Equivalence of Sobolev Norms with Respect to Weighted Gaussian Measures

We consider the spaces \({\text {L}}^p(X,\nu ;V)\) ${\text{L}}^p(X,\nu ;V)$ , where X is a separable Banach space, \(\mu \) $\mu$ is a centred non-degenerate Gaussian measure, \(\nu :=Ke^{-U}\mu \) $\nu :=K{e}^{-U}\mu$ with normalizing factor K and V is a separable Hilbert space. In this paper we prove a vector-valued Poincaré inequality for functions \(F\in W^{1,p}(X,\nu ;V)\) $F\in W^{1,p}(X,\nu ;V)$ , which allows us to show that for every \(p\in (1,\infty )\) $p\in (1,\infty )$ and every \(k\in \mathbb {N}\) $k\in \mathbb{N}$ the norm in \(W^{k,p}(X,\nu )\) $W^{k,p}(X,\nu )$ is equivalent to the graph norm of \(D_H^{k}\) $D_H^k$ (the k-th Malliavin derivative) in \({\text {L}}^p(X,\nu )\) ${\text{L}}^p(X,\nu )$ . To conclude, we show exponential decay estimates for the V-valued perturbed Ornstein-Uhlenbeck semigroup \((T^V(t))_{t\ge 0}\) ${(T^V(t))}_{t\ge 0}$ , defined in Section 2.6, as t goes to infinity. Useful tools are the study of the asymptotic behaviour of the scalar perturbed Ornstein-Uhlenbeck \((T(t))_{t\ge 0}\) ${(T(t))}_{t\ge 0}$ , and pointwise estimates for \(|D_HT(t)f|_H^p\) $|D_H{T(t)f|}_H^p$ by means of both \(T(t)|D_Hf|^p_H\) $T(t)|D_H{f|}_H^p$ and \(T(t)|f|^p\) ${T(t)|f|}^p$ .