### Article

 dc.creator Lu, Guozhen es dc.creator Pérez Moreno, Carlos es dc.date.accessioned 2016-11-14T10:49:16Z dc.date.available 2016-11-14T10:49:16Z dc.date.issued 2002-01 dc.identifier.citation Lu, G. y Pérez Moreno, C. (2002). L1 → Lq Poincaré inequalities for 0 < q < 1 imply representation formulas. Acta Mathematica Sinica, 18 (1), 1-20. dc.identifier.issn 1439-8516 es dc.identifier.issn 1439-7617 es dc.identifier.uri http://hdl.handle.net/11441/48523 dc.description.abstract Given two doubling measures μ and ν in a metric space (S, ρ) of homogeneous type, let B0⊂S be a given ball. It has been a well-known result by now (see [1–4]) that the validity of an L1→L1 Poincaré inequality of the following form: ∫B|f−fB|dv⩽cr(B)∫Bgdμ, for all metric balls B⊂B0⊂S, implies a variant of representation formula of fractional integral type: for ν-a.e. x∈B0, es |f(x)−fB0|⩽C∫B0g(y)ρ(x,y)μ(B(x,ρ(x,y)))dμ(y)+Cr(B0)μ(B0)∫B0g(y)dμ(y). One of the main results of this paper shows that an L1 to Lq Poincaré inequality for some 0 < q < 1, i.e., (∫B|f−fB|qdv)1/q⩽cr(B)∫Bgdμ, for all metric balls B⊂B0, will suffice to imply the above representation formula. As an immediate corollary, we can show that the weak-type condition, supλ>0λν({x∈B:|f(x)−fB|>λ})ν(B)⩽Cr(B)∫Bgdμ, also implies the same formula. Analogous theorems related to high-order Poincaré inequalities and Sobolev spaces in metric spaces are also proved. dc.description.sponsorship National Science Foundation es dc.description.sponsorship Dirección General de Investigación Científica y Técnica es dc.description.sponsorship North Atlantic Treaty Organization es dc.format application/pdf es dc.language.iso eng es dc.publisher Springer es dc.relation.ispartof Acta Mathematica Sinica, 18 (1), 1-20. dc.rights Attribution-NonCommercial-NoDerivatives 4.0 Internacional * dc.rights.uri http://creativecommons.org/licenses/by-nc-nd/4.0/ * dc.subject Sobolev spaces es dc.subject Representation formulas es dc.subject High-order derivatives es dc.subject Vector fields es dc.subject Metric spaces es dc.subject Polynomials es dc.subject Doubling measures es dc.subject Poincaré inequalities es dc.title L1 → Lq Poincaré inequalities for 0 < q < 1 imply representation formulas es dc.type info:eu-repo/semantics/article es dcterms.identifier https://ror.org/03yxnpp24 dc.type.version info:eu-repo/semantics/submittedVersion es dc.rights.accessrights info:eu-repo/semantics/openAccess es dc.contributor.affiliation Universidad de Sevilla. Departamento de Análisis Matemático es dc.relation.projectID DMS96-22996 es dc.relation.projectID DMS99-70352 es dc.relation.projectID PB940192 es dc.relation.projectID 972144 es dc.relation.publisherversion https://dx.doi.org/10.1007/s101140100154 es dc.identifier.doi 10.1007/s101140100154 es dc.contributor.group Universidad de Sevilla. FQM354: Análisis Real es idus.format.extent 23 p. es dc.journaltitle Acta Mathematica Sinica es dc.publication.volumen 18 es dc.publication.issue 1 es dc.publication.initialPage 1 es dc.publication.endPage 20 es dc.identifier.idus https://idus.us.es/xmlui/handle/11441/48523
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