The classical Gaussian hypercontractivity is stated as following: Suppose $\xi$ is a Gaussian variable and $H_n(\xi)$ is the space of n-th homogeneous Wiener chaos constructed from $\xi$, then for any $X\in H_n(\xi)$ and $p \geq 1$ one has the following estimate $$\mathbb{E}[|X|^{2p}] \leq (2p-1)^{np}\mathbb{E}[X^2]^p$$
I'm wondering if there is any generalization of this into the case where $\xi$ is a Gaussian vector with values in some Hilbert spaces, with some suitable definition of homogeneous Wiener chaos vector, one could have something looks like $$\mathbb{E}[||X||^{2p}] \leq C(n,p)\mathbb{E}[||X||^2]^p$$ note that the absolute value of random variable in previous case in now changed into the norm of Hilbert space.
Is there a generalization like this? Is there any good references on this generalization if exsits?
