The Coordinate Fields of a Coupling, and the Second Moment as a Lipschitz Function of the Wasserstein Distance

In the setting of Plans, Marginals, Vector Fields and Symmetric Matrices on the Wasserstein Space: Standing Notation, let $m\in\mathbb{N}$ satisfy $1\le m$, and let $\mu,\nu\in\mathcal{P}_{2}(\mathbb{R}^{m})$, the second moment $M_{2}$, the Wasserstein space $\mathcal{P}_{2}(\mathbb{R}^{m})$, the couplings $\Pi(\mu,\nu)$ with their quadratic cost $I$, the distance $W_{2}$ and the spaces $L^{2}(\rho;\mathbb{R}^{m})$ being those fixed in Plans, Marginals, Vector Fields and Symmetric Matrices on the Wasserstein Space: Standing Notation §dimensions and Plans, Marginals, Vector Fields and Symmetric Matrices on the Wasserstein Space: Standing Notation §fields. Write $\mathrm{pr}_{1}^{m},\mathrm{pr}_{2}^{m}:\mathbb{R}^{m+m}\to\mathbb{R}^{m}$ for the coordinate projections of $\mathbb{R}^{m+m}$. Then the following hold.

1. (The coordinate fields of a coupling)¶ Let $\pi\in\Pi(\mu,\nu)$. Then the classes of $\mathrm{pr}_{1}^{m}$ and $\mathrm{pr}_{2}^{m}$ belong to $L^{2}(\pi;\mathbb{R}^{m})$, and

2. (Lipschitz bound)¶

3. (Bound on a Wasserstein ball)¶ If $R\in\mathbb{R}$ is nonnegative and $W_{2}(\mu,\nu)\le R$, then