The Mean-Square Norm of a Lipschitz Function Depends Lipschitz-Continuously on the Measure for the Wasserstein Distance

In the setting of Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation, whose probability space $(\Omega,\mathcal{F},P)$ is not used (the letters $P$ and $P'$ below denote probability measures on $\mathbb{R}^{n}$), let $n\in\mathbb{N}$, let $L\in\mathbb{R}$ be nonnegative, let $\Phi:\mathbb{R}^{n}\to\mathbb{R}$ be Lipschitz with constant $L$ for the Euclidean distance and the metric of The Absolute Value Metric on the Real Line, and let $P,P'\in\mathcal{P}_{2}(\mathbb{R}^{n})$, the set of The Second Moment of a Probability Measure on Euclidean Space and the Probability Measures with Finite Second Moment §space, with $W_{2}$ the Wasserstein distance.

(Bound)¶ $\Phi$ is Borel and $\Phi^{2}$ is integrable with respect to $P$ and to $P'$, so that $\Phi$ lies in $L^{2}(P;\mathbb{R})$ and in $L^{2}(P';\mathbb{R})$ of Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation §fields; with $\lVert\Phi\rVert_{P}$ and $\lVert\Phi\rVert_{P'}$ the norms of that clause, which are the nonnegative square roots of $\int_{\mathbb{R}^{n}}\Phi^{2}\,dP$ and $\int_{\mathbb{R}^{n}}\Phi^{2}\,dP'$,