{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/71271"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/71271","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Lattice Properties and Interpolation Theory of the Spaces Lambda(psi,q) and M(psi)","abstract":"Since Lorentz introduced Lorentz space, there have been several generalizations of this space. Hunt and Cwikel studied Lorentz L$\\sb{\\rm p,q}$ spaces and showed some basic properties such as the characterization of the dual space of L$\\sb{\\rm p,q}$. Sharpley's version of Lorentz space is the space $\\Lambda\\sb\\alpha$(X); he extended Calderon's interpolation theory of Lorentz L$\\sb{\\rm p,q}$ spaces to the spaces $\\Lambda\\sb\\alpha$(X).","abstract_html":"Since Lorentz introduced Lorentz space, there have been several generalizations of this space. Hunt and Cwikel studied Lorentz L$\\sb{\\rm p,q}$ spaces and showed some basic properties such as the characterization of the dual space of L$\\sb{\\rm p,q}$. Sharpley&#x27;s version of Lorentz space is the space <span class=\"etd-inline-math\">\\Lambda\\sb&alpha;</span>(X); he extended Calderon&#x27;s interpolation theory of Lorentz L$\\sb{\\rm p,q}$ spaces to the spaces <span class=\"etd-inline-math\">\\Lambda\\sb&alpha;</span>(X).","abstract_has_math":true,"creators":["Lee, Chongsung"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Peck, N. Tenney,"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2014,"date_issued":"2014-12-16T06:18:27Z","date_published":"2014-12-16T06:18:27Z","updated_at":"2026-07-22T22:26:04Z","subjects":["Mathematics"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["(UMI)AAI8908741"],"render_values":[{"text":"(UMI)AAI8908741","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/71271","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Peck, N. Tenney,"]},{"key":"dc:creator","label":"Author","values":["Lee, Chongsung"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2014-12-16T06:18:27Z","10000-01-01","1988"]},{"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":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/71271","(UMI)AAI8908741"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["Since Lorentz introduced Lorentz space, there have been several generalizations of this space. Hunt and Cwikel studied Lorentz L$\\sb{\\rm p,q}$ spaces and showed some basic properties such as the characterization of the dual space of L$\\sb{\\rm p,q}$. Sharpley's version of Lorentz space is the space $\\Lambda\\sb\\alpha$(X); he extended Calderon's interpolation theory of Lorentz L$\\sb{\\rm p,q}$ spaces to the spaces $\\Lambda\\sb\\alpha$(X).","In this thesis, we take Sharpley's Lorentz space $\\Lambda\\sb\\alpha$(X) with minor modifications and define a Lorentz space $\\Lambda\\sb{\\psi,{\\rm q}}$. From its definition, it is easily observed that $\\Lambda\\sb{\\psi,{\\rm q}}$ is a symmetric space. Some geometrical properties of symmetric spaces are related to the growth rate of their fundamental functions which is always quasiconcave. We define the notion of p-power quasiconcavity to clarify this relation. We show that if the lower index of a given quasiconcave function $\\psi(t)$ is strictly greater than zero, there exists p such that $\\psi(t)$ is p-power quasiconcave. With the help of this notion, we extend some properties of Lorentz L$\\sb{\\rm p,q}$ space which were shown by Creekmore to the spaces $\\Lambda\\sb{\\psi,{\\rm q}}$. We also show the existence of bounded lattice isomorphisms from the Banach lattices $\\ell\\sb{\\rm p}$, $\\ell\\sb\\infty$ and L$\\sb{\\rm p}$ onto closed sublattices of Marcinkiewicz space.","The well known K-method of Peetre allows us to construct interpolation spaces. One question is whether all interpolation spaces can be constructed by the Peetre K-method. Cwikel and Peetre showed that if a given Banach couples A is a K-monotone space, all interpolation spaces can be constructed by the Peetre K-method. But, they really show only that all interpolation cones can be constructed by the Peetre K-method, rather than interpolation spaces; when they wrote their paper, an important result of Brudnyi and Krugljak was not available to them. We study this question when the given Banach couples