Integrals and potentials of differential 1-forms on the Sierpinski gasket

We provide a definition of the integral, along paths in the Sierpinski gasket KK, for differential smooth 1-forms associated to the standard Dirichlet form EE on KK. We show how this tool can be used to study the potential theory on KK. In particular, we prove: (i) a de Rham reconstruction of a 1-form from its periods around lacunas in KK; (ii) a Hodge decomposition of 1-forms with respect to the Hilbertian energy norm; (iii) the existence of potentials of smooth 1-forms on a suitable covering space of KK. We finally show that this framework provides versions of the de Rham duality theorem for the fractal KK.