Weighted Banach spaces of Lipschitz functions

Abstract

Given a pointed metric space $X$ and a weight $v$ on $\tilde{X}$ (the complement of the diagonal set in $X\times X$), let ${\mathrm{Lip}}_v\left(X\right)$ and ${\mathrm{lip}}_v\left(X\right)$ denote the Banach spaces of all scalar-valued Lipschitz functions $f$ on $X$ vanishing at the basepoint such that $v\Phi \left(f\right)$ is bounded and $v\Phi \left(f\right)$ vanishes at infinity on $\tilde{X}$, respectively, where $\Phi \left(f\right)$ is the de Leeuw’s map of $f$ on $\tilde{X}$, under the weighted Lipschitz norm. The space ${\mathrm{Lip}}_v\left(X\right)$ has an isometric predual ${\mathcal{F}}_v\left(X\right)$ and it is proved that $\left({\mathrm{Lip}}_v\right(X),{\tau }_{{bw}^{\ast }})=\left({\mathcal{F}}_v\right(X{)}^{\ast },{\tau }_c)$ and ${\mathcal{F}}_v\left(X\right)=\left(\right({\mathrm{Lip}}_v\left(X\right),{\tau }_{{bw}^{\ast }})\text{'},{\tau }_c)$, where ${\tau }_{{bw}^{\ast }}$ denotes the bounded weak∗ topology and ${\tau }_c$ the topology of uniform convergence on compact sets. The linearization of the elements of ${\mathrm{Lip}}_v\left(X\right)$ is also tackled. Assuming that $X$ is compact, we address the question as to when ${\mathrm{Lip}}_v\left(X\right)$ is canonically isometrically isomorphic to ${\mathrm{lip}}_v(X{)}^{\ast \ast }$, and we show that this is the case whenever ${\mathrm{lip}}_v\left(X\right)$ is an M-ideal in ${\mathrm{Lip}}_v\left(X\right)$ and the so-called associated weights ${\tilde{v}}_L$ and ${\tilde{v}}_l$ coincide.