Let $\Omega$ be a non-empty, simply connected, and open subset of $\mathbb{R}^d$ for some positive integer $d$. Let $k$ be a non-negative integer. Consider the Banach space $C^{k,1}_0(\Omega)$ equipped with its normed by $$ \|f\|_{C^{k,1}} := \max_{|\beta|\leq k} \max_{x_1,x_2\in \Omega}\, \|D^{\beta}f(x_1)-D^{\beta}f(x_2)\| + \max_{|\beta|=k} \sup_{x_1,x_2;\,x_1\neq x_2}\, \frac{\|D^{\beta}f(x_1) - D^{\beta}f(x_2)\|}{\|x_1-x_2\|}. $$ Now, consider the closure of the set of point-evaluation maps $\delta_x:f\mapsto f(x)$ in the dual space of $C_0^{k,1}$ with respect to the dual norm $$ D(F,G) := \sup_{f\in C_0^{k,1}(\Omega),\,\|f\|_{C^{k,1}}\leq 1}\, (F-G)(f); $$ we denote this closure by $\mathcal{P}^{k,1}_0(\Omega)$.
If $k=0$, $\Omega=\mathbb{R}^d$, then $\mathcal{P}_0^{k,1}(\Omega)$ is the 1-Wasserstein space and there is a nice transport-type duality describing $D|_{\mathcal{P}^{k,1}_0(\Omega)}$ in terms of optimal couplings.
However, what about the general case; especially for $k>0$. Is there an optimal transport type description of $D|_{\mathcal{P}^{k,1}_0(\Omega)}$? That is, I am looking for a Kantorovich-esque duality for $D|_{\mathcal{P}^{k,1}_0(\Omega)}$ on $\mathcal{P}^{k,1}_0(\Omega)$
Relevant Literature
