Linear and Quadratic Kernel Functionals of a Measure Are Test Functions on the Wasserstein Space

In the setting of Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation, assume that $(\Omega,\mathcal{F},P)$ is rich. Test functions on $\mathcal{P}_{2}(\mathbb{R}^{d})$, their intrinsic gradients and translation Hessians are those of that definition; the translations $\tau_{a}$ and push-forwards (Probability Measures on Euclidean Space and Random Vectors: Standing Notation (iv)) are as fixed there, as are the coordinate projections $\mathrm{pr}_{1},\mathrm{pr}_{2}$ of $\mathbb{R}^{d+d}$ and the product measure $\mu\boxtimes\mu$; a function on $\mathcal{P}_{2}(\mathbb{R}^{d})$ is Lipschitz with a constant for the distance $W_{2}$; for a Borel map $\xi:\mathbb{R}^{d}\to\mathbb{R}^{d}$ with $\int\lVert\xi\rVert^{2}\,d\mu<\infty$ its class in $L^{2}(\mu;\mathbb{R}^{d})$ is again written $\xi$; the gradient $Df(x)$ and Hessian matrix $D^{2}f(x)$ of a function $f$ of class $C^{2}$ on $\mathbb{R}^{d}$ are those of Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation (ii), $\mathbb{R}^{d}$ being open by claim 1 of Euclidean Space Is Open in Itself; and $0_{d}$ denotes the $d\times d$ matrix all of whose entries are $0$, an element of the set $\mathcal{S}(d)$ of Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation (i). Let $M$ be a nonnegative real number.

1. (Linear functionals)(i) Let $f:\mathbb{R}^{d}\to\mathbb{R}$ be of class $C^{2}$ on $\mathbb{R}^{d}$ with $|\partial_{i}f(x)|\le M$ and $|\partial_{j}\partial_{i}f(x)|\le M$ for all $x\in\mathbb{R}^{d}$ and $i,j\in[d]$; then $f$ is integrable with respect to every $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ by The Lift of a Linear Functional of the Measure: Integrability, L-Gradient, Lipschitz Gradient Map and Translation Laplacian (i). The function

$$u_{f}:\mathcal{P}_{2}(\mathbb{R}^{d})\to\mathbb{R},\qquad u_{f}(\mu)=\int_{\mathbb{R}^{d}}f\,d\mu,$$

is a test function on $\mathcal{P}_{2}(\mathbb{R}^{d})$ and is Lipschitz with constant $\sqrt{d}\,M$. At $\mu$ its intrinsic gradient is the class of the gradient map $x\mapsto Df(x)$, and its translation Hessian is the matrix whose entry in row $i$ and column $j$ is $\int_{\mathbb{R}^{d}}\partial_{j}\partial_{i}f\,d\mu$, each $\partial_{j}\partial_{i}f$ being bounded and Borel.

2. (Quadratic kernel functionals)(ii) Let $K:\mathbb{R}^{d}\to\mathbb{R}$ be of class $C^{2}$ on $\mathbb{R}^{d}$ and even, $K(-x)=K(x)$, with $|K(x)|\le M$, $|\partial_{i}K(x)|\le M$ and $|\partial_{j}\partial_{i}K(x)|\le M$ for all $x\in\mathbb{R}^{d}$ and $i,j\in[d]$. Then $\partial_{i}K(-x)=-\partial_{i}K(x)$ for all $x\in\mathbb{R}^{d}$ and $i\in[d]$. Let $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$. The function $z\mapsto K(\mathrm{pr}_{1}(z)-\mathrm{pr}_{2}(z))$ on $\mathbb{R}^{d+d}$ is bounded and Borel; for every $x\in\mathbb{R}^{d}$ the function $y\mapsto K(x-y)$ is bounded and Borel, and the function $K*\mu:\mathbb{R}^{d}\to\mathbb{R}$, $(K*\mu)(x)=\int_{\mathbb{R}^{d}}K(x-y)\,\mu(dy)$, is of class $C^{2}$ on $\mathbb{R}^{d}$ with

$$\partial_{i}(K*\mu)(x)=\int_{\mathbb{R}^{d}}\partial_{i}K(x-y)\,\mu(dy),\qquad\partial_{j}\partial_{i}(K*\mu)(x)=\int_{\mathbb{R}^{d}}\partial_{j}\partial_{i}K(x-y)\,\mu(dy),$$

and $|K*\mu|\le M$, $|\partial_{i}(K*\mu)|\le M$, $|\partial_{j}\partial_{i}(K*\mu)|\le M$ on $\mathbb{R}^{d}$. The function

$$\mathcal{K}_{K}:\mathcal{P}_{2}(\mathbb{R}^{d})\to\mathbb{R},\qquad\mathcal{K}_{K}(\mu)=\int_{\mathbb{R}^{d+d}}K\bigl(\mathrm{pr}_{1}(z)-\mathrm{pr}_{2}(z)\bigr)\,(\mu\boxtimes\mu)(dz)=\int_{\mathbb{R}^{d}}K*\mu\,d\mu,$$

is a test function on $\mathcal{P}_{2}(\mathbb{R}^{d})$, satisfies $\mathcal{K}_{K}((\tau_{a})_{\#}\mu)=\mathcal{K}_{K}(\mu)$ for all $\mu$ and all $a\in\mathbb{R}^{d}$, and is Lipschitz with constant $2\sqrt{d}\,M$. At $\mu$ its intrinsic gradient is the class of the map $x\mapsto2\,D(K*\mu)(x)$ and its translation Hessian is $0_{d}$.