# Does the Gaussian Poincare inequality hold for infinite dimensional measure metric spaces?

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 $$$$\int \lvert F(v) \rvert d \mu(v), \int \lVert DF(v) \rVert^* d \mu(v) < \infty \text{ with } \int F(v) d \mu(v)=0.$$$$ Here, $$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 $$1 \leq p,q, < \infty$$? $$$$\Bigl(\int \lvert F(v) \rvert^p d \mu(v)\Bigr)^{1/p} \leq C_{p,q, \mu} \Bigl( \int [\lVert DF(v) \rVert^*] ^q d \mu(v) \Bigr)^{1/q}$$$$ where $$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?

• The usual approach is that you start with the case where $F$ is a smooth cylinder function. Since such $F$ only depends on finitely many dimensions, the integrals on both sides are equivalent to integrals with respect to a finite-dimensional Gaussian measure, so the desired Poincare inequality holds, hopefully with a constant not depending on the dimension. Then you pass to the limit, approximating your desired test functions by cylinder functions in some appropriate mode of convergence, where normally your space of test functions is defined so as to make this work. Commented May 16, 2023 at 16:43
• Could you give a reference for the Silva notions of smoothness and differentiation? I'm not familiar with them, though I am guessing they are similar to the Malliavin notions. Commented May 16, 2023 at 16:44
• Sorry for the lack of reference. It is given in Part I of "Differential Calculus and Holomorphy"(1982) by Colombeau. Commented May 16, 2023 at 17:04
• So, according to your argument, it seems that even the range $\mathbb{R}$ can be generalized to any (separable) Banach space as well? Commented May 16, 2023 at 17:05
• The case $q=p$ is Theorem 5.5.11 in Bogachev's 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. Commented May 16, 2023 at 17:19