Weighted Banach spaces of Lipschitz functions

Abstract

Given a pointed metric space X and a weight v on \widetilde{X} (the complement of the diagonal set in X x X), let Lip_v(X) and lip_v(X) denote the Banach spaces of all scalar-valued Lipschitz functions f on X vanishing at the basepoint such that v\Phi(f) is bounded and v\Phi(f) vanishes at infi nity on \widetilde{X}, respectively, where \Phi(f) is the de Leeuw's map of f on \widetilde{X}, under the weighted Lipschitz norm. The space Lip_v(X) has an isometric predual F_v(X) and it is proved that (Lip_v(X); \tau_{bw*} ) = (F_v(X),\tau_c) and F_v(X) = ((Lip_v(X),\tau_{bw*})',\tau_c), where \tau_{bw*} denotes the bounded weak* topology and \tau_c the topology of uniform convergence on compact sets. The linearization of the elements of Lip_v(X) is also tackled. Assuming that X is compact, we address the question as to when Lip_v(X) is canonically isometrically isomorphic to lip_v(X)**, and we show that this is the case whenever lip_v(X) is an M-ideal in Lip_v(X) and the so-called associated weights \widetilde{v}_L and \widetilde{v}_l coincide.