Integrals of Functions with Bounded First and Second Derivatives are Intrinsic Test Functions on the Wasserstein Space

In the setting of The Intrinsic Calculus on the Wasserstein Space: Standing Notation. Intrinsic test functions on a subset of $\mathcal{P}_{2}(\mathbb{R}^{d})$ and their translation Hessians $H_{\varphi}(\mu)$ are those of that definition, and $\nabla\varphi(\mu)$ is the gradient along couplings. Being of class $C^{2}$ on $\mathbb{R}^{d}$, the gradient $Df(x)$ and the Hessian matrix $D^{2}f(x)$ are those of Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation §derivatives, $\mathbb{R}^{d}$ being open by claim 1 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous. Let $M\in\mathbb{R}$ be nonnegative and 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 the following hold.

1. (Integrability)¶ $f$ is Borel and integrable with respect to every $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$, so that

is defined. Each entry of $D^{2}f$ is Borel and bounded, and for $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ we write $\int_{\mathbb{R}^{d}}D^{2}f\,d\mu$ for the matrix each of whose entries is the $\mu$-integral of the corresponding entry of $D^{2}f$; it lies in $\mathcal{S}(d)$.

2. (Test function)¶ $\varphi_{f}$ is an intrinsic test function on $\mathcal{P}_{2}(\mathbb{R}^{d})$. For every $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ its gradient along couplings $\nabla\varphi_{f}(\mu)$ is the class $Df$ in $L^{2}(\mu;\mathbb{R}^{d})$ of the gradient map of $f$, as in Gradients of Functions with Bounded Derivatives Belong to the Tangent Space; Constants Are Tangent; the Score Identity; the Score Has Mean Zero; Translation of Tangent Fields §tangent, and its translation Hessian is