Open Access
January 2018 Weighted Banach spaces of Lipschitz functions
A. Jiménez-Vargas
Banach J. Math. Anal. 12(1): 240-257 (January 2018). DOI: 10.1215/17358787-2017-0030


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.


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A. Jiménez-Vargas. "Weighted Banach spaces of Lipschitz functions." Banach J. Math. Anal. 12 (1) 240 - 257, January 2018.


Received: 12 April 2017; Accepted: 7 July 2017; Published: January 2018
First available in Project Euclid: 19 December 2017

zbMATH: 06841274
MathSciNet: MR3745583
Digital Object Identifier: 10.1215/17358787-2017-0030

Primary: ‎46E15
Secondary: 46A20

Keywords: Duality , Lipschitz function , little Lipschitz function , weighted Banach space

Rights: Copyright © 2018 Tusi Mathematical Research Group

Vol.12 • No. 1 • January 2018
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