Proof of Continuity, Semicontinuity and Lipschitz Bounds Pass from the Centred Heat Gauge to the Wasserstein Distance

Each result cited is universally quantified over the data in its own statement and is applied here to the data named in the statement of the lemma. The real line carries the metric $d_{\mathbb{R}}$ with $d_{\mathbb{R}}(s,t)=|s-t|$ of The Absolute Value Metric on the Real Numbers.

A comparison of radii. Since $0\le C_{\rho}$ and $0<1$ (claim 6 of Elementary Order Arithmetic in an Ordered Field), the compatibility of the order with addition (an axiom of Ordered Field) gives $1\le1+C_{\rho}$, so $1+C_{\rho}$ is positive by mixed transitivity (claim 2 of Elementary Order Arithmetic in an Ordered Field) and its inverse $(1+C_{\rho})^{-1}$ is positive by claim 7 of that lemma. Strict compatibility of the order with addition (claim 1 of Elementary Order Arithmetic in an Ordered Field), applied to $0<1$, gives $C_{\rho}<1+C_{\rho}$, and multiplying by the positive $(1+C_{\rho})^{-1}$ (claim 10 of Elementary Order Arithmetic in an Ordered Field) yields

$$C_{\rho}\,(1+C_{\rho})^{-1}<1 .$$

Let $r\in\mathbb{R}$ be positive and put $r'=r\,(1+C_{\rho})^{-1}$, which is positive by claim 5 of Elementary Order Arithmetic in an Ordered Field. Let $\sigma,\tau\in\mathcal{P}_{2}(\mathbb{R}^{d})$ satisfy $W_{2}(\sigma,\tau)<r'$. Multiplying $W_{2}(\sigma,\tau)\le r'$ by the nonnegative $C_{\rho}$ (claim 5 of Elementary Arithmetic in an Ordered Field) gives $C_{\rho}W_{2}(\sigma,\tau)\le C_{\rho}r'$, while multiplying the displayed inequality by the positive $r$ (claim 10 of Elementary Order Arithmetic in an Ordered Field) gives $C_{\rho}r'=r\,C_{\rho}(1+C_{\rho})^{-1}<r$, the identification of the two products being the commutativity and associativity of multiplication (axioms of Field). With the hypothesis $\rho(\sigma,\tau)\le C_{\rho}W_{2}(\sigma,\tau)$ and the transitivity of $\le$ (Total Order on a Set) we obtain $\rho(\sigma,\tau)\le C_{\rho}r'$, and then $\rho(\sigma,\tau)<r$ by mixed transitivity. We record this as

$$(\ast)\qquad W_{2}(\sigma,\tau)<r\,(1+C_{\rho})^{-1}\ \Longrightarrow\ \rho(\sigma,\tau)<r\qquad\bigl(\sigma,\tau\in\mathcal{P}_{2}(\mathbb{R}^{d}),\ r\text{ positive}\bigr),$$

together with the positivity of $r(1+C_{\rho})^{-1}$.

Claim 1. Let $\varepsilon\in\mathbb{R}$ be positive. Since $f$ is continuous at $\mu$ relative to $A$ in $(\mathcal{P}_{2}(\mathbb{R}^{d}),\rho)$, Continuous Map Between Metric Spaces provides a positive $r\in\mathbb{R}$ such that every $\sigma\in A$ with $\rho(\mu,\sigma)<r$ satisfies $d_{\mathbb{R}}(f(\sigma),f(\mu))<\varepsilon$. By $(\ast)$, every $\sigma\in A$ with $W_{2}(\mu,\sigma)<r(1+C_{\rho})^{-1}$ satisfies $\rho(\mu,\sigma)<r$ and hence $d_{\mathbb{R}}(f(\sigma),f(\mu))<\varepsilon$. As $r(1+C_{\rho})^{-1}$ is positive and $\varepsilon$ was arbitrary, $f$ is continuous at $\mu$ relative to $A$ in $(\mathcal{P}_{2}(\mathbb{R}^{d}),W_{2})$, again by Continuous Map Between Metric Spaces.

Claim 2. Let $\mu\in A$ and let $\varepsilon\in\mathbb{R}$ be positive. Since $f$ is upper semicontinuous at $\mu$ relative to $A$ in $(\mathcal{P}_{2}(\mathbb{R}^{d}),\rho)$, Upper Semicontinuous Function on a Subset of a Metric Space provides a positive $r\in\mathbb{R}$ such that every $\sigma\in A$ with $\rho(\mu,\sigma)<r$ satisfies $f(\sigma)<f(\mu)+\varepsilon$. By $(\ast)$ the same conclusion holds for every $\sigma\in A$ with $W_{2}(\mu,\sigma)<r(1+C_{\rho})^{-1}$, so $f$ is upper semicontinuous at $\mu$ relative to $A$ in $(\mathcal{P}_{2}(\mathbb{R}^{d}),W_{2})$. As $\mu\in A$ was arbitrary, claim 2 follows.

Claim 3. Identical to claim 2, with Lower Semicontinuous Function on a Subset of a Metric Space in place of Upper Semicontinuous Function on a Subset of a Metric Space and the conclusion $f(\mu)-\varepsilon<f(\sigma)$ in place of $f(\sigma)<f(\mu)+\varepsilon$.

Claim 4. Let $\mu,\nu\in A$. Multiplying $\rho(\mu,\nu)\le C_{\rho}W_{2}(\mu,\nu)$ by the nonnegative $L$ (claim 5 of Elementary Arithmetic in an Ordered Field) gives

$$L\,\rho(\mu,\nu)\le L\,C_{\rho}W_{2}(\mu,\nu)=C_{\rho}L\,W_{2}(\mu,\nu),$$

the last identity by the commutativity and associativity of multiplication. Combining with the hypothesis $|f(\mu)-f(\nu)|\le L\rho(\mu,\nu)$ and the transitivity of $\le$ (Total Order on a Set) gives $|f(\mu)-f(\nu)|\le C_{\rho}L\,W_{2}(\mu,\nu)$, which is claim 4.