Equivalence of Sobolev Norms with Respect to Weighted Gaussian Measures

Abstract

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