We consider the spaces Lp(X, ν; V), where X is a separable Banach space, μ is a centred non-degenerate Gaussian measure, ν := Ke−U μ with normalizing factor K and V is a separable Hilbert space. In this paper we prove a vector-valued Poincaré inequality for functions F ∈ W 1,p(X, ν; V), which allows us to show that for every p ∈ (1,+∞) and every k ∈ N the norm in W k,p(X, ν) is equivalent to the graph norm of DkH (the k-th Malliavin derivative) in Lp(X, ν). To conclude, we show exponential decay estimates for (TV (t))t≥0 as t → +∞. Useful tools are the study of the asymptotic behaviour of the scalar perturbed Ornstein-Uhlenbeck (T(t))t≥0, and pointwise estimates for |DHT(t)f |pH by means both of T(t)|DHf |pH and of T(t)|f |p.
SubjClass[2010]: 28C20; 46E35; 47D06