Points of differentiability of the norm in Lipschitz-free spaces

We consider convex series of molecules in Lipschitz-free spaces, i.e. elements of the form μ=∑nλnδxn−δynd(xn,yn) such that ‖μ‖=∑n|λn|. We characterise these elements in terms of geometric conditions on the points xn, yn of the underlying metric space, and determine when they are points of Gâteaux di...

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Bibliographic Details
Published in:Journal of mathematical analysis and applications 2020-09, Vol.489 (2), p.124171, Article 124171
Main Authors: Aliaga, Ramón J., Rueda Zoca, Abraham
Format: Article
Language:English
Subjects:
Fréchet differentiability
Gâteaux differentiability
Lipschitz-free space
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Summary:We consider convex series of molecules in Lipschitz-free spaces, i.e. elements of the form μ=∑nλnδxn−δynd(xn,yn) such that ‖μ‖=∑n|λn|. We characterise these elements in terms of geometric conditions on the points xn, yn of the underlying metric space, and determine when they are points of Gâteaux differentiability of the norm. In particular, we show that Gâteaux and Fréchet differentiability are equivalent for finitely supported elements of Lipschitz-free spaces over uniformly discrete and bounded metric spaces, and that their tensor products with Gâteaux (resp. Fréchet) differentiable elements of a Banach space are Gâteaux (resp. Fréchet) differentiable in the corresponding projective tensor product.
ISSN:0022-247X
1096-0813
DOI:10.1016/j.jmaa.2020.124171