Translation and Mollification Estimates in the Integrable Norm for a Density of Finite Fisher Information, and the Mollified Density Cost

In the setting of Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation, with the space $L^{2}(\mu;\mathbb{R}^{d})$, its inner product $\langle\cdot,\cdot\rangle_{\mu}$ and its norm $\lVert\cdot\rVert_{\mu}$; finite Fisher information, the set $\mathcal{P}_{2}^{\mathcal{I}}(\mathbb{R}^{d})$, the score $\xi_{\mu}$ and the Fisher information $\mathcal{I}(\mu)=\lVert\xi_{\mu}\rVert_{\mu}^{2}$ are those of that definition, and $\sqrt{t}$ is the nonnegative square root of a nonnegative real $t$, as in Probability Measures on Euclidean Space and Random Vectors: Standing Notation §spaces. $\lambda_{d}$ is Lebesgue measure on $\mathcal{B}(\mathbb{R}^{d})$, and $\int_{\mathbb{R}^{d}}f(y)\,dy$ denotes $\int_{\mathbb{R}^{d}}f\,d\lambda_{d}$; Borel is as in Probability Measures on Euclidean Space and Random Vectors: Standing Notation §borel-maps, and integrable, for $\lambda_{d}$ or for a probability measure, is as in Measure Spaces and the Lebesgue Integral: Standing Notation §integral. Absolute continuity is that of that definition, and densities with respect to $\lambda_{d}$ are those of The Radon-Nikodym Theorem for a Finite Measure and a Sigma-Finite Measure, and Uniqueness of Densities. Mollifier kernels on $\mathbb{R}^{d}$ are those of that definition; convex Lipschitz integrands are those of that definition; the density cost $\mathcal{G}_{\Phi}$, defined on the set $\mathcal{P}_{2}^{\mathrm{ac}}(\mathbb{R}^{d})$ of absolutely continuous members of $\mathcal{P}_{2}(\mathbb{R}^{d})$, is that of that definition; and the mollified density cost $\mathcal{G}_{\Phi,\varepsilon}$ is that of that definition.

Let $\mu\in\mathcal{P}_{2}^{\mathcal{I}}(\mathbb{R}^{d})$ be absolutely continuous, and let $\rho:\mathbb{R}^{d}\to\mathbb{R}$ be a density of $\mu$ with respect to $\lambda_{d}$; one exists by The Radon-Nikodym Theorem for a Finite Measure and a Sigma-Finite Measure, and Uniqueness of Densities §existence, $\lambda_{d}$ being $\sigma$-finite by Lebesgue Measure on Euclidean Space is Sigma-Finite §sigma-finite, and it is Borel and nonnegative.

1. (Translation)¶ For every $z\in\mathbb{R}^{d}$ the function $y\mapsto\rho(y-z)-\rho(y)$ is Borel and integrable with respect to $\lambda_{d}$, and

2. (Mollification)¶ Let $\eta$ be a mollifier kernel of radius $1$ on $\mathbb{R}^{d}$ and let $\varepsilon\in\mathbb{R}$ be positive. Let $\eta_{\varepsilon}:\mathbb{R}^{d}\to\mathbb{R}$, $\eta_{\varepsilon}(y)=(\varepsilon^{-1})^{d}\,\eta(\varepsilon^{-1}y)$, be the rescaled kernel, a mollifier kernel of radius $\varepsilon$ by Rescaling a Mollifier Kernel, and let $\eta_{\varepsilon}*\mu:\mathbb{R}^{d}\to\mathbb{R}$,

be the mollified density entering The Mollified Density Cost of a Probability Measure with Finite Second Moment §cost. Then $\eta_{\varepsilon}*\mu$ is continuous, nonnegative, bounded and Borel; for every $y\in\mathbb{R}^{d}$ the function $z\mapsto\eta_{\varepsilon}(z)\rho(y-z)$ is integrable with respect to $\lambda_{d}$ and

and the function $y\mapsto(\eta_{\varepsilon}*\mu)(y)-\rho(y)$ is integrable with respect to $\lambda_{d}$, with

3. (Cost)¶ Let $\eta$, $\varepsilon$ and $\eta_{\varepsilon}$ be as in clause 2, let $L\in\mathbb{R}$ be nonnegative, and let $\Phi$ be a convex Lipschitz integrand with constant $L$. Then $\mu\in\mathcal{P}_{2}^{\mathrm{ac}}(\mathbb{R}^{d})$, and the density cost $\mathcal{G}_{\Phi}(\mu)$ and the mollified density cost $\mathcal{G}_{\Phi,\varepsilon}(\mu)$ built from the kernel $\eta$ satisfy