I am thinking about the martingale version of Monge-Kantorovich Problem.
Let $\mu(x)$ and $\nu(y)$ denote two density laws on $\mathbb{R}$, and define $M(\mu,\nu)$ the set of densities $f(x,y)$ on $\mathbb{R}^2$ with marginal distributions $\mu$ and $\nu$. Then for a cost function $c(x,y)$, set
$$L(f)=\int_{\mathbb{R}^2}c(x,y)f(x,y)dxdy,\ \ f\in M(\mu,\nu)$$
Then the MKP can be formulated as
$$\sup_{f\in M(\mu,\nu)}L(f)$$
Then we study its martingale version, define
$$\mathcal{M}(\mu,\nu)=\{f\in M(\mu,\nu): \frac{\int_{\mathbb{R}}yf(x,y)dy}{\int_{\mathbb{R}}f(x,y)dy}=x,\ \ \forall x\in\mathbb{R}\}$$
I would like to estimate for $c(x,y)=|x-y|^p$, $p\geq 2$
$$\sup_{f\in\mathcal{M}(\mu,\nu)}L(f)$$
In other words, I look for some function $K(\mu,\nu)$ such that
$$\sup_{f\in\mathcal{M}(\mu,\nu)}L(f)\leq K(\mu,\nu)$$
Does someone have an idea? Many thanks!