论文标题
相对于加权高斯措施的Sobolev规范的等效性
Equivalence of Sobolev norms with respect to weighted Gaussian measures
论文作者
论文摘要
我们考虑$ l^p（x，ν; v）$的空间，其中$ x $是可分离的banach空间，$μ$是一种中心的非降级高斯措施，$ν：= ke^{ - u}μ$，具有归一化因子$ k $，$ v $是可分离的Hilbert空间。在本文中，我们证明了w^{1，p}（x，x，v; v）$ f \ in w^{1，x，v; v）$的矢量价值不平等，这使我们可以证明，每一个$ p \ in（1，+\ infty）$中的每一个$ p \ in（+\ infty）$和每个$ k \ in \ mathb n $ in \ mathb n $ in \ mathb n $ sorm in $ w^^w^， $ d_h^k $（$ k $ -th malliavin derivative）中的$ l^p（x，ν）$。总而言之，我们将$（t^v（t））的指数衰减估计值_ {t \ geq0} $作为$ t \ rightarrow+\ infty $。有用的工具是对标量扰动的Ornstein-uhlenbeck $（t（t））_ {t \ geq0} $的渐近行为的研究，以及$ | d_ht（t）f | _h^p $的点估计，$ | d_ht（t）f | _h^p $ y MAST $ t（t）
We consider the spaces $L^p(X,ν;V)$, 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 $F\in W^{1,p}(X,ν;V)$, which allows us to show that for every $p\in(1,+\infty)$ and every $k\in\mathbb N$ the norm in $W^{k,p}(X,ν)$ is equivalent to the graph norm of $D_H^k$ (the $k$-th Malliavin derivative) in $L^p(X,ν)$. To conclude, we show exponential decay estimates for $(T^V(t))_{t\geq0}$ as $t\rightarrow+\infty$. Useful tools are the study of the asymptotic behaviour of the scalar perturbed Ornstein-Uhlenbeck $(T(t))_{t\geq0}$, and pointwise estimates for $|D_HT(t)f|_H^p$ by means both of $T(t)|D_Hf|^p_H$ and of $T(t)|f|^p$.
