WEAK SOBOLEV SPACES AND MARKOV UNIQUENESS OF OPERATORS

We introduce the notion of a generalized differential on an abstract m easure space and construct corresponding strong and weak Sobolev space s. We show that the validity of a ''weak equals strong''-theorem, i.e. the coincidence of the weak and strong Sobolev space, implies Markov uniqueness for the diffusion operator corresponding to the generalized differential. Applications include a necessary and sufficient conditi on for Markov uniqueness for finite-dimensional, locally strictly elli ptic diffusion operators, and an abstract condition for Markov uniquen ess for Ornstein-Uhlenbeck operators on path spaces of compact Riemann ian manifolds.