In one of the problems that I am working on, I came across the topic of smoothness of optimal transport for Gaussian mixtures. In particular, let $P=P_\theta=\sum_{i=1}^k \frac{1}{k}\mathcal{N}(x| \theta_i, \sigma^2 I)$ be a Gaussian $k$-mixture distribution where each $\theta_i \in \mathbb{R}^d$ and $\sigma>0$ . Let the target distribution $Q=\sum_{i=1}^k \frac{1}{k}\mathcal{N}(x| \mu_i, \sigma^2 I)$ also be a $k$-Gaussian mixture. We know that the standard Wasserstein-$2$ loss is given by $$ W_2^2(P_\theta, Q)= \inf_{T: T\#P_\theta = Q}E\|X-T(X)\|^2= \sup_{f,g:f(x)+g(y) \leq \|x-y\|^2} E_{P_\theta}[f(X)]+E_Q[g(Y)]. $$ Furthermore, we know that the optimal transport map $T$ and the optimal dual function $f$ are related to the convex Kantorovich potential $u$ via $$ T(x) =\nabla u(x)= x- \frac{\nabla f(x)}{2}. $$
So my question is regarding the smoothness of the map $T$. In particular, I am looking for bounds on the spectral norm of the Jacobian of $T$, or equivalently Hessian of the convex potential $u$: $$ \| E_X[\nabla T(X)] \|_2=\| E_X[\nabla^2 u(X)] \|_2 \leq \text{ something}, \quad X \sim \mathcal{N}(x| \theta_i, \sigma^2 I), i \in [k]. $$
To the best of my knowledge, regularity of the transport maps are only known in the settings where the distributions $P$ and $Q$ have bounded support and also are log-concave distributions, for example equation (1.1) in http://www.math.ias.edu/~yjhaveri/documents/lipcov.pdf. So I am wondering if any such smoothness results are known in my setting. Please point me to any relevant references or techniques that you think might be useful.
