{"id":{"repo_id":"ghent","oai_identifier":"oai:archive.ugent.be:8686773"},"canonical_url":"https://search.dev.ndltd.org/etd/ghent/oai:archive.ugent.be:8686773","repository":{"repo_id":"ghent","name":"Ghent University","base_url":"https://biblio.ugent.be/oai"},"display":{"title":"Basic functional and geometric inequalities for the fractional order operators on homogenous Lie groups","abstract":"In this PhD dissertation, we study functional and geometric inequalities on homo- geneous Lie groups. As for direct inequalities we obtain fractional Hardy, Sobolev, Hardy-Sobolev, Gagliardo-Nirenberg, Caffarelli-Kohn-Nirenberg, logarithmic inequal- ities, Hardy-Littlewood-Sobolev and Stein-Weiss inequalities on homogeneous Lie groups. Also, we obtain the integer order Sobolev-Folland-Stein inequality on strati- fied groups. For reverse inequalities, we prove reverse integral Hardy inequalities with pa- rametersq<0,p∈(0,1)and−∞<q≤p<0. Also,weshowreverseinte- gral Hardy inequalities on homogeneous Lie groups, hyperbolic spaces and Cartan- Hadamard manifolds with q < 0, p ∈ (0,1). As consequences, we show reverse Hardy-Littlewood-Sobolev, Stein-Weiss and improved version Stein-Weiss inequali- tiesforcasesq<0,p∈(0,1)and−∞<q≤p<0. Inaddition,weobtain reverse Hardy, Lp-Sobolev and Lp- Caffarelli-Kohn-Nirenberg inequalities with radial derivative on homogeneous Lie groups. Then we show some applications of these inequalities for linear and nonlinear PDEs on homogeneous groups. Also, we consider one-dimensional functional inequalities in Euclidean settings. We establish fractional Hardy, Poincar ́e type, Gagliardo-Nirenberg type and Caffarelli- Kohn-Nirenberg inequalities for fractional order differential operators as Caputo, Riemann-Liouville and Hadamard fractional derivatives. Also, we discuss applica- tions of these inequalities. In addition, we show Lyapunov and Hartman-Wintner- type inequalities for a fractional partial differential equation with Dirichlet condition, we give an application of these inequalities for the first eigenvalue and we show a de La Vall ́ee Poussin-type inequality for fractional elliptic boundary value problem.","abstract_html":"In this PhD dissertation, we study functional and geometric inequalities on homo- geneous Lie groups. As for direct inequalities we obtain fractional Hardy, Sobolev, Hardy-Sobolev, Gagliardo-Nirenberg, Caffarelli-Kohn-Nirenberg, logarithmic inequal- ities, Hardy-Littlewood-Sobolev and Stein-Weiss inequalities on homogeneous Lie groups. Also, we obtain the integer order Sobolev-Folland-Stein inequality on strati- fied groups. For reverse inequalities, we prove reverse integral Hardy inequalities with pa- rametersq&lt;0,p∈(0,1)and−∞&lt;q≤p&lt;0. Also,weshowreverseinte- gral Hardy inequalities on homogeneous Lie groups, hyperbolic spaces and Cartan- Hadamard manifolds with q &lt; 0, p ∈ (0,1). As consequences, we show reverse Hardy-Littlewood-Sobolev, Stein-Weiss and improved version Stein-Weiss inequali- tiesforcasesq&lt;0,p∈(0,1)and−∞&lt;q≤p&lt;0. Inaddition,weobtain reverse Hardy, Lp-Sobolev and Lp- Caffarelli-Kohn-Nirenberg inequalities with radial derivative on homogeneous Lie groups. Then we show some applications of these inequalities for linear and nonlinear PDEs on homogeneous groups. Also, we consider one-dimensional functional inequalities in Euclidean settings. We establish fractional Hardy, Poincar ́e type, Gagliardo-Nirenberg type and Caffarelli- Kohn-Nirenberg inequalities for fractional order differential operators as Caputo, Riemann-Liouville and Hadamard fractional derivatives. Also, we discuss applica- tions of these inequalities. In addition, we show Lyapunov and Hartman-Wintner- type inequalities for a fractional partial differential equation with Dirichlet condition, we give an application of these inequalities for the first eigenvalue and we show a de La Vall ́ee Poussin-type inequality for fractional elliptic boundary value problem.","abstract_has_math":false,"creators":["Kassymov, Aidyn"],"institution":"Al-Farabi Kazakh National University. Faculty of Mechanics and Mathematics ; Ghent University. Faculty of Sciences","degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":null,"school":null,"contributors":["Ruzhansky, Michael","Suragan, Durvudkhan"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2020,"date_issued":"2020","date_published":"2020","updated_at":"2026-07-24T02:23:00Z","subjects":["Mathematics and Statistics"],"languages":["eng"],"rights":["info:eu-repo/semantics/openAccess"],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["https://biblio.ugent.be/publication/8686773","https://biblio.ugent.be/publication/8686773/file/8686774"],"render_values":[{"text":"https://biblio.ugent.be/publication/8686773","href":"https://biblio.ugent.be/publication/8686773","code":true},{"text":"https://biblio.ugent.be/publication/8686773/file/8686774","href":"https://biblio.ugent.be/publication/8686773/file/8686774","code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/1854/LU-8686773","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Ruzhansky, Michael","Suragan, Durvudkhan"]},{"key":"dc:creator","label":"Author","values":["Kassymov, Aidyn"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2020"]},{"key":"dc:publisher","label":"Institution","values":["Al-Farabi