The Kantorovich Dual of the 1-Wasserstein distance $W_1(p,q)$ between two densities $p(x), q(x)$ is given by $$W_1(p,q) = \sup_{|f|_L\leq 1} \int f(x)(p(x)-q(x))dx$$
with $|f|_L$ denoting the Lipschitz constant of $f$. The function $$f_*^{p,q}=\arg\sup_{|f|_L\leq 1} \int f(x)(p(x)-q(x))dx$$ attaining this supremum is called the Kantorovich potential.
To examine the asymptotics of estimators asymptotically minimizing a Wasserstein distance, I want to understand if there's a way to characterize the pathwise derivative $$\nabla_{p\to p'} f_*^{p,q}(x)=\frac{d}{d\tau} f_*^{p+\tau p',q}(x)|_{\tau=0}$$ in some direction $p'$ at $p=q$, under sufficiently strong regularity conditions (identical support + smoothness of $p,q$, some tie-braker rule to get uniqueness of $f_*$ if necessary, ...). I found a paper characterizing a representation of $f_*^{p,q}$ (which we could differentiate wrt $p$) when the supports of $p,q$ are disjoint, but that doesn't help me since the asymptotics of an estimator depend on what happens at $p\approx q$.
Any ideas/pointers? Or do you expect the Kantorovich potential to be nondifferentiable wrt to the densities at $p=q$?