Capacities, removable sets and L^p-uniqueness on Wiener spaces

2018/05/10 by Hinz, Michael, Kang, Seunghyun
#28A12 #31C15 #31E05 #35J25 #35J40 #47B25 #47B38 #47N30 #60H07 #60J45 #FOS: Mathematics #Functional Analysis (math.FA) #Probability (math.PR)

paper · doi:10.48550/arxiv.1805.03764

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 Σ of zero Gaussian measure. To prove the equivalence we show the W^r,p(B,μ)-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 Σ must have sufficiently large codimension, in the case of the Ornstein-Uhlenbeck operator for instance at least 2p.

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