Banach Journal of Mathematical Analysis

Weighted Banach spaces of Lipschitz functions

A. Jiménez-Vargas

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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.

Article information

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

Received: 12 April 2017
Accepted: 7 July 2017
First available in Project Euclid: 19 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Lipschitz function little Lipschitz function duality weighted Banach space


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.

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