{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/20716"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/20716","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Some sharp inequalities for conditionally symmetric martingales","abstract":"Let f be a conditionally symmetric martingale taking values in a Hilbert space $\\rm I\\!H$ and let S(f) be its square function. If $\\nu\\sb{\\rm p}$ is the smallest positive zero of the confluent hypergeometric function and $\\mu\\sb{\\rm p}$ is the largest positive zero of the parabolic cylinder function of parameter p, then(UNFORMATTED TABLE OR EQUATION FOLLOWS)$$\\eqalign{\\rm\\Vert f\\Vert\\sb{p} \\leq \\nu\\sb{p}\\Vert S(f)\\Vert\\sb{p}\\quad&\\rm if\\quad 0 < p \\le 2,\\cr\\rm\\Vert f\\Vert\\sb{p} \\leq \\mu\\sb{p}\\Vert S(f)\\Vert\\sb{p}\\quad&\\rm if\\quad p \\ge 3,\\cr\\rm\\nu\\sb{p}\\Vert S(f)\\Vert\\sb{p} \\leq \\Vert f\\Vert\\sb{p}\\quad&\\rm if\\quad p \\geq 2,\\cr}$$(TABLE/EQUATION ENDS)and the above inequalities are sharp.","abstract_html":"Let f be a conditionally symmetric martingale taking values in a Hilbert space $\\rm I\\!H$ and let S(f) be its square function. If $\\nu\\sb{\\rm p}$ is the smallest positive zero of the confluent hypergeometric function and <span class=\"etd-inline-math\">&mu;\\sb{\\rm p}</span> is the largest positive zero of the parabolic cylinder function of parameter p, then(UNFORMATTED TABLE OR EQUATION FOLLOWS)$<span class=\"etd-inline-math\">\\eqalign{\\rm\\Vert f\\Vert\\sb{p} \\leq \\nu\\sb{p}\\Vert S(f)\\Vert\\sb{p}\\quad&amp;\\rm if\\quad 0 &lt; p \\le 2,\\cr\\rm\\Vert f\\Vert\\sb{p} \\leq &mu;\\sb{p}\\Vert S(f)\\Vert\\sb{p}\\quad&amp;\\rm if\\quad p \\ge 3,\\cr\\rm\\nu\\sb{p}\\Vert S(f)\\Vert\\sb{p} \\leq \\Vert f\\Vert\\sb{p}\\quad&amp;\\rm if\\quad p \\geq 2,\\cr}</span>$(TABLE/EQUATION ENDS)and the above inequalities are sharp.","abstract_has_math":true,"creators":["Wang, Gang"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2011,"date_issued":"2011-05-07T12:47:14Z","date_published":"2011-05-07T12:47:14Z","updated_at":"2026-07-22T22:25:16Z","subjects":["Mathematics"],"languages":["eng"],"rights":["Copyright 1989 Wang, Gang"],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["AAI8924965","(UMI)AAI8924965"],"render_values":[{"text":"AAI8924965","href":null,"code":true},{"text":"(UMI)AAI8924965","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/20716","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["Wang, Gang"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2011-05-07T12:47:14Z","10000-01-01","1989"]},{"key":"dc:type","label":"Dc Type","values":["text"]},{"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":["Mathematics"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["eng"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 1989 Wang, Gang"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["AAI8924965","(UMI)AAI8924965","http://hdl.handle.net/2142/20716"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["Let f be a conditionally symmetric martingale taking values in a Hilbert space $\\rm I\\!H$ and let S(f) be its square function. If $\\nu\\sb{\\rm p}$ is the smallest positive zero of the confluent hypergeometric function