{"id":{"repo_id":"alabama","oai_identifier":"oai:ir.ua.edu:123456789/8380"},"canonical_url":"https://search.dev.ndltd.org/etd/alabama/oai:ir.ua.edu:123456789/8380","repository":{"repo_id":"alabama","name":"University of Alabama","base_url":"https://ir-api.ua.edu/oai/request"},"display":{"title":"Weighted Norm Inequalities for the Maximal Operator on Variable Lebesgue Spaces Over Spaces of Homogeneous Type","abstract":"Given a space of homogeneous type $(X,\\mu,d)$, we prove strong-type weighted norm inequalities for the Hardy-Littlewood maximal operator over the variable exponent Lebesgue spaces $L^\\pp$. We prove that the variable Muckenhoupt condition $\\App$ is necessary and sufficient for the strong type inequality if $\\pp$ satisfies log-H\\\"older continuity conditions and $1 < p_- \\leq p_+ < \\infty$. Our results generalize to spaces of homogeneous type the analogous results in Euclidean space proved in [14].","abstract_html":"Given a space of homogeneous type <span class=\"etd-inline-math\">(X,&mu;,d)</span>, we prove strong-type weighted norm inequalities for the Hardy-Littlewood maximal operator over the variable exponent Lebesgue spaces <span class=\"etd-inline-math\">L<sup>\\</sup>pp</span>. We prove that the variable Muckenhoupt condition $\\App$ is necessary and sufficient for the strong type inequality if $\\pp$ satisfies log-H\\&quot;older continuity conditions and <span class=\"etd-inline-math\">1 &lt; p<sub>-</sub> \\leq p<sub>+</sub> &lt; \\infty</span>. Our results generalize to spaces of homogeneous type the analogous results in Euclidean space proved in [14].","abstract_has_math":true,"creators":["Cummings, Jeremy"],"institution":"University of Alabama Libraries","degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":null,"school":null,"contributors":["Ferguson, Tim","Moen, Kabe","Rodney, Scott"],"advisors":["Cruz-Uribe, David"],"committee_chairs":[],"committee_members":[],"year":2020,"date_issued":"2020","date_published":"2020","updated_at":"2026-07-27T18:44:16Z","subjects":["Harmonic analysis","Maximal operator","Variable Lebesgue Spaces","Weighted estimates"],"languages":["en_US","English"],"rights":["All rights reserved by the author unless otherwise indicated."],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier.other","label":"Dc Identifier Other","values":["http://purl.lib.ua.edu/176881","u0015_0000001_0003675","Cummings_alatus_0004M_14220"],"render_values":[{"text":"http://purl.lib.ua.edu/176881","href":"http://purl.lib.ua.edu/176881","code":true},{"text":"u0015_0000001_0003675","href":null,"code":true},{"text":"Cummings_alatus_0004M_14220","href":null,"code":true}]}]},"links":{"outbound_url":"https://ir.ua.edu/handle/123456789/8380","outbound_label":"Repository record","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Ferguson, Tim","Moen, Kabe","Rodney, Scott"]},{"key":"dc:contributor.advisor","label":"Advisor","values":["Cruz-Uribe, David"]},{"key":"dc:contributor.other","label":"Dc Contributor Other","values":["University of Alabama Tuscaloosa"]},{"key":"dc:creator","label":"Author","values":["Cummings, Jeremy"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2022-04-13T20:33:44Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2022-04-13T20:33:44Z"]},{"key":"dc:date.issued","label":"Date","values":["2020"]},{"key":"dc:publisher","label":"Institution","values":["University of Alabama Libraries"]},{"key":"dc:type","label":"Dc Type","values":["thesis","text"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Harmonic analysis","Maximal operator","Variable Lebesgue Spaces","Weighted estimates"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["English"]},{"key":"dc:language.iso","label":"Language (ISO)","values":["en_US"]},{"key":"dc:rights","label":"Dc Rights","values":["All rights reserved by the author unless otherwise indicated."]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.other","label":"Dc Identifier Other","values":["http://purl.lib.ua.edu/176881","u0015_0000001_0003675","Cummings_alatus_0004M_14220"]},{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://ir.ua.edu/handle/123456789/8380"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["Electronic Thesis or Dissertation"]},{"key":"dc:description.abstract","label":"Abstract","values":["Given a space of homogeneous type $(X,\\mu,d)$, we prove strong-type weighted norm inequalities for the Hardy-Littlewood maximal operator over the variable exponent Lebesgue spaces $L^\\pp$. We prove that the variable Muckenhoupt condition $\\App$ is necessary and sufficient for the strong type inequality if $\\pp$ satisfies log-H\\\"older continuity conditions and $1 < p_- \\leq p_+ < \\infty$. Our results generalize to spaces of homogeneous type the analogous results in Euclidean space proved in [14]."]},{"key":"dc:format.medium","label":"Dc Format Medium","values":["electronic"]},{"key":"dc:format.mimetype","label":"Dc Format Mimetype","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Weighted Norm Inequalities for the Maximal Operator on Variable Lebesgue Spaces Over Spaces of Homogeneous Type"]}]}],"canonical_facts":{"dc:contributor":["Ferguson, Tim","Moen, Kabe","Rodney, Scott"],"dc:contributor.advisor":["Cruz-Uribe, David"],"dc:contributor.other":["University of Alabama Tuscaloosa"],"dc:creator":["Cummings, Jeremy"],"dc:date.accessioned":["2022-04-13T20:33:44Z"],"dc:date.available":["2022-04-13T20:33:44Z"],"dc:date.issued":["2020"],"dc:description":["Electronic Thesis or Dissertation"],"dc:description.abstract":["Given a space of homogeneous type $(X,\\mu,d)$, we prove strong-type weighted norm inequalities for the Hardy-Littlewood maximal operator over the variable exponent Lebesgue spaces $L^\\pp$. We prove that the variable Muckenhoupt condition $\\App$ is necessary and sufficient for the strong type inequality if $\\pp$ satisfies log-H\\\"older continuity conditions and $1 < p_- \\leq p_+ < \\infty$. Our results generalize to spaces of homogeneous type the analogous results in Euclidean space proved in [14]."],"dc:format.medium":["electronic"],"dc:format.mimetype":["application/pdf"],"dc:identifier.other":["http://purl.lib.ua.edu/176881","u0015_0000001_0003675","Cummings_alatus_0004M_14220"],"dc:identifier.uri":["https://ir.ua.edu/handle/123456789/8380"],"dc:language":["English"],"dc:language.iso":["en_US"],"dc:publisher":["University of Alabama Libraries"],"dc:rights":["All rights reserved by the author unless otherwise indicated."],"dc:subject":["Harmonic analysis","Maximal operator","Variable Lebesgue Spaces","Weighted estimates"],"dc:title":["Weighted Norm Inequalities for the Maximal Operator on Variable Lebesgue Spaces Over Spaces of Homogeneous Type"],"dc:type":["thesis","text"]},"updated_at":"2026-07-27T18:44:16Z"}