Conical square functions for degenerate elliptic operators
The aim of the present paper is to study the boundedness of different conical square functions that arise naturally from second order divergence form degenerate elliptic operators. More precisely, let Lw = - w- 1 div(wA▽), where w∈A2 and A is an n × n bounded, complex-valued, uniformly elliptic matr...
| Autores: | , , |
|---|---|
| Tipo de recurso: | artículo |
| Fecha de publicación: | 2017 |
| País: | España |
| Institución: | Universidad Autónoma de Madrid |
| Repositorio: | Biblos-e Archivo. Repositorio Institucional de la UAM |
| Idioma: | inglés |
| OAI Identifier: | oai:repositorio.uam.es:10486/696901 |
| Acceso en línea: | http://hdl.handle.net/10486/696901 https://dx.doi.org/10.1515/acv-2016-0062 |
| Access Level: | acceso abierto |
| Palabra clave: | Conical square functions Muckenhoupt weights Change of angle formulas Degenerate elliptic operators Heat and Poisson semigroups Off-diagonal estimates Matemáticas |
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Conical square functions for degenerate elliptic operatorsChen, LiMartell, José MaríaPrisuelos-Arribas, CruzConical square functionsMuckenhoupt weightsChange of angle formulasDegenerate elliptic operatorsHeat and Poisson semigroupsOff-diagonal estimatesMatemáticasThe aim of the present paper is to study the boundedness of different conical square functions that arise naturally from second order divergence form degenerate elliptic operators. More precisely, let Lw = - w- 1 div(wA▽), where w∈A2 and A is an n × n bounded, complex-valued, uniformly elliptic matrix. D. Cruz-Uribe and Rios solved the L2(w)-Kato square root problem obtaining that √Lw is equivalent to the gradient on L2(w). The same authors in collaboration with the second named author of this paper studied the Lp(w)-boundedness of operators that are naturally associated with Lw, such as the functional calculus, Riesz transforms, and vertical square functions. The theory developed admitted also weighted estimates (i.e., estimates in Lp(v dw) for v∈ A∞(w)), and in particular a class of "degeneracy" weights w was found in such a way that the classical L2 -Kato problem can be solved. In this paper, continuing this line of research, and also that originated in some recent results by the second and third named authors of the current paper, we study the boundedness on Lp(w) and on Lp(v dw), with v∈ A∞(w), of the conical square functions that one can construct using the heat or Poisson semigroup associated with Lw. As a consequence of our methods, we find a class of degeneracy weights w for which L2-estimates for these conical square functions hold. This opens the door to the study of weighted and unweighted Hardy spaces and of boundary value problems associated with LwThe research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ ERC agreement no. 615112 HAPDEGMT. The authors also acknowledge financial support from the Spanish Ministry of Economy and Competitiveness, through the “Severo Ochoa Programme for Centres of Excellence in R&D” (SEV-2015-0554)De GruyterDepartamento de MatemáticasFacultad de Ciencias20172017-12-08research articlehttp://purl.org/coar/resource_type/c_2df8fbb1AMhttp://purl.org/coar/version/c_ab4af688f83e57aainfo:eu-repo/semantics/articleapplication/pdfhttp://hdl.handle.net/10486/696901https://dx.doi.org/10.1515/acv-2016-0062reponame:Biblos-e Archivo. Repositorio Institucional de la UAMinstname:Universidad Autónoma de MadridInglésengEuropean Commission http://dx.doi.org/10.13039/501100000780 Framework Programme Seven 615112open accesshttp://purl.org/coar/access_right/c_abf2info:eu-repo/semantics/openAccessoai:repositorio.uam.es:10486/6969012026-06-23T12:46:27Z |
| dc.title.none.fl_str_mv |
Conical square functions for degenerate elliptic operators |
| title |
Conical square functions for degenerate elliptic operators |
| spellingShingle |
Conical square functions for degenerate elliptic operators Chen, Li Conical square functions Muckenhoupt weights Change of angle formulas Degenerate elliptic operators Heat and Poisson semigroups Off-diagonal estimates Matemáticas |
| title_short |
Conical square functions for degenerate elliptic operators |
