We define $$ \text{Lip}_0(\mathbb R^n)=\{f:\mathbb R^n\rightarrow \mathbb R, \text{such that $f(0)=0$ and } \sup_{x\not=y}\frac{\vert f(x)-f(y)\vert}{\vert x-y\vert}<+\infty. \} $$ It is well-known that $\text{Lip}_0(\mathbb R^n)$ is a Banach space, which is the dual space of the so-called $\mathcal F(\mathbb R^n)$, a.k.a. the Lipschitz-free space of $\mathbb R^n$.

Claim: $\text{Lip}_0(\mathbb R^n)$ is the dual space of $X/N$ where $X$ is the space of $L^{1}({\mathbb R}^{n})$ vector fields and $N$ is the subspace of vector fields with null divergence. In other words, with $$ X=(L^{1}({\mathbb R}^{n}))^{n},\quad N=\{(f_{j})_{1\le j\le n}\in X, \ \sum_{1\le j\le n}\frac{\partial f_{j}}{\partial x_{j}}=0\}, $$ we have $ \text{Lip}_0(\mathbb R^n)=(X/N)^{*}. $ Note that in the easy case $n=1$, we find the familiar $\mathcal F(\mathbb R)=L^1(\mathbb R)$. The derivatives above are taken in the distribution sense.

Questions. (1) Is the statement of this claim well-known? (2) Could it be useful to describe more explicitly the properties of $\mathcal F(\mathbb R^n)$ when $n\ge 2$?