Equivalence of Sobolev Norms with Respect to Weighted Gaussian Measures

Abstract

AbstractWe consider the spaces $${\text {L}}^p(X,\nu ;V)$$ L p ( X , ν ; V ) , where X is a separable Banach space, $$\mu $$ μ is a centred non-degenerate Gaussian measure, $$\nu :=Ke^{-U}\mu $$ ν : = K e - 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 $$F\in W^{1,p}(X,\nu ;V)$$ F ∈ W 1 , p ( X , ν ; V ) , which allows us to show that for every $$p\in (1,\infty )$$ p ∈ ( 1 , ∞ ) and every $$k\in \mathbb {N}$$ k ∈ N the norm in $$W^{k,p}(X,\nu )$$ W k , p ( X , ν ) is equivalent to the graph norm of $$D_H^{k}$$ D H k (the k-th Malliavin derivative) in $${\text {L}}^p(X,\nu )$$ L p ( X , ν ) . 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 ≥ 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 ≥ 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 .