Weighted Banach spaces of Lipschitz functions

Abstract

Given a pointed metric space X and a weight v on X˜ (the complement of the diagonal set in X×X), let Lipv(X) and lipv(X) denote the Banach spaces of all scalar-valued Lipschitz functions f on X vanishing at the basepoint such that vΦ(f) is bounded and vΦ(f) vanishes at infinity on X˜, respectively, where Φ(f) is the de Leeuw’s map of f on X˜, under the weighted Lipschitz norm. The space Lipv(X) has an isometric predual Fv(X) and it is proved that (Lipv(X),τbw∗)=(Fv(X)∗,τc) and Fv(X)=((Lipv(X),τbw∗)',τc), where τbw∗ denotes the bounded weak∗ topology and τc the topology of uniform convergence on compact sets. The linearization of the elements of Lipv(X) is also tackled. Assuming that X is compact, we address the question as to when Lipv(X) is canonically isometrically isomorphic to lipv(X)∗∗, and we show that this is the case whenever lipv(X) is an M-ideal in Lipv(X) and the so-called associated weights v˜L and v˜l coincide.