Differential forms for fractal subspaces and finite energy coordinates

This paper introduces a notion of differential forms on closed, potentially fractal, subsets of Euclidean space by defining pointwise cotangent spaces using the restriction of $C^1$ functions to this set. Aspects of cohomology are developed: it is shown that the differential forms are a Banach algebra and it is possible to integrate these forms along rectifiable paths. These definitions are connected to the theory of differential forms on Dirichlet spaces by considering fractals with finite energy coordinates. In this situation, the $C^1$ differential forms project onto the space of Dirichlet differential forms. Further, it is shown that if the intrinsic metric of a Dirichlet form is a length space, then the image of any rectifiable path through a finite energy coordinate sequence is also rectifiable. The example of the harmonic Sierpinski gasket is worked out in detail.