Kazakh National University. Faculty of Mechanics and Mathematics ; Ghent University. Faculty of Sciences"]},{"key":"dc:type","label":"Dc Type","values":["dissertation","info:eu-repo/semantics/doctoralThesis","info:eu-repo/semantics/publishedVersion"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Mathematics and Statistics"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["eng"]},{"key":"dc:rights","label":"Dc Rights","values":["info:eu-repo/semantics/openAccess"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://biblio.ugent.be/publication/8686773","http://hdl.handle.net/1854/LU-8686773","https://biblio.ugent.be/publication/8686773/file/8686774"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["In this PhD dissertation, we study functional and geometric inequalities on homo- geneous Lie groups. As for direct inequalities we obtain fractional Hardy, Sobolev, Hardy-Sobolev, Gagliardo-Nirenberg, Caffarelli-Kohn-Nirenberg, logarithmic inequal- ities, Hardy-Littlewood-Sobolev and Stein-Weiss inequalities on homogeneous Lie groups. Also, we obtain the integer order Sobolev-Folland-Stein inequality on strati- fied groups. For reverse inequalities, we prove reverse integral Hardy inequalities with pa- rametersq<0,p∈(0,1)and−∞<q≤p<0. Also,weshowreverseinte- gral Hardy inequalities on homogeneous Lie groups, hyperbolic spaces and Cartan- Hadamard manifolds with q < 0, p ∈ (0,1). As consequences, we show reverse Hardy-Littlewood-Sobolev, Stein-Weiss and improved version Stein-Weiss inequali- tiesforcasesq<0,p∈(0,1)and−∞<q≤p<0. Inaddition,weobtain reverse Hardy, Lp-Sobolev and Lp- Caffarelli-Kohn-Nirenberg inequalities with radial derivative on homogeneous Lie groups. Then we show some applications of these inequalities for linear and nonlinear PDEs on homogeneous groups. Also, we consider one-dimensional functional inequalities in Euclidean settings. We establish fractional Hardy, Poincar ́e type, Gagliardo-Nirenberg type and Caffarelli- Kohn-Nirenberg inequalities for fractional order differential operators as Caputo, Riemann-Liouville and Hadamard fractional derivatives. Also, we discuss applica- tions of these inequalities. In addition, we show Lyapunov and Hartman-Wintner- type inequalities for a fractional partial differential equation with Dirichlet condition, we give an application of these inequalities for the first eigenvalue and we show a de La Vall ́ee Poussin-type inequality for fractional elliptic boundary value problem."]},{"key":"dc:format","label":"Dc Format","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Basic functional and geometric inequalities for the fractional order operators on homogenous Lie groups"]}]}],"canonical_facts":{"dc:contributor":["Ruzhansky, Michael","Suragan, Durvudkhan"],"dc:creator":["Kassymov, Aidyn"],"dc:date":["2020"],"dc:description":["In this PhD dissertation, we study functional and geometric inequalities on homo- geneous Lie groups. As for direct inequalities we obtain fractional Hardy, Sobolev, Hardy-Sobolev, Gagliardo-Nirenberg, Caffarelli-Kohn-Nirenberg, logarithmic inequal- ities, Hardy-Littlewood-Sobolev and Stein-Weiss inequalities on homogeneous Lie groups. Also, we obtain the integer order Sobolev-Folland-Stein inequality on strati- fied groups. For reverse inequalities, we prove reverse integral Hardy inequalities with pa- rametersq<0,p∈(0,1)and−∞<q≤p<0. Also,weshowreverseinte- gral Hardy inequalities on homogeneous Lie groups, hyperbolic spaces and Cartan- Hadamard manifolds with q < 0, p ∈ (0,1). As consequences, we show reverse Hardy-Littlewood-Sobolev, Stein-Weiss and improved version Stein-Weiss inequali- tiesforcasesq<0,p∈(0,1)and−∞<q≤p<0. Inaddition,weobtain reverse Hardy, Lp-Sobolev and Lp- Caffarelli-Kohn-Nirenberg inequalities with radial derivative on homogeneous Lie groups. Then we show some applications of these inequalities for linear and nonlinear PDEs on homogeneous groups. Also, we consider one-dimensional functional inequalities in Euclidean settings. We establish fractional Hardy, Poincar ́e type, Gagliardo-Nirenberg type and Caffarelli- Kohn-Nirenberg inequalities for fractional order differential operators as Caputo, Riemann-Liouville and Hadamard fractional derivatives. Also, we discuss applica- tions of these inequalities. In addition, we show Lyapunov and Hartman-Wintner- type inequalities for a fractional partial differential equation with Dirichlet condition, we give an application of these inequalities for the first eigenvalue and we show a de La Vall ́ee Poussin-type inequality for fractional elliptic boundary value problem."],"dc:format":["application/pdf"],"dc:identifier":["https://biblio.ugent.be/publication/8686773","http://hdl.handle.net/1854/LU-8686773","https://biblio.ugent.be/publication/8686773/file/8686774"],"dc:language":["eng"],"dc:publisher":["Al-Farabi Kazakh National University. Faculty of Mechanics and Mathematics ; Ghent University. Faculty of Sciences"],"dc:rights":["info:eu-repo/semantics/openAccess"],"dc:subject":["Mathematics and Statistics"],"dc:title":["Basic functional and geometric inequalities for the fractional order operators on homogenous Lie groups"],"dc:type":["dissertation","info:eu-repo/semantics/doctoralThesis","info:eu-repo/semantics/publishedVersion"]},"updated_at":"2026-07-24T02:23:00Z"}