Let $\mathcal{P}(\mathbb{R}^n)$ be the set of Borel probability measures on the Euclidean space $\mathbb{R}^n$ and consider thereof consisting of all probability measures $\mathbb{P}$ satisfying $\int\, \|x\|\mathbb{P}(dx)<\infty$. One can identify this as a subspace of the pre-dual of $\operatorname{Lip}(\mathbb{R}^n)$ (via $\mathbb{P}\mapsto \mathbb{P}-\delta_0$) normed by $\|f\|_{\operatorname{Lip}}:=\sup_{x\in \mathbb{R}^n}\, \|f(x)\| + \sup_{x_1,x_2\in \mathbb{R}^n}\, \frac{|f(x)-f(x_2)|}{\|x_1-x_2\|}$ which we metrize using the restriction of the dual-norm $$ d(\mathbb{P},\mathbb{Q}) := \sup_{\|f\|_{Lip}\leq 1}\,\int f d\mathbb{P} - \int f d\mathbb{Q}. $$ What's rather nice is that the Kantorovich duality (essentially) identifies this norm with the Wasserstein 1 distance from optimal transport on the set $\mathcal{P}(\mathbb{R}^n)$.
Suppose that $k,p\geq 1, k\in \mathbb{N}$ and consider the Sobolev space $W^{k,p}(\mathbb{R}^n)$. Define the norm $$ d_{k,p}(\mathbb{P},\mathbb{Q}) := \sup_{\|f\|_{W^{k,p}}\leq 1}\, \int f d\mathbb{P} - \int f d\mathbb{Q}, $$ Is there a (somewhat well-studied or classical) metric $\tilde{d}_{k,p}$ on a (reasonable) subset of $\mathcal{P}(\mathbb{R}^n)$ for which there is a similar duality; or at-least $$ d_{k,p}\leq \tilde{d}_{k,p}? $$
