Representation of Lipschitz Maps and Metric Coordinate Systems

Roger Arnau, Jose M. Calabuig and Enrique A. Sánchez Pérez ()
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Roger Arnau: Instituto Universitario de Matemática Pura y Aplicada, Universitat Politècnica de València, Camino de Vera s/n, 46022 Valencia, Spain
Jose M. Calabuig: Instituto Universitario de Matemática Pura y Aplicada, Universitat Politècnica de València, Camino de Vera s/n, 46022 Valencia, Spain
Enrique A. Sánchez Pérez: Instituto Universitario de Matemática Pura y Aplicada, Universitat Politècnica de València, Camino de Vera s/n, 46022 Valencia, Spain

Mathematics, 2022, vol. 10, issue 20, 1-23

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 L 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.

Keywords: Lipschitz; operator; metric space; extension; metric coordinates (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
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