of finite dimensional matrix algebras, and discrete graphs. By generalizing the monotone metrics in the space of quantum states, we develop a deeper understanding of $f$-Sobolev inequalities in the noncommutative setting.","abstract_has_math":true,"creators":["Li, Haojian"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Junge, Marius","Boca, Florin","Oikhberg, Timur","Leditzky, Felix"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2022,"date_issued":"2022-01-12T22:35:19Z","date_published":"2022-01-12T22:35:19Z","updated_at":"2026-07-22T22:24:53Z","subjects":["Sobolev inequalities","noncommutative analysis","open quantum system"],"languages":["en"],"rights":["Copyright 2021 Haojian Li"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/2142/113210","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Junge, Marius","Boca, Florin","Oikhberg, Timur","Leditzky, Felix"]},{"key":"dc:creator","label":"Author","values":["Li, Haojian"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2022-01-12T22:35:19Z","2024-01-12T22:35:30Z","2021-07-16","2021-08"]},{"key":"dc:type","label":"Dc Type","values":["text","Thesis"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Sobolev inequalities","noncommutative analysis","open quantum system"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 2021 Haojian Li"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/113210"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["In this thesis, we study Sobolev type inequalities in the noncommutative (quantum) settings. We establish the abstract Bakry-\\'Emery criterion for the operator-valued $f$-Sobolev inequalities on the derivation triple. We recapture the celebrated Bakry-\\'Emery theorem for operator-valued functions on Riemannian manifolds. We discuss examples including noncommutative $f$-Sobolev inequalities on the intervals, Lindblad operators of finite dimensional matrix algebras, and discrete graphs. By generalizing the monotone metrics in the space of quantum states, we develop a deeper understanding of $f$-Sobolev inequalities in the noncommutative setting.","Submission published under a 24 month embargo labeled 'U of I Access', the embargo will last until 2023-08-01","The student, Haojian Li, accepted the attached license on 2021-07-15 at 19:32.","The student, Haojian Li, submitted this Dissertation for approval on 2021-07-15 at 19:37.","This Dissertation was approved for publication on 2021-07-16 at 13:22.","DSpace SAF Submission Ingestion Package generated from Vireo submission #16972 on 2022-01-12 at 12:55:25","Made available in DSpace on 2022-01-12T22:35:19Z (GMT). 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We establish the abstract Bakry-\\'Emery criterion for the operator-valued $f$-Sobolev inequalities on the derivation triple. We recapture the celebrated Bakry-\\'Emery theorem for operator-valued functions on Riemannian manifolds. We discuss examples including noncommutative $f$-Sobolev inequalities on the intervals, Lindblad operators of finite dimensional matrix algebras, and discrete graphs. By generalizing the monotone metrics in the space of quantum states, we develop a deeper understanding of $f$-Sobolev inequalities in the noncommutative setting.","Submission published under a 24 month embargo labeled 'U of I Access', the embargo will last until 2023-08-01","The student, Haojian Li, accepted the attached license on 2021-07-15 at 19:32.","The student, Haojian Li, submitted this Dissertation for approval on 2021-07-15 at 19:37.","This Dissertation was approved for publication on 2021-07-16 at 13:22.","DSpace SAF Submission Ingestion Package generated from Vireo submission #16972 on 2022-01-12 at 12:55:25","Made available in DSpace on 2022-01-12T22:35:19Z (GMT). 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