Infinitesimally Lipschitz functions on metric spaces

2009/01/21 by E. Durand, Durand, E., J. A. Jaramillo +1
Mathematics · #46E15 #46E35 #FOS: Mathematics #Metric Geometry (math.MG) #math.MG #msc:46E15 #msc:46E35

paper · pdf · doi:10.48550/arxiv.0901.3236

28 pages, 2 figures

arxiv created 2009/01/21 · arxiv updated 2009/12/01

Abstract

For a metric space X, we study the space D^∞(X) of bounded functions on X whose infinitesimal Lipschitz constant is uniformly bounded. D^∞(X) is compared with the space \LIP^∞(X) of bounded Lipschitz functions on X, in terms of different properties regarding the geometry of X. We also obtain a Banach-Stone theorem in this context. In the case of a metric measure space, we also compare D^∞(X) with the Newtonian-Sobolev space N^{1, ∞}(X). In particular, if X supports a doubling measure and satisfies a local Poincaré inequality, we obtain that D^∞(X)=N^{1, ∞}(X).

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