and $\\mu\\sb{\\rm p}$ is the largest positive zero of the parabolic cylinder function of parameter p, then(UNFORMATTED TABLE OR EQUATION FOLLOWS)$$\\eqalign{\\rm\\Vert f\\Vert\\sb{p} \\leq \\nu\\sb{p}\\Vert S(f)\\Vert\\sb{p}\\quad&\\rm if\\quad 0 < p \\le 2,\\cr\\rm\\Vert f\\Vert\\sb{p} \\leq \\mu\\sb{p}\\Vert S(f)\\Vert\\sb{p}\\quad&\\rm if\\quad p \\ge 3,\\cr\\rm\\nu\\sb{p}\\Vert S(f)\\Vert\\sb{p} \\leq \\Vert f\\Vert\\sb{p}\\quad&\\rm if\\quad p \\geq 2,\\cr}$$(TABLE/EQUATION ENDS)and the above inequalities are sharp.","Made available in DSpace on 2011-05-07T12:47:14Z (GMT). No. of bitstreams: 2 license.txt: 4922 bytes, checksum: 910b249b4beec47e7ab768910c8f966f (MD5) 8924965.pdf: 2010283 bytes, checksum: 46249149966b414fd25d3e8259eefadb (MD5) Previous issue date: 1989","Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Howard Ding (hding2@illinois.edu) on 2011-05-07T14:45:47Z Item is restricted indefinitely.","Restriction data tranferred 2014-07-01T11:20:21-05:00 Original Data Group with Access UIUC Users [automated] Release Date: none Reason: ETDs are only available to UIUC Users without author permission","ETDs are only available to UIUC Users without author permission","U of I Only"]},{"key":"dc:title","label":"Title","values":["Some sharp inequalities for conditionally symmetric martingales"]}]}],"canonical_facts":{"dc:creator":["Wang, Gang"],"dc:date":["2011-05-07T12:47:14Z","10000-01-01","1989"],"dc:description":["Let f be a conditionally symmetric martingale taking values in a Hilbert space $\\rm I\\!H$ and let S(f) be its square function. If $\\nu\\sb{\\rm p}$ is the smallest positive zero of the confluent hypergeometric function and $\\mu\\sb{\\rm p}$ is the largest positive zero of the parabolic cylinder function of parameter p, then(UNFORMATTED TABLE OR EQUATION FOLLOWS)$$\\eqalign{\\rm\\Vert f\\Vert\\sb{p} \\leq \\nu\\sb{p}\\Vert S(f)\\Vert\\sb{p}\\quad&\\rm if\\quad 0 < p \\le 2,\\cr\\rm\\Vert f\\Vert\\sb{p} \\leq \\mu\\sb{p}\\Vert S(f)\\Vert\\sb{p}\\quad&\\rm if\\quad p \\ge 3,\\cr\\rm\\nu\\sb{p}\\Vert S(f)\\Vert\\sb{p} \\leq \\Vert f\\Vert\\sb{p}\\quad&\\rm if\\quad p \\geq 2,\\cr}$$(TABLE/EQUATION ENDS)and the above inequalities are sharp.","Made available in DSpace on 2011-05-07T12:47:14Z (GMT). No. of bitstreams: 2 license.txt: 4922 bytes, checksum: 910b249b4beec47e7ab768910c8f966f (MD5) 8924965.pdf: 2010283 bytes, checksum: 46249149966b414fd25d3e8259eefadb (MD5) Previous issue date: 1989","Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Howard Ding (hding2@illinois.edu) on 2011-05-07T14:45:47Z Item is restricted indefinitely.","Restriction data tranferred 2014-07-01T11:20:21-05:00 Original Data Group with Access UIUC Users [automated] Release Date: none Reason: ETDs are only available to UIUC Users without author permission","ETDs are only available to UIUC Users without author permission","U of I Only"],"dc:identifier":["AAI8924965","(UMI)AAI8924965","http://hdl.handle.net/2142/20716"],"dc:language":["eng"],"dc:rights":["Copyright 1989 Wang, Gang"],"dc:subject":["Mathematics"],"dc:title":["Some sharp inequalities for conditionally symmetric martingales"],"dc:type":["text"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:25:16Z"}