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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Bibliographic Details
Published inMathematics (Basel) Vol. 10; no. 20; p. 3867
Main Authors Arnau, Roger, Calabuig, Jose M., Sánchez Pérez, Enrique A.
Format Journal Article
LanguageEnglish
Published Basel MDPI AG 01.10.2022
Subjects
Algebra
Coordinates
extension
Lipschitz
Mappings (Mathematics)
Mathematical research
metric coordinates
Metric space
operator
Operators (mathematics)
Representations
Subspaces
United States
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Summary: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.
ISSN:2227-7390
2227-7390
DOI:10.3390/math10203867