Representation of Lipschitz Maps and Metric Coordinate Systems
Here, we prove some general results that allow us to ensure that specific representations (as well as extensions) of certain Lipschitz operators exist, provided we have some additional information about the underlying space, in the context of what we call enriched metric spaces. In this conceptual f...
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| Published in | Mathematics (Basel) Vol. 10; no. 20; p. 3867 |
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| Main Authors | , , |
| Format | Journal Article |
| Language | English |
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01.10.2022
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| Abstract | Here, we prove some general results that allow us to ensure that specific representations (as well as extensions) of certain Lipschitz operators exist, provided we have some additional information about the underlying space, in the context of what we call enriched metric spaces. In this conceptual framework, we introduce some new classes of Lipschitz operators whose definition depends on the notion of metric coordinate system, which are defined by specific dominance inequalities involving summations of distances between certain points in the space. We analyze “Pietsch Theorem inspired factorizations" through subspaces of ℓ∞ and L1, which are proved to characterize when a given metric space is Lipschitz isomorphic to a metric subspace of these spaces. As an application, extension results for Lipschitz maps that are obtained by a coordinate-wise adaptation of the McShane–Whitney formulas, are also given. |
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| AbstractList | Here, we prove some general results that allow us to ensure that specific representations (as well as extensions) of certain Lipschitz operators exist, provided we have some additional information about the underlying space, in the context of what we call enriched metric spaces. In this conceptual framework, we introduce some new classes of Lipschitz operators whose definition depends on the notion of metric coordinate system, which are defined by specific dominance inequalities involving summations of distances between certain points in the space. We analyze “Pietsch Theorem inspired factorizations" through subspaces of ℓ∞ and L1, which are proved to characterize when a given metric space is Lipschitz isomorphic to a metric subspace of these spaces. As an application, extension results for Lipschitz maps that are obtained by a coordinate-wise adaptation of the McShane–Whitney formulas, are also given. Here, we prove some general results that allow us to ensure that specific representations (as well as extensions) of certain Lipschitz operators exist, provided we have some additional information about the underlying space, in the context of what we call enriched metric spaces. In this conceptual framework, we introduce some new classes of Lipschitz operators whose definition depends on the notion of metric coordinate system, which are defined by specific dominance inequalities involving summations of distances between certain points in the space. We analyze “Pietsch Theorem inspired factorizations" through subspaces of ℓ[sub.∞] and L[sup.1], which are proved to characterize when a given metric space is Lipschitz isomorphic to a metric subspace of these spaces. As an application, extension results for Lipschitz maps that are obtained by a coordinate-wise adaptation of the McShane–Whitney formulas, are also given. |
| Audience | Academic |
| Author | Calabuig, Jose M Sánchez Pérez, Enrique A Arnau, Roger |
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| Cites_doi | 10.1016/j.na.2012.04.044 10.1016/j.jmaa.2015.11.050 10.1007/978-3-662-03961-8 10.1090/S0002-9947-1934-1501735-3 10.1353/ajm.2007.0000 10.4064/fm-22-1-77-108 10.3390/math10020220 10.1090/S0002-9939-2011-11140-7 10.1090/S0002-9939-09-09865-7 10.1007/s43037-020-00060-3 10.1016/j.jmaa.2009.10.025 10.1090/S0002-9904-1934-05978-0 10.1016/j.jfa.2011.03.013 10.1007/978-3-030-16489-8 10.1017/CBO9780511526138 10.1215/20088752-3720614 10.1515/crelle-2018-0037 |
