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

TL;DR

This paper develops a framework for differential forms on non-smooth, possibly fractal metric spaces with local Dirichlet forms, generalizing classical localization and approximation results to complex, irregular spaces.

Contribution

It introduces a localization result for antisymmetric functions on diagonal neighborhoods and constructs a chain map linking Kolmogorov-Alexander-Spanier and de Rham complexes on such spaces.

Findings

Localization of antisymmetric functions to differential forms on fractal spaces

Construction of a chain map between different differential complexes

Generalization of classical localization and approximation results

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.