1. Capacities, removable sets and $L^p$-uniqueness on Wiener spaces
- Author
-
Hinz, Michael and Kang, Seunghyun
- Subjects
Mathematics - Functional Analysis ,Mathematics - Probability ,28A12, 31E05, 31C15, 35J25, 35J40, 47B25, 47B38, 47N30, 60H07, 60J45 - Abstract
We prove the equivalence of two different types of capacities in abstract Wiener spaces. This yields a criterion for the $L^p$-uniqueness of the Ornstein-Uhlenbeck operator and its integer powers defined on suitable algebras of functions vanishing in a neighborhood of a given closed set $\Sigma$ of zero Gaussian measure. To prove the equivalence we show the $W^{r,p}(B,\mu)$-boundedness of certain smooth nonlinear truncation operators acting on potentials of nonnegative functions. We also give connections to Gaussian Hausdorff measures. Roughly speaking, if $L^p$-uniqueness holds then the 'removed' set $\Sigma$ must have sufficiently large codimension, in the case of the Ornstein-Uhlenbeck operator for instance at least $2p$.
- Published
- 2018