{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/132507"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/132507","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Riesz capacity: Hausdorff measure and extremal ratios","abstract":"Riesz capacity measures the size of a set in $\\mathbb{R}^n$ in terms of a pairwise interaction kernel $|x-y|^{-p}$ with exponent $p In the first part of the dissertation, the decay rate of Riesz capacity as the exponent $p$ increases to $n$ is shown to yield the Hausdorff measure of the set. The result applies to strongly rectifiable sets, and so in particular to submanifolds of Euclidean space. For strictly self-similar fractals, a one-sided decay estimate is found. How does the capacity change when the exponent $p$ increases? A longstanding conjecture by P\\'olya and Szeg\\H{o} claims that the ball maximizes the ratio of $q$-capacity over $p$-capacity when $q>p>0$. In the second part of the dissertation, we investigate the capacity ratio when $p","abstract_html":"Riesz capacity measures the size of a set in <span class=\"etd-inline-math\">\\mathbb{R}<sup>n</sup></span> in terms of a pairwise interaction kernel <span class=\"etd-inline-math\">|x-y|<sup>-p</sup></span> with exponent $p In the first part of the dissertation, the decay rate of Riesz capacity as the exponent $p$ increases to $n$ is shown to yield the Hausdorff measure of the set. The result applies to strongly rectifiable sets, and so in particular to submanifolds of Euclidean space. For strictly self-similar fractals, a one-sided decay estimate is found. How does the capacity change when the exponent $p$ increases? A longstanding conjecture by P\\&#x27;olya and Szeg\\H{o} claims that the ball maximizes the ratio of $q$-capacity over $p$-capacity when $q&gt;p&gt;0$. In the second part of the dissertation, we investigate the capacity ratio when $p","abstract_has_math":true,"creators":["Fan, Qiuling"],"institution":"University of Illinois Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Laugesen, Richard","Tyson, Jeremy","Song, Renming","Zharnitsky, Vadim"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2025,"date_issued":"2025-12","date_published":"2025-12","updated_at":"2026-07-22T22:25:07Z","subjects":["Riesz capacity","Hausdorff measure","extremal ratios"],"languages":["en"],"rights":["Copyright 2025 Qiuling Fan"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://hdl.handle.net/2142/132507","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Laugesen, Richard","Tyson, Jeremy","Song, Renming","Zharnitsky, Vadim"]},{"key":"dc:creator","label":"Author","values":["Fan, Qiuling"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2025-12","2025-12-01"]},{"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 Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Riesz capacity","Hausdorff measure","extremal ratios"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 2025 Qiuling Fan"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://hdl.handle.net/2142/132507"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["Riesz capacity measures the size of a set in $\\mathbb{R}^n$ in terms of a pairwise interaction kernel $|x-y|^{-p}$ with exponent $p In the first part of the dissertation, the decay rate of Riesz capacity as the exponent $p$ increases to $n$ is shown to yield the Hausdorff measure of the set. The result applies to strongly rectifiable sets, and so in particular to submanifolds of Euclidean space. For strictly self-similar fractals, a one-sided decay estimate is found. How does the capacity change when the exponent $p$ increases? A longstanding conjecture by P\\'olya and Szeg\\H{o} claims that the ball maximizes the ratio of $q$-capacity over $p$-capacity when $q>p>0$. In the second part of the dissertation, we investigate the capacity ratio when $p","Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2026-02-19 without embargo terms","The student, Qiuling Fan, accepted the attached license on 2025-11-19 at 04:41.","The student, Qiuling Fan, submitted this Dissertation for approval on 2025-11-19 at 04:47.","This Dissertation was approved for publication on 2025-12-01 at 14:40.","DSpace SAF Submission Ingestion Package generated from Vireo submission #22891 on 2026-02-19 at 18:24:59"]},{"key":"dc:format","label":"Dc Format","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Riesz capacity: Hausdorff measure and extremal ratios"]}]}],"canonical_facts":{"dc:contributor":["Laugesen, Richard","Tyson, Jeremy","Song, Renming","Zharnitsky, Vadim"],"dc:creator":["Fan, Qiuling"],"dc:date":["2025-12","2025-12-01"],"dc:description":["Riesz capacity measures the size of a set in $\\mathbb{R}^n$ in terms of a pairwise interaction kernel $|x-y|^{-p}$ with exponent $p In the first part of the dissertation, the decay rate of Riesz capacity as the exponent $p$ increases to $n$ is shown to yield the Hausdorff measure of the set. The result applies to strongly rectifiable sets, and so in particular to submanifolds of Euclidean space. For strictly self-similar fractals, a one-sided decay estimate is found. How does the capacity change when the exponent $p$ increases? A longstanding conjecture by P\\'olya and Szeg\\H{o} claims that the ball maximizes the ratio of $q$-capacity over $p$-capacity when $q>p>0$. In the second part of the dissertation, we investigate the capacity ratio when $p","Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2026-02-19 without embargo terms","The student, Qiuling Fan, accepted the attached license on 2025-11-19 at 04:41.","The student, Qiuling Fan, submitted this Dissertation for approval on 2025-11-19 at 04:47.","This Dissertation was approved for publication on 2025-12-01 at 14:40.","DSpace SAF Submission Ingestion Package generated from Vireo submission #22891 on 2026-02-19 at 18:24:59"],"dc:format":["application/pdf"],"dc:identifier":["https://hdl.handle.net/2142/132507"],"dc:language":["en"],"dc:rights":["Copyright 2025 Qiuling Fan"],"dc:subject":["Riesz capacity","Hausdorff measure","extremal ratios"],"dc:title":["Riesz capacity: Hausdorff measure and extremal ratios"],"dc:type":["text","Thesis"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois Urbana-Champaign"]},"updated_at":"2026-07-22T22:25:07Z"}