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

Each result cited is universally quantified over the data in its own statement. The results on couplings are used with $d=n$, allowed since $1\le n$.

Step 1 ($\Phi$ is Borel, and a pointwise bound). By hypothesis $|\Phi(u)-\Phi(v)|\le L\,d_{E}(u,v)=L\lVert u-v\rVert$ for all $u,v\in\mathbb{R}^{n}$, the metric on $\mathbb{R}$ being $d_{\mathbb{R}}(s,t)=|s-t|$ of The Absolute Value Metric on the Real Line and $d_{E}(u,v)=\lVert u-v\rVert$ by claim 2 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$. A Lipschitz map is continuous by A Lipschitz Map is Uniformly Continuous, so $\Phi$ is Borel by Probability Measures on Euclidean Space and Random Vectors: Standing Notation §borel-maps. Both sides of the Lipschitz bound are nonnegative (claim 1 of Properties of the Absolute Value in an Ordered Field; claim 1 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ with claim 5 of Elementary Arithmetic in an Ordered Field, as $0\le L$), and for real $y$ one has $|y|^{2}=y^{2}$, because $|y|$ is $y$ or $-y$ (claim 1 of Properties of the Absolute Value in an Ordered Field) and $(-y)(-y)=yy$ (claim 2 of Zero Products and Elementary Identities in a Field). Hence claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field gives

Step 2 (integrability of $\Phi^{2}$). Let $R\in\{P,P'\}$; then $(\mathbb{R}^{n},\mathcal{B}(\mathbb{R}^{n}),R)$ is a probability space. Let $Y$ be the constant $\Phi(0_{\mathbb{R}^{n}})$ and $X=\Phi-Y$; both are random variables on it by claims 1 and 2 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions. By (1) with $v=0_{\mathbb{R}^{n}}$, $X(u)^{2}\le L^{2}\lVert u\rVert^{2}$ for every $u$, so claim 1 of Linearity and Monotonicity of the Lebesgue Integral (monotonicity and homogeneity for nonnegative functions) gives

since $R\in\mathcal{P}_{2}(\mathbb{R}^{n})$ (The Second Moment of a Probability Measure on Euclidean Space and the Probability Measures with Finite Second Moment §space); and $\mathbb{E}[Y^{2}]=\Phi(0_{\mathbb{R}^{n}})^{2}R(\mathbb{R}^{n})=\Phi(0_{\mathbb{R}^{n}})^{2}$ by The Integral of an Indicator Function is the Measure of the Set and the same claim. So $X$ and $Y$ are square-integrable, hence so is $\Phi=X+Y$ by that definition: $\int_{\mathbb{R}^{n}}\Phi^{2}\,dR<\infty$, so the nonnegative Borel function $\Phi^{2}$ (claim 3 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions) is integrable with respect to $R$ by Measure Spaces and the Lebesgue Integral: Standing Notation §integral. This gives the membership of $\Phi$ in $L^{2}(P;\mathbb{R})$ and $L^{2}(P';\mathbb{R})$ described in Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation §fields, with $\lVert\Phi\rVert_{P}=\sqrt{\int\Phi^{2}\,dP}$ and $\lVert\Phi\rVert_{P'}=\sqrt{\int\Phi^{2}\,dP'}$ as in the statement.

Step 3 (one coupling). Let $\pi\in\Pi(P,P')$. By Couplings on Euclidean Space: Product Coupling, Swap, Finiteness of the Cost, Push-Forward Couplings, Modifying One Marginal, Quantisation, Gluing over a Finitely Supported Measure, and the Lipschitz Bound §cost-finite, $I(\pi)$ is a nonnegative real number. On the probability space $(\mathbb{R}^{n+n},\mathcal{B}(\mathbb{R}^{n+n}),\pi)$ let $U=\Phi\circ\mathrm{pr}_{1}$ and $V=\Phi\circ\mathrm{pr}_{2}$, random variables because the projections are Borel (Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §projections) and compositions of Borel maps are Borel (Probability Measures on Euclidean Space and Random Vectors: Standing Notation §borel-maps). Since $(\mathrm{pr}_{1})_{\#}\pi=P$ and $(\mathrm{pr}_{2})_{\#}\pi=P'$ (Couplings of Two Probability Measures on Euclidean Space and Their Quadratic Cost §coupling), the change-of-variables formula of Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pushforward, applied to the nonnegative Borel function $\Phi^{2}$, gives

