Let $X$ be a compact pointed metric subspace of the $d$-dimensional Euclidean space $(\mathbb{R}^d,d\_E)$ and let $AE(X)$ denote its Arens-Eells space. Then a result of [Nik Weaver](https://www.researchgate.net/scientific-contributions/9128689_Nik_Weaver) shows that for every Lipschitz map $f:X\rightarrow E$ into a separable Banach space, there exists a unique continuous linear extension $F:AE(X)\rightarrow E$ satisfying
$$
F\circ \delta = f,
$$
where $\delta$ is the canonical isometric embedding of $X$ in $AE(X)$. *(See [Nik's book](https://rads.stackoverflow.com/amzn/click/com/B07CWSG9YP) for more details).*


**Question:**




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Is there a *concrete* description of what $F$ is or how to explicitly construct it? I would like to use it for computations...



Concrete description of lift in Arens-Eells space
$AE(X)$ is the completion of the space of "molecules", i.e., the finitely supported functions $m: X \to \mathbb{R}$ which satisfy $\sum\_{p \in X}m(p) = 0$. The extension $F$ of $f: X \to E$ satisfies $F(m) = \sum\_{p \in X} m(p)f(p)$. (BTW $E$ need not be separable.)


