Optimal exponents in weighted estimates without examples

Autores
Luque, Teresa Guadalupe; Pérez Moreno, Carlos; Rela, Ezequiel
Año de publicación
2015
Idioma
inglés
Tipo de recurso
artículo
Estado
versión publicada
Descripción
We present a general approach for proving the optimality of the exponents on weighted estimates. We show that if an operator T satisfies a bound like ∥ T ∥ Lp(w) ≤ c [w]βAp w ε Ap, then the optimal lower bound for β is closely related to the asymptotic behaviour of the unweighted Lp norm ∥ T ∥ Lp(Rn) as p goes to 1 and +∞. By combining these results with the known weighted inequalities, we derive the sharpness of the exponents, without building any specific example, for a wide class of operators including maximaltype, Caldeŕon-Zygmund and fractional operators. In particular, we obtain a lower bound for the best possible exponent for Bochner- Riesz multipliers.We also present a new result concerning a continuum family of maximal operators on the scale of logarithmic Orlicz functions. Further, our method allows to consider in a unified way maximal operators defined over very general Muckenhoupt bases.
Fil: Luque, Teresa Guadalupe. University of Birmingham; Reino Unido
Fil: Pérez Moreno, Carlos. Universidad del País Vasco; España
Fil: Rela, Ezequiel. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentina. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina
Materia
Muckenhoupt weights
Calderon-Zygmund operators
Maximal functions
Nivel de accesibilidad
acceso abierto
Condiciones de uso
https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
Repositorio
CONICET Digital (CONICET)
Institución
Consejo Nacional de Investigaciones Científicas y Técnicas
OAI Identificador
oai:ri.conicet.gov.ar:11336/160873
Acceder
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oai_identifier_str oai:ri.conicet.gov.ar:11336/160873
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repository_id_str 3498
network_name_str CONICET Digital (CONICET)
spelling Optimal exponents in weighted estimates without examplesLuque, Teresa GuadalupePérez Moreno, CarlosRela, EzequielMuckenhoupt weightsCalderon-Zygmund operatorsMaximal functionshttps://purl.org/becyt/ford/1.1https://purl.org/becyt/ford/1We present a general approach for proving the optimality of the exponents on weighted estimates. We show that if an operator T satisfies a bound like ∥ T ∥ Lp(w) ≤ c [w]βAp w ε Ap, then the optimal lower bound for β is closely related to the asymptotic behaviour of the unweighted Lp norm ∥ T ∥ Lp(Rn) as p goes to 1 and +∞. By combining these results with the known weighted inequalities, we derive the sharpness of the exponents, without building any specific example, for a wide class of operators including maximaltype, Caldeŕon-Zygmund and fractional operators. In particular, we obtain a lower bound for the best possible exponent for Bochner- Riesz multipliers.We also present a new result concerning a continuum family of maximal operators on the scale of logarithmic Orlicz functions. Further, our method allows to consider in a unified way maximal operators defined over very general Muckenhoupt bases.Fil: Luque, Teresa Guadalupe. University of Birmingham; Reino UnidoFil: Pérez Moreno, Carlos. Universidad del País Vasco; EspañaFil: Rela, Ezequiel. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentina. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; ArgentinaInternational Press Boston2015-04-13info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionhttp://purl.org/coar/resource_type/c_6501info:ar-repo/semantics/articuloapplication/pdfapplication/pdfapplication/pdfhttp://hdl.handle.net/11336/160873Luque, Teresa Guadalupe; Pérez Moreno, Carlos; Rela, Ezequiel; Optimal exponents in weighted estimates without examples; International Press Boston; Mathematical Research Letters; 22; 1; 13-4-2015; 183-2011073-27801945-001XCONICET DigitalCONICETenginfo:eu-repo/semantics/altIdentifier/doi/10.4310/MRL.2015.v22.n1.a10info:eu-repo/semantics/altIdentifier/url/https://www.intlpress.com/site/pub/pages/journals/items/mrl/content/vols/0022/0001/a010/index.phpinfo:eu-repo/semantics/openAccesshttps://creativecommons.org/licenses/by-nc-sa/2.5/ar/reponame:CONICET Digital (CONICET)instname:Consejo Nacional de Investigaciones Científicas y Técnicas2025-09-03T10:05:05Zoai:ri.conicet.gov.ar:11336/160873instacron:CONICETInstitucionalhttp://ri.conicet.gov.ar/Organismo científico-tecnológicoNo correspondehttp://ri.conicet.gov.ar/oai/requestdasensio@conicet.gov.ar; lcarlino@conicet.gov.arArgentinaNo correspondeNo correspondeNo correspondeopendoar:34982025-09-03 10:05:05.814CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicasfalse
dc.title.none.fl_str_mv Optimal exponents in weighted estimates without examples
title Optimal exponents in weighted estimates without examples
spellingShingle Optimal exponents in weighted estimates without examples
Luque, Teresa Guadalupe
Muckenhoupt weights
Calderon-Zygmund operators
Maximal functions