| title_full |
Conical square functions for degenerate elliptic operators |
| title_fullStr |
Conical square functions for degenerate elliptic operators |
| title_full_unstemmed |
Conical square functions for degenerate elliptic operators |
| title_sort |
Conical square functions for degenerate elliptic operators |
| dc.creator.none.fl_str_mv |
Chen, Li Martell, José María Prisuelos-Arribas, Cruz |
| author |
Chen, Li |
| author_facet |
Chen, Li Martell, José María Prisuelos-Arribas, Cruz |
| author_role |
author |
| author2 |
Martell, José María Prisuelos-Arribas, Cruz |
| author2_role |
author author |
| dc.contributor.none.fl_str_mv |
Departamento de Matemáticas Facultad de Ciencias |
| dc.subject.none.fl_str_mv |
Conical square functions Muckenhoupt weights Change of angle formulas Degenerate elliptic operators Heat and Poisson semigroups Off-diagonal estimates Matemáticas |
| topic |
Conical square functions Muckenhoupt weights Change of angle formulas Degenerate elliptic operators Heat and Poisson semigroups Off-diagonal estimates Matemáticas |
| description |
The aim of the present paper is to study the boundedness of different conical square functions that arise naturally from second order divergence form degenerate elliptic operators. More precisely, let Lw = - w- 1 div(wA▽), where w∈A2 and A is an n × n bounded, complex-valued, uniformly elliptic matrix. D. Cruz-Uribe and Rios solved the L2(w)-Kato square root problem obtaining that √Lw is equivalent to the gradient on L2(w). The same authors in collaboration with the second named author of this paper studied the Lp(w)-boundedness of operators that are naturally associated with Lw, such as the functional calculus, Riesz transforms, and vertical square functions. The theory developed admitted also weighted estimates (i.e., estimates in Lp(v dw) for v∈ A∞(w)), and in particular a class of "degeneracy" weights w was found in such a way that the classical L2 -Kato problem can be solved. In this paper, continuing this line of research, and also that originated in some recent results by the second and third named authors of the current paper, we study the boundedness on Lp(w) and on Lp(v dw), with v∈ A∞(w), of the conical square functions that one can construct using the heat or Poisson semigroup associated with Lw. As a consequence of our methods, we find a class of degeneracy weights w for which L2-estimates for these conical square functions hold. This opens the door to the study of weighted and unweighted Hardy spaces and of boundary value problems associated with Lw |
| publishDate |
2017 |
| dc.date.none.fl_str_mv |
2017 2017-12-08 |
| dc.type.none.fl_str_mv |
research article http://purl.org/coar/resource_type/c_2df8fbb1 AM http://purl.org/coar/version/c_ab4af688f83e57aa |
| dc.type.openaire.fl_str_mv |
info:eu-repo/semantics/article |
| format |
article |
| dc.identifier.none.fl_str_mv |
http://hdl.handle.net/10486/696901 https://dx.doi.org/10.1515/acv-2016-0062 |
| url |
http://hdl.handle.net/10486/696901 https://dx.doi.org/10.1515/acv-2016-0062 |
| dc.language.none.fl_str_mv |
Inglés eng |
| language_invalid_str_mv |
Inglés |
| language |
eng |
| dc.relation.none.fl_str_mv |
European Commission http://dx.doi.org/10.13039/501100000780 Framework Programme Seven 615112 |
| dc.rights.none.fl_str_mv |
open access http://purl.org/coar/access_right/c_abf2 |
| dc.rights.openaire.fl_str_mv |
info:eu-repo/semantics/openAccess |
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open access http://purl.org/coar/access_right/c_abf2 |
| eu_rights_str_mv |
openAccess |
| dc.format.none.fl_str_mv |
application/pdf |
| dc.publisher.none.fl_str_mv |
De Gruyter |
| publisher.none.fl_str_mv |
De Gruyter |
| dc.source.none.fl_str_mv |
reponame:Biblos-e Archivo. Repositorio Institucional de la UAM instname:Universidad Autónoma de Madrid |
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Universidad Autónoma de Madrid |
| reponame_str |
Biblos-e Archivo. Repositorio Institucional de la UAM |
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Biblos-e Archivo. Repositorio Institucional de la UAM |
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