Hypercontractivity of the Mehler Semigroup with Noise Weights (Nelson's Theorem)

In the setting of Gaussian Analysis Relative to a Diagonal Gaussian Reference Measure with Noise Weights: Standing Notation, with the norms $\lVert\cdot\rVert_{p}$ of $L^{p}(\gamma_{c})$ of Gaussian Analysis Relative to a Diagonal Gaussian Reference Measure with Noise Weights: Standing Notation §lebesgue, the set $\mathcal{F}C_{\mathrm{pol}}(X)$ of continuous cylindrical functions of polynomial growth and the Mehler semigroup $P_{t}$ with noise weights $a$; every $F\in\mathcal{F}C_{\mathrm{pol}}(X)$ and $P_{t}F$ lie in $L^{p}(\gamma_{c})$ for every real $p\ge1$ by The Mehler Semigroup: Cylindrical Form, the Semigroup Law, Invariance, Positivity, Contraction, Symmetry, Hermite Eigenfunctions, Commutation with Partial Derivatives, and the Generator §class and The Mehler Semigroup: Cylindrical Form, the Semigroup Law, Invariance, Positivity, Contraction, Symmetry, Hermite Eigenfunctions, Commutation with Partial Derivatives, and the Generator §cylindrical. $\exp$ is the exponential function. Let $\kappa\in\mathbb{R}$ be positive with $c_{k}\le\kappa\,a_{k}$ for every $k\in\mathbb{N}$.

(Hypercontractivity)¶ Let $p,q,t\in\mathbb{R}$ with $1<p\le q$, $0\le t$ and $q-1\le\exp(2t/\kappa)\,(p-1)$. Then for every $F\in\mathcal{F}C_{\mathrm{pol}}(X)$,

$$\lVert P_{t}F\rVert_{q}\le\lVert F\rVert_{p}.$$