A and B are different. In this case, we need a stronger condition, the strong $\\lambda$-K-monotone property. We also show that every intermediate space A of the Banach couple A = ($\\Lambda\\sb{\\varphi\\sb0,1}, \\Lambda\\sb{\\varphi\\sb1,1}$) is a strong $\\lambda$-K-monotone space with respect to A = ($\\Lambda\\sb{\\varphi\\sb0,1}, \\Lambda\\sb{\\varphi\\sb1,1}$) and B = (M$\\sb{\\psi\\sb0}$,M$\\sb{\\psi\\sb1}$).","Made available in DSpace on 2014-12-16T06:18:27Z (GMT). No. of bitstreams: 1 8908741.pdf: 2852909 bytes, checksum: 9bbff633153e5c2fe3b885f3be0a0c33 (MD5) Previous issue date: 1988","Embargo set by: Seth Robbins for item 71437 Lift date: Forever Reason: Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","U of I Only","98 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1988."]},{"key":"dc:title","label":"Title","values":["Lattice Properties and Interpolation Theory of the Spaces Lambda(psi,q) and M(psi)"]}]}],"canonical_facts":{"dc:contributor":["Peck, N. Tenney,"],"dc:creator":["Lee, Chongsung"],"dc:date":["2014-12-16T06:18:27Z","10000-01-01","1988"],"dc:description":["Since Lorentz introduced Lorentz space, there have been several generalizations of this space. Hunt and Cwikel studied Lorentz L$\\sb{\\rm p,q}$ spaces and showed some basic properties such as the characterization of the dual space of L$\\sb{\\rm p,q}$. Sharpley's version of Lorentz space is the space $\\Lambda\\sb\\alpha$(X); he extended Calderon's interpolation theory of Lorentz L$\\sb{\\rm p,q}$ spaces to the spaces $\\Lambda\\sb\\alpha$(X).","In this thesis, we take Sharpley's Lorentz space $\\Lambda\\sb\\alpha$(X) with minor modifications and define a Lorentz space $\\Lambda\\sb{\\psi,{\\rm q}}$. From its definition, it is easily observed that $\\Lambda\\sb{\\psi,{\\rm q}}$ is a symmetric space. Some geometrical properties of symmetric spaces are related to the growth rate of their fundamental functions which is always quasiconcave. We define the notion of p-power quasiconcavity to clarify this relation. We show that if the lower index of a given quasiconcave function $\\psi(t)$ is strictly greater than zero, there exists p such that $\\psi(t)$ is p-power quasiconcave. With the help of this notion, we extend some properties of Lorentz L$\\sb{\\rm p,q}$ space which were shown by Creekmore to the spaces $\\Lambda\\sb{\\psi,{\\rm q}}$. We also show the existence of bounded lattice isomorphisms from the Banach lattices $\\ell\\sb{\\rm p}$, $\\ell\\sb\\infty$ and L$\\sb{\\rm p}$ onto closed sublattices of Marcinkiewicz space.","The well known K-method of Peetre allows us to construct interpolation spaces. One question is whether all interpolation spaces can be constructed by the Peetre K-method. Cwikel and Peetre showed that if a given Banach couples A is a K-monotone space, all interpolation spaces can be constructed by the Peetre K-method. But, they really show only that all interpolation cones can be constructed by the Peetre K-method, rather than interpolation spaces; when they wrote their paper, an important result of Brudnyi and Krugljak was not available to them. We study this question when the given Banach couples A and B are different. In this case, we need a stronger condition, the strong $\\lambda$-K-monotone property. We also show that every intermediate space A of the Banach couple A = ($\\Lambda\\sb{\\varphi\\sb0,1}, \\Lambda\\sb{\\varphi\\sb1,1}$) is a strong $\\lambda$-K-monotone space with respect to A = ($\\Lambda\\sb{\\varphi\\sb0,1}, \\Lambda\\sb{\\varphi\\sb1,1}$) and B = (M$\\sb{\\psi\\sb0}$,M$\\sb{\\psi\\sb1}$).","Made available in DSpace on 2014-12-16T06:18:27Z (GMT). No. of bitstreams: 1 8908741.pdf: 2852909 bytes, checksum: 9bbff633153e5c2fe3b885f3be0a0c33 (MD5) Previous issue date: 1988","Embargo set by: Seth Robbins for item 71437 Lift date: Forever Reason: Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","U of I Only","98 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1988."],"dc:identifier":["http://hdl.handle.net/2142/71271","(UMI)AAI8908741"],"dc:subject":["Mathematics"],"dc:title":["Lattice Properties and Interpolation Theory of the Spaces Lambda(psi,q) and M(psi)"],"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":"2026-07-22T22:26:04Z"}