both finite by Step 2. So $U,V$ are square-integrable with mean-square norms (Square-Integrable Random Variables and the Mean-Square Inner Product) $\lVert U\rVert_{2}=\lVert\Phi\rVert_{P}$ and $\lVert V\rVert_{2}=\lVert\Phi\rVert_{P'}$. By the same definition $W=U-V$ and $-W$ are square-integrable, and $\lVert-W\rVert_{2}=\lVert W\rVert_{2}$ since $(-W)^{2}=W^{2}$ pointwise (claim 2 of Zero Products and Elementary Identities in a Field). By (1), $W(z)^{2}\le L^{2}\lVert\mathrm{pr}_{1}(z)-\mathrm{pr}_{2}(z)\rVert^{2}$ for every $z$, so claim 1 of Linearity and Monotonicity of the Lebesgue Integral and Couplings of Two Probability Measures on Euclidean Space and Their Quadratic Cost §cost give $\lVert W\rVert_{2}^{2}=\mathbb{E}[W^{2}]\le L^{2}I(\pi)$. As $L\sqrt{I(\pi)}$ is nonnegative with square $L^{2}I(\pi)$ (Existence and Uniqueness of the Nonnegative Square Root), claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field gives $\lVert W\rVert_{2}\le L\sqrt{I(\pi)}$.

Since $U=W+V$ and $V=(-W)+U$ pointwise, claim 2 of Cauchy-Schwarz and Triangle Inequalities for the Mean-Square Norm gives $\lVert U\rVert_{2}\le\lVert W\rVert_{2}+\lVert V\rVert_{2}$ and $\lVert V\rVert_{2}\le\lVert W\rVert_{2}+\lVert U\rVert_{2}$. Writing $\Delta=\bigl|\lVert\Phi\rVert_{P}-\lVert\Phi\rVert_{P'}\bigr|$, these two inequalities with claim 3 of Elementary Arithmetic in an Ordered Field and claim 4 of Elementary Order Arithmetic in an Ordered Field give $-\lVert W\rVert_{2}\le\lVert U\rVert_{2}-\lVert V\rVert_{2}\le\lVert W\rVert_{2}$, so claim 6 of Properties of the Absolute Value in an Ordered Field and the previous paragraph give

Step 4 (passage to $W_{2}$). The set $\Pi(P,P')$ is nonempty by Couplings on Euclidean Space: Product Coupling, Swap, Finiteness of the Cost, Push-Forward Couplings, Modifying One Marginal, Quantisation, Gluing over a Finitely Supported Measure, and the Lipschitz Bound §product; fix one of its members for use in (2). By The Quadratic Wasserstein Distance on Euclidean Space §distance, $W_{2}(P,P')$ is the nonnegative square root of $m=\inf\{I(\pi):\pi\in\Pi(P,P')\}$, so $W_{2}(P,P')^{2}=m$ by Existence and Uniqueness of the Nonnegative Square Root; and $0\le\Delta$ by claim 1 of Properties of the Absolute Value in an Ordered Field.

If $L=0$, then (2) for the fixed coupling gives $\Delta\le0$, so $\Delta=0=L\,W_{2}(P,P')$ by antisymmetry of the order and claim 1 of Zero Products and Elementary Identities in a Field.

If $0<L$, then $0<L^{-1}$ by claim 7 of Elementary Order Arithmetic in an Ordered Field, and multiplying (2) by $L^{-1}$ (claim 5 of Elementary Arithmetic in an Ordered Field) gives $0\le L^{-1}\Delta\le\sqrt{I(\pi)}$ for every $\pi\in\Pi(P,P')$; by claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field and Existence and Uniqueness of the Nonnegative Square Root, $(L^{-1}\Delta)^{2}\le I(\pi)$ for every such $\pi$. Thus $(L^{-1}\Delta)^{2}$ is a lower bound of $\{I(\pi):\pi\in\Pi(P,P')\}$, hence $(L^{-1}\Delta)^{2}\le m=W_{2}(P,P')^{2}$ by condition (ii) of Lower Bound and Greatest Lower Bound. Both $L^{-1}\Delta$ and $W_{2}(P,P')$ are nonnegative, so claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field gives $L^{-1}\Delta\le W_{2}(P,P')$, and multiplying by $0\le L$ (claim 5 of Elementary Arithmetic in an Ordered Field) gives $\Delta\le L\,W_{2}(P,P')$.

In both cases $\bigl|\lVert\Phi\rVert_{P}-\lVert\Phi\rVert_{P'}\bigr|\le L\,W_{2}(P,P')$.