Smoothing operators in multi-marginal Optimal Transport

Given $N$ absolutely continuous probabilities ${\rho }_1,\dots ,{\rho }_N$ over ${\mathbb{R}}^d$ which have Sobolev regularity, and given a transport plan $P$ with marginals ${\rho }_1,\dots ,{\rho }_N$, we provide a universal technique to approximate $P$ with Sobolev regular transport plans with the same marginals. Moreover, we prove a sharp control of the energy and some continuity property of the approximating family.