While I don't work on the regularity theory for the optimal transport map, I was curious about the open problem 1.28 listed in Ambrosio and Gigli's User's guide: the problem to determine whether we have $T\in W^{1,1}$ for the transport map $T$ is still open?

While I don't work on the regularity theory for the optimal transport map, I was curious about the open problem 1.28 listed in Ambrosio and Gigli's User's guide: the problem to determine whether we have $T\in W^{1,1}$ for the transport map $T$ is still open?

More precisely, assume $\Omega_1,\Omega_2\subset \mathbb{R}^d$ are two bounded and connected open sets, $\mu=\rho \mathcal{L}^d |_{\Omega_1}, \nu=\eta \mathcal{L}^d |_{\Omega_2}$ with $0<c\leqslant \rho,\eta \leqslant C$ for some $c,C\in\mathbb{R}$. Assume also that $\Omega_2$ is convex, then the optimal transport map $T\in W^{1,1}(\Omega_1)$?

I didn't work on the regularity theory for optimal transport map, while I was curious if the open problem 1.28: the transport map $T\in W^{1,1}$ is still open, listed in the user's guide by Ambrosio and Gigli.

While I don't work on the regularity theory for the optimal transport map, I was curious about the open problem 1.28 listed in Ambrosio and Gigli's User's guide: the problem to determine whether we have $T\in W^{1,1}$ for the transport map $T$ is still open?