## 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.

## Citation

A. Jiménez-Vargas. "Weighted Banach spaces of Lipschitz functions." Banach J. Math. Anal. 12 (1) 240 - 257, January 2018. https://doi.org/10.1215/17358787-2017-0030

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