This is a question subsequent to the one: Does the Gaussian Poincare inequality hold for $p=1$ as well as $p=2$?

There, I received a very helpful answer that the Gaussian poincare inequality for any pair $(p,q)$ holds when the underlying metric measure space is finite-dimensional.

Now, I wonder if this result also holds for infinite dimensional cases.

More specifically, let $V:=C^\infty(\mathbb{T})$ be the Frechet space of real-valued smooth periodic functions on $\mathbb{R}$; here $\mathbb{T}$ is the $1$-dimensional torus. Next, assume that a Gaussian measure $\mu$ of mean zero is given on this space.

Let us denote its covariance as $B_\mu : V^* \times V^* \to \mathbb{R}$, where $V^*$ is the continuous dual space of $V$ equipped with the strong dual topology.

Now, for a smooth real-valued mapping $F : V \to \mathbb{R}$, where smoothness is in Silva's sense, $DF$ denoting the Silva "gradient", assume that \begin{equation} \int \lvert F(v) \rvert d \mu(v), \int \lVert DF(v) \rVert^* d \mu(v) < \infty \text{ with } \int F(v) d \mu(v)=0. \end{equation} Here, $DF(v) : V \to \mathbb{R}$ is a bounded linear map for each $v \in V$. Also $\lVert DF(v) \rVert^*:= \sup_{\lVert w \rVert_2 \leq 1 } \lvert [DF(v)](w) \rvert$.

Then, do we have the following $(p,q)$ Poincare inequalities for all $1 \leq p,q, < \infty$? \begin{equation} \Bigl(\int \lvert F(v) \rvert^p d \mu(v)\Bigr)^{1/p} \leq C_{p,q, \mu} \Bigl( \int [\lVert DF(v) \rVert^*] ^q d \mu(v) \Bigr)^{1/q} \end{equation} where $C_{p,q,\mu}$ depends only on $p,q$ and $B_{\mu}$.

I tried to start from finite dimensions and use Kolmogorov extension theorem to establish the above inequality, but I cannot find a way to obtain some $C_{p,q,\mu}$ in the limit.

Could anyone please provide any insight?

Gaussian Measures, at least for test functions in the Gaussian Sobolev space. Connecting it to your desired Silva-smooth test functions could take some work. Bogachev discusses other notions of differentiability earlier in Chapter 5. $\endgroup$1more comment