Intrinsic metrics and Lipschitz functions (Q1565967)

The author studies the notion of measurable metrics and Lipschitz functions w.r.t. such metrics -- originally introduced by \textit{N. Weaver} [Stud. Math. 115, 277-289 (1995; Zbl 0839.46014)] -- on Dirichlet structures and, in particular, abstract Wiener spaces. In the latter case it is shown that the intrinsic metric [in the sense of \textit{M. Biroli} and \textit{U. Mosco}, Ann. Math. Pura Appl. 169, 125-181 (1995; Zbl 0851.31008)] coincides with Weaver's measurable metric. For more general local Dirichlet structures supporting a carré du champ operator this is still an open problem. In the setting of such Dirichlet spaces the Lipschitz-continuous functions in \(L^2\) and \(L^\infty\) are characterized in terms of the carré du champ operator which, in the case of abstract Wiener spaces, leads to a new characterization of Enchev-and-Stroock's H-Lipschitz functions [\textit{O. Enchev} and \textit{D. W. Stroock}, Ann. Probab. 21, 25-33 (1993; Zbl 0773.60042)].