A Bound on the Noise Ornstein-Uhlenbeck Functional by Noise Gradients Gives Finite Weighted Fisher Information

In the setting of A Diagonal Gaussian Reference Measure on the Noise Wasserstein Space, Rescaled Heads and Gaussian Tails: Standing Notation, let $\mu\in\mathcal{P}_{2}(X)$, the set of The Second Moment of a Borel Probability Measure on a Hilbert Space and the Probability Measures with Finite Second Moment §space, and let $R\in\mathbb{R}$ be nonnegative. For $n\in\mathbb{N}$, $C^{2}_{b}(\mathbb{R}^{n})$ is the set of Bounded Twice Continuously Differentiable Functions with Bounded First and Second Partial Derivatives on Euclidean Space §bounded with $q=n$; for $g\in C^{2}_{b}(\mathbb{R}^{n})$, the function $g\circ p_{n}$ is a bounded $C^{2}$ cylindrical function, hence a bounded $C^{1}$ cylindrical function by Noise Gradients of Bounded C^2 Cylindrical Functions Are Dense in the Noise Tangent Space §inclusion; the class of its noise gradient in $L^{2}(\mu;X^{a})$ is again written $\nabla_{a}(g\circ p_{n})$, $\lVert\cdot\rVert_{\mu}$ is the norm of $L^{2}(\mu;X^{a})$, and $L^{a}_{\mu}(g)$ is the noise Ornstein-Uhlenbeck functional of $\mu$ at $g$. Suppose that

$$\bigl|L^{a}_{\mu}(g)\bigr|\le R\,\lVert\nabla_{a}(g\circ p_{n})\rVert_{\mu}\qquad\text{for every }n\in\mathbb{N}\text{ and every }g\in C^{2}_{b}(\mathbb{R}^{n}).$$

(Finite weighted Fisher information)¶ Then $\mu$ has a relative score with respect to $\gamma_{c}$ and finite Fisher information relative to $\gamma_{c}$ with weights $a$, and $\mathcal{I}_{a}(\mu\,|\,\gamma_{c})\le R^{2}$.