Banach Journal of Mathematical Analysis

Weighted Banach spaces of Lipschitz functions

A. Jiménez-Vargas

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

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

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

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

Permanent link to this document
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

Keywords
Lipschitz function little Lipschitz function duality weighted Banach space

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


Export citation

References

  • [1] E. M. Alfsen and E. G. Effros,Structure in real Banach spaces, I, Ann. of Math. (2)96(1972), 98–173.
  • [2] H. Berninger and D. Werner,Lipschitz spaces and M-ideals, Extracta Math.18(2003), no. 1, 33–56.
  • [3] K. D. Bierstedt and W. H. Summers,Biduals of weighted Banach spaces of analytic functions, J. Aust. Math. Soc.54(1993), no. 1, 70–79.
  • [4] C. Boyd and P. Rueda,The biduality problem and M-ideals in weighted spaces of holomorphic functions, J. Convex Anal.18(2011), no. 4, 1065–1074.
  • [5] H. G. Dales, F. K. Dashiell, Jr., A.T.-M. Lau, and D. Strauss,Banach Spaces of Continuous Functions as Dual Spaces, CMS Books Math., Springer, Cham, 2016.
  • [6] R. B. Fraser,Banach spaces of functions satisfying a modulus of continuity condition, Studia Math.32(1969), 277–283.
  • [7] G. Godefroy, “Existence and uniqueness of isometric preduals: A survey” inBanach Space Theory (Iowa City, 1987), Contemp. Math.85, Amer. Math. Soc., Providence, 1989, 131–193.
  • [8] P. Harmand, D. Werner, and W. Werner,$M$-ideals in Banach Spaces and Banach Algebras, Lecture Notes in Math.1547, Springer, Berlin, 1993.
  • [9] J. A. Johnson,Banach spaces of Lipschitz functions and vector-valued Lipschitz functions, Trans. Amer. Math. Soc.148(1970), 147–169.
  • [10] N. J. Kalton,Spaces of Lipschitz and Hölder functions and their applications, Collect. Math.55(2004), no. 2, 171–217.
  • [11] J. L. Kelley,General Topology, Grad. Texts in Math.27. Springer, New York, 1975.
  • [12] R. E. Megginson,An Introduction to Banach Space Theory, Grad. Texts in Math.183, Springer, New York, 1998.
  • [13] J. Mujica,A Banach–Dieudonné theorem for germs of holomorphic functions, J. Funct. Anal.57(1984), no. 1, 31–48.
  • [14] K. F. Ng,On a theorem of Dixmier, Math. Scand.29(1971), 279–280.
  • [15] N. Weaver,Lipschitz Algebras, World Scientific, River Edge, NJ, 1999.