Sharp constants related to the triangle inequality in Lorentz spaces

We study the Lorentz spaces $ L^{p,s}(R,\mu)$ in the range $ 1<p<s\le \infty$, for which the standard functional $\displaystyle \vert\vert f\vert\vert _{p,s}=\left(\int_0^\infty (t^{1/p}f^*(t))^s\frac{dt}{t}\right)^{1/s} $ is only a quasi-norm. We find the optimal constant in the triangle ineq...

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Detalles Bibliográficos
Autores: Barza, Sorina, Kolyada, Viktor, Soria de Diego, F. Javier
Tipo de recurso: artículo
Estado:Versión publicada
Fecha de publicación:2009
País:España
Institución:Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya)
Repositorio:Recercat. Dipósit de la Recerca de Catalunya
OAI Identifier:oai:recercat.cat:2445/95822
Acceso en línea:https://hdl.handle.net/2445/95822
Access Level:acceso abierto
Palabra clave:Anàlisi funcional
Espais de Lorentz
Functional analysis
Lorentz spaces
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spelling Sharp constants related to the triangle inequality in Lorentz spacesBarza, SorinaKolyada, ViktorSoria de Diego, F. JavierAnàlisi funcionalEspais de LorentzFunctional analysisLorentz spacesWe study the Lorentz spaces $ L^{p,s}(R,\mu)$ in the range $ 1<p<s\le \infty$, for which the standard functional $\displaystyle \vert\vert f\vert\vert _{p,s}=\left(\int_0^\infty (t^{1/p}f^*(t))^s\frac{dt}{t}\right)^{1/s} $ is only a quasi-norm. We find the optimal constant in the triangle inequality for this quasi-norm, which leads us to consider the following decomposition norm: $\displaystyle \vert\vert f\vert\vert _{(p,s)}=\inf\bigg\{\sum_{k}\vert\vert f_k\vert\vert _{p,s}\bigg\}, $ where the infimum is taken over all finite representations $ f=\sum_{k}f_k. $ We also prove that the decomposition norm and the dual norm $\displaystyle \vert\vert f\vert\vert _{p,s}'= \sup\left\{ \int_R fg d\mu: \vert\vert g\vert\vert _{p',s'}=1\right\}$ agree for all values of $ p,s>1$.American Mathematical Society (AMS)2016201620092016info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersion20 p.application/pdfhttps://hdl.handle.net/2445/95822Articles publicats en revistes (Matemàtiques i Informàtica)reponame:Recercat. Dipósit de la Recerca de Catalunyainstname:Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya)InglésReproducció del document publicat a: http://dx.doi.org/10.1090/S0002-9947-09-04739-4Transactions of the American Mathematical Society, 2009, vol. 361, num. 10, p. 5555-5574http://dx.doi.org/10.1090/S0002-9947-09-04739-4(c) American Mathematical Society (AMS), 2009info:eu-repo/semantics/openAccessoai:recercat.cat:2445/958222026-05-29T05:05:01Z
dc.title.none.fl_str_mv Sharp constants related to the triangle inequality in Lorentz spaces
title Sharp constants related to the triangle inequality in Lorentz spaces
spellingShingle Sharp constants related to the triangle inequality in Lorentz spaces
Barza, Sorina
Anàlisi funcional
Espais de Lorentz
Functional analysis
Lorentz spaces
title_short Sharp constants related to the triangle inequality in Lorentz spaces
title_full Sharp constants related to the triangle inequality in Lorentz spaces
title_fullStr Sharp constants related to the triangle inequality in Lorentz spaces
title_full_unstemmed Sharp constants related to the triangle inequality in Lorentz spaces
title_sort Sharp constants related to the triangle inequality in Lorentz spaces
dc.creator.none.fl_str_mv Barza, Sorina
Kolyada, Viktor
Soria de Diego, F. Javier
author Barza, Sorina
author_facet Barza, Sorina
Kolyada, Viktor
Soria de Diego, F. Javier
author_role author
author2 Kolyada, Viktor
Soria de Diego, F. Javier
author2_role author
author
dc.subject.none.fl_str_mv Anàlisi funcional
Espais de Lorentz
Functional analysis
Lorentz spaces
topic Anàlisi funcional
Espais de Lorentz
Functional analysis
Lorentz spaces
description We study the Lorentz spaces $ L^{p,s}(R,\mu)$ in the range $ 1<p<s\le \infty$, for which the standard functional $\displaystyle \vert\vert f\vert\vert _{p,s}=\left(\int_0^\infty (t^{1/p}f^*(t))^s\frac{dt}{t}\right)^{1/s} $ is only a quasi-norm. We find the optimal constant in the triangle inequality for this quasi-norm, which leads us to consider the following decomposition norm: $\displaystyle \vert\vert f\vert\vert _{(p,s)}=\inf\bigg\{\sum_{k}\vert\vert f_k\vert\vert _{p,s}\bigg\}, $ where the infimum is taken over all finite representations $ f=\sum_{k}f_k. $ We also prove that the decomposition norm and the dual norm $\displaystyle \vert\vert f\vert\vert _{p,s}'= \sup\left\{ \int_R fg d\mu: \vert\vert g\vert\vert _{p',s'}=1\right\}$ agree for all values of $ p,s>1$.
publishDate 2009
dc.date.none.fl_str_mv 2009
2016
2016
2016
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/publishedVersion
format article
status_str publishedVersion
dc.identifier.none.fl_str_mv https://hdl.handle.net/2445/95822
url https://hdl.handle.net/2445/95822
dc.language.none.fl_str_mv Inglés
language_invalid_str_mv Inglés
dc.relation.none.fl_str_mv Reproducció del document publicat a: http://dx.doi.org/10.1090/S0002-9947-09-04739-4
Transactions of the American Mathematical Society, 2009, vol. 361, num. 10, p. 5555-5574
http://dx.doi.org/10.1090/S0002-9947-09-04739-4
dc.rights.none.fl_str_mv (c) American Mathematical Society (AMS), 2009
info:eu-repo/semantics/openAccess
rights_invalid_str_mv (c) American Mathematical Society (AMS), 2009
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv 20 p.
application/pdf
dc.publisher.none.fl_str_mv American Mathematical Society (AMS)
publisher.none.fl_str_mv American Mathematical Society (AMS)
dc.source.none.fl_str_mv Articles publicats en revistes (Matemàtiques i Informàtica)
reponame:Recercat. Dipósit de la Recerca de Catalunya
instname:Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya)
instname_str Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya)
reponame_str Recercat. Dipósit de la Recerca de Catalunya
collection Recercat. Dipósit de la Recerca de Catalunya
repository.name.fl_str_mv
repository.mail.fl_str_mv
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score 15.811543

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