The Mollified Density Cost is Bounded, Lipschitz for the Wasserstein Distance, and Below the Density Cost

In the setting of Probability Measures on Euclidean Space and Random Vectors: Standing Notation, let $d$, $\lambda_{d}$, $\mathcal{P}_{2}(\mathbb{R}^{d})$, the mollifier kernel $\eta$ of radius $1$, the scale $\varepsilon$ with $0<\varepsilon$, the nonnegative real $L$ and the convex Lipschitz integrand $\Phi$ with constant $L$ be as in The Mollified Density Cost of a Probability Measure with Finite Second Moment, and let $\mathcal{G}_{\Phi,\varepsilon}$ be the mollified density cost on $\mathcal{P}_{2}(\mathbb{R}^{d})$. Let $W_{2}$ be the quadratic Wasserstein distance on $\mathcal{P}_{2}(\mathbb{R}^{d})$, a metric by The Quadratic Wasserstein Distance is a Metric on the Wasserstein Space §metric; let $\kappa_{d}=\lambda_{d}(\bar{B}(0,1))$ be the normalising constant of closed balls; and let $\mathcal{P}_{2}^{\mathrm{ac}}(\mathbb{R}^{d})$ and the density cost $\mathcal{G}_{\Phi}$ be those of The Density Cost of a Convex Lipschitz Integrand. Natural numbers occurring as real factors are read through the canonical map into $\mathbb{R}$, powers with natural exponent are those of Natural Number Power of an Element of a Field, $t^{-1}$ is the multiplicative inverse of a real $t\ne0$, $2=1+1$, and the partial derivatives $\partial_{i}\eta$, $i\in[d]$, are those of Mollifying a Probability Measure on Euclidean Space: Kernel Bounds, the Translation Estimate, the Mollified Density and Its Cost §kernel. Then the following hold.

1. (Bounds)¶ For every $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$, $0\le\mathcal{G}_{\Phi,\varepsilon}(\mu)\le L$.

2. (Lipschitz continuity)¶ Let $D$ be a nonnegative real number with $|\partial_{i}\eta(z)|\le D$ for every $z\in\mathbb{R}^{d}$ and every $i\in[d]$; such a $D$ exists by Mollifying a Probability Measure on Euclidean Space: Kernel Bounds, the Translation Estimate, the Mollified Density and Its Cost §kernel. Then for all $\mu,\nu\in\mathcal{P}_{2}(\mathbb{R}^{d})$,

that is, $\mathcal{G}_{\Phi,\varepsilon}$ is Lipschitz with constant $K$ from the metric space $(\mathcal{P}_{2}(\mathbb{R}^{d}),W_{2})$ to $\mathbb{R}$ with the absolute-value metric, and $K$ depends only on $L$, $\varepsilon$, $d$ and the kernel $\eta$ (through $D$).

3. (Comparison with the density cost)¶ For every $\mu\in\mathcal{P}_{2}^{\mathrm{ac}}(\mathbb{R}^{d})$,