{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/19178"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/19178","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Inequalities for the differential subordinates of Martingales, harmonic functions and Ito processes","abstract":"In Chapter 1 we sharpen Burkholder's inequality $\\mu(\\vert v\\vert\\geq1)\\leq2\\Vert u\\Vert\\sb1$ for two harmonic functions u and v by adjoining an extra assumption. That is, we prove the weak-type inequality $\\mu(\\vert v\\vert\\geq1)\\leq K\\Vert u\\Vert\\sb1$ under the assumptions that $\\vert v(\\xi)\\vert\\leq\\vert u(\\xi)\\vert, \\vert\\nabla v\\vert\\leq\\vert\\nabla u\\vert$ and the extra assumption that $\\nabla u\\cdot\\nabla v$ = 0. Here $\\mu$ is the harmonic measure with respect to $\\xi$ and the constant 1 $<K<$ 2, found by Davis, is the best constant in Kolmogorov's weak-type inequality for conjugate functions.","abstract_html":"In Chapter 1 we sharpen Burkholder&#x27;s inequality <span class=\"etd-inline-math\">&mu;(\\vert v\\vert\\geq1)\\leq2\\Vert u\\Vert\\sb1</span> for two harmonic functions u and v by adjoining an extra assumption. That is, we prove the weak-type inequality <span class=\"etd-inline-math\">&mu;(\\vert v\\vert\\geq1)\\leq K\\Vert u\\Vert\\sb1</span> under the assumptions that $\\vert v(\\xi)\\vert\\leq\\vert u(\\xi)\\vert, \\vert\\nabla v\\vert\\leq\\vert\\nabla u\\vert$ and the extra assumption that $\\nabla u\\cdot\\nabla v$ = 0. Here <span class=\"etd-inline-math\">&mu;</span> is the harmonic measure with respect to $\\xi$ and the constant 1 $&lt;K&lt;$ 2, found by Davis, is the best constant in Kolmogorov&#x27;s weak-type inequality for conjugate functions.","abstract_has_math":true,"creators":["Choi, Changsun"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Ruan, Zhong-Jin"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2011,"date_issued":"2011-05-07T11:59:20Z","date_published":"2011-05-07T11:59:20Z","updated_at":"2026-07-22T22:25:12Z","subjects":["Mathematics"],"languages":["eng"],"rights":["Copyright 1995 Choi, Changsun"],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["AAI9624313","(UMI)AAI9624313"],"render_values":[{"text":"AAI9624313","href":null,"code":true},{"text":"(UMI)AAI9624313","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/19178","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Ruan, Zhong-Jin"]},{"key":"dc:creator","label":"Author","values":["Choi, Changsun"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2011-05-07T11:59:20Z","10000-01-01","1995"]},{"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 1995 Choi, Changsun"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["AAI9624313","(UMI)AAI9624313","http://hdl.handle.net/2142/19178"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["In Chapter 1 we sharpen Burkholder's inequality $\\mu(\\vert v\\vert\\geq1)\\leq2\\Vert u\\Vert\\sb1$ for two harmonic functions u and v by adjoining an extra assumption. That is, we prove the weak-type inequality $\\mu(\\vert v\\vert\\geq1)\\leq K\\Vert u\\Vert\\sb1$ under the assumptions that $\\vert v(\\xi)\\vert\\leq\\vert u(\\xi)\\vert, \\vert\\nabla v\\vert\\leq\\vert\\nabla u\\vert$ and the extra assumption that $\\nabla u\\cdot\\nabla v$ = 0. Here $\\mu$ is the harmonic measure with respect to $\\xi$ and the constant 1 $<K<$ 2, found by Davis, is the best constant in Kolmogorov's weak-type inequality for conjugate functions.","In Chapter 2 we get norm inequalities. Let ($\\Omega,{\\cal F},P$) be a probability space with filtration (${\\cal F}\\sb{n}).$ Let f be a nonnegative submartingale and g be an adapted sequence. Let d be the difference sequence of f and e of $g{:} f\\sb{n}=\\sum\\limits\\sbsp{k=0}{n}\\ d\\sb{k}$ and $g\\sb{n}=\\sum\\limits\\sbsp{k=0}{n}\\ e\\sb{k}, n\\ge 0.