## Banach Journal of Mathematical Analysis

### Weighted Banach spaces of Lipschitz functions

A. Jiménez-Vargas

#### Abstract

Given a pointed metric space $X$ and a weight $v$ on $\widetilde{X}$ (the complement of the diagonal set in $X\times X$), let $\mathrm{Lip}_{v}(X)$ and $\mathrm{lip}_{v}(X)$ 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 infinity 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 $\mathrm{Lip}_{v}(X)$ has an isometric predual $\mathcal{F}_{v}(X)$ and it is proved that $(\mathrm{Lip}_{v}(X),\tau_{\operatorname{bw}^{*}})=(\mathcal{F}_{v}(X)^{*},\tau_{c})$ and $\mathcal{F}_{v}(X)=((\mathrm{Lip}_{v}(X),\tau_{\operatorname{bw}^{*}})',\tau_{c})$, where $\tau_{\operatorname{bw}^{*}}$ denotes the bounded weak∗ topology and $\tau_{c}$ the topology of uniform convergence on compact sets. The linearization of the elements of $\mathrm{Lip}_{v}(X)$ is also tackled. Assuming that $X$ is compact, we address the question as to when $\mathrm{Lip}_{v}(X)$ is canonically isometrically isomorphic to $\mathrm{lip}_{v}(X)^{**}$, and we show that this is the case whenever $\mathrm{lip}_{v}(X)$ is an M-ideal in $\mathrm{Lip}_{v}(X)$ and the so-called associated weights $\widetilde{v}_{L}$ and $\widetilde{v}_{l}$ coincide.

#### Article information

Source
Banach J. Math. Anal., Volume 12, Number 1 (2018), 240-257.

Dates
Accepted: 7 July 2017
First available in Project Euclid: 19 December 2017

https://projecteuclid.org/euclid.bjma/1513674119

Digital Object Identifier
doi:10.1215/17358787-2017-0030

Mathematical Reviews number (MathSciNet)
MR3745583

Zentralblatt MATH identifier
06841274

Subjects
Primary: 46E15: Banach spaces of continuous, differentiable or analytic functions
Secondary: 46A20: Duality theory

#### Citation

Jiménez-Vargas, A. Weighted Banach spaces of Lipschitz functions. Banach J. Math. Anal. 12 (2018), no. 1, 240--257. doi:10.1215/17358787-2017-0030. https://projecteuclid.org/euclid.bjma/1513674119

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