Points of differentiability of the norm in Lipschitz-free spaces

Abstract

We consider convex series of molecules in Lipschitz-free spaces, i.e. elements of the form $μ=\sum_n λ_n \frac{δ_{x_n}-δ_{y_n}}{d(x_n,y_n)}$ such that $\|μ\|=\sum_n |λ_n |$. We characterise these elements in terms of geometric conditions on the points $x_n$, $y_n$ 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.