Let $\mu$ and $\nu$ be two probability measures with finite moments (in $\mathcal{P}_2(\mathbb{R})$) equipped with 2-Wasserstein distance. Let $F_\mu$, $F_\nu$ be their cumulative distribution functions respectively. Then it is well known that the optimal transport map from $\mu$ to $\nu$ is given by $T_\mu(x):=F^{-1}_\nu(F_\mu(x))$ provided $\mu$ is absolutely continuous w.r.t. Lebesgue measure. Consider the map $T^\sigma_\mu(x):=F^{-1}_{\nu_\sigma}(F_{\mu_\sigma}(x))$ where $\mu_\sigma:=\mu*N_\sigma$ and $N_\sigma$ is the normal distribution with zero mean and variance $\sigma^2$.
If $X$ has the law $\mu_\sigma$, then the law of $T^\sigma_\mu(X+N_\sigma)$ is $\nu_\sigma$ which implies that $$\int_{\mathbb{R}}|T^\sigma_\mu(x)|^2\mu_\sigma(dx)=\int_{\mathbb{R}}|y|^2\nu_\sigma(dy)$$
Let $z \in \mathbb{R}$, can we have some estimate on the term
$$\int_{\mathbb{R}}|T^\sigma_\mu(x+z)|^2\mu_\sigma(dx)?$$ I believe there should be some results at least for some small $z$ due to the stability of the map $T^\sigma_\mu$?