title_short Optimal exponents in weighted estimates without examples
title_full Optimal exponents in weighted estimates without examples
title_fullStr Optimal exponents in weighted estimates without examples
title_full_unstemmed Optimal exponents in weighted estimates without examples
title_sort Optimal exponents in weighted estimates without examples
dc.creator.none.fl_str_mv Luque, Teresa Guadalupe
Pérez Moreno, Carlos
Rela, Ezequiel
author Luque, Teresa Guadalupe
author_facet Luque, Teresa Guadalupe
Pérez Moreno, Carlos
Rela, Ezequiel
author_role author
author2 Pérez Moreno, Carlos
Rela, Ezequiel
author2_role author
author
dc.subject.none.fl_str_mv Muckenhoupt weights
Calderon-Zygmund operators
Maximal functions
topic Muckenhoupt weights
Calderon-Zygmund operators
Maximal functions
purl_subject.fl_str_mv https://purl.org/becyt/ford/1.1
https://purl.org/becyt/ford/1
dc.description.none.fl_txt_mv We present a general approach for proving the optimality of the exponents on weighted estimates. We show that if an operator T satisfies a bound like ∥ T ∥ Lp(w) ≤ c [w]βAp w ε Ap, then the optimal lower bound for β is closely related to the asymptotic behaviour of the unweighted Lp norm ∥ T ∥ Lp(Rn) as p goes to 1 and +∞. By combining these results with the known weighted inequalities, we derive the sharpness of the exponents, without building any specific example, for a wide class of operators including maximaltype, Caldeŕon-Zygmund and fractional operators. In particular, we obtain a lower bound for the best possible exponent for Bochner- Riesz multipliers.We also present a new result concerning a continuum family of maximal operators on the scale of logarithmic Orlicz functions. Further, our method allows to consider in a unified way maximal operators defined over very general Muckenhoupt bases.
Fil: Luque, Teresa Guadalupe. University of Birmingham; Reino Unido
Fil: Pérez Moreno, Carlos. Universidad del País Vasco; España
Fil: Rela, Ezequiel. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentina. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina
description We present a general approach for proving the optimality of the exponents on weighted estimates. We show that if an operator T satisfies a bound like ∥ T ∥ Lp(w) ≤ c [w]βAp w ε Ap, then the optimal lower bound for β is closely related to the asymptotic behaviour of the unweighted Lp norm ∥ T ∥ Lp(Rn) as p goes to 1 and +∞. By combining these results with the known weighted inequalities, we derive the sharpness of the exponents, without building any specific example, for a wide class of operators including maximaltype, Caldeŕon-Zygmund and fractional operators. In particular, we obtain a lower bound for the best possible exponent for Bochner- Riesz multipliers.We also present a new result concerning a continuum family of maximal operators on the scale of logarithmic Orlicz functions. Further, our method allows to consider in a unified way maximal operators defined over very general Muckenhoupt bases.
publishDate 2015
dc.date.none.fl_str_mv 2015-04-13
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/publishedVersion
http://purl.org/coar/resource_type/c_6501
info:ar-repo/semantics/articulo
format article
status_str publishedVersion
dc.identifier.none.fl_str_mv http://hdl.handle.net/11336/160873
Luque, Teresa Guadalupe; Pérez Moreno, Carlos; Rela, Ezequiel; Optimal exponents in weighted estimates without examples; International Press Boston; Mathematical Research Letters; 22; 1; 13-4-2015; 183-201
1073-2780
1945-001X
CONICET Digital
CONICET
url http://hdl.handle.net/11336/160873
identifier_str_mv Luque, Teresa Guadalupe; Pérez Moreno, Carlos; Rela, Ezequiel; Optimal exponents in weighted estimates without examples; International Press Boston; Mathematical Research Letters; 22; 1; 13-4-2015; 183-201
1073-2780
1945-001X
CONICET Digital
CONICET
dc.language.none.fl_str_mv eng
language eng
dc.relation.none.fl_str_mv info:eu-repo/semantics/altIdentifier/doi/10.4310/MRL.2015.v22.n1.a10
info:eu-repo/semantics/altIdentifier/url/https://www.intlpress.com/site/pub/pages/journals/items/mrl/content/vols/0022/0001/a010/index.php
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
eu_rights_str_mv openAccess
rights_invalid_str_mv https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
dc.format.none.fl_str_mv application/pdf
application/pdf
application/pdf
dc.publisher.none.fl_str_mv International Press Boston
publisher.none.fl_str_mv International Press Boston
dc.source.none.fl_str_mv reponame:CONICET Digital (CONICET)
instname:Consejo Nacional de Investigaciones Científicas y Técnicas
reponame_str CONICET Digital (CONICET)
collection CONICET Digital (CONICET)
instname_str Consejo Nacional de Investigaciones Científicas y Técnicas
repository.name.fl_str_mv CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicas
repository.mail.fl_str_mv dasensio@conicet.gov.ar; lcarlino@conicet.gov.ar
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score 13.13397

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