McCann's Jacobian Equation Along an Optimal Map Between Absolutely Continuous Measures

In the setting of Probability Measures on Euclidean Space and Random Vectors: Standing Notation and Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation, let $d\in\mathbb{N}$ satisfy $1\le d$, let $\lambda_{d}$ be Lebesgue measure on $\mathcal{B}(\mathbb{R}^{d})$, and let densities with respect to $\lambda_{d}$ be those of The Radon-Nikodym Theorem for a Finite Measure and a Sigma-Finite Measure, and Uniqueness of Densities. Let $\mu,\nu$ belong to the set $\mathcal{P}_{2}(\mathbb{R}^{d})$ of probability measures with finite second moment and be absolutely continuous, with densities $\rho$ and $\rho_{1}$ with respect to $\lambda_{d}$. Let $T$ be an optimal map from $\mu$ to $\nu$ and $S$ an optimal map from $\nu$ to $\mu$. Let $G,G'\subseteq\mathbb{R}^{d}$ be open and convex with $\mu(G)=1$ and $\nu(G')=1$, let $\varphi:G\to\mathbb{R}$ be convex on $G$ and $\psi:G'\to\mathbb{R}$ convex on $G'$, with subdifferentials $\partial_{G}\varphi$ and $\partial_{G'}\psi$, and let $D,D'\in\mathcal{B}(\mathbb{R}^{d})$ satisfy $D\subseteq G$, $D'\subseteq G'$, $\mu(D)=1$, $\nu(D')=1$, $\partial_{G}\varphi(x)=\{T(x)\}$ for $x\in D$ and $\partial_{G'}\psi(y)=\{S(y)\}$ for $y\in D'$. Twice differentiability at a point and the Hessian $D^{2}\varphi(x)$ are as fixed there, and $\det$ is the determinant.

1. (Jacobian equation)¶ There is $X\in\mathcal{B}(\mathbb{R}^{d})$ with $X\subseteq D$ and $\mu(X)=1$ such that every $x\in X$ has the properties listed in Along an Optimal Map between Absolutely Continuous Measures the Hessians of the Two Convex Potentials are Inverse Matrices §hessians and satisfies