$h$-Laplacians on singular sets

Abstract

Until now, the correspondence between the Alexander–Kolmogorov complex, and the de Rham one, by means of a small scale parameter, has not gone that far as passing to the limit of the resolvent of the associated Laplacian, when the small parameter tends towards zero. Along these lines, a result proving a complete Hodge decomposition was missing. We bridge this gap by means of our own rescaled $h$-cohomology, $h$ being a very small parameter. Passing to the limit of the resolvent enables us to consider the extension to singular spaces, in particular, our $h$-differential operators also enable us to also make the connection with those of analysis on fractals, as introduced by Jun Kigami, and taken up by Robert S. Strichartz.