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

In the setting of Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation, let $(\mathcal{P}_{2}(\mathbb{R}^{d}),W_{2})$ be the Wasserstein space with its distance, a metric space as fixed there, and let $\rho$ be the centred heat gauge, a metric on $\mathcal{P}_{2}(\mathbb{R}^{d})$ by The Centred Heat Gauge: Metric Properties, Comparison with the Wasserstein Distance, Behaviour Under Translations, the Squared Gauge as a Test Function, and the Polarisation Inequality (i), satisfying

$$\rho(\mu,\nu)\le C_{\rho}\,W_{2}(\mu,\nu)\qquad\bigl(\mu,\nu\in\mathcal{P}_{2}(\mathbb{R}^{d})\bigr),$$

where $C_{\rho}$ is the nonnegative real number of The Centred Heat Gauge: Metric Properties, Comparison with the Wasserstein Distance, Behaviour Under Translations, the Squared Gauge as a Test Function, and the Polarisation Inequality (i). Let $A\subseteq\mathcal{P}_{2}(\mathbb{R}^{d})$ and let $f:A\to\mathbb{R}$, the real line carrying the metric of the absolute value. Continuity at a point relative to $A$ and upper and lower semicontinuity on $A$ are read in the metric space named in each claim. Then the following hold.

1. (Continuity)(i) Let $\mu\in A$. If $f$ is continuous at $\mu$ relative to $A$ in $(\mathcal{P}_{2}(\mathbb{R}^{d}),\rho)$, then $f$ is continuous at $\mu$ relative to $A$ in $(\mathcal{P}_{2}(\mathbb{R}^{d}),W_{2})$.

2. (Upper semicontinuity)(ii) If $f$ is upper semicontinuous on $A$ in $(\mathcal{P}_{2}(\mathbb{R}^{d}),\rho)$, then $f$ is upper semicontinuous on $A$ in $(\mathcal{P}_{2}(\mathbb{R}^{d}),W_{2})$.

3. (Lower semicontinuity)(iii) If $f$ is lower semicontinuous on $A$ in $(\mathcal{P}_{2}(\mathbb{R}^{d}),\rho)$, then $f$ is lower semicontinuous on $A$ in $(\mathcal{P}_{2}(\mathbb{R}^{d}),W_{2})$.

4. (Lipschitz bounds)(iv) Let $L\in\mathbb{R}$ be nonnegative and suppose that $|f(\mu)-f(\nu)|\le L\,\rho(\mu,\nu)$ for all $\mu,\nu\in A$. Then

$$|f(\mu)-f(\nu)|\le C_{\rho}L\,W_{2}(\mu,\nu)\qquad(\mu,\nu\in A).$$