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
- Example 3.10 of Weaver's bookWeaver'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 3.10Example 3.10 and Theorem 3.13Theorem 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
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 $\alpha \in (0,1)$)?
