Gaussian Integration by Parts for Products of Cylindrical Functions, and Closability of the Noise Gradient in the Gaussian Lebesgue Space

In the setting of Gaussian Analysis Relative to a Diagonal Gaussian Reference Measure with Noise Weights: Standing Notation, with $\mathcal{F}C^{1}_{b}(X)$, the partial derivatives $\partial_{k}$ and the noise gradients $\nabla_{a}F$ of Gaussian Analysis Relative to a Diagonal Gaussian Reference Measure with Noise Weights: Standing Notation §cylindrical; for $F\in\mathcal{F}C^{1}_{b}(X)$ the class of $F$ lies in $L^{2}(\gamma_{c})$ by Bounded C^1 Cylindrical Functions: Linear Structure, the Gradient and the Partial Derivatives, Bounds and Integrability, and Density in the Square-Integrable Functions §square-integrable, and the class of $\nabla_{a}F$ lies in $L^{2}(\gamma_{c};X^{a})$ by The Noise Gradient of a Cylindrical Function and the Noise Tangent Space at a Probability Measure on a Hilbert Space §gradient; both are again written $F$ and $\nabla_{a}F$.

1. (Integration by parts for products)¶ For all $F,\varphi\in\mathcal{F}C^{1}_{b}(X)$ and $k\in\mathbb{N}$, the functions $\partial_{k}F\,\varphi$ and $x\mapsto F(x)\bigl(x_{k}\varphi(x)/c_{k}-\partial_{k}\varphi(x)\bigr)$ are integrable with respect to $\gamma_{c}$, and

$$\int_{X}\partial_{k}F\,\varphi\,d\gamma_{c}=\int_{X}F(x)\Bigl(\frac{x_{k}}{c_{k}}\,\varphi(x)-\partial_{k}\varphi(x)\Bigr)\gamma_{c}(dx).$$

2. (Closability)¶ Let $(F_{j})_{j\in\mathbb{N}}$ be a sequence in $\mathcal{F}C^{1}_{b}(X)$ and let $G\in L^{2}(\gamma_{c};X^{a})$ be such that $(\lVert F_{j}\rVert_{2})_{j\in\mathbb{N}}$ and $(\lVert\nabla_{a}F_{j}-G\rVert_{\gamma_{c}})_{j\in\mathbb{N}}$ converge to $0$. Then $G$ is the zero vector of $L^{2}(\gamma_{c};X^{a})$.