The Gaussian Sobolev Space: Linear Structure, Closedness, Hermite Polynomials and the Number-Operator Form of the Dirichlet Form, and the Log-Sobolev and Poincare Inequalities

In the setting of Gaussian Analysis Relative to a Diagonal Gaussian Reference Measure with Noise Weights: Standing Notation, with the Gaussian Sobolev space $\mathbb{D}^{1,2}_{a}$, the noise gradient $\nabla_{a}F$ of $F\in\mathbb{D}^{1,2}_{a}$ of The Gaussian Sobolev Space of the Noise Gradient, and the Dirichlet form $\mathcal{E}_{a}$; the coordinate of an element of $L^{2}(\gamma_{c};X^{a})$ along $f_{k}$, an element of $L^{2}(\gamma_{c})$, is that of The Space of Square-Integrable Hilbert-Valued Maps is a Real Hilbert Space: Coordinates and Synthesis §coordinates; the rates $\theta_{k}$ of Gaussian Analysis Relative to a Diagonal Gaussian Reference Measure with Noise Weights: Standing Notation §rates and $\theta=(\theta_{k})_{k\in\mathbb{N}}$; the multi-indices $\alpha\in\mathcal{A}$ with $\alpha!$, $c^{\alpha}$, $\theta\cdot\alpha$, $\varepsilon_{k}$ and $\alpha-\varepsilon_{k}$, and the cylindrical Hermite polynomials $H_{\alpha}$; and, $\gamma_{c}$ being a probability measure, the entropy $\operatorname{Ent}_{\gamma_{c}}$, with the function $\phi$ of The Function $s\log s$: Continuity, Young's Inequality and Lower Bounds, with the Elementary Bounds for the Exponential and the Logarithm.

1. (Linear structure)¶ $\mathbb{D}^{1,2}_{a}$ is a linear subspace of $L^{2}(\gamma_{c})$, and $F\mapsto\nabla_{a}F$ is linear on it.

2. (Closedness)¶ Let $(F_{j})_{j\in\mathbb{N}}$ be a sequence in $\mathbb{D}^{1,2}_{a}$, $F\in L^{2}(\gamma_{c})$ and $G\in L^{2}(\gamma_{c};X^{a})$ with $\lVert F_{j}-F\rVert_{2}\to0$ and $\lVert\nabla_{a}F_{j}-G\rVert_{\gamma_{c}}\to0$. Then $F\in\mathbb{D}^{1,2}_{a}$ and $\nabla_{a}F=G$.

3. (Hermite polynomials)¶ For every $\alpha\in\mathcal{A}$, the class of $H_{\alpha}$, which lies in $L^{2}(\gamma_{c})$ by Cylindrical Hermite Polynomials: Growth and Integrability, Partial Derivatives, Orthogonality, and Monomials as Hermite Combinations §cylindrical and is again written $H_{\alpha}$, belongs to $\mathbb{D}^{1,2}_{a}$, and for every $k\in\mathbb{N}$ the coordinate of $\nabla_{a}H_{\alpha}$ along $f_{k}$ is $a_{k}^{1/2}\alpha_{k}H_{\alpha-\varepsilon_{k}}$ if $\alpha_{k}\ge1$ and $0$ if $\alpha_{k}=0$.

4. (Number-operator form)¶ For all $\alpha,\beta\in\mathcal{A}$,

$$\mathcal{E}_{a}(H_{\alpha},H_{\beta})=\begin{cases}(\theta\cdot\alpha)\,\alpha!\,c^{\alpha}&\text{if }\alpha=\beta,\\0&\text{if }\alpha\ne\beta.\end{cases}$$

5. (Log-Sobolev inequality)¶ Let $\kappa\in\mathbb{R}$ be positive with $c_{k}\le\kappa\,a_{k}$ for every $k\in\mathbb{N}$, and let $F\in\mathbb{D}^{1,2}_{a}$ with a Borel representative $f$. Then $f^{2}$ and $\phi\circ f^{2}$ are integrable with respect to $\gamma_{c}$, and

$$\operatorname{Ent}_{\gamma_{c}}(f^{2})\le2\kappa\,\mathcal{E}_{a}(F,F).$$

6. (Poincar'e inequality)¶ For $\kappa$, $F$ and $f$ as in claim 5, $f$ is integrable with respect to $\gamma_{c}$, and

$$\int_{X}f^{2}\,d\gamma_{c}-\Bigl(\int_{X}f\,d\gamma_{c}\Bigr)^{2}\le\kappa\,\mathcal{E}_{a}(F,F).$$