$ We prove $\\Vert g\\Vert\\sb{p}\\le(r-1)\\Vert f\\Vert\\sb{p}$ under the assumption that $\\vert e\\sb{n}\\vert\\le\\vert d\\sb{n}\\vert$ for $n\\ge 0$ and $\\vert{\\rm I\\!E}(e\\sb{n}\\ \\mid\\ {\\cal F}\\sb{n-1})\\vert\\le\\alpha\\vert{\\rm I\\!E}(d\\sb{n}\\ \\mid\\ {\\cal F}\\sb{n-1})\\vert$ for $n\\ge 1.$ Here 0 $\\le\\alpha\\le$ 1, 1 $<p<\\infty$ are constants, $\\Vert f\\Vert\\sb{p}={\\rm sup}\\Vert f\\sb{n}\\Vert\\sb{p}$ and $r=\\max\\{(\\alpha+1)p,p/(p-1)\\} .$ We also get similar inequalities $\\Vert v\\Vert\\sb{p}\\le(r-1)\\Vert u\\Vert\\sb{p}$ and $\\Vert\\vert Y\\Vert\\vert\\sb{p}\\le(r-1)\\Vert\\vert X\\Vert\\vert\\sb{p}$ where u, v are smooth functions and X, Y are Ito processes.","Made available in DSpace on 2011-05-07T11:59:20Z (GMT). No. of bitstreams: 2 license.txt: 4922 bytes, checksum: 910b249b4beec47e7ab768910c8f966f (MD5) 9624313.pdf: 1508665 bytes, checksum: 319c96e239a6704c6759240300eeef2c (MD5) Previous issue date: 1995","Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Howard Ding (hding2@illinois.edu) on 2011-05-07T14:35:11Z Item is restricted indefinitely.","Restriction data tranferred 2014-07-01T11:13:46-05:00 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":["Inequalities for the differential subordinates of Martingales, harmonic functions and Ito processes"]}]}],"canonical_facts":{"dc:contributor":["Ruan, Zhong-Jin"],"dc:creator":["Choi, Changsun"],"dc:date":["2011-05-07T11:59:20Z","10000-01-01","1995"],"dc:description":["In Chapter 1 we sharpen Burkholder's inequality $\\mu(\\vert v\\vert\\geq1)\\leq2\\Vert u\\Vert\\sb1$ for two harmonic functions u and v by adjoining an extra assumption. That is, we prove the weak-type inequality $\\mu(\\vert v\\vert\\geq1)\\leq K\\Vert u\\Vert\\sb1$ under the assumptions that $\\vert v(\\xi)\\vert\\leq\\vert u(\\xi)\\vert, \\vert\\nabla v\\vert\\leq\\vert\\nabla u\\vert$ and the extra assumption that $\\nabla u\\cdot\\nabla v$ = 0. Here $\\mu$ is the harmonic measure with respect to $\\xi$ and the constant 1 $<K<$ 2, found by Davis, is the best constant in Kolmogorov's weak-type inequality for conjugate functions.","In Chapter 2 we get norm inequalities. Let ($\\Omega,{\\cal F},P$) be a probability space with filtration (${\\cal F}\\sb{n}).$ Let f be a nonnegative submartingale and g be an adapted sequence. Let d be the difference sequence of f and e of $g{:} f\\sb{n}=\\sum\\limits\\sbsp{k=0}{n}\\ d\\sb{k}$ and $g\\sb{n}=\\sum\\limits\\sbsp{k=0}{n}\\ e\\sb{k}, n\\ge 0.$ We prove $\\Vert g\\Vert\\sb{p}\\le(r-1)\\Vert f\\Vert\\sb{p}$ under the assumption that $\\vert e\\sb{n}\\vert\\le\\vert d\\sb{n}\\vert$ for $n\\ge 0$ and $\\vert{\\rm I\\!E}(e\\sb{n}\\ \\mid\\ {\\cal F}\\sb{n-1})\\vert\\le\\alpha\\vert{\\rm I\\!E}(d\\sb{n}\\ \\mid\\ {\\cal F}\\sb{n-1})\\vert$ for $n\\ge 1.$ Here 0 $\\le\\alpha\\le$ 1, 1 $<p<\\infty$ are constants, $\\Vert f\\Vert\\sb{p}={\\rm sup}\\Vert f\\sb{n}\\Vert\\sb{p}$ and $r=\\max\\{(\\alpha+1)p,p/(p-1)\\} .$ We also get similar inequalities $\\Vert v\\Vert\\sb{p}\\le(r-1)\\Vert u\\Vert\\sb{p}$ and $\\Vert\\vert Y\\Vert\\vert\\sb{p}\\le(r-1)\\Vert\\vert X\\Vert\\vert\\sb{p}$ where u, v are smooth functions and X, Y are Ito processes.","Made available in DSpace on 2011-05-07T11:59:20Z (GMT). No. of bitstreams: 2 license.txt: 4922 bytes, checksum: 910b249b4beec47e7ab768910c8f966f (MD5) 9624313.pdf: 1508665 bytes, checksum: 319c96e239a6704c6759240300eeef2c (MD5) Previous issue date: 1995","Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Howard Ding (hding2@illinois.edu) on 2011-05-07T14:35:11Z Item is restricted indefinitely.","Restriction data tranferred 2014-07-01T11:13:46-05:00 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":["AAI9624313","(UMI)AAI9624313","http://hdl.handle.net/2142/19178"],"dc:language":["eng"],"dc:rights":["Copyright 1995 Choi, Changsun"],"dc:subject":["Mathematics"],"dc:title":["Inequalities for the differential subordinates of Martingales, harmonic functions and Ito processes"],"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":"2026-07-22T22:25:12Z"}