Consider an infinite-dimensional Gaussian random vector $X$, and a positive random variable $f(X) \in L^p, p > 1$. Let $f(X) \sim \sum_n f_n(X)$ be its (formal) chaos expansion. Let $(U_\rho, \rho \in (0,1])$ be the Ornstein-Uhlenbeck semigroup parametrized by correlation $\rho$, i.e. $U_\rho f(X) \sim \sum_n \rho^n f_n(X)$.
Let $Q(\rho) := \sup\{q \mid U_\rho f(X) \in L^q \}$.
My question is: what are the possible functions $Q: (0,1] \to (1,\infty]$?
It follows from the hypercontractivity theorem for $U$ that the function $\rho \mapsto \rho^2 (Q(\rho) - 1)$ is decreasing. The question is whether there are other constraints on $Q$ - say, any kind of convexity? What if we know the value of $Q$ at two points - say, $1$ and $\rho_0$, and it happens that $Q(\rho_0)$ is better than the hypercontractivity bound - does this imply any nontrivial bounds for $Q$ on the segment $[\rho_0, 1]$?
