Equivalence of Sobolev norms with respect to weighted Gaussian measures

Abstract

We consider the spaces $L^p(X,\nu ;V)$, where $X$ is a separable Banach space, $\mu$ is a centred non-degenerate Gaussian measure, $\nu :=K{e}^{-U}\mu$ with normalizing factor $K$ and $V$ is a separable Hilbert space. In this paper we prove a vector-valued Poincaré inequality for functions $F\in W^{1,p}(X,\nu ;V)$, 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 $L^p(X,\nu )$. To conclude, we show exponential decay estimates for $(T^V(t){)}_{t\ge 0}$ as $t\to +\infty$. Useful tools are the study of the asymptotic behaviour of the scalar perturbed Ornstein-Uhlenbeck $(T(t){)}_{t\ge 0}$, and pointwise estimates for $|D_{H}T(t)f{|}_H^p$ by means both of $T(t)|D_{H}f{|}_H^p$ and of $T(t)|f{|}^p$.

arXiv commentThe paper contains many mistakes and it will be completely changed