Pointwise Lipschitz functions on metric spaces

For a metric space X, we study the space D-infinity(X) of bounded functions on X whose pointwise Lipschitz constant is uniformly bounded. D-infinity(X) is compared with the space LIP infinity(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-infinity(X) with the Newtonian-Sobolev space N-1,N-infinity(X). In particular, if X Supports a doubling measure and satisfies a local Poincare inequality, we obtain that D-infinity(X) = N-1,N-infinity(X). (C) 2009 Elsevier Inc. All rights reserved.