Known Lipschitz-free spaces

Question

The Lipschitz-Free space (also known as Arens-Eells spaces) $\mathcal{F}(X,d)$ over a pointed metric space $(X,d)$ is a well-studied object. In many instances, we have "concrete" representations of it.

Example

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• Example 3.10 of Weaver's book: $$ \mathcal{F}(\mathbb{N},d_{discrete})\cong \ell^1(\mathbb{N}) , $$ a more general version of this result still holds true when $\mathbb{N}$ is replaced by any set metrized by the discrete metric and in the case where $\mathbb{N}$ is replaced by a countable tree with the graph metric (see Example 3.10 and Theorem 3.13, respectively).
• For $\mathbb{R}^d$ it is known that $$ \mathcal{F}(\mathbb{R}^d,d_{Euclidean})\cong L^1(\mathbb{R}^d;\mathbb{R}^d), $$ and more generally any convex domain can be represented as the quotient of $L^1(\mathbb{R}^d;\mathbb{R}^d)$ with respect to a subspace of $(L^1)-$vector fields with divergence 0.

Question

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Are there known concrete representations the spaces $\mathcal{F}_{\omega}(X)$ studied by Kalton; where $$ \mathcal{F}_{\omega}(X,d)\triangleq \mathcal{F}(X,\omega\circ d), $$ and $\omega$ is a suitable gauge?
Specifically in the case where $(X,d)=\mathbb{R}$ and $\omega(t)=t^{\alpha}$ or $\omega(t)=\max(t,t^{\alpha})$ (and $\alpha \in (0,1)$)?

Answer 1

Let me try again, at least for the case of metric spaces. In my opinion, the question has been answered completely but I don‘t think you will like the description. The sloppy way to state it is that these free spaces are always spaces of measures on the underlying space. Thus for your example of a discrete space, the space of bounded, Lipschitz functions is just $\ell^\infty$. Its unit ball has a natural compact topology, pointwise convergence, and so its dual, not as a Banach space but as a Waelbroeck space (I will add references later), is $\ell^1$ i.e., the space of bounded, radon measures on the metric space, trivially in this case.

Before turning to metric spaces, I think that it would be clearer if I jump to the more general case of uniform spaces: Pachl has introduced the space $M^u$ of uniform measures and they have the following universal property: the uniform space $X$ embeds into it in the usual way and every bounded, uniformly continuous mapping from it into a Banach space has a unique extension to a continuous linear mapping on $M^u$. In this sense, it is the free lcs over $X$. It can be constructed in several ways, one of them being that it is the dual of the space of bounded, uniformly continuous functions on $X$, not as a Banach space—-that would be the measures on the Samuel compactifucation—— but for a coarser structure.

Returning to metric spaces, there are three cases to consider——pointed spaces with radius $\leq 1$, spaces with bounded metric and the rest. The second case can essentially be reduced to the first one (fix an arbitrary base point and rescale). The second one requires one to take care of the fact that Lipschitz functions need not be bounded—-one has to use a supremum norm in addition to the Lipschitz constant. In order to keep this contribution manageable, I will return to the first one. Here, the space of Lipschitz functions is not just a Banach space, but is Waelbroek and its dual in this sense has precisely the property that you (I hope) are looking for. The metric space embeds isometrically into it and every Lipschitz map from the space into a Banach space lifts to a unique continuous linear mapping.

The original paper of Arens and Eells in Pac. Jour. (1956), as I recall it, doesn‘t mention the crucial universal property. The first reference on free locally convex spaces to my knowledge is Raikov „Free locally convex spaces for uniform spaces“ in Math. Sb. (1964). There are also two articles on this subject by Tomášek (who called them $\Lambda$ structures) in the Czech. Math. J. (1970). The Banach-Waelbroeck duality is due to Waelbroeck and Buchwalter—-probably the most accessible version is the Cigler-Michor-Losert monograph on categories of Banach spaces. Pachl‘s theory of uniform measures is in his text „Uniform spaces and measures“. The treatment of free Banach spaces and uniform measures using duality is in the article I already mentioned —-„Uniform measures and Co-Saks spaces“ in Springer Lecture Notes, 843 (1978).

I have no information to offer on the more general situation investigated by Kalton that you mention.