Let $X=\mathbb{R}^n$; I am also interested in the general case $X$ is a metric space but for simplicity let's focus on Euclidean space. Let $\mathcal{P}(X)$ denote the space of Borel probability measures on $X$. I am interested in maps $f:(\mathcal{P}(X), W_q) \to (\mathcal{P}(X), TV)$ and $f:(\mathcal{P}(X), TV) \to (\mathcal{P}(X), W_q)$, where $W_q$ is the $q$th-order Wasserstein distance and $TV$ is the strong topology induced by the total variation norm on measures. In other words, the domain is given the weak (probabilist's) topology and the codomain is given the strong topology (or vice versa).
I want to study the smoothness properties of such functions $f$. I am particularly interested in computing moduli of continuity and/or differentiability of $f$. I am being deliberately vague here: Since we are dealing with a map between two metric spaces, differentiability isn't even well-defined, although one can imagine reasonable notions such as metric derivatives along paths or even $$ f'(P) := \lim_{Q\to P} \frac{TV(f(P),f(Q))}{W_q(P,Q)}. $$ I am curious if smoothness/regularity of functions in this setting (or something closely related) has been studied before, and if so, what references/techniques there are. For example, what are reasonable notions of "differentiability" for such functions, and if so, can they be related to Lipschitzness/moduli of continuity?