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| References | Achour (ref_4) 2017; 8 Ambrosio (ref_8) 2020; 764 (ref_5) 2011; 261 Brudnyi (ref_9) 2007; 129 ref_13 ref_12 ref_11 McShane (ref_14) 1934; 40 ref_10 ref_20 (ref_6) 2012; 140 Whitney (ref_15) 1934; 36 Chen (ref_7) 2012; 75 ref_19 Botelho (ref_21) 2010; 365 Farmer (ref_1) 2009; 137 ref_17 Achour (ref_2) 2020; 14 Achour (ref_3) 2016; 436 Calcaterra (ref_18) 2009; 6 Kirszbraun (ref_16) 1934; 22 |
| References_xml | – volume: 75 start-page: 5270 year: 2012 ident: ref_7 article-title: Lipschitz p-integral operators and Lipschitz p-nuclear operators publication-title: Nonlinear Anal. Theory Methods Appl. doi: 10.1016/j.na.2012.04.044 contributor: fullname: Chen – volume: 436 start-page: 217 year: 2016 ident: ref_3 article-title: Lipschitz operator ideals and the approximation property publication-title: J. Math. Anal. Appl. doi: 10.1016/j.jmaa.2015.11.050 contributor: fullname: Achour – ident: ref_19 doi: 10.1007/978-3-662-03961-8 – volume: 6 start-page: 79 year: 2009 ident: ref_18 article-title: Metric Coordinate Systems publication-title: Commun. Math. Anal. contributor: fullname: Calcaterra – volume: 36 start-page: 63 year: 1934 ident: ref_15 article-title: Analytic extensions of differentiable functions defined in closed sets publication-title: Trans. Am. Math. Soc. doi: 10.1090/S0002-9947-1934-1501735-3 contributor: fullname: Whitney – ident: ref_10 – ident: ref_11 – volume: 129 start-page: 217 year: 2007 ident: ref_9 article-title: Metric spaces with linear extensions preserving Lipschitz condition publication-title: Am. J. Math. doi: 10.1353/ajm.2007.0000 contributor: fullname: Brudnyi – volume: 22 start-page: 77 year: 1934 ident: ref_16 article-title: Über die zusammenziehende und Lipschitzsche Transformationen publication-title: Fund. Math. doi: 10.4064/fm-22-1-77-108 contributor: fullname: Kirszbraun – ident: ref_12 doi: 10.3390/math10020220 – volume: 140 start-page: 3101 year: 2012 ident: ref_6 article-title: Lipschitz (q,p)-mixing operators publication-title: Proc. Am. Math. Soc. doi: 10.1090/S0002-9939-2011-11140-7 – ident: ref_17 – volume: 137 start-page: 2989 year: 2009 ident: ref_1 article-title: Lipschitz p-summing operators publication-title: Proc. Am. Math. Soc. doi: 10.1090/S0002-9939-09-09865-7 contributor: fullname: Farmer – volume: 14 start-page: 1241 year: 2020 ident: ref_2 article-title: The Lipschitz injective hull of Lipschitz operator ideals and applications publication-title: Banach J. Math. Anal. doi: 10.1007/s43037-020-00060-3 contributor: fullname: Achour – volume: 365 start-page: 269 year: 2010 ident: ref_21 article-title: A unified Pietsch domination theorem publication-title: J. Math. Anal. Appl. doi: 10.1016/j.jmaa.2009.10.025 contributor: fullname: Botelho – volume: 40 start-page: 837 year: 1934 ident: ref_14 article-title: Extension of range of functions publication-title: Bull. Am. Math. Soc. doi: 10.1090/S0002-9904-1934-05978-0 contributor: fullname: McShane – volume: 261 start-page: 387 year: 2011 ident: ref_5 article-title: Duality for Lipschitz p-summing operators publication-title: Ann. Funct. Anal. doi: 10.1016/j.jfa.2011.03.013 – ident: ref_13 doi: 10.1007/978-3-030-16489-8 – ident: ref_20 doi: 10.1017/CBO9780511526138 – volume: 8 start-page: 38 year: 2017 ident: ref_4 article-title: (p, σ)-Absolutely Lipschitz operators publication-title: Ann. Funct. Anal. doi: 10.1215/20088752-3720614 contributor: fullname: Achour – volume: 764 start-page: 1 year: 2020 ident: ref_8 article-title: Linear extension operators between spaces of Lipschitz maps and optimal transport publication-title: J. Reine Angew. Math. doi: 10.1515/crelle-2018-0037 contributor: fullname: Ambrosio |
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| SubjectTerms | Algebra Coordinates extension Lipschitz Mappings (Mathematics) Mathematical research metric coordinates Metric space operator Operators (mathematics) Representations Subspaces |
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| Title | Representation of Lipschitz Maps and Metric Coordinate Systems |
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