{"id":{"repo_id":"wayne-thes","oai_identifier":"oai:digitalcommons.wayne.edu:oa_dissertations-1810"},"canonical_url":"https://search.dev.ndltd.org/etd/wayne-thes/oai:digitalcommons.wayne.edu:oa_dissertations-1810","repository":{"repo_id":"wayne-thes","name":"Wayne State University","base_url":"https://digitalcommons.wayne.edu/do/oai/"},"display":{"title":"Discrete Littlewood-Paley-Stein Theory And Wolff Potentials On Homogeneous Spaces And Multi-Parameter Hardy Spaces","abstract":"<p>This dissertation consists of two parts:</p> <p>In part I, We establish a new atomic decomposition of the multi-parameter Hardy spaces of homogeneous type and obtain the associated $H^p-L^p$ and $H^p-H^p$ boundedness criterions for singular integral operators. On the other hand, we compare the Wolff and Riesz potentials on spaces of homogenous type, followed by a Hardy-Littlewood-Sobolev type inequality. Then we drive integrability estimates of positive solutions to the Lane-Emden type integral systems on spaces of homogeneous type.</p> <p>In part II, We establish a $(p,2)$-atomic decomposition of the Hardy space associated with different homogeneities for $0<p\\le 1$. In addition, We characterize the dual spaces of the weighted multi-parameter Hardy spaces associated with Zygmund dilations, i. e. $(H^p_{\\mathcal Z}(w))^*=CMO^p_{\\mathcal Z}(w)$ for $w\\in A_{\\infty}(\\mathcal Z)$ and $0<p\\le 1$.</p>","abstract_html":"&lt;p&gt;This dissertation consists of two parts:&lt;/p&gt; &lt;p&gt;In part I, We establish a new atomic decomposition of the multi-parameter Hardy spaces of homogeneous type and obtain the associated <span class=\"etd-inline-math\">H<sup>p</sup>-L<sup>p</sup></span> and <span class=\"etd-inline-math\">H<sup>p</sup>-H<sup>p</sup></span> boundedness criterions for singular integral operators. On the other hand, we compare the Wolff and Riesz potentials on spaces of homogenous type, followed by a Hardy-Littlewood-Sobolev type inequality. Then we drive integrability estimates of positive solutions to the Lane-Emden type integral systems on spaces of homogeneous type.&lt;/p&gt; &lt;p&gt;In part II, We establish a $(p,2)$-atomic decomposition of the Hardy space associated with different homogeneities for $0&lt;p\\le 1$. In addition, We characterize the dual spaces of the weighted multi-parameter Hardy spaces associated with Zygmund dilations, i. e. <span class=\"etd-inline-math\">(H<sup>p</sup><sub>\\mathcal Z</sub>(w))<sup>*</sup>=CMO<sup>p</sup><sub>\\mathcal Z</sub>(w)</span> for <span class=\"etd-inline-math\">w\\in A<sub>\\infty</sub>(\\mathcal Z)</span> and $0&lt;p\\le 1$.&lt;/p&gt;","abstract_has_math":true,"creators":["Xiao, Yayuan"],"institution":null,"degree_name":"Ph.D.","degree_level":"Open Access Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Guozhen Lu"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2013,"date_issued":"2013-01-02T08:00:00Z","date_published":"2013-01-02T08:00:00Z","updated_at":"2026-07-24T05:59:26Z","subjects":["Atomic decomposition","Discrete Littlewood-Paley-Stein theory","Hardy space","Multi-parameter","Wolff potential","Zygmund dilation","Mathematics"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://digitalcommons.wayne.edu/oa_dissertations/811","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Guozhen Lu"]},{"key":"dc:creator","label":"Author","values":["Xiao, Yayuan"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.available","label":"Dc Date Available","values":["2013-01-01T08:00:00Z"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Open Access Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Atomic decomposition","Discrete Littlewood-Paley-Stein theory","Hardy space","Multi-parameter","Wolff potential","Zygmund dilation","Mathematics"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://digitalcommons.wayne.edu/oa_dissertations/811"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["<p>This dissertation consists of two parts:</p> <p>In part I, We establish a new atomic decomposition of the multi-parameter Hardy spaces of homogeneous type and obtain the associated $H^p-L^p$ and $H^p-H^p$ boundedness criterions for singular integral operators. On the other hand, we compare the Wolff and Riesz potentials on spaces of homogenous type, followed by a Hardy-Littlewood-Sobolev type inequality. Then we drive integrability estimates of positive solutions to the Lane-Emden type integral systems on spaces of homogeneous type.</p> <p>In part II, We establish a $(p,2)$-atomic decomposition of the Hardy space associated with different homogeneities for $0<p\\le 1$. In addition, We characterize the dual spaces of the weighted multi-parameter Hardy spaces associated with Zygmund dilations, i. e. $(H^p_{\\mathcal Z}(w))^*=CMO^p_{\\mathcal Z}(w)$ for $w\\in A_{\\infty}(\\mathcal Z)$ and $0<p\\le 1$.</p>"]},{"key":"dc:title","label":"Title","values":["Discrete Littlewood-Paley-Stein Theory And Wolff Potentials On Homogeneous Spaces And Multi-Parameter Hardy Spaces"]}]}],"canonical_facts":{"dc:contributor":["Guozhen Lu"],"dc:creator":["Xiao, Yayuan"],"dc:date.available":["2013-01-01T08:00:00Z"],"dc:description.abstract":["<p>This dissertation consists of two parts:</p> <p>In part I, We establish a new atomic decomposition of the multi-parameter Hardy spaces of homogeneous type and obtain the associated $H^p-L^p$ and $H^p-H^p$ boundedness criterions for singular integral operators. On the other hand, we compare the Wolff and Riesz potentials on spaces of homogenous type, followed by a Hardy-Littlewood-Sobolev type inequality. Then we drive integrability estimates of positive solutions to the Lane-Emden type integral systems on spaces of homogeneous type.</p> <p>In part II, We establish a $(p,2)$-atomic decomposition of the Hardy space associated with different homogeneities for $0<p\\le 1$. In addition, We characterize the dual spaces of the weighted multi-parameter Hardy spaces associated with Zygmund dilations, i. e. $(H^p_{\\mathcal Z}(w))^*=CMO^p_{\\mathcal Z}(w)$ for $w\\in A_{\\infty}(\\mathcal Z)$ and $0<p\\le 1$.</p>"],"dc:identifier":["https://digitalcommons.wayne.edu/oa_dissertations/811"],"dc:subject":["Atomic decomposition","Discrete Littlewood-Paley-Stein theory","Hardy space","Multi-parameter","Wolff potential","Zygmund dilation","Mathematics"],"dc:title":["Discrete Littlewood-Paley-Stein Theory And Wolff Potentials On Homogeneous Spaces And Multi-Parameter Hardy Spaces"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Open Access Dissertation"],"thesis:degree_name":["Ph.D."]},"updated_at":"2026-07-24T05:59:26Z"}