Differential complexes for local Dirichlet spaces, and non-local-to-local approximations

Abstract: We study differential $p$-forms on non-smooth and possibly fractal metric measure spaces, endowed with a local Dirichlet form. Using this local Dirichlet form, we prove a result on the localization of antisymmetric functions of $p+1$ variables on diagonal neighborhoods to differential $p$-forms. This result generalizes both the well-known classical localization on smooth Riemannian manifolds and the well-known semigroup approximation for quadratic forms. We observe that a related localization map taking functions into forms is well-defined and induces a chain map from a differential complex of Kolmogorov-Alexander-Spanier type onto a differential complex of deRham type.