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Resumen:
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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 N1,?(X). In particular, if X supports a doubling measure and satisfies a local Poincaré inequality, we obtain that D?(X) = N1,?